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\begin{document}

\title{A Coupled Chemo-Thermo-Poroelastic System: Wellposedness and Approximation for Control Applications}
\author{Tetyana Malysheva}
\date{March 2015}

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{ %\large
UNIVERSITY OF OKLAHOMA
\par
\vspace{0.16in}
GRADUATE COLLEGE
\par
\vspace{1.2in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
A COUPLED CHEMO-THERMO-POROELASTIC SYSTEM:
\par
\vspace{0.17in}
WELL-POSEDNESS AND APPROXIMATION
\par
\vspace{0.17in}
FOR CONTROL APPLICATIONS
\par
\vspace{1.2in}
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A DISSERTATION
\par
\vspace{0.17in}
SUBMITTED TO THE GRADUATE FACULTY
\par
\vspace{0.17in}
in partial fulfillment of the requirements for the
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Degree of
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DOCTOR OF PHILOSOPHY
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By
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\vspace{0.17in}
TETYANA M. MALYSHEVA 
\par
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Norman, Oklahoma
\par
%\vspace{0.05in}
2015
}
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{
A COUPLED CHEMO-THERMO-POROELASTIC SYSTEM:
\par
WELL-POSEDNESS AND APPROXIMATION
\par
FOR CONTROL APPLICATIONS
\par
\vspace{0.5in}
A DISSERTATION APPROVED FOR THE
\par
DEPARTMENT OF MATHEMATICS
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BY
\par
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\hline
\  \  \  \  \  \  \  & Dr. Luther White, Chair\\
 \\ &  \\
  \\ &  \\
\hline
  &  Dr. Musharraf Zaman\\
\\ & \\
 \\ &  \\
\hline
 &  Dr. Kevin Grasse\\
\\   & \\
 \\ &  \\
\hline
 & Dr. John Albert \\
 \\ &  \\
  \\ &  \\
\hline
   & Dr. Christian Remling \\
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}
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\copyright \    Copyright by TETYANA M. MALYSHEVA \   2015

All Rights Reserved.
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\chapter*{Acknowledgements}

First and foremost, I would like to express my sincere and deepest gratitude to my Ph.D. advisor, Professor Luther W. White, for his professional and inspirational guidance, unconditional support and dedication, continuous encouragement, and believing in me throughout my Ph.D. studies. I am very grateful for having had the chance to work with him.\\
\indent
I would like to specially thank Dr. Kevin Grasse, Dr. John Albert, Dr. Christian Remling, and Dr. Musharraf Zaman for agreeing to serve on my Doctoral Advisory Committee, their valuable time, and support.\\
\indent
I am also thankful to all the faculty and staff members of the Department of Mathematics at the University of Oklahoma for providing a supportive environment for my study. I am especially grateful to Dr. Andy Miller, Dr. Kevin Grasse, and Dr. Keri Kornelson for everything they did to help me with my professional growth.\\
\indent
Last but not least, I would like to wholeheartedly thank my mother, Liudmyla Popovichieva, and my husband, Yuri Malyshev, for their unconditional and endless love, incredible support, constant encouragement, and unwavering confidence in my ability to achieve this goal.

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\chapter*{Abstract}
%
This dissertation addresses well-posedness and approximation problems for coupled parabolic-elliptic systems with applications to geomechanics. This work is motivated by problems of borehole stability in porous formations that involve the modeling of fully coupled thermal, chemical, hydraulic, and mechanical processes. The underlying thermo-chemo-poroelastic (TCPu) model is a system of time dependent parabolic equations modeling thermal, solute, and fluid diffusions coupled with Navier-type elliptic equations that attempt to capture the elastic behavior of rock around the borehole. The obtained results are fundamental to posing optimal control problems using borehole parameters as controls to achieve desired stress distributions in the neighborhood of the borehole.\\
\indent
Sufficient conditions for well-posedness (in the sense of Hadamard) of the coupled parabolic-elliptic initial-boundary value problems are obtained for the two- and three-dimensional TCPu models. These results provide a mathematical basis for the general theory of fully coupled chemical thermo-poroelasticity in homogeneous isotropic porous media saturated by a fluid.\\
\indent 
Numerical methods for solving fully coupled parabolic-elliptic initial-boundary value problems for two- and three-dimensional TCPu processes, including radially non-symmetric processes in an inclined borehole, are developed. These methods are based on a hybrid Fourier-finite-element approximation technique proposed in this research. The technique also involves a boundary penalization strategy and captures borehole geometry and mechanics. The numerical methods allow numerical analysis of a control problem in which the temperature and pressure on the borehole boundary are considered as control parameters used to achieve desirable stresses in the neighborhood of the borehole. 
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\chapter {Introduction}
Natural systems are usually complex and composed of numerous interacting parts. Such interactions require coupled models that, in turn, create a need for advanced mathematical analysis and control techniques. In this dissertation, we consider initial-boundary value problems for coupled parabolic-elliptic systems with applications to geomechanics. This work is motivated by the problems of borehole stability in 
porous formations saturated by a fluid that involve the modeling of fully coupled thermal, chemical, hydraulic, and mechanical (elastic) processes. \\
\indent
The poroelasticity theory describing the coupled processes of elastic deformation and pore fluid diffusion in fluid-saturated isothermal porous media can be traced back to the pioneering works of von Terzaghi \cite{Terzaghi1, Terzaghi2} (1925, 1943) and Biot \cite{Biot1} (1941). In 1956, Biot \cite{Biot2} pointed out a complete mathematical analogy between poroelasticity and thermoelasticity with the temperature playing the same role as the fluid pressure and heat conduction corresponding to fluid flow. A complete mathematical analysis of the well-posedness of a general initial-boundary value problem for a system of coupled partial differential equations that describes the Biot consolidation model \cite{Biot1} in poroelasticity, as well as a coupled quasi-static problem in thermoelasticity, has been carried out by Showalter \cite{Showalter2, Showalter3}. The corresponding Biot consolidation model has the form
\begin{gather}
-(\lambda + \mu) \nabla (\nabla\cdot \mathbf{u}(x,t)) - \mu\Delta\mathbf{u}(x,t)+\alpha\nabla p(x,t)=\mathbf{f}(x,t) \label{eqn:intro1} \\
\frac{\partial}{\partial t} (c_0p(x,t)+\alpha \nabla \cdot \mathbf{u}(x,t)) - \nabla \cdot k\nabla p(x,t) = h(x,t) \label{eqn:intro2}
\end{gather}
consisting of the equilibrium equation for momentum (\ref{eqn:intro1}) and the diffusion equation for Darcy flow (\ref{eqn:intro2}). Here $~\mathbf{u}(x,t)$ is the displacement of the stucture; $~p(x,t)$ is the fluid pore pressure; the positive Lam\'e constants $~\lambda$ and $~\mu$ are the dilation and shear moduli of elasticity, respectively; $~\alpha>0$ is the Biot-Willis constant that accounts for the pressure-deformation coupling; the constant $~c_0 \geq 0$ combines the porosity of the medium and the compressibility of the fluid; $~k>0$ involves the permeability of the medium and the viscosity of the fluid; $~f$ and $~g$ are given data, typically the volume-distributed external forces and the volume-distributed source density, respectively. In the context of the coupled thermoelasticity system, $~p(x,t)$ denotes the temperature; $~c_0>0$ is the specific heat of the medium; and $~k>0$ is the conductivity.\\
\indent
Based on the theory of linear degenerate evolution equations in Hilbert spaces, the existence and uniqueness of strong and weak solutions, as well as regularity theory for the system (\ref{eqn:intro1})-(\ref{eqn:intro2}) supplemented by initial and boundary conditions, were developed. The earlier works on the mathematical issues of analysis and numerical approximation for the Biot consolidation model include \cite{Auriault, Biot3, Biot4, Mourits, Yoloo, Zenisek, Zienkiewicz}.  The a priori analysis of Euler-Galerkin approximations for the Biot consolidation problem is presented in the works of Murad, Loula, and Thom\'ee \cite{Murad1, Murad2, Murad3}, including the semi-discrete and fully discrete cases and long-time behavior; and the a posteriori error analysis was developed by Ern and Meunier \cite{Ern}.\\
\indent
In 1971, Schiffman \cite{Schiffman} first extended the Biot theory for non-isothermal systems to account for the effects of thermal expansion of both the pore fluid and the elastic matrix. Since then, a substantial literature on thermo-poroelasticity theory and the modeling of the coupled hydro-thermo-mechanical behavior of a fluid-saturated porous media has been developed for geomechanics, including petroleum and geothermal borehole stability \cite{Abousleiman, Belotserkovets, Coussy, Diek4, Kodashima1, Kodashima2,Kurashige, McTigue, Pao, Tao, Wang}. Considerable attention has been recently placed on the impact of chemical processes in porous media on drilling and borehole stability.  Ghassemi and Diek \cite{Ghassemi3} developed a linear chemo-poroelastic model that couples solute transfer in the mud/shale system to stress and pore pressure within the framework of a Biot-like isotropic poroelastic theory. Thereafter, Zhou and Ghassemi \cite{Zhou} and Rafieepour et al. \cite{Rafieepour} presented linear and nonlinear chemo-thermo-poroelastic models, respectively. However, all those models are not rigorously developed from the first and second laws of thermodynamics. In 2011, Diek et al. \cite{Diek2, Diek3, Diek1} first presented fully coupled thermo-poroelastic and chemo-thermo-poroelastic models for drilling in geological formations that satisfy the first and second laws of thermodynamics.\\
\indent
Due to the complexity of cross-coupling mechanisms involved in thermo-poro-elastic and chemo-thermo-poroelastic models, the majority of corresponding initial-boundary value problems are solved by various numerical and computational techniques \cite{Ghassemi1, Ghassemi2, Ghassemi3, Koshelev, Rafieepour}. Very few analytical solutions of the fully coupled thermo-poroelastic and chemo-poroelastic equations are currently available. Typically, the solutions are derived under the assumptions that some of the couplings can be neglected \cite{Coussy, Kodashima1, Kodashima2, McTigue}. For instance, in the works of Kodashima and Kurashige \cite{Kodashima1, Kodashima2}, the displacement field is decoupled from the temperature and pore fluid pressure fields; and in Coussy \cite{Coussy}, an analytical solution is presented under the assumption that the temperature equation can be decoupled and approximated by a purely diffusive equation. Belotserkovets and Prevost \cite{Belotserkovets}, using the Laplace transformation and the residue theorem, developed an analytical method and presented an exact unique solution for a specific case of the fully coupled thermo-hydro-mechanical response of a fluid saturated porous sphere under mechanical pulse load.  \\[12pt]
\indent
In this dissertation, we study the model that is based on equations derived by Diek \cite{Diek1} and constitutes the general theory of fully coupled chemo-thermo-poroelasticity in homogeneous isotropic porous media saturated by a fluid. The theory satisfies the first and second laws of thermodynamics and is based on concepts of irreversible thermodynamics, a novel rock constitutive relation, and Onsager's transport phenomenology \cite{Diek1}. Since the model is derived from physical laws, it requires a mathematical basis for its well-posedness. On the other hand, the geomechanical nature of the problem demands analytical and numerical methods that capture both the geometry and the physical background of the problem. With this in mind, the aims of the dissertation are to establish the mathematical theory of well-posedness for coupled linear parabolic-elliptic systems modeling fully coupled mechanical, hydraulic, chemical, and thermal processes in homogeneous isotropic media and to develop and experimentally validate numerical methods for approximation and boundary control problems in geomechanical applications.\\[12pt]
\indent
The dissertation is organized in chapters as follows. Chapter 2 describes the underlying fully coupled thermo-chemo-poroelastic (TCPu) model for fluid-saturated porous rock formations and formulates the corresponding coupled parabolic-elliptic initial-boundary value problems for two- and three-dimensional regions.  Chapter 3 addresses the well-posedness theory for the initial-boundary value problems introduced in Chapter 2. Chapter 4 develops numerical methods for approximation and boundary control problems for the two- and three-dimensional TCPu models. Chapter 5 presents experimental validation of the numerical methods developed in Chapter 4. Chapter 6 contains the conclusions and potential directions for further research.\\ 
\newpage
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Chapter 2: Underlying Equations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter {Coupled TCPu System: Underlying Equations}
\indent 
%
The underlying TCPu model \cite{Diek1} is a system of time-dependent parabolic partial differential equations (PDEs) coupled with Navier-type elliptic PDEs with time, $t \in (0, t_f)$, as a parameter. The parabolic equations represent heat, solute, and fluid diffusions, and the Navier-type elliptic equations attempt to capture the elastic behavior of rock, while incorporating thermal, chemical, and porous media effects. The equations are developed in terms of the following independent variables: the absolute temperature $T$, the solute mass fraction $C$, the pore pressure $p$, and the vector of solid displacements $\vec{u}$. The coupled partial differential equations supplemented by the appropriate initial and boundary conditions constitute an initial-boundary value problem defined in a region exterior to the borehole. Although this region is infinite, it is specified with a far-field radius such that, without loss of generality, we may assume that, on the far-field boundary, the absolute temperature, solute mass fraction, pore pressure, and displacements are time-independent, and the displacement is negligibly small. \\[\baselineskip]
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2D Underlying equations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section {Two-dimensional Model}
%
The initial-boundary value problem that constitutes our model is defined in a region $~\Omega$ in $~\reals ^2$ exterior to the borehole. We assume that $~\Omega$ is a bounded open domain with a sufficiently smooth boundary $~\Gamma$. The boundary $~\Gamma$ is expressed as the union of disjoint sets $~\Gamma_B$ and $~\Gamma_F$ having nonempty interiors relative to the set $~\Gamma$. Specifically, $~\Omega \subset \mathbb R^{2}$ is an annular region with an inner radius $~R_{B}$ and an outer radius $~R_{F}$. 
%
\newpage
Let $(X,Y)$ be a borehole Cartesian coordinate system such that\\
\indent \indent
$x$ -axis is in the direction of maximum horizontal principal stress $\sigma_H$ and\\
\indent \indent 
$y$ -axis is in the direction of minimum horizontal principal stress $\sigma_h$.\\
%
\indent
A point $\bar{x}$ in $\reals ^2$ is associated with position vector $~\bar{x}=x\vec{i}+y\vec{j}$, where $~\vec{i}$ and $~\vec{j}$ are the basis vectors of the Cartesian coordinate system $(X,Y)$. Displacements of points within $\Omega$ are described in terms of a vector-valued displacement function of the position $\bar{x}$ and time $t$ as $~\vec{u}(\bar{x},t)=u(\bar{x},t)\vec{i}+v(\bar{x},t)\vec{j}$.\\
%
\indent
The boundary $\Gamma$ of $\Omega$ is $\Gamma = \Gamma_B\cup\Gamma_F$, where
\begin{gather*}
\Gamma_B = \left\{ \bar{x}=(x,y) \in \mathbb R^{2}: \sqrt{x^2+y^2}=R_B \right\}\\
\Gamma_F = \left\{ \bar{x}=(x,y) \in \mathbb R^{2}: \sqrt{x^2+y^2}=R_F \right\}
\end{gather*}
Figure ~\ref{2Dregion} shows the two-dimensional region $\Omega$ in $\reals ^2$ exterior to the borehole.\\
%
\vspace{-0.2 in}
\begin{figure}[ht]
\centering
\includegraphics[scale=0.46]{borehole_2D.png}
\caption{Two-dimensional region exterior to the borehole}
\label{2Dregion}
\end{figure}

%
The initial-boundary value problem that constitutes the two-dimensional radially nonsymmetric TCPu model is the following.\\
Thermal diffusion:
\begin{equation}
\Lambda\dot{T}+\Sigma\dot{C}+\Phi\dot{p}-\frac{k^T}{T_F}\nabla^2T-\frac{\rho_fRT_FD^T}{M^s(1-C_F)}\nabla^2C+K^T\nabla^2p=-\zeta(\nabla\cdot\dot{\vec{u}}) \qquad \text{in } \Omega \times(0,t_f) \label{eq:uneq2d1}
\end{equation}
with boundary conditions
\begin{gather}
T(\bar{x},t)=T_B(t) \qquad \text{on } \Gamma_B\times[0,t_f) \label{eq:uneq2d2}\\
T(\bar{x},t)=T_F(\bar{x}) \qquad \text{on } \Gamma_F\times[0,t_f) \label{eq:uneq2d3}
\end{gather}
and initial conditions
\begin{equation}
T(\bar{x},0)=\tilde{T}_I(\bar{x}) \qquad \text{in } \Omega \label{eq:uneq2d4}
\end{equation}
%
Solute diffusion:
\begin{equation}
\phi\dot{C}-C_FD^T\nabla^2T-D\nabla^2C+\frac{k\mathfrak{R}}{\eta}\nabla^2p=0 \qquad \text{in } \Omega \times(0,t_f) \label{eq:uneq2d5}
\end{equation}
with boundary conditions
\begin{gather}
C(\bar{x},t)=C_B(t) \qquad \text{on } \Gamma_B\times[0,t_f) \label{eq:uneq2d6}\\
C(\bar{x},t)=C_F(\bar{x}) \qquad \text{on } \Gamma_F\times[0,t_f) \label{eq:uneq2d7}
\end{gather}
and initial conditions
\begin{equation}
C(\bar{x},0)=\tilde{C}_I(\bar{x}) \qquad \text{in } \Omega \label{eq:uneq2d8}
\end{equation}
%
Fluid diffusion:
\begin{equation}
\Gamma\dot{T}+\chi\dot{C}+\Psi\dot{p}+K^T\nabla^2T+\frac{\rho_f\mathfrak{R}\Omega k}{\eta}\nabla^2C-\frac{k}{\eta}\nabla^2p=-\alpha(\nabla\cdot\dot{\vec{u}}) \qquad \text{in } \Omega \times(0,t_f) \label{eq:uneq2d9}
\end{equation}
with boundary conditions
\begin{gather}
p(\bar{x},t)=p_B(t) \qquad \text{on } \Gamma_B\times[0,t_f) \label{eq:uneq2d10}\\
p(\bar{x},t)=p_F(\bar{x}) \qquad \text{on } \Gamma_F\times[0,t_f) \label{eq:uneq2d11}
\end{gather}
and initial conditions
\begin{equation}
p(\bar{x},0)=\tilde{p}_I(\bar{x}) \qquad \text{in } \Omega \label{eq:uneq2d12}
\end{equation}
%
Navier-type elastic equations:
\begin{equation}
\Big(K+\frac{G}{3}\Big)\nabla(\nabla\cdot\dot{\vec{u}})+G\nabla^2\dot{\vec{u}}-\tilde{\zeta}\nabla\dot{T}+\xi\nabla\dot{C}-\tilde{\alpha}\nabla\dot{p}=0 \qquad \text{in } \Omega \times(0,t_f) \label{eq:uneq2d13}
\end{equation}
with boundary conditions 
\begin{gather}
\vec{u}(\bar{x},t) \approx 0 \qquad \text{on } \Gamma_F\times(0,t_f) \label{eq:uneq2d14} \\
\dot{\vec{u}}(\bar{x},t) \approx 0 \qquad \text{on } \Gamma_F\times(0,t_f) \label{eq:uneq2d15}
\end{gather}
and initial conditions
\begin{equation}
\vec{u}(\bar{x},0)= 0 \qquad \text{in } \bar{\Omega} \label{eq:uneq2d16}
\end{equation}
%
The description and values of physical constants are presented in Appendix A.\\ 
%
\indent
It is convenient to consider the thermal diffusion, solute diffusion, and fluid diffusion as a system (TCP system). Hence, we introduce the following matrices and vectors:\\
\begin{equation}
M=\begin{bmatrix} \\[-2.4em]
\Lambda & \Sigma & \Phi \\[-6pt]
0 & \phi & 0 \\[-6pt]
\Gamma & \chi & \Psi
\end{bmatrix}, \quad   
A=\begin{bmatrix} \\[-1.6em]
\dfrac{k^T}{T_F} & \dfrac{\rho_fRT_FD^T}{M^s(1-C_F)} & -K^T \\[9pt]
C_FD^T & D & -\dfrac{k\mathfrak{R}}{\eta} \\[9pt]
-K^T & -\dfrac{\rho_f\mathfrak{R}\Omega k}{\eta} & \dfrac{k}{\eta}
\end{bmatrix}, \quad 
\vec{b}_0=\begin{bmatrix} \\[-2.4em]
\zeta \\[-6pt]
0 \\[-6pt]
\alpha
\end{bmatrix} \label{eq:uneq2d17}
\end{equation}\\[-6pt]
%
\begin{equation}
\bar{V}(\bar{x},t)=\begin{bmatrix} \\[-2.4em]
T \\[-6pt]
C \\[-6pt]
p
\end{bmatrix}(\bar{x},t), \qquad 
V_B=\begin{bmatrix} \\[-2.4em]
T_B \\[-6pt]
C_B \\[-6pt]
p_B
\end{bmatrix}, \qquad 
V_F=\begin{bmatrix} \\[-2.4em]
T_F \\[-6pt]
C_F \\[-6pt]
p_F
\end{bmatrix}  \label{eq:uneq2d18}
\end{equation}\\
%
\indent
With the notations (\ref{eq:uneq2d17}) and (\ref{eq:uneq2d18}), the initial-boundary value problem (\ref{eq:uneq2d1})-(\ref{eq:uneq2d12}) has the form
%
\begin{equation} 
M \dot{\bar{V}} - A \nabla^{2} \bar{V}=-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f) \label{eq:uneq2d19}
\end{equation}
%
with boundary conditions
\begin{equation} \label{eq:uneq2d20}
\bar{V}(\bar{x},t)=
\begin{cases}\\[-3em]
V_B(t) & \text{on } \Gamma_B \times [0,t_f)\\
V_F(\bar{x}) & \text{on } \Gamma_F \times [0,t_f)
\end{cases}
\end{equation}
%
and initial conditions 
\begin{equation} \label{eq:uneq2d21}
\bar{V}(\bar{x},0)=\tilde{V}_I(\bar{x})\qquad \text{in } \Omega
\end{equation} 
%
Note that $\vec{b}_0$ is the factor that couples the displacement velocity to the absolute temperature, the pore pressure, the solute mass fraction (TCP), and their velocities.\\
%
\indent
By the principle of minimum total potential energy, the region $\Omega$ shall displace to a position that minimizes the total potential energy; it is the stable configuration for equilibrium.\\
%
\indent
The total potential energy of our system is
\begin{equation}
\mathcal{V}(\vec{u})=\mathcal{V}_S(\vec{u})-W_b(\vec{u})-W_S(\vec{u}) \label{eq:uneq2d22}
\end{equation}
where $~~\mathcal{V}_S(\vec{u})$ is the elastic energy of the system; $~~W_b(\vec{u})$ is work done by body forces due to the absolute temperature, the solute mass fraction, and the pore pressure; and $~~W_S(\vec{u})$ is work done by applied boundary stress.\\ 
%
\indent
Now we specify $~\mathcal{V}_S(\vec{u})$, $~W_b(\vec{u})$, and $~W_S(\vec{u})$. Here and in the following we will suppress the time dependence of the displacement vector for the sake of brevity. \\
%
\indent
Given the displacement
\begin{equation*}
\vec{u}=u\vec{i}+v\vec{j} 
\end{equation*}
the linearized strain is a second order symmetric tensor \\[-6pt]
\begin{equation*}
\varepsilon=\begin{bmatrix} \\[-2.5em]
\varepsilon_{11}&\varepsilon_{12} \\[-6pt]
\varepsilon_{21}&\varepsilon_{22}
\end{bmatrix}=\frac{1}{2}\big(\nabla\vec{u}+\nabla\vec{u}^T\big)
\end{equation*}
where
\begin{equation*}
\varepsilon_{11}=u_x, \qquad \varepsilon_{12}=\varepsilon_{21}=\frac{1}{2}\big(u_y+v_x\big), \qquad \varepsilon_{22}=v_y 
\end{equation*}
%
\newpage
\indent
In the linear elasticity theory, a relation between the stress tensor $\tau$ and the linearized strain tensor $~\varepsilon(\vec{u})$ is given by
\begin{equation}
\tau_{ij}(\vec{u})=a_{ijkh}\varepsilon_{kh}(\vec{u}) \label{eq:uneq2d23}
\end{equation}  
where $~a_{ijkh}$ are the coefficients of elasticity, independent of the strain tensor.\\
\indent
The coefficients of elasticity have properties of symmetry
\begin{equation}
a_{ijkh}=a_{jihk}=a_{khij} \label{eq:uneq2d24}
\end{equation}
and of ellipticity: There exists a constant $~\alpha_1>0$ such that
\begin{equation}
a_{ijkh}\varepsilon_{ij}\varepsilon_{kh} \geq \alpha_1 \varepsilon_{ij}\varepsilon_{ij}, \qquad \forall~\varepsilon_{ij}. \label{eq:uneq2d25}
\end{equation}
In equations (\ref{eq:uneq2d23})-(\ref{eq:uneq2d25}), we made use of the Einstein summation convention concerning repeated indices.\\
%
\indent
For a homogeneous elastic isotropic medium, the stress-strain relation (\ref{eq:uneq2d23}) in terms of the bulk modulus $K$ and the shear modulus $G$ has the form
\begin{equation}
\tau=\begin{bmatrix} \\[-2.5em]
\tau_{11}&\tau_{12} \\[-6pt]
\tau_{21}&\tau_{22}
\end{bmatrix} = 2G\varepsilon + \big(K-\frac{2}{3}G\big)(\text{tr}\varepsilon)I \label{eq:uneq2d26}
\end{equation}
%
That is,
\begin{align}
\tau_{11}&=2Gu_x+\big(K-\frac{2}{3}G\big)\big(u_x+v_y\big) \nonumber \\
\tau_{12}&=\tau_{21}=G\big(u_y+v_x\big) \label{eq:uneq2d27} \\
\tau_{22}&=2Gv_y+\big(K-\frac{2}{3}G\big)\big(u_x+v_y\big) \nonumber
\end{align} 
%
\indent
The elastic strain energy of the system is
\begin{equation}
\mathcal{V}_S(\vec{u})= \frac{1}{2}\int_\Omega\tau_{11}(\vec{u})\varepsilon_{11}(\vec{u})+2\tau_{12}(\vec{u})\varepsilon_{12}(\vec{u})+\tau_{22}(\vec{u})\varepsilon_{22}(\vec{u})\ud \Omega \label{eq:uneq2d28}
\end{equation}
%
\newpage
The work done by body forces due to the absolute temperature, the solute mass fraction, and the pore pressure is 
\begin{equation}
W_b(\vec{u})=\int_\Omega f\big(\text{tr}\varepsilon(\vec{u})\big)\ud \Omega \label{eq:uneq2d29}
\end{equation}
where 
\begin{equation}
f=\vec{b}_1^T\bar{V}, \qquad \vec{b}_1=[\tilde{\zeta} \quad -\xi \quad \tilde{\alpha}]^T, \qquad \bar{V}=[T \quad C \quad p]^T \label{eq:uneq2d30}
\end{equation}
%
Note that $~\vec{b}_1$ is the factor that couples the absolute temperature, the pore pressure, and the solute mass fraction (TCP) to the displacement.\\ 
%
\indent
The work done by applied boundary stress is
\begin{equation}
W_S(\vec{u})=\int_{\Gamma}(\hat{\sigma}\vec{n})\cdot\vec{u}\ud \Gamma \label{eq:uneq2d31}
\end{equation}
where $~\hat{\sigma}$ is applied boundary stress and $~\vec{n}$ is the outward unit normal vector on the boundary.\\
%
\indent
Applying (\ref{eq:uneq2d28}), (\ref{eq:uneq2d29}), and (\ref{eq:uneq2d31}) to (\ref{eq:uneq2d22}), we have
\begin{align}
\mathcal{V}(\vec{u})&=\frac{1}{2}\int_\Omega\tau_{11}(\vec{u})\varepsilon_{11}(\vec{u})+2\tau_{12}(\vec{u})\varepsilon_{12}(\vec{u})+\tau_{22}(\vec{u})\varepsilon_{22}(\vec{u})\ud \Omega \nonumber \\[6pt]
&\quad-\int_\Omega f\big(\text{tr}\varepsilon(\vec{u})\big)\ud \Omega - \int_{\Gamma}(\hat{\sigma}\vec{n})\cdot\vec{u}\ud \Gamma \label{eq:uneq2d32}
\end{align}
%
\indent
To aid in giving a mathematical formulation, let
\begin{gather*}
H=L^2\big(\Omega; \reals ^2\big)=\big\{L^2\big(\Omega; \reals\big)\big\}^2 \\
V=H^1\big(\Omega; \reals ^2\big)=\big\{H^1\big(\Omega; \reals\big)\big\}^2 \\
V_0=\big\{\varphi\in V:\varphi\big|_{\Gamma_F} =0\big\} \\
\vec{\Phi}(\bar{x})=\phi(\bar{x})\vec{i}+\psi(\bar{x})\vec{j}
\end{gather*}
%
\indent
At this point, we wish to express the elastic strain energy $\mathcal{V}_S(\vec{u})$ as a functional on $V$. To that end, define the bilinear form on $V$ by \\
\vspace{-0.2in}
\begin{equation}
a_E(\vec{u},\vec{\Phi})=\int_\Omega\tau_{11}(\vec{u})\varepsilon_{11}(\vec{\Phi})+2\tau_{12}(\vec{u})\varepsilon_{12}(\vec{\Phi})+\tau_{22}(\vec{u})\varepsilon_{22}(\vec{\Phi})\ud \Omega, \qquad \forall ~\vec{u}, \vec{\Phi} \in V \label{eq:uneq2d33}
\end{equation}
%
Then from (\ref{eq:uneq2d28}) and (\ref{eq:uneq2d33}),
\begin{equation}
\mathcal{V}_S(\vec{u})=\frac{1}{2}a_E(\vec{u},\vec{u}) \label{eq:uneq2d34}
\end{equation}
and, from (\ref{eq:uneq2d32}) and (\ref{eq:uneq2d34}), the total potential energy has the form 
\begin{equation}
\mathcal{V}(\vec{u})=\frac{1}{2}a_E(\vec{u},\vec{u})-\int_\Omega f\big(\text{tr}\varepsilon(\vec{u})\big)\ud \Omega - \int_{\Gamma}(\hat{\sigma}\vec{n})\cdot\vec{u}\ud \Gamma \label{eq:uneq2d35}
\end{equation}
%
\indent
Define the following vectors:
\begin{equation*}
\vec{\tau}_1=[\tau_{11} \quad \tau_{12}]^T, \qquad \vec{\tau}_2=[\tau_{21} \quad \tau_{22}]^T
\end{equation*}
%
Then 
\begin{align}
a_E(\vec{u},\vec{\Phi})&=\int_\Omega\tau_{11}\phi_x+\tau_{12}\phi_y+\tau_{21}\psi_x+\tau_{22}\psi_y \ud \Omega \nonumber \\[6pt]
&=\int_\Omega \vec{\tau}_1 \cdot \nabla \phi + \vec{\tau}_2 \cdot \nabla \psi \ud \Omega \nonumber \\[6pt]
&=\int_\Omega \nabla \cdot (\phi \vec{\tau}_1) -\phi \nabla \cdot \vec{\tau}_1 + \nabla \cdot (\psi \vec{\tau}_2) -\psi \nabla \cdot \vec{\tau}_2 \ud \Omega \nonumber \\[6pt]
&=\int_{\Gamma} \phi (\vec{\tau}_1 \cdot \vec{n})+\psi (\vec{\tau}_2 \cdot \vec{n}) \ud \Gamma - \int_\Omega \phi \nabla \cdot \vec{\tau}_1+\psi \nabla \cdot \vec{\tau}_2 \ud \Omega \nonumber \\[6pt]
&=\int_{\Gamma} (\tau\vec{n}) \cdot \vec{\Phi}\ud \Gamma - \int_\Omega [\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2]^T \cdot \vec{\Phi} \ud \Omega, \qquad \forall ~\vec{\Phi} \in V \label{eq:uneq2d36}
\end{align}
%
Thus, we obtain the following Green's formula:
\begin{equation}
a_E(\vec{u},\vec{\Phi})=\int_{\Gamma} (\tau\vec{n}) \cdot \vec{\Phi}\ud \Gamma - \int_\Omega [\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2]^T \cdot \vec{\Phi} \ud \Omega, \qquad \forall ~\vec{\Phi} \in V \label{eq:uneq2d37}
\end{equation}
%
\indent
Referring to (\ref{eq:uneq2d14}) and (\ref{eq:uneq2d15}), we assume that $\vec{u}$ and $\dot{\vec{u}}$ are negligible on $\Gamma_F$ and therefore, the set of admissible displacements is 
\begin{equation*}
\mathcal{U}_{ad}=\big\{\vec{u} \in V_0:\dot{\vec{u}} \in V_0\big\}
\end{equation*}
%
\indent
Using the principle of minimum total potential energy, the displacement $\vec{u}$ that the region $\Omega$ undergoes is given by
\begin{equation}
D\mathcal{V}(\vec{u})\vec{\Phi}=0, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:uneq2d38}
\end{equation} 
where 
\begin{equation*}
D\mathcal{V}(\vec{u})\vec{\Phi}=\frac{d}{d\delta}\mathcal{V}(\vec{u}+\delta\vec{\Phi})\Big\vert_{\delta=0}
\end{equation*}\\[-6pt]
is the G\^{a}teaux differential of $~\mathcal{V}$ with increment $~\vec{\Phi}$.
\\
%
From (\ref{eq:uneq2d35}), (\ref{eq:uneq2d38})
\begin{align}
a_E(\vec{u},\vec{\Phi})-\int_\Omega f\big(\text{tr}\varepsilon(\vec{\Phi})\big)\ud \Omega - \int_{\Gamma_B}(\hat{\sigma}\vec{n})\cdot\vec{\Phi}\ud \Gamma = 0, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:uneq2d39}
\end{align}
%
From Green's formula (\ref{eq:uneq2d37}) we have 
\begin{equation}
a_E(\vec{u},\vec{\Phi})=\int_{\Gamma_B} (\tau\vec{n}) \cdot \vec{\Phi}\ud \Gamma - \int_\Omega [\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2]^T \cdot \vec{\Phi} \ud \Omega, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:uneq2d40}
\end{equation}
%
Now 
\begin{align}
\int_\Omega f\big(\text{tr}\varepsilon(\vec{\Phi})\big)\ud \Omega &=\int_\Omega f (\nabla \cdot \vec{\Phi}) \ud \Omega \nonumber \\[6pt]
&=\int_\Omega\nabla \cdot (f\vec{\Phi})-\nabla f \cdot \vec{\Phi} \ud \Omega \nonumber \\[6pt]
&=\int_{\Gamma_B} f\vec{\Phi} \cdot \vec{n} \ud \Gamma - \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega \nonumber \\[6pt]
&=\int_{\Gamma_B} (fI\vec{n}) \cdot \vec{\Phi} \ud \Gamma - \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:uneq2d41}
\end{align}
%
Applying (\ref{eq:uneq2d40}) and (\ref{eq:uneq2d41}) to (\ref{eq:uneq2d39}), we have
\begin{gather*}
\int_{\Gamma_B} (\tau\vec{n}) \cdot \vec{\Phi}\ud \Gamma - \int_\Omega [\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2]^T \cdot \vec{\Phi} \ud \Omega - \\[6pt]
-\int_{\Gamma_B} (fI\vec{n}) \cdot \vec{\Phi} \ud \Gamma + \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega -\int_{\Gamma_B} (\hat{\sigma}\vec{n}) \cdot \vec{\Phi} \ud \Gamma =0, \qquad \Rightarrow
\end{gather*}
%
\begin{equation*}
\int_{\Gamma_B} (\tau-fI-\hat{\sigma})\vec{n} \cdot \vec{\Phi} \ud \Gamma + \int_\Omega \big(-[\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2]^T+\nabla f\big) \cdot \vec{\Phi} \ud \Omega =0, \qquad \forall ~\vec{\Phi} \in V_0 
\vspace{0.1in}
\end{equation*}
%
It follows that 
\begin{align}
[\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2]^T = \nabla f \qquad &\text{in } \Omega \label{eq:uneq2d42} \\[6pt]
\tau \vec{n}=(fI+\hat{\sigma})\vec{n} \qquad &\text{on } \Gamma_B \label{eq:uneq2d43} 
\end{align} 
%
From (\ref{eq:uneq2d27}),
\begin{align*}
\nabla \cdot \vec{\tau}_1=\tau_{11_x}+\tau_{12_y}&=\big(K+\frac{G}{3}\big)u_{xx}+\big(K-\frac{2G}{3}\big)v_{yx}+Gu_{xx}+Gu_{yy}+Gv_{xy} \\[6pt]
&=\big(K+\frac{G}{3}\big)u_{xx}+\big(K+\frac{G}{3}\big)v_{yx}+Gu_{xx}+Gu_{yy} \\[6pt]
\nabla \cdot \vec{\tau}_2=\tau_{21_x}+\tau_{22_y}&=Gu_{yx}+Gv_{xx}+\big(K+\frac{G}{3}\big)v_{yy}+\big(K-\frac{2G}{3}\big)u_{xy}+Gv_{yy} \\[6pt]
&=\big(K+\frac{G}{3}\big)u_{xy}+\big(K+\frac{G}{3}\big)v_{yy}+Gv_{xx}+Gv_{yy} \qquad \Rightarrow \\[-6pt]
\end{align*} 
%
\begin{equation*}
\begin{bmatrix} \\[-2.5em]
\nabla \cdot \vec{\tau}_1 \\[-6pt]
\nabla \cdot \vec{\tau}_2
\end{bmatrix} = \big(K+\frac{G}{3}\big) \begin{bmatrix} \\[-2.5em]
(u_x+v_y)_x \\[-6pt]
(u_x+v_y)_y 
\end{bmatrix} + G\begin{bmatrix} \\[-2.5em]
(u_{xx}+u_{yy}) \\[-6pt]
(v_{xx}+v_{yy}) 
\end{bmatrix} \qquad \Rightarrow 
\end{equation*} \\[-12pt]
%
\begin{equation}
[\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2]^T = \big(K+\frac{G}{3}\big)\nabla(\nabla \cdot \vec{u})+G\nabla^2\vec{u} \label{eq:uneq2d44}
\end{equation} 
%
From (\ref{eq:uneq2d42})-(\ref{eq:uneq2d44}) and (\ref{eq:uneq2d15}),
\begin{align}
\big(K+\frac{G}{3}\big)\nabla(\nabla \cdot \vec{u})+G\nabla^2\vec{u} = \nabla f \qquad &\text{in } \Omega \times (0,t_f)\label{eq:uneq2d45} \\[6pt]
\tau \vec{n}=(fI+\hat{\sigma})\vec{n} \qquad &\text{on } \Gamma_B \times (0,t_f) \tag{\ref{eq:uneq2d43}}\\[6pt]
\vec{u} \approx 0 \qquad \qquad \qquad &\text{on } \Gamma_F \times (0,t_f) \tag{\ref{eq:uneq2d14}}
\end{align} 
%
Differentiating (\ref{eq:uneq2d45}) with respect to time and applying (\ref{eq:uneq2d30}), we obtain the system (\ref{eq:uneq2d13}). On the other hand, since $\dot{\vec{u}} \in V_0,$ from (\ref{eq:uneq2d39}) we have \\
\vspace{-0.2in}
\begin{align}
a_E(\dot{\vec{u}},\vec{\Phi})-\int_\Omega f\big(\text{tr}\varepsilon(\vec{\Phi})\big)\ud \Omega - \int_{\Gamma_B}(\hat{\sigma}\vec{n})\cdot\vec{\Phi}\ud \Gamma = 0, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:uneq2d46}
\end{align}
Using the same argument as above and applying (\ref{eq:uneq2d30}), we obtain the equivalence of systems (\ref{eq:uneq2d45}), (\ref{eq:uneq2d43}), (\ref{eq:uneq2d14}) and (\ref{eq:uneq2d13})-(\ref{eq:uneq2d15}).
%
\begin{remark} [\cite{Barber}]  \label{Rem1}
The condition for mechanical equilibrium is 
\begin{equation*}
\nabla \cdot \sigma=0
\end{equation*}
where $~\sigma=\tau-fI$ is the poroelastic stress tensor. \\
%
From this relation, (\ref{eq:uneq2d27}), and (\ref{eq:uneq2d30}), we obtain the system (\ref{eq:uneq2d45}) that constitutes the equations of equilibrium.
\end{remark}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3D Underlying Equations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.6in}
\section {Three-dimensional Model}
%
The initial-boundary value problem that constitutes our model is defined in a bounded open domain $~\Omega$ in $~\reals ^3$ exterior to the borehole, with a sufficiently smooth boundary $~\Gamma$. Specifically, $~\Omega \subset \reals^3$ is an inclined cylindrical region with an inner radius $~R_{B}$ and an outer radius $~R_{F}$. \\
%
\indent
Let $(X,Y,Z)$ be a Cartesian coordinate system such that\\
\indent \indent
$x$ -axis is in the direction of maximum horizontal principal stress $\sigma_H(z),$\\
\indent \indent 
$y$ -axis is in the direction of minimum horizontal principal stress $\sigma_h(z),$ and\\
\indent \indent
$z$ -axis is in the direction of overburden stress $\sigma_V$ and \\
\noindent
$(X_{b}, Y_{b}, Z_{b})$ be a borehole Cartesian coordinate system. \\
%
\indent
A point $~\bar{x}$ in $\reals ^3$ is associated with position vector $~\bar{x}=x\vec{i}+y\vec{j}+z\vec{k}$, where $~\vec{i}$, $~\vec{j}$, and $~\vec{k}$ are the basis vectors of the borehole Cartesian coordinate system $~(X_{b}, Y_{b}, Z_{b})$. Displacements of points within $~\Omega$ are described in terms of a vector-valued displacement function of the position $~\bar{x}$ and time $~t$ as $~\vec{u}(\bar{x},t)=u(\bar{x},t)\vec{i}+v(\bar{x},t)\vec{j}+w(\bar{x},t)\vec{k}$.\\
\indent
The boundary $~\Gamma$ of $~\Omega$ is $~\Gamma = \Gamma_B\cup\Gamma_F\cup\tilde{\Gamma}_T\cup\tilde{\Gamma}_B$, where
\begin{gather*}
\Gamma_B = \left\{ (x,y,z) \in \reals^3: \sqrt{x^2+y^2}=R_B, ~z_T \leq z \leq z_B \right\} \\[6pt]
\Gamma_F = \left\{ (x,y,z) \in \reals^3: \sqrt{x^2+y^2}=R_F, ~z_T \leq z \leq z_B \right\} \\[6pt]
\tilde{\Gamma}_T = \left\{ (x,y,z_T) \in \reals^3: R_B \leq \sqrt{x^2+y^2} \leq R_F \right\} \\[6pt]
\tilde{\Gamma}_B = \left\{ (x,y,z_B) \in \reals^3: R_B \leq \sqrt{x^2+y^2} \leq R_F \right\}
\end{gather*}
%
Figure ~\ref{3Dregion} shows the two-dimensional region $\Omega$ in $\reals ^3$ exterior to the borehole.\\
%
\vspace{-0.2 in}
\begin{figure}[ht]
\centering
\includegraphics[scale=0.48]{borehole_3D.png}
\caption{Three-dimensional region exterior to the borehole}
\label{3Dregion}
\end{figure}\\
\indent
Now we formulate the initial-boundary value problem that constitutes the three-dimensional radially nonsymmetric TCPu model. The thermal diffusion, solute diffusion, and fluid diffusion equations, as well as the Navier-type elastic equations for the three-dimensional problem are the same as in the two-dimensional case and thus, the problem is the following. \\\\ 
%
Thermal diffusion:
\begin{equation}
\Lambda\dot{T}+\Sigma\dot{C}+\Phi\dot{p}-\frac{k^T}{T_F}\nabla^2T-\frac{\rho_fRT_FD^T}{M^s(1-C_F)}\nabla^2C+K^T\nabla^2p=-\zeta(\nabla\cdot\dot{\vec{u}}) \qquad \text{in } \Omega \times(0,t_f) \label{eq:uneq3d1}
\end{equation}
with boundary conditions
\begin{subequations} \label{eq:uneq3d2}
\begin{alignat}{2}
T(\bar{x},t) &=T_B(z,t) \qquad &\text{on } \Gamma_B\times[0,t_f) \\
T(\bar{x},t) &=T_F(\bar{x}) \qquad &\text{on } \Gamma_F\times[0,t_f) \\
T(\bar{x},t) &=\tilde{T}_T(x,y,t) \qquad &\text{on } \tilde{\Gamma}_T \times [0,t_f) \\
T(\bar{x},t) &=\tilde{T}_B(x,y,t) \qquad &\text{on } \tilde{\Gamma}_B \times [0,t_f)
\end{alignat}
\end{subequations}
compatibility conditions
\begin{subequations} \label{eq:uneq3d3}
\begin{align}
\tilde{T}_T(t)\big|_{\Gamma_B} &=T_B(t) \big|_{\tilde{\Gamma}_T} \\
\tilde{T}_T(t)\big|_{\Gamma_F} &=T_F \big|_{\tilde{\Gamma}_T} \\
\tilde{T}_B(t)\big|_{\Gamma_B} &=T_B(t) \big|_{\tilde{\Gamma}_B} \\
\tilde{T}_B(t)\big|_{\Gamma_F} &=T_F \big|_{\tilde{\Gamma}_T} 
\end{align}
\end{subequations}
and initial conditions
\begin{equation}
T(\bar{x},0)=\tilde{T}_I(\bar{x}) \qquad \text{in } \Omega \label{eq:uneq3d4}
\end{equation}
%
Solute diffusion:
\begin{equation}
\phi\dot{C}-C_FD^T\nabla^2T-D\nabla^2C+\frac{k\mathfrak{R}}{\eta}\nabla^2p=0 \qquad \text{in } \Omega \times(0,t_f) \label{eq:uneq3d5}
\end{equation}
with boundary conditions
\begin{subequations} \label{eq:uneq3d6}
\begin{alignat}{2}
C(\bar{x},t) &=C_B(z,t) \qquad &\text{on } \Gamma_B\times[0,t_f) \\
C(\bar{x},t) &=C_F(\bar{x}) \qquad &\text{on } \Gamma_F\times[0,t_f) \\
C(\bar{x},t) &=\tilde{C}_T(x,y,t) \qquad &\text{on } \tilde{\Gamma}_T \times [0,t_f)\\ 
C(\bar{x},t) &=\tilde{C}_B(x,y,t) \qquad &\text{on } \tilde{\Gamma}_B \times [0,t_f) 
\end{alignat}
\end{subequations}
compatibility conditions
\begin{subequations} \label{eq:uneq3d7}
\begin{align}
\tilde{C}_T(t)\big|_{\Gamma_B}&=C_B(t)\big|_{\tilde{\Gamma}_T} \\
\tilde{C}_T(t)\big|_{\Gamma_F}&=C_F\big|_{\tilde{\Gamma}_T} \\
\tilde{C}_B(t)\big|_{\Gamma_B}&=C_B(t)\big|_{\tilde{\Gamma}_B} \\
\tilde{C}_B(t)\big|_{\Gamma_F}&=C_F\big|_{\tilde{\Gamma}_T}
\end{align}
\end{subequations}
and initial conditions
\begin{equation}
C(\bar{x},0)=\tilde{C}_I(\bar{x}) \qquad \text{in } \Omega \label{eq:uneq3d8}
\end{equation}
%
Fluid diffusion:
\begin{equation}
\Gamma\dot{T}+\chi\dot{C}+\Psi\dot{p}+K^T\nabla^2T+\frac{\rho_f\mathfrak{R}\Omega k}{\eta}\nabla^2C-\frac{k}{\eta}\nabla^2p=-\alpha(\nabla\cdot\dot{\vec{u}}) \quad \text{in } \Omega \times (0,t_f) \label{eq:uneq3d9}
\end{equation}
with boundary conditions
\begin{subequations} \label{eq:uneq3d10}
\begin{alignat}{2}
p(\bar{x},t) &=p_B(z,t) \qquad &\text{on } \Gamma_B\times[0,t_f) \\
p(\bar{x},t) &=p_F(\bar{x}) \qquad &\text{on } \Gamma_F\times[0,t_f) \\
p(\bar{x},t) &=\tilde{p}_T(x,y,t) \qquad &\text{on } \tilde{\Gamma}_T \times [0,t_f)\\ 
p(\bar{x},t) &=\tilde{p}_B(x,y,t) \qquad &\text{on } \tilde{\Gamma}_B \times [0,t_f) 
\end{alignat}
\end{subequations}
compatibility conditions
\begin{subequations} \label{eq:uneq3d11}
\begin{align}
\tilde{p}_T(t)\big|_{\Gamma_B}&=p_B(t)\big|_{\tilde{\Gamma}_T} \\
\tilde{p}_T(t)\big|_{\Gamma_F}&=p_F\big|_{\tilde{\Gamma}_T} \\
\tilde{p}_B(t)\big|_{\Gamma_B}&=p_B(t)\big|_{\tilde{\Gamma}_B} \\
\tilde{p}_B(t)\big|_{\Gamma_F}&=p_F\big|_{\tilde{\Gamma}_T}
\end{align}
\end{subequations}
and initial conditions
\begin{equation}
p(\bar{x},0)=\tilde{p}_I(\bar{x}) \qquad \text{in } \Omega \label{eq:uneq3d12}
\end{equation}
%
Navier-type elastic equations:
\begin{equation}
\Big(K+\frac{G}{3}\Big)\nabla(\nabla\cdot\dot{\vec{u}})+G\nabla^2\dot{\vec{u}}-\tilde{\zeta}\nabla\dot{T}+\xi\nabla\dot{C}-\tilde{\alpha}\nabla\dot{p}=0 \qquad \text{in } \Omega \times(0,t_f) \label{eq:uneq3d13}
\end{equation}
with boundary conditions 
\begin{gather}
\vec{u}(\bar{x},t) \approx 0 \qquad \text{on } \Gamma_F\times(0,t_f) \label{eq:uneq3d14} \\
\dot{\vec{u}}(\bar{x},t) \approx 0 \qquad \text{on } \Gamma_F\times(0,t_f) \label{eq:uneq3d15}
\end{gather}
and initial conditions
\begin{equation}
\vec{u}(\bar{x},0)= 0 \qquad \text{in } \bar{\Omega}=\Omega \cup\Gamma \label{eq:uneq3d16}
\end{equation}
%
The description and values of physical constants are presented in Appendix A.\\ 
%
\indent
Using the matrices and vectors (\ref{eq:uneq2d17}) and (\ref{eq:uneq2d18}) defined for the two-dimensional case, we have the following TCP initial-boundary value problem:
%
\begin{equation} 
M \dot{\bar{V}} - A \nabla^{2} \bar{V}=-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f) \label{eq:uneq3d17}
\end{equation}
with boundary conditions
\begin{equation} \label{eq:uneq3d18}
\bar{V}(\bar{x},t)=
\begin{cases}
V_B(z,t) & \text{on } \Gamma_B \times [0,t_f)\\
V_F(\bar{x}) & \text{on } \Gamma_F \times [0,t_f)\\
\tilde{V}_T(x,y,t) & \text{on } \tilde{\Gamma}_T \times [0,t_f)\\
\tilde{V}_B(x,y,t) & \text{on } \tilde{\Gamma}_B \times [0,t_f)\\
\end{cases}
\end{equation}
%
compatibility conditions
\begin{subequations} \label{eq:uneq3d19}
\begin{align}
\tilde{V}_T(t)\big|_{\Gamma_B} &=V_B(t)\big|_{\tilde{\Gamma}_T} \\
\tilde{V}_T(t)\big|_{\Gamma_F} &=V_F\big|_{\tilde{\Gamma}_T} \\
\tilde{V}_B(t)\big|_{\Gamma_B} &=V_B(t)\big|_{\tilde{\Gamma}_B} \\
\tilde{V}_B(t)\big|_{\Gamma_F} &=V_F\big|_{\tilde{\Gamma}_T}
\end{align}
\end{subequations}
%
and initial conditions
\begin{equation} 
\bar{V}(\bar{x},0)=\tilde{V}_I(\bar{x}) \qquad \text{in } \Omega \label{eq:uneq3d20}
\end{equation}
%
As before, $~\vec{b}_0$ is the factor that couples the displacement velocity to the absolute temperature, the pore pressure, the solute mass fraction (TCP), and their velocities.\\
%
\indent
Next, using the principle of minimum total potential energy, we will show that the three-dimensional Navier-type elastic system (\ref{eq:uneq3d13})-(\ref{eq:uneq3d15}) is equivalent to the system 
%
\begin{align}
\big(K+\frac{G}{3}\big)\nabla(\nabla \cdot \vec{u})+G\nabla^2\vec{u} = \nabla f \qquad &\text{in } \Omega \times (0,t_f) \label{eq:uneq3d21} \\[6pt]
\tau \vec{n}=(fI+\hat{\sigma})\vec{n} \qquad &\text{on } \Gamma \setminus \Gamma_F \times (0,t_f) \label{eq:uneq3d22} \\[6pt]
\vec{u} \approx 0 \qquad \qquad \qquad &\text{on } \Gamma_F \times (0,t_f) \tag{\ref{eq:uneq3d14}}
\end{align}
where $~f~$ is given by (\ref{eq:uneq2d30}). \\
%
\indent
Here and in the following we will suppress the time dependence of the displacement vector for the sake of brevity. Given the displacement
\begin{equation*}
\vec{u}=u\vec{i}+v\vec{j}+w\vec{k} 
\end{equation*}
the linearized strain is a second order symmetric tensor \\[-6pt]
\begin{equation*}
\varepsilon=\begin{bmatrix} \\[-2.5em]
\varepsilon_{11}&\varepsilon_{12}&\varepsilon_{13} \\[-6pt]
\varepsilon_{21}&\varepsilon_{22}&\varepsilon_{23} \\[-6pt]
\varepsilon_{31}&\varepsilon_{32}&\varepsilon_{33} 
\end{bmatrix}=\frac{1}{2}\big(\nabla\vec{u}+\nabla\vec{u}^T\big)
\end{equation*}
where
\begin{gather*}
\varepsilon_{11}=u_x, \qquad \varepsilon_{12}=\varepsilon_{21}=\frac{1}{2}\big(u_y+v_x\big), \qquad \varepsilon_{22}=v_y \\
\varepsilon_{13}=\varepsilon_{31}=\frac{1}{2}\big(u_z+w_x\big), \qquad \varepsilon_{23}=\varepsilon_{32}=\frac{1}{2}\big(v_z+w_y\big), \qquad \varepsilon_{33}=w_z
\end{gather*}
%
\indent
For a homogeneous elastic isotropic medium, the stress-strain relation (\ref{eq:uneq2d23}) in terms of the bulk modulus $K$ and the shear modulus $G$ has the form\\
\begin{equation}
\tau=\begin{bmatrix} \\[-2.5em]
\tau_{11}&\tau_{12}&\tau_{13} \\[-6pt]
\tau_{21}&\tau_{22}&\tau_{23} \\[-6pt]
\tau_{13}&\tau_{23}&\tau_{33}
\end{bmatrix} = 2G\varepsilon + \big(K-\frac{2}{3}G\big)(\text{tr}\varepsilon)I \label{eq:uneq3d23}
\end{equation}
%
That is,
\begin{align}
\tau_{11}&=2Gu_x+\big(K-\frac{2}{3}G\big)\big(u_x+v_y+w_z\big) \nonumber \\
\tau_{12}&=\tau_{21}=G\big(u_y+v_x\big) \nonumber \\
\tau_{22}&=2Gv_y+\big(K-\frac{2}{3}G\big)\big(u_x+v_y+w_z\big) \nonumber\\
\tau_{13}&=\tau_{31}=G\big(u_z+w_x\big) \nonumber \\
\tau_{23}&=\tau_{32}=G\big(v_z+w_y\big) \nonumber \\
\tau_{33}&=2Gw_z+\big(K-\frac{2}{3}G\big)\big(u_x+v_y+w_z\big) \label{eq:uneq3d24}
\end{align} 
%
\indent
The total potential energy of our system is
\begin{equation}
\mathcal{V}(\vec{u})=\mathcal{V}_S(\vec{u})-W_b(\vec{u})-W_S(\vec{u}) \label{eq:uneq3d25}
\end{equation}
where
\begin{align}
\mathcal{V}_S(\vec{u}) = \frac{1}{2}&\int_\Omega \tau_{11}(\vec{u})\varepsilon_{11}(\vec{u})+2\tau_{12}(\vec{u})\varepsilon_{12}(\vec{u})+\tau_{22}(\vec{u})\varepsilon_{22}(\vec{u})+ \nonumber \\[6pt]
& +2\tau_{13}(\vec{u})\varepsilon_{13}(\vec{u})+2\tau_{23}(\vec{u})\varepsilon_{23}(\vec{u})+\tau_{33}(\vec{u})\varepsilon_{33}(\vec{u}) \ud \Omega \label{eq:uneq3d26}
\end{align}
is the elastic strain energy of the system;\\
%
\begin{equation}
W_b(\vec{u})=\int_\Omega f\big(\text{tr}\varepsilon(\vec{u})\big)\ud \Omega \label{eq:uneq3d27}
\end{equation}
with $~f~$ given by (\ref{eq:uneq2d30}), is work done by body forces due to the absolute temperature, the solute mass fraction, and the pore pressure ($~\vec{b}_1$ is the factor that couples the absolute temperature, the pore pressure, and the solute mass fraction (TCP) to the displacement); and \\
\begin{equation}
W_S(\vec{u})=\int_{\Gamma}(\hat{\sigma}\vec{n})\cdot\vec{u}\ud \Gamma \label{eq:uneq3d28}
\end{equation}
is the work done by applied boundary stress $~\hat{\sigma}$. Here $~\vec{n}$ is the outward unit normal vector on the boundary.\\
%
\indent
Applying (\ref{eq:uneq3d26}), (\ref{eq:uneq3d27}), and (\ref{eq:uneq3d28}) to (\ref{eq:uneq3d25}), we have
\begin{align}
\mathcal{V}(\vec{u}) &=\frac{1}{2}\int_\Omega \tau_{11}(\vec{u})\varepsilon_{11}(\vec{u})+2\tau_{12}(\vec{u})\varepsilon_{12}(\vec{u})+\tau_{22}(\vec{u})\varepsilon_{22}(\vec{u})+ \nonumber \\[6pt]
&\qquad \quad +2\tau_{13}(\vec{u})\varepsilon_{13}(\vec{u})+2\tau_{23}(\vec{u})\varepsilon_{23}(\vec{u})+\tau_{33}(\vec{u})\varepsilon_{33}(\vec{u}) \ud \Omega \nonumber \\[6pt]
&\quad -\int_\Omega f\big(\text{tr}\varepsilon(\vec{u})\big)\ud \Omega - \int_{\Gamma}(\hat{\sigma}\vec{n})\cdot\vec{u}\ud \Gamma \label{eq:uneq3d29}
\end{align}
%
\indent
To aid in giving a mathematical formulation, let
\begin{gather*}
H=L^2\big(\Omega; \reals ^3\big)=\big\{L^2\big(\Omega; \reals\big)\big\}^3 \\
V=H^1\big(\Omega; \reals ^3\big)=\big\{H^1\big(\Omega; \reals\big)\big\}^3 \\
V_0=\big\{\varphi\in V:\varphi\big|_{\Gamma_F} =0\big\} \\
\vec{\Phi}(\bar{x})=\phi(\bar{x})\vec{i}+\psi(\bar{x})\vec{j}+\omega(\bar{x})\vec{k}
\end{gather*}
%
\indent
Define the bilinear form on $V$ by 
\begin{align}
a_E(\vec{u},\vec{\Phi}) &= \int_\Omega\tau_{11}(\vec{u})\varepsilon_{11}(\vec{\Phi})+2\tau_{12}(\vec{u})\varepsilon_{12}(\vec{\Phi})+\tau_{22}(\vec{u})\varepsilon_{22}(\vec{\Phi})+2\tau_{13}(\vec{u})\varepsilon_{13}(\vec{\Phi})\nonumber \\[6pt]
& \quad +2\tau_{23}(\vec{u})\varepsilon_{23}(\vec{\Phi})+\tau_{33}(\vec{u})\varepsilon_{33}(\vec{\Phi}) \ud \Omega, \qquad \forall ~\vec{u}, \vec{\Phi} \in V \label{eq:uneq3d30}
\end{align}
%
Then from (\ref{eq:uneq3d26}) and (\ref{eq:uneq3d30})
\begin{equation}
\mathcal{V}_S(\vec{u})=\frac{1}{2}a_E(\vec{u},\vec{u}) \label{eq:uneq3d31}
\end{equation}
and, from (\ref{eq:uneq3d29}) and (\ref{eq:uneq3d31}), the total potential energy has the form 
\begin{align}
\mathcal{V}(\vec{u})&=\frac{1}{2}a_E(\vec{u},\vec{u})-\int_\Omega f\big(\text{tr}\varepsilon(\vec{u})\big)\ud \Omega - \int_{\Gamma}(\hat{\sigma}\vec{n})\cdot\vec{u}\ud \Gamma \label{eq:uneq3d32}
\end{align}
%
\indent
Define the following vectors:
\begin{equation*}
\vec{\tau}_1=[\tau_{11} \quad \tau_{12} \quad \tau_{13}]^T, \quad \vec{\tau}_2=[\tau_{21} \quad \tau_{22} \quad \tau_{23}]^T, \quad \vec{\tau}_3=[\tau_{31} \quad \tau_{32} \quad \tau_{33}]^T
\end{equation*}
%
Then 
\begin{align}
a_E(\vec{u},\vec{\Phi})&=\int_\Omega\tau_{11}\phi_x+\tau_{12}\phi_y+\tau_{13}\phi_z+\tau_{21}\psi_x+\tau_{22}\psi_y+\tau_{23}\psi_z+\nonumber \\[6pt]
&\qquad + \tau_{31}\omega_x+\tau_{32}\omega_y+\tau_{33}\omega_z\ud \Omega \nonumber \\[6pt]
&=\int_\Omega \vec{\tau}_1 \cdot \nabla \phi + \vec{\tau}_2 \cdot \nabla \psi + \vec{\tau}_3 \cdot \nabla \omega \ud \Omega \nonumber \\[6pt]
&=\int_\Omega \nabla \cdot (\phi \vec{\tau}_1) -\phi \nabla \cdot \vec{\tau}_1 + \nabla \cdot (\psi \vec{\tau}_2) -\psi \nabla \cdot \vec{\tau}_2 + \nabla \cdot (\omega \vec{\tau}_3) -\omega \nabla \cdot \vec{\tau}_3 \ud\Omega \nonumber \\[6pt]
&=\int_{\Gamma} \phi (\vec{\tau}_1 \cdot \vec{n})+\psi (\vec{\tau}_2 \cdot \vec{n})+\omega (\vec{\tau}_3 \cdot \vec{n}) \ud \Gamma - \nonumber \\[6pt]
& \qquad \qquad \qquad -\int_\Omega \phi \nabla \cdot \vec{\tau}_1+\psi \nabla \cdot \vec{\tau}_2+\omega \nabla \cdot \vec{\tau}_3\ud \Omega \nonumber \\[6pt]
&=\int_{\Gamma} (\tau\vec{n}) \cdot \vec{\Phi}\ud \Gamma - \int_\Omega [\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2 \quad \nabla \cdot \vec{\tau}_3]^T \cdot \vec{\Phi} \ud \Omega, \qquad \forall ~\vec{u}, ~\vec{\Phi} \in V \label{eq:uneq3d33}
\end{align}
%
Thus, we obtain the following Green's formula:
\begin{equation}
a_E(\vec{u},\vec{\Phi})=\int_{\Gamma} (\tau\vec{n}) \cdot \vec{\Phi}\ud \Gamma - \int_\Omega [\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2 \quad \nabla \cdot \vec{\tau}_3]^T \cdot \vec{\Phi} \ud \Omega, \qquad \forall ~\vec{u}, ~\vec{\Phi} \in V \label{eq:uneq3d34}
\end{equation}
%
\indent
Referring to (\ref{eq:uneq3d14}) and (\ref{eq:uneq3d15}), we assume that $~\vec{u}$ and $~\dot{\vec{u}}$ are negligible on $~\Gamma_F$ and therefore, the set of admissible displacements is 
\begin{equation*}
\mathcal{U}_{ad}=\big\{\vec{u} \in V_0:\dot{\vec{u}} \in V_0\big\}
\end{equation*}
%
\indent
Using the principle of minimum total potential energy, the displacement $~\vec{u}~$ that the region $~\Omega~$ undergoes is given by
\begin{equation}
D\mathcal{V}(\vec{u})\vec{\Phi}=0, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:uneq3d35}
\end{equation}
where 
\begin{equation*}
D\mathcal{V}(\vec{u})\vec{\Phi}=\frac{d}{d\delta}\mathcal{V}(\vec{u}+\delta\vec{\Phi})\Big\vert_{\delta=0}
\end{equation*}\\[-6pt]
is the G\^{a}teaux differential of $~\mathcal{V}$ with increment $~\vec{\Phi}$.\\
%
From (\ref{eq:uneq3d32}), (\ref{eq:uneq3d35})
\begin{equation}
a_E(\vec{u},\vec{\Phi})-\int_\Omega f\big(\text{tr}\varepsilon(\vec{\Phi})\big)\ud \Omega - \int_{\Gamma \setminus \Gamma_F}(\hat{\sigma}\vec{n})\cdot\vec{\Phi}\ud \Gamma = 0, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:uneq3d36}
\end{equation}
%
From Green's formula (\ref{eq:uneq3d34}) we have 
\begin{equation}
a_E(\vec{u},\vec{\Phi})=\int_{\Gamma \setminus \Gamma_F} (\tau\vec{n}) \cdot \vec{\Phi}\ud \Gamma - \int_\Omega [\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2 \quad \nabla \cdot \vec{\tau}_3]^T \cdot \vec{\Phi} \ud \Omega, \quad \forall ~\vec{\Phi} \in V_0 \label{eq:uneq3d37}
\end{equation}
%
Now 
\begin{align}
\int_\Omega f\big(\text{tr}\varepsilon(\vec{\Phi})\big)\ud \Omega &=\int_\Omega f (\nabla \cdot \vec{\Phi}) \ud \Omega \nonumber \\[6pt]
&=\int_\Omega\nabla \cdot (f\vec{\Phi})-\nabla f \cdot \vec{\Phi} \ud \Omega \nonumber \\[6pt]
&=\int_{\Gamma \setminus \Gamma_F} f\vec{\Phi} \cdot \vec{n} \ud \Gamma - \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega \nonumber \\[6pt]
&=\int_{\Gamma \setminus \Gamma_F} (fI\vec{n}) \cdot \vec{\Phi} \ud \Gamma - \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:uneq3d38}
\end{align}
%
Applying (\ref{eq:uneq3d37}) and (\ref{eq:uneq3d38}) to (\ref{eq:uneq3d36}), we have
\begin{gather*}
\int_{\Gamma \setminus \Gamma_F} (\tau\vec{n}) \cdot \vec{\Phi}\ud \Gamma - \int_\Omega [\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2 \quad \nabla \cdot \vec{\tau}_3]^T \cdot \vec{\Phi} \ud \Omega \\[6pt]
-\int_{\Gamma \setminus \Gamma_F} (fI\vec{n}) \cdot \vec{\Phi} \ud \Gamma + \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega -\int_{\Gamma \setminus \Gamma_F} (\hat{\sigma}\vec{n}) \cdot \vec{\Phi} \ud \Gamma =0
\end{gather*}
and so,
\begin{equation*}
\int_{\Gamma \setminus \Gamma_F} (\tau-fI-\hat{\sigma})\vec{n} \cdot \vec{\Phi} \ud \Gamma + \int_\Omega \big(-[\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2 \quad \nabla \cdot \vec{\tau}_3]^T+\nabla f\big) \cdot \vec{\Phi} \ud \Omega =0, \quad \forall ~\vec{\Phi} \in V_0 
\end{equation*}
%
It follows that 
\begin{align}
[\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2 \quad \nabla \cdot \vec{\tau}_3]^T = \nabla f \qquad &\text{in } \Omega \label{eq:uneq3d39} \\[6pt]
\tau \vec{n}=(fI+\hat{\sigma})\vec{n} \qquad &\text{on } \Gamma \setminus \Gamma_F \label{eq:uneq3d40} 
\end{align} 
%
From (\ref{eq:uneq3d24}),
\begin{align*}
\nabla \cdot \vec{\tau}_1&=\tau_{11_x}+\tau_{12_y}+\tau_{13_z} \\[6pt]
&=\big(K+\frac{G}{3}\big)u_{xx}+\big(K-\frac{2G}{3}\big)(v_{yx}+w_{zx})+ \\
& \quad +Gu_{xx}+Gu_{yy}+Gv_{xy}+Gu_{zz}+Gw_{xz} \\[6pt]
&=\big(K+\frac{G}{3}\big)u_{xx}+\big(K+\frac{G}{3}\big)(v_{yx}+w_{zx})+G(u_{xx}+u_{yy}+u_{zz}) \\[6pt]
&=\big(K+\frac{G}{3}\big)(\nabla \cdot \vec{u})_x+G\nabla^2u
\end{align*} 
%
\begin{align*}
\nabla \cdot \vec{\tau}_2&=\tau_{21_x}+\tau_{22_y}+\tau_{23_z} \\[6pt]
&=G(u_{yx}+v_{xx})+\big(K+\frac{G}{3}\big)v_{yy}+\big(K-\frac{2G}{3}\big)(u_{xy}+w_{zy})+ \\
& \quad +Gv_{yy}+G(v_{zz}+w_{yz}) \\[6pt]
&=\big(K+\frac{G}{3}\big)(u_{xy}+v_{yy}+w_{zy})+G(v_{xx}+v_{yy}+v_{zz}) \\[6pt]
&=\big(K+\frac{G}{3}\big)(\nabla \cdot \vec{u})_y+G\nabla^2v
\end{align*}
%
\begin{align*}
\nabla \cdot \vec{\tau}_3&=\tau_{31_x}+\tau_{32_y}+\tau_{33_z} \\[6pt]
&=G(u_{zx}+w_{xx})+G(v_{zy}+w_{yy})+\big(K+\frac{G}{3}\big)w_{zz}+ \\
& \quad +\big(K-\frac{2G}{3}\big)(u_{xz}+v_{yz})+Gw_{zz} \\[6pt]
&=\big(K+\frac{G}{3}\big)(u_{xz}+v_{yz}+w_{zz})+G(w_{xx}+w_{yy}+w_{zz}) \\[6pt]
&=\big(K+\frac{G}{3}\big)(\nabla \cdot \vec{u})_z+G\nabla^2w
\end{align*}
%
It follows that 
\begin{equation*}
\begin{bmatrix} \\[-2.4em]
\nabla \cdot \vec{\tau}_1 \\[-6pt]
\nabla \cdot \vec{\tau}_2 \\[-6pt]
\nabla \cdot \vec{\tau}_3
\end{bmatrix} = \big(K+\frac{G}{3}\big) \begin{bmatrix} \\[-2.4em]
(\nabla \cdot \vec{u})_x \\[-6pt]
(\nabla \cdot \vec{u})_y \\[-6pt]
(\nabla \cdot \vec{u})_z  
\end{bmatrix} + G\begin{bmatrix} \\[-2.4em]
\nabla^2 u \\[-6pt]
\nabla^2 v \\[-6pt]
\nabla^2 w
\end{bmatrix} \qquad \Rightarrow 
\end{equation*} 
%
\begin{equation}
[\nabla \cdot \vec{\tau}_1 \quad \nabla \cdot \vec{\tau}_2 \quad \nabla \cdot \vec{\tau}_3]^T = \big(K+\frac{G}{3}\big)\nabla(\nabla \cdot \vec{u})+G\nabla^2\vec{u} \label{eq:uneq3d41}
\end{equation} 
%
From (\ref{eq:uneq3d39})-(\ref{eq:uneq3d41}) and (\ref{eq:uneq3d14})
\begin{align}
\big(K+\frac{G}{3}\big)\nabla(\nabla \cdot \vec{u})+G\nabla^2\vec{u} = \nabla f \qquad &\text{in } \Omega \times (0,t_f)\label{eq:uneq3d42} \\[6pt]
\tau \vec{n}=(fI+\hat{\sigma})\vec{n} \qquad &\text{on } \Gamma \setminus \Gamma_F \times (0,t_f) \tag{\ref{eq:uneq3d40}} \\[6pt]
\vec{u} \approx 0 \qquad \qquad \qquad &\text{on }\Gamma_F \times (0,t_f) \tag{\ref{eq:uneq3d14}}
\end{align} 
%
Differentiating (\ref{eq:uneq3d42}) with respect to time and applying (\ref{eq:uneq2d30}), we obtain the system (\ref{eq:uneq3d13}). On the other hand, since $\dot{\vec{u}} \in V_0,$ from (\ref{eq:uneq3d36}) we have 
\begin{equation}
a_E(\dot{\vec{u}},\vec{\Phi})-\int_\Omega f\big(\text{tr}\varepsilon(\vec{\Phi})\big)\ud \Omega - \int_{\Gamma \setminus \Gamma_F}(\hat{\sigma}\vec{n})\cdot\vec{\Phi}\ud \Gamma = 0, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:uneq3d43}
\end{equation}
Using the same argument as above and applying (\ref{eq:uneq2d30}), we obtain the equivalence of systems (\ref{eq:uneq3d42}), (\ref{eq:uneq3d40}), (\ref{eq:uneq3d14}) and (\ref{eq:uneq3d13})-(\ref{eq:uneq3d15}).
%
\begin{remark} 
From the condition of mechanical equilibrium given in Remark \ref{Rem1}, (\ref{eq:uneq3d24}), and (\ref{eq:uneq2d30}), it follows that the system (\ref{eq:uneq3d42}) constitutes the equation of equilibrium. 
\end{remark}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Diagonalized TCP system %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\vspace{0.6in}
\section {Diagonalized TCP System with Homogeneous Boundary Conditions}
%
In the next chapter, we will derive the weak formulation of the TCP initial-boundary value problem. To aid in the formulation of the weak TCP problem, in this section we obtain the diagonalized two- and three-dimensional TCP systems with homogeneous boundary conditions.\\
\indent
First, we obtain the two-dimensional TCP system with homogeneous boundary conditions that is equivalent to (\ref{eq:uneq2d19})-(\ref{eq:uneq2d21}). To that end, we define the function
\begin{equation}
W(\bar{x},t)=V_B(t)+\frac{|\bar{x}|-R_B}{R_F-R_B}\big(V_F(\bar{x})-V_B(t)\big) \label{eq:dtcp1}
\end{equation}
and the vector
\begin{equation}
V(\bar{x},t)=\bar{V}(\bar{x},t)-W(\bar{x},t) \label{eq:dtcp2}
\end{equation}
%
Then (\ref{eq:uneq2d19})-(\ref{eq:uneq2d21}) has the form
\begin{equation*} 
M \dot{V} - A \nabla^{2} V=-M \dot{W}+A \nabla^{2} W-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f) 
\end{equation*}
%
with boundary conditions
\begin{equation*} 
V(\bar{x},t)= 0 \qquad \text{on } \Gamma \times [0,t_f) 
\end{equation*}
%
and initial conditions 
\begin{equation*} 
V(\bar{x},0)= V_0(\bar{x}) \qquad \text{in } \Omega 
\end{equation*} 
where
\begin{equation} 
V_0(\bar{x})= \tilde{V}_I(\bar{x})-\dfrac{R_F-|\bar{x}|}{R_F-R_B}V_B(0)-\dfrac{|\bar{x}|-R_B}{R_F-R_B}V_F(\bar{x}) \label{eq:dtcp3}
\vspace{0.2in}
\end{equation} 
%
\indent
Now we obtain the three-dimensional TCP system with homogeneous boundary conditions that is equivalent to (\ref{eq:uneq3d17})-(\ref{eq:uneq3d20}). Define the function
\begin{equation}
W^0(\bar{x},t)=\frac{R_F-\sqrt{x^2+y^2}}{R_F-R_B}V_B(z,t)+\frac{\sqrt{x^2+y^2}-R_B}{R_F-R_B}V_F(\bar{x}) \label{eq:dtcp4}
\end{equation}
and the vector
\begin{equation}
\bar{V}^0(\bar{x},t)=\bar{V}(\bar{x},t)-W^0(\bar{x},t) \label{eq:dtcp5}
\end{equation}
%
Then (\ref{eq:uneq3d17})-(\ref{eq:uneq3d20}) has the form
\begin{equation*} 
M \dot{\bar{V}}^0 - A \nabla^{2} \bar{V}^0=-M \dot{W}^0+A \nabla^{2}W^0-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f)
\end{equation*}
%
with boundary conditions
\begin{equation*} 
\bar{V}^0(\bar{x},t)=
\begin{cases}
\qquad 0 & \text{on } (\Gamma_B \cup \Gamma_F) \times [0,t_f)\\
\tilde{V}_T(x,y,t)-W^0(x,y,z_T,t) & \text{on } \tilde{\Gamma}_T \times [0,t_f)\\
\tilde{V}_B(x,y,t)-W^0(x,y,z_B,t) & \text{on } \tilde{\Gamma}_B \times [0,t_f) \\
\end{cases} 
\end{equation*}\\[-6pt]
%
and initial conditions 
\begin{equation*} 
\bar{V}^0(\bar{x},0)=\tilde{V}_I(\bar{x})-W^0(\bar{x},0) \qquad \text{in } \Omega 
\end{equation*}
%
Next define
\begin{align} 
\hat{V}_T(x,y,t) & =\tilde{V}_T(x,y,t)-W^0(x,y,z_T,t)  \label{eq:dtcp6} \\
\hat{V}_B(x,y,t) & =\tilde{V}_B(x,y,t)-W^0(x,y,z_B,t) \label{eq:dtcp7}
\end{align}
%
and 
\begin{equation} 
W^z(\bar{x},t) =\frac{z-z_T}{z_B-z_T}\hat{V}_B(x,y,t)+\frac{z_B-z}{z_B-z_T}\hat{V}_T(x,y,t) \label{eq:dtcp8}
\end{equation}
\begin{align}
V & =\bar{V}^0-W^z=\bar{V}-W^0-W^z, \quad \text{so} \nonumber \\
V & =\bar{V}-W, \qquad \text{where} ~~W=W^0+W^z  \label{eq:dtcp9}
\end{align}
%
Then (\ref{eq:uneq3d17})-(\ref{eq:uneq3d20}) has the form 
\begin{equation*} 
M \dot{V} - A \nabla^{2} V =-M \dot{W}+A \nabla^{2}W-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f)
\end{equation*}
%
with boundary conditions
\begin{equation*}
V(\bar{x},t)=0 \qquad \text{on } \Gamma \times[0,t_f) 
\end{equation*}
%
and initial conditions
\begin{equation*}
V(\bar{x},0)=V_0(\bar{x}) \qquad \text{in } \Omega 
\end{equation*}
where
\begin{equation}
V_0(\bar{x})=\tilde{V}_I(\bar{x})-W(\bar{x},0) \label{eq:dtcp10}
\end{equation}
%
Note that both the two-dimensional TCP system ~(\ref{eq:uneq2d19})-(\ref{eq:uneq2d21}) and the three-dimensional TCP system ~(\ref{eq:uneq3d17})-(\ref{eq:uneq3d20}) are equivalent to the following parabolic system with homogeneous boundary conditions:
%
\begin{equation} \label{eq:dtcp11}
M \dot{V} - A \nabla^{2} V =-M \dot{W}+A \nabla^{2}W-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f)
\end{equation}
%
with boundary conditions
\begin{equation}
V(\bar{x},t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:dtcp12}
\end{equation}
%
and initial conditions
\begin{equation}
V(\bar{x},0)=V_0(\bar{x}) \qquad \text{in } \Omega \label{eq:dtcp13}
\end{equation}
%
\indent
Now we diagonalize the TCP system (\ref{eq:dtcp11})-(\ref{eq:dtcp13}). To that end, we make the following assumption.
%
\begin{assumption} \label{assm:ch1assm1}
The matrix $M$ is nonsingular and the matrix 
\begin{equation}
\mathcal{A}=M^{-1}A \label{eq:dtcp14}
\end{equation} 
has three distinct positive eigenvalues $~~\lambda_1>\lambda_2>\lambda_3>0$.
\end{assumption}
\vspace{0.1in}
%
\begin{remark}
Under Assumption \ref{assm:ch1assm1}, the matrix $\mathcal{A}$ is diagonalizable:
\begin{equation}
\mathcal{A}=PDP^{-1} \label{eq:dtcp15}
\end{equation}
where 
\begin{equation*}
D=\begin{bmatrix} \\[-2.5em]
\lambda_1 & 0 & 0 \\[-6pt]
0 & \lambda_2 & 0 \\[-6pt]
0 & 0 & \lambda_3  
\end{bmatrix}, \qquad P=[\vec{e}_1 \quad \vec{e}_2 \quad \vec{e}_3],
\end{equation*}
$\vec{e}_i$ is the eigenvector of $\mathcal{A}$ corresponding to the eigenvalue $\lambda_i$, $||\vec{e}_i||=1, \quad 1\leq i \leq 3.$ 
\end{remark}
%
Let
\begin{equation}
U=\begin{bmatrix} \\[-2.5em]
U_1 \\[-6pt]
U_2 \\[-6pt]
U_3 
\end{bmatrix} =P^{-1}V  \label{eq:dtcp16}
\end{equation}\\
%
Using (\ref{eq:dtcp14})-(\ref{eq:dtcp16}), the TCP system (\ref{eq:dtcp11})-(\ref{eq:dtcp13}) has the form
\begin{equation}
\dot{U} - D \nabla^{2} U=-P^{-1}\dot{W}+DP^{-1}\nabla^{2} W-P^{-1}M^{-1}\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f) \label{eq:dtcp17}
\end{equation}
%
with boundary conditions
\begin{equation} 
U(\bar{x},t)= 0 \qquad \text{on } \Gamma \times [0,t_f) \label{eq:dtcp18}
\end{equation}
%
and initial conditions 
\begin{equation} 
U(\bar{x},0)=P^{-1}V_0(\bar{x}) \qquad \text{in } \Omega \label{eq:dtcp19}
\end{equation}
%
Define 
\begin{gather}
F_U=-P^{-1}\dot{W}+DP^{-1}\nabla^{2} W-P^{-1}M^{-1}\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \label{eq:dtcp20} \\[6pt]
U_0(\bar{x})= U(\bar{x},0)=P^{-1}V_0(\bar{x}) \label{eq:dtcp21}
\end{gather}
%
Then we have the following diagonalized TCP system:
\begin{equation}
\dot{U} - D \nabla^{2} U=F_U \qquad \text{in }  \Omega \times (0, t_f) \label{eq:dtcp22}
\end{equation}
%
with homogeneous boundary conditions
\begin{equation} 
U(\bar{x},t)= 0 \qquad \text{on } \Gamma \times [0,t_f) \label{eq:dtcp23}
\end{equation}
%
and initial conditions 
\begin{equation} 
U(\bar{x},0)=U_0(\bar{x})\qquad \text{in } \Omega \label{eq:dtcp24}
\end{equation}
\\[-6pt]
\indent
In this chapter, we presented a fully coupled parabolic-elliptic initial-boundary value problem that models fully coupled thermal, chemical, hydraulic, and mechanical processes in homogeneous isotropic chemically active porous media saturated by a fluid. It was shown that, under appropriate assumptions on the matrices of diffusion coefficients, the parabolic initial-boundary value problem is equivalent to the diagonalized one with homogeneous boundary conditions. Using the principle of the minimum total potential energy, we derived the equivalent formulation of the Navier-type elliptic initial-boundary value problem. While the original elliptic system is developed in terms of velocity, the derived system is formulated in terms of the displacement vector and contains compatibility boundary conditions on stresses. The obtained results play a key role in well-posedness analysis presented in the next chapter.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Well-posedness of the TCPu problem %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\chapter {Well-Posedness of the Coupled TCPu Problem}
We begin by defining what we mean by a well-posed problem.  
\begin{definition}[R. Showalter ~\cite{Showalter}]
A problem is called well-posed if for each set of data there exists exactly one solution and this dependence of the solution on the data is continuous. 
\end{definition}
%
First, we will discuss the well-posedness of the parabolic initial-boundary value problem for the TCP system and the elliptic initial-boundary value problem for the elastic system considering coupling terms as data. Then we will investigate the existence and uniqueness of a solution to the coupled TCPu problem. \\[\baselineskip]
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Weak formulation of the TCP problem %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section {TCP Problem}
%
To aid in giving a mathematical formulation of the problem, we make the following assumption.
\begin{assumption} \label{assm:ch2assm1}
$V_B(t)$ and $~\dot{V}_B(t)$ are $~H^1\big([0, t_f]; \reals^3\big)$-functions; $V_B(z,t)$ and $~\dot{V}_B(z,t)$ are $~H^1\big(\Gamma_B \times [0, t_f]; \reals^3\big)$-functions; $~V_F(\bar{x})$ is an $H^2(\Omega;\reals^3)$-function; $~\tilde{V}_T(\bar{x},t)$ and $~\tilde{V}_B(\bar{x},t)$ are $~L^2$-integrable functions from $~[0,t_f]$ into $H^2(\Omega;\reals^3)$; ~and $~\dot{\tilde{V}}_T(\bar{x},t)$ and $~\dot{\tilde{V}}_B(\bar{x},t)$ are $~L^2$-integrable functions from $~[0,t_f]$ into $L^2(\Omega;\reals^3).$
\vspace{0.1in}
\end{assumption}
%
We start with the weak formulation of the system (\ref{eq:dtcp22})-(\ref{eq:dtcp24}). Suppose that $U$ is a classical solution of the system (\ref{eq:dtcp22})-(\ref{eq:dtcp24}), say $U \in C^2(\bar{\Omega}\times [0,t_f])$. \\[6pt]
% 
Let $\varphi(\bar{x})=\begin{bmatrix} \\[-2.5em]
\varphi_1 \\[-6pt]
\varphi_2 \\[-6pt]
\varphi_3 
\end{bmatrix}(\bar{x}) \in C_0^\infty (\Omega;\reals ^3)$. Then from (\ref{eq:dtcp22}) we have 
\vspace{1pt}
%
\begin{gather*}
\int_\Omega \dot{U}\cdot \varphi \ud \Omega - \int_\Omega (D\nabla^2U)\cdot\varphi\ud\Omega = \int_\Omega F_U\cdot\varphi \ud\Omega \qquad \Rightarrow \\[6pt]
\int_\Omega \dot{U}\cdot \varphi \ud \Omega - \sum_{k=1}^3\int_\Omega \lambda_k\nabla^2U_k\varphi_k \ud\Omega = \int_\Omega F_U\cdot\varphi \ud\Omega 
\end{gather*}
%
Applying Green's first identity, 
\begin{equation*}
\int_\Omega \dot{U}\cdot \varphi \ud \Omega + \sum_{k=1}^3\int_\Omega \lambda_k(\nabla U_k\cdot \nabla\varphi_k) \ud\Omega = \int_\Omega F_U\cdot\varphi \ud\Omega
\end{equation*}
%
Since
\begin{equation*}
\int_\Omega \dot{U}\cdot \varphi \ud \Omega = \frac{d}{dt}\int_\Omega U\cdot \varphi \ud \Omega,
\end{equation*}
%
it follows that
\begin{equation}
\frac{d}{dt}\int_\Omega U\cdot \varphi \ud \Omega+ \sum_{k=1}^3\int_\Omega \lambda_k(\nabla U_k\cdot \nabla\varphi_k) \ud\Omega = \int_\Omega F_U\cdot\varphi \ud\Omega \label{eq:wptcp1}
\end{equation}
%
This equation holds $~\forall ~\varphi \in C_0^\infty (\Omega;\reals ^3)$ and, by continuity, it holds $~\forall ~\varphi \in H_0^1 (\Omega;\reals ^3)$.\\
\indent
Define the following spaces:
\begin{equation*}
\mathcal{H}=L^2(\Omega;\reals^3)=\big\{L^2(\Omega)\big\}^3\\
\end{equation*} 
with the scalar product
\begin{equation}
(u,v)_{\mathcal{H}}=\sum_{k=1}^3(u_k,v_k)_{L^2(\Omega)}=\int_{\Omega} u\cdot v \ud\Omega, \qquad u=\begin{bmatrix} \\[-2.5em]
u_1 \\[-6pt]
u_2 \\[-6pt]
u_3 
\end{bmatrix}, v=\begin{bmatrix} \\[-2.5em]
v_1 \\[-6pt]
v_2 \\[-6pt]
v_3 
\end{bmatrix} \in \mathcal{H} \label{eq:wptcp2}
\end{equation}
and the norm $~\displaystyle{||u||_{\mathcal{H}}=(u,u)_{\mathcal{H}}^{1/2}}~$ and
% 
\begin{equation*}
\mathcal{V}_0=H_0^1(\Omega;\reals^3)=\big\{H_0^1(\Omega)\big\}^3\\
\end{equation*} 
with the scalar product
\begin{equation}
(u,v)_{\mathcal{V}_0}=\sum_{k=1}^3(u_k,v_k)_{H_0^1(\Omega)}=\sum_{k=1}^3\int_{\Omega} \nabla u_k\cdot \nabla v_k \ud\Omega, \quad u=\begin{bmatrix} \\[-2.5em]
u_1 \\[-6pt]
u_2 \\[-6pt]
u_3 
\end{bmatrix}, v=\begin{bmatrix} \\[-2.5em]
v_1 \\[-6pt]
v_2 \\[-6pt]
v_3 
\end{bmatrix} \in \mathcal{V}_0 \label{eq:wptcp3}
\end{equation}
and the norm $~\displaystyle{||u||_{\mathcal{V}_0}=(u,u)_{\mathcal{V}_0}^{1/2}}.$\\[6pt]
%
\indent
The space $~\mathcal{H}$ is the Cartesian product of three separable Hilbert spaces $~L^2(\Omega)$ and the space $~\mathcal{V}_0$ is the Cartesian product of three separable Hilbert spaces $~H_0^1(\Omega)$, therefore $~\mathcal{H}$ and $~\mathcal{V}_0$ are also separable Hilbert spaces. Since $~H_0^1(\Omega)$ is dense in $~L^2(\Omega)$ and the injection $~H_0^1(\Omega) \hookrightarrow L^2(\Omega)$ is continuous, it follows that $~\mathcal{V}_0$ is dense in $~\mathcal{H}$ and the injection $~\mathcal{V}_0 \hookrightarrow \mathcal{H}$ is continuous. Also, $~L^2(\Omega)$ is identified with its dual space $~L^2(\Omega)'$ and $~L^2(\Omega)'$ is dense in $~H_0^1(\Omega)'=H^{-1}(\Omega)$. It follows that $~\mathcal{H}$ is identified with its dual space $~\mathcal{H}'$ and $~\mathcal{H}'$ is dense in $~\mathcal{V}_0'=\big\{H^{-1}(\Omega)\big\}^3=H^{-1}(\Omega;\reals^3)$. We have
%
\begin{equation}
\mathcal{V}_0 \subset \mathcal{H}=\mathcal{H}' \subset \mathcal{V}_0' \label{eq:wptcp4}
\end{equation}
%
Also, the scalar product in $~\mathcal{H}$ of $~u \in\mathcal{H}$ and $~v \in\mathcal{V}_0$ is the same as the scalar product of $~u$ and $~v$ in the duality between $~\mathcal{V}_0'$ and $~\mathcal{V}_0$, denoted by $~<u,v>$:
%
\begin{equation}
<u,v>=(u,v)_{\mathcal{H}}, \qquad u\in\mathcal{H}, v\in\mathcal{V}_0 \label{eq:wptcp5}
\end{equation}
%
\indent
Define the bilinear form $~a:\mathcal{V}_0 \times \mathcal{V}_0 \rightarrow \reals$ by
\begin{equation}
a(u,v)=\sum_{k=1}^3 \lambda_k \int_{\Omega} \nabla u_k \cdot\nabla v_k \ud\Omega \label{eq:wptcp6}
\end{equation}
%
$a(\cdot,\cdot)$ is symmetric: 
\begin{equation}
a(u,v)=a(v,u) \label{eq:wptcp7}
\end{equation}
%
\begin{gather*}
a(u,u)=\sum_{k=1}^3\int_{\Omega}\lambda_k \nabla u_k \cdot\nabla u_k \ud\Omega = \sum_{k=1}^3\int_{\Omega}\lambda_k |\nabla u_k|^2 \ud\Omega \geq \\[6pt]
\geq \big\{\underset{k}{min}\lambda_k\big\}\sum_{k=1}^3\int_{\Omega}|\nabla u_k|^2 \ud\Omega = \lambda_3 ||u||_{\mathcal{V}_0}^2
\end{gather*}
Thus,
\begin{equation}
a(u,u)\geq \lambda_3 ||u||_{\mathcal{V}_0}^2, \qquad \lambda_3=\underset{k}{min}\lambda_k >0 \label{eq:wptcp8}
\end{equation}
Also,
\vspace{1pt}
\begin{gather*}
|a(u,v)|=\left|\sum_{k=1}^3\int_{\Omega}\lambda_k \nabla u_k \cdot\nabla v_k \ud\Omega\right| \leq \big\{\underset{k}{max}\lambda_k\big\}\sum_{k=1}^3\big|(u_k,v_k)_{H_0^1(\Omega)}\big| \\[6pt]
\leq \lambda_1 \sum_{k=1}^3||u_k||_{H_0^1(\Omega)}||v_k||_{H_0^1(\Omega)}\leq \lambda_1 \sum_{i=1}^3||u_i||_{H_0^1(\Omega)} \sum_{j=1}^3||v_j||_{H_0^1(\Omega)} \\[6pt]
\leq 3\lambda_1||u||_{\mathcal{V}_0}||v||_{\mathcal{V}_0}, \qquad \text{so}
\end{gather*}
%
\begin{equation}
|a(u,v)|\leq 3\lambda_1||u||_{\mathcal{V}_0}||v||_{\mathcal{V}_0} \label{eq:wptcp9}
\end{equation}
%
From (\ref{eq:wptcp7})-(\ref{eq:wptcp9}) it follows that $~a(\cdot,\cdot)$ defines a scalar product on $\mathcal{V}_0$. We denote this scalar product by 
\begin{equation}
(u,v)_a=a(u,v), \qquad u,v \in \mathcal{V}_0 \label{eq:wptcp10}
\end{equation}
and the corresponding (energy) norm by $~\displaystyle{||u||_a=(u,u)_a^{1/2}}.~$ Furthermore, the norm $||\cdot||_a$ is equivalent to the norm $||\cdot||_{\mathcal{V}_0}$.\\
%
\indent
For each $u\in\mathcal{V}_0$, the form $~u\in\mathcal{V}_0 \rightarrow (u,v)_a \in\reals~$ is linear and continuous on $\mathcal{V}_0$. Therefore, there exists an element $Au\in \mathcal{V}_0'$ such that\\
\vspace{-0.2in}
\begin{equation}
(u,v)_a=<Au,v>, \qquad \forall ~v\in\mathcal{V}_0 \label{eq:wptcp11}
\end{equation}
%
From (\ref{eq:wptcp9})-(\ref{eq:wptcp11}), the mapping $u\mapsto Au$ is linear and continuous, that is, $A\in\mathcal{L}(\mathcal{V}_0,\mathcal{V}_0')$. Also, by the Riesz representation theorem, for every $Au\in\mathcal{V}_0'$, there exists a unique $u\in\mathcal{V}_0$ such that (\ref{eq:wptcp11}) holds. Thus, $A:\mathcal{V}_0 \rightarrow \mathcal{V}_0'$ is an isomorphism from $\mathcal{V}_0$ onto $\mathcal{V}_0'$.\\
%
\indent
Let $\mathrm{X}$ be a Hilbert space. We introduce the space $L^2(0,t_f;\mathrm{X})$ of $L^2$-integrable functions from $[0,t_f]$ into $\mathrm{X}$ with the norm
\begin{equation*}
||u||_{L^2(0,t_f;\mathrm{X})}=\bigg[\int_0^{t_f}||u(t)||_{\mathrm{X}}^2\ud t\bigg]^{1/2}
\end{equation*}
%
Note that if $~u\in L^2(0,t_f;\mathcal{V}_0)$, then $~t \mapsto Au(t)\in\mathcal{V}_0'$, that is, 
\begin{equation*}
A\in\mathcal{L}\big(L^2(0,t_f;\mathcal{V}_0), L^2(0,t_f;\mathcal{V}_0')\big)
\end{equation*}
%
From Assumption \ref{assm:ch2assm1}, (\ref{eq:wptcp1}), (\ref{eq:wptcp2}), (\ref{eq:wptcp5}), (\ref{eq:wptcp6}), and (\ref{eq:wptcp10}), we obtain the following weak formulation of the problem (\ref{eq:dtcp22})-(\ref{eq:dtcp24}):\\[6pt]
%
Given $~F_U\in L^2(0,t_f;\mathcal{V}_0')$ and $~U_0\in\mathcal{H},~$ find $~U\in L^2(0,t_f;\mathcal{V}_0)$ such that 
\begin{equation}
\frac{d}{dt}(U,\varphi)_{\mathcal{H}}+(U,\varphi)_a = <F_U,\varphi>, \qquad \forall ~\varphi\in\mathcal{V}_0 \label{eq:wptcp12}
\end{equation}
and 
\begin{equation}
U(0)=U_0 \qquad \text{in } \mathcal{H} \label{eq:wptcp13}
\end{equation}
The meaning of (\ref{eq:wptcp13}) will be explained below.\\
%
\indent
By (\ref{eq:wptcp5}) and (\ref{eq:wptcp10}), the equation (\ref{eq:wptcp12}) can be written as 
\begin{equation*}
\frac{d}{dt}<U,\varphi> + <AU,\varphi> = <F_U,\varphi>, \qquad \forall ~\varphi \in\mathcal{V}_0
\end{equation*}
and so
\begin{equation}
\frac{d}{dt}<U,\varphi> = <F_U-AU,\varphi>, \qquad \forall ~\varphi \in\mathcal{V}_0 \label{eq:wptcp14}
\end{equation}
%
Since $~F_U\in L^2(0,t_f;\mathcal{V}_0')$ and $~AU \in L^2(0,t_f;\mathcal{V}_0')$, it follows that $~F_U-AU\in L^2(0,t_f;\mathcal{V}_0')$. From ~\cite{Temam},
\begin{equation}
\frac{d}{dt}<U,\varphi> = <\dot{U},\varphi>, \qquad \forall ~\varphi\in\mathcal{V}_0 \label{eq:wptcp15}
 \end{equation}
in the scalar distribution sense on $(0,t_f)$ and, from (\ref{eq:wptcp14}) and (\ref{eq:wptcp15}), $\dot{U}\in L^2(0,t_f;\mathcal{V}_0')$.\\
\indent
We introduce the following Hilbert space ~\cite{Temam, Lions1}:
\begin{equation}
\mathcal{W}(0,t_f;\mathcal{V}_0)=\big\{u:~u\in L^2(0,t_f;\mathcal{V}_0), ~\dot{u}\in L^2(0,t_f;\mathcal{V}_0')\big\} \label{eq:wptcp16}
\end{equation}
with the norm
\begin{equation}
||u||_{\mathcal{W}}=\bigg[\int_0^{t_f} ||u(t)||_{\mathcal{V}_0}^2 \ud t + \int_0^{t_f} ||\dot{u}(t)||_{\mathcal{V}_0'}^2 \ud t\bigg]^{1/2} \label{eq:wptcp17} 
\vspace{0.3in}
\end{equation}
%
\begin{remark} [\cite{Temam}]
Each $u\in \mathcal{W}(0,t_f;\mathcal{V}_0)$ is almost everywhere equal to a function continuous from $[0,t_f]$ into $\mathcal{H}$, and the following inequality holds in the scalar distribution sense on $(0,t_f)$:
\begin{equation}
2<\dot{u},u> = \frac{d}{dt}||u||_{\mathcal{H}}^2 \label{eq:wptcp18}
\end{equation} 
The initial condition (\ref{eq:wptcp13}) now makes sense. 
\vspace{0.1in}
\end{remark}
%
From (\ref{eq:wptcp14}) and (\ref{eq:wptcp15}),
\begin{equation*}
<\dot{U},\varphi> = <F_U-AU,\varphi>, \qquad \forall ~\varphi\in\mathcal{V}_0, \qquad \text{so }
\end{equation*} 
%
\begin{equation*}
\dot{U}+AU = F_U
\end{equation*}
%
Thus, the alternative formulation of the weak problem (\ref{eq:wptcp12})-(\ref{eq:wptcp13}) is the following.\\
Given $~F_U\in L^2(0,t_f;\mathcal{V}_0')$ and $~U_0\in\mathcal{H}$, find $~U\in \mathcal{W}(0,t_f;\mathcal{V}_0)$ such that
\begin{equation}
\dot{U}+AU = F_U \qquad \text{on } (0,t_f) \label{eq:wptcp19}
\end{equation}
and 
\begin{equation*}
U(0)=U_0 \qquad \text{in } \mathcal{H} \tag{\ref{eq:wptcp13}} \\
\end{equation*}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Well-posedness of the TCP system %%%%%%%%%%%%%%%%%%%%%%%%%%
%
\indent
The well-posedness of the problem (\ref{eq:wptcp19}), (\ref{eq:wptcp13}) is proved in ~\cite{Temam, Lions1} and we have the following result.
\begin{theorem}
Problem (\ref{eq:wptcp19}), (\ref{eq:wptcp13}) admits a unique solution in $~\mathcal{W}(0,t_f;\mathcal{V}_0)$. Furthermore, the solution depends continuously on the data: the bilinear map
\begin{equation*}
F_U,U_0 \mapsto U
\end{equation*} 
is continuous from $~L^2(0,t_f;\mathcal{V}_0') \times \mathcal{H}$ to $~\mathcal{W}(0,t_f;\mathcal{V}_0)$. 
\end{theorem} 
%
\begin{corollary} \label{cor:TCP1}
The TCP system (\ref{eq:uneq2d19})-(\ref{eq:uneq2d21}) admits a unique weak solution in $L^2\big(0,t_f;H^1(\Omega,\reals^3)\big)$ and this solution depends continuously on the data.
\end{corollary}
\begin{proof}
The result follows immediately from Assumption \ref{assm:ch2assm1}, (\ref{eq:dtcp1}), (\ref{eq:dtcp2}), and (\ref{eq:dtcp16}). 
\end{proof}
%
\begin{corollary} \label{cor:TCP2}
The TCP system (\ref{eq:uneq3d17})-(\ref{eq:uneq3d19}) admits a unique weak solution in $L^2\big(0,t_f;H^1(\Omega,\reals^3)\big)$ and this solution depends continuously on the data.
\end{corollary}
\begin{proof}
The result follows immediately from Assumption \ref{assm:ch2assm1}, (\ref{eq:dtcp4})-(\ref{eq:dtcp9}), and (\ref{eq:dtcp16}). 
\end{proof}
%
We finish this section with the following theorem which will be needed later. 
\begin{theorem} \label{thm:wptcp1}
If $~F_U \in L^2(0,t_f;\mathcal{H})$ and $~U_0\in \mathcal{V}_0$, then $\dot{U}\in L^2(0,t_f,\mathcal{H})$ and
\begin{equation}
||\dot{U}||_{L^2(0,t_f,\mathcal{H})} \leq \sqrt{\lambda_1}||U_0||_{\mathcal{V}_0} + ||F_U||_{L^2(0,t_f,\mathcal{H})} \label{eq:wptcp20}
\end{equation}
\end{theorem} 
%
\begin{proof}
Since $\mathcal{V}_0$ is separable, there exists a countable set $w_1, \ldots, w_m, \ldots$ which is dense in $\mathcal{V}_0$. For each $m$, define an approximate solution $U_m(\bar{x},t)$ as
\begin{equation}
U_m(\bar{x},t) = \sum_{i=1}^m g_{im}(t)w_i(\bar{x}) \label{eq:wptcp21}
\end{equation}
where $~g_{im}(t), ~~ 1\leq i \leq m,~$ are the scalar functions defined on $~[0,t_f]$, $~U_m(\bar{x},t)$ satisfies  
\begin{equation}
(\dot{U}_m,w_j)_{\mathcal{H}}+(U_m,w_j)_a = <F_U,w_j>, \qquad j=1,\ldots,m \label{eq:wptcp22}
\end{equation}
%
\begin{equation}
U_m(\bar{x},0) = U_{0_m} = \sum_{i=1}^m \xi_{im}w_i(\bar{x}) \underset{m\rightarrow \infty}{\longrightarrow} U_0 \qquad \text{in } \mathcal{H} \label{eq:wptcp23}
\end{equation} 
and
\begin{equation}
U_m(0) = U_{0_m} \label{eq:wptcp24}
\end{equation}
where $U_{0_m}$ is the orthogonal projection in $~\mathcal{H}$ of $~U_0$ on the space spanned by $w_1, \ldots, w_m.$ \\
We multiply (\ref{eq:wptcp22}) by $~\dot{g}_{jm}(t)$ and add for $j$:
\begin{equation*}
||\dot{U}_m(t)||_{\mathcal{H}}^2 + (U_m(t),\dot{U}_m(t))_a = <F_U(t),\dot{U}_m(t)>
\end{equation*} 
Using (\ref{eq:wptcp23}), (\ref{eq:wptcp24}), and 
\begin{equation*}
(U_m(t),\dot{U}_m(t))_a = \frac{1}{2}\frac{d}{dt}||U_m(t)||_a^2
\end{equation*}
and integrating over $(0,t_f)$, we have
\begin{gather}
2\int_0^{t_f}||\dot{U}_m(t)||_{\mathcal{H}}^2 \ud t + \int_0^{t_f}\frac{d}{dt}||U_m(t)||_a^2 \ud t = \int_0^{t_f} 2<F_U(t),\dot{U}_m(t)> \ud t \qquad \Rightarrow \nonumber \\[12pt]
2\int_0^{t_f}||\dot{U}_m(t)||_{\mathcal{H}}^2 \ud t  + ||U_m(t_f)||_a^2 -||U_m(0)||_a^2 \leq \nonumber \\[6pt]
\leq \int_0^{t_f} ||F_U(t)||_{\mathcal{H}}^2 \ud t + \int_0^{t_f}||\dot{U}_m(t)||_{\mathcal{H}}^2 \ud t \qquad \Rightarrow \nonumber \\[12pt]
||\dot{U}_m||_{L^2(0,t_f;\mathcal{H})}^2 \leq ||U_{0_m}||_a^2 + ||F||_{L^2(0,t_f;\mathcal{H})}^2 \label{eq:wptcp25}
\end{gather}
Note that 
\begin{equation*}
||U||_a^2 = a(U,U)=\sum_{k=1}^3\int_{\Omega}\lambda_k|\nabla U_k|^2 \ud\Omega \leq \big\{\underset{k}{max}\lambda_k\big\} \sum_{k=1}^3\int_{\Omega}|\nabla U_k|^2 \ud\Omega = \lambda_1||U||_{\mathcal{V}_0}^2
\end{equation*}
and so
\begin{equation}
||U||_a \leq \sqrt{\lambda_1}||U||_{\mathcal{V}_0} \label{eq:wptcp26}
\end{equation}
Applying (\ref{eq:wptcp26}) to (\ref{eq:wptcp25}), we obtain
\begin{equation*}
||\dot{U}_m||_{L^2(0,t_f;\mathcal{H})}^2 \leq \lambda_1 ||U_0||_{\mathcal{V}_0}^2 + ||F||_{L^2(0,t_f;\mathcal{H})}^2
\end{equation*}
Thus, 
\begin{equation*}
||\dot{U}_m||_{L^2(0,t_f;\mathcal{H})} \leq \bigg[\lambda_1 ||U_0||_{\mathcal{V}_0}^2 + ||F||_{L^2(0,t_f;\mathcal{H})}^2\bigg]^{1/2}
\end{equation*} 
This shows that the sequence $~\dot{U}_m$ ranges in a bounded set of $~L^2(0,t_f;\mathcal{H})$. Therefore, $~\dot{U} \in L^2(0,t_f;\mathcal{H})$ and 
\begin{equation*}
||\dot{U}||_{L^2(0,t_f,\mathcal{H})} \leq \sqrt{\lambda_1}||U_0||_{\mathcal{V}_0} + ||F_U||_{L^2(0,t_f,\mathcal{H})}
\end{equation*}
\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Weak formulation of the elastic system %%%%%%%%%%%%%%%%%%%%%%%%
%
\vspace{0.4in}
\section {Elastic Problem}
%
Using notations from Chapter 2, let
\begin{gather*}
H=L^2\big(\Omega; \reals ^n\big)=\big\{L^2\big(\Omega; \reals\big)\big\}^n \\
V=H^1\big(\Omega; \reals ^n\big)=\big\{H^1\big(\Omega; \reals\big)\big\}^n \\
V_0=\big\{\varphi\in V:\varphi\big|_{\Gamma_F} =0\big\} 
\end{gather*}
where $~n=2, 3$ is the spatial dimensionality of the problem. \\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2D elastic system %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%
\indent
We start with the weak formulation of the two-dimensional Navier-type elastic system 
\begin{align*}
\big(K+\frac{G}{3}\big)\nabla(\nabla \cdot \vec{u})+G\nabla^2\vec{u} = \nabla f \qquad &\text{in } \Omega \times (0,t_f)\tag{\ref{eq:uneq2d45}} \\[6pt]
\tau \vec{n}=(fI+\hat{\sigma})\vec{n} \qquad &\text{on } \Gamma_B \times (0,t_f) \tag{\ref{eq:uneq2d43}}\\[6pt]
\vec{u} \approx 0 \qquad \qquad \qquad &\text{on } \Gamma_F \times (0,t_f) \tag{\ref{eq:uneq2d14}}
\end{align*} 
%
\indent
Take an arbitrary $~\vec{\Phi}(\bar{x})=\phi(\bar{x})\vec{i}+\psi(\bar{x})\vec{j} \in V_0$. From (\ref{eq:uneq2d27}) and (\ref{eq:uneq2d45}) we have
\begin{gather*}
\big(K+\frac{G}{3}\big)\int_\Omega \nabla(\nabla \cdot \vec{u})\cdot \vec{\Phi} \ud \Omega +G\int_\Omega(\nabla^2\vec{u})\cdot \vec{\Phi} \ud \Omega=\int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega \qquad \Rightarrow \\ \\
%
\big(K+\frac{G}{3}\big)\int_\Omega (u_x+v_y)_x\phi+(u_x+v_y)_y\psi \ud \Omega+ \\[6pt]
+G\int_\Omega (u_{xx}+u_{yy})\phi+(v_{xx}+v_{yy})\psi \ud \Omega = \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega \qquad \Rightarrow \\ \\
%
\int_\Omega \bigg[\big(K+\frac{G}{3}\big)u_{xx}+\big(K-\frac{2G}{3}\big)v_{yx}+Gu_{xx}+Gu_{yy}+Gv_{xy}\bigg] \phi +\\[6pt]
+\bigg[\big(K+\frac{G}{3}\big)v_{yy}+\big(K-\frac{2G}{3}\big)u_{xy}+Gv_{yy}+Gv_{xx}+Gu_{yx}\bigg] \psi \ud \Omega = \\ 
=\int_\Omega\Bigg\{\Big[\big(K+\frac{G}{3}\big)u_x+\big(K-\frac{2G}{3}\big)v_y+Gu_x\Big]_x+G(u_y+v_x)_y\Bigg\}\phi +\\[6pt]
+\Bigg\{\Big[\big(K+\frac{G}{3}\big)v_y+\big(K-\frac{2G}{3}\big)u_x+Gv_y\Big]_y+G(u_y+v_x)_x\Bigg\}\psi \ud\Omega = \\[12pt]
=\int_\Omega (\tau_{{11}_x}+\tau_{{12}_y})\phi+(\tau_{{21}_x}+\tau_{{22}_y})\psi \ud \Omega = \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega \qquad \Rightarrow \\ \\
%
\int_\Omega (\nabla\cdot\vec{\tau}_1)\phi+(\nabla\cdot\vec{\tau}_2)\psi \ud \Omega = \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega \qquad \Rightarrow \\ \\
%
\int_\Omega \nabla\cdot(\phi\vec{\tau}_1)-\vec{\tau}_1 \cdot\nabla\phi+\nabla\cdot(\psi\vec{\tau}_2)-\vec{\tau}_2 \cdot\nabla\psi \ud\Omega = \int_\Omega \nabla\cdot(f\vec{\Phi})-f\nabla\cdot\vec{\Phi} \ud\Omega \quad \Rightarrow \\ \\
%
\int_{\Gamma_B} (\phi\vec{\tau}_1\cdot\vec{n}+\psi\vec{\tau}_2\cdot\vec{n}) \ud \Gamma -\int_\Omega \vec{\tau}_1\cdot\nabla\phi+\vec{\tau}_2\cdot\nabla\psi \ud\Omega = \int_{\Gamma_B} f\vec{\Phi}\cdot\vec{n} \ud\Gamma-\int_\Omega f\nabla\cdot\vec{\Phi} \ud\Omega \\ \\
%
\Rightarrow \qquad \int_{\Gamma_B} (\tau\vec{n})\cdot\vec{\Phi} \ud\Gamma-\int_\Omega (\tau_{11}\phi_x+\tau_{12}\phi_y+\tau_{21}\psi_x+\tau_{22}\psi_y)\ud\Omega = \\[6pt]
=\int_{\Gamma_B} (f\vec{\Phi})\cdot \vec{n} \ud\Gamma-\int_\Omega f(\phi_x+\psi_y) \ud\Omega \quad \Rightarrow \\ \\
\int_\Omega (\tau_{11}\phi_x+\tau_{12}(\phi_y+\psi_x)+\tau_{22}\psi_y)\ud\Omega-\int_\Omega f(\phi_x+\psi_y) \ud\Omega +\\[6pt]
\Big[ -\int_{\Gamma_B} (\tau\vec{n})\cdot\vec{\Phi} \ud\Gamma + \int_{\Gamma_B} (fI\vec{n})\cdot \vec{\Phi} \ud\Gamma \Big] = 0
\end{gather*}
%
Using (\ref{eq:uneq2d32}), (\ref{eq:uneq2d38}) and (\ref{eq:uneq2d43}),
\begin{gather*}
\int_\Omega (\tau_{11}\phi_x+\tau_{12}(\phi_y+\psi_x)+\tau_{22}\psi_y)\ud\Omega-\int_\Omega f(\phi_x+\psi_y) \ud\Omega - \\[6pt]
-\int_{\Gamma_B} (\hat{\sigma}\vec{n})\cdot\vec{\Phi} \ud\Gamma= 0 \quad \Rightarrow
\end{gather*}
%
\begin{gather*}
\int_\Omega \tau_{11}\varepsilon_{11}(\vec{\Phi})+2\tau_{12}\varepsilon_{12}(\vec{\Phi})+\tau_{22}\varepsilon_{22}(\vec{\Phi})\ud\Omega -\int_\Omega f \text{tr}\varepsilon(\vec{\Phi}) \ud\Omega -\int_{\Gamma_B} (\hat{\sigma}\vec{n})\cdot\vec{\Phi} \ud\Gamma= 0 \\[12pt]
\Rightarrow \qquad D\mathcal{V}(\vec{u})\vec{\Phi}=0, \qquad \forall ~\vec{\Phi} \in V_0
\end{gather*}
%
Thus,
\begin{equation}
D\mathcal{V}(\vec{u})\vec{\Phi}=0, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:wpes1}
\end{equation}
is a weak formulation of the elastic system (\ref{eq:uneq2d43}) and (\ref{eq:uneq2d45}). As before, $~D\mathcal{V}(\vec{u})\vec{\Phi}~$ is the G\^{a}teaux differential of $~\mathcal{V}$ with increment $~\vec{\Phi}$. \\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3D elastic system %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\indent
Now we obtain the weak formulation of the three-dimensional Navier-type elastic system
\begin{align*}
\big(K+\frac{G}{3}\big)\nabla(\nabla \cdot \vec{u})+G\nabla^2\vec{u} = \nabla f \qquad &\text{in } \Omega \times (0,t_f)\tag{\ref{eq:uneq3d43}} \\[6pt]
\tau \vec{n}=(fI+\hat{\sigma})\vec{n} \qquad &\text{on } \Gamma \setminus \Gamma_F \times (0,t_f)\tag{\ref{eq:uneq3d41}} \\[6pt]
\vec{u} \approx 0 \qquad \qquad \qquad &\text{on } \Gamma_F \times (0,t_f) \tag{\ref{eq:uneq3d14}}
\end{align*} 
%
\indent
Take an arbitrary $~\vec{\Phi}(\bar{x})=\phi(\bar{x})\vec{i}+\psi(\bar{x})\vec{j}+\omega(\bar{x})\vec{k} \in V_0$. From (\ref{eq:uneq3d43}) we have\\
%
\vspace{-0.2in}
\begin{gather*}
\big(K+\frac{G}{3}\big)\int_\Omega \nabla(\nabla \cdot \vec{u})\cdot \vec{\Phi} \ud \Omega +G\int_\Omega(\nabla^2\vec{u})\cdot \vec{\Phi} \ud \Omega = \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega \qquad \Rightarrow \\ \\
%
\big(K+\frac{G}{3}\big)\int_\Omega (u_x+v_y+w_z)_x\phi+(u_x+v_y+w_z)_y\psi+(u_x+v_y+w_z)_z\omega \ud \Omega+ \\[6pt]
+G\int_\Omega (u_{xx}+u_{yy}+u_{zz})\phi+(v_{xx}+v_{yy}+v_{zz})\psi+(w_{xx}+w_{yy}+w_{zz})\omega \ud \Omega = \\[6pt]
= \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega \qquad \Rightarrow \\ \\
%
\int_\Omega \bigg[\big(K+\frac{G}{3}\big)u_{xx}+\big(K-\frac{2G}{3}\big)(v_y+w_z)_x+G(u_{xx}+v_{xy}+w_{zx}+u_{yy}+u_{zz})\bigg] \phi +\\[6pt]
+\bigg[\big(K+\frac{G}{3}\big)v_{yy}+\big(K-\frac{2G}{3}\big)(u_x+w_z)_y+G(v_{yy}+u_{xy}+w_{zy}+v_{xx}+v_{zz})\bigg] \psi + \\[6pt] 
+\bigg[\big(K+\frac{G}{3}\big)w_{zz}+\big(K-\frac{2G}{3}\big)(u_x+v_y)_z+G(w_{zz}+u_{xz}+v_{yz}+w_{xx}+w_{zz})\bigg] \omega \ud \Omega = \\ \\
%
\int_\Omega\Bigg\{\Big[\big(K+\frac{G}{3}\big)u_x+\big(K-\frac{2G}{3}\big)(v_y+w_z)+Gu_x\Big]_x + \\[6pt]
+G(u_y+v_x)_y+G(w_x+u_z)_z\Bigg\}\phi +\\[6pt]
+\Bigg\{\Big[\big(K+\frac{G}{3}\big)v_y+\big(K-\frac{2G}{3}\big)(u_x+w_z)+Gv_y\Big]_y+ \\[6pt]
+G(u_y+v_x)_x+G(w_y+v_z)_z\Bigg\}\psi +\\[6pt]
+\Bigg\{\Big[\big(K+\frac{G}{3}\big)w_z+\big(K-\frac{2G}{3}\big)(u_x+v_y)+Gw_z\Big]_z+ \\[6pt]
+G(u_z+w_x)_x+G(v_z+w_y)_y\Bigg\}\omega \ud\Omega = \\[12pt]
%
=\int_\Omega (\tau_{{11}_x}+\tau_{{12}_y}+\tau_{{13}_z})\phi+(\tau_{{21}_x}+\tau_{{22}_y}+\tau_{{23}_z})\psi+(\tau_{{31}_x}+\tau_{{32}_y}+\tau_{{33}_z})\omega \ud \Omega =\\[6pt]
%
=\int_\Omega (\nabla\cdot\vec{\tau}_1)\phi+(\nabla\cdot\vec{\tau}_2)\psi+(\nabla\cdot\vec{\tau}_3)\omega \ud \Omega = \int_\Omega \nabla f \cdot \vec{\Phi} \ud \Omega \qquad \Rightarrow \\ \\
%
\int_\Omega \nabla\cdot(\phi\vec{\tau}_1)-\vec{\tau}_1 \cdot\nabla\phi+\nabla\cdot(\psi\vec{\tau}_2)-\vec{\tau}_2 \cdot\nabla\psi+\nabla\cdot(\omega\vec{\tau}_3)-\vec{\tau}_3 \cdot\nabla\omega \ud\Omega =\\[6pt]
= \int_\Omega \nabla\cdot(f\vec{\Phi})-f\nabla\cdot\vec{\Phi} \ud\Omega \quad \Rightarrow \\ \\
%
\int_{\Gamma \setminus \Gamma_F} (\phi\vec{\tau}_1\cdot\vec{n}+\psi\vec{\tau}_2\cdot\vec{n}+\omega\vec{\tau}_3\cdot\vec{n}) \ud \Gamma -\int_\Omega \vec{\tau}_1\cdot\nabla\phi+\vec{\tau}_2\cdot\nabla\psi+\vec{\tau}_3\cdot\nabla\omega \ud\Omega = \\[6pt]
= \int_{\Gamma \setminus \Gamma_F} f\vec{\Phi}\cdot\vec{n} \ud\Gamma-\int_\Omega f\nabla\cdot\vec{\Phi} \ud\Omega \qquad \Rightarrow \\ 
%
\int_{\Gamma \setminus \Gamma_F} (\tau\vec{n})\cdot\vec{\Phi} \ud\Gamma-\int_\Omega (\tau_{11}\phi_x+\tau_{12}\phi_y+\tau_{13}\phi_z + \\[6pt]
+\tau_{21}\psi_x+\tau_{22}\psi_y+\tau_{23}\psi_z+\tau_{31}\omega_x+\tau_{32}\omega_y+\tau_{33}\omega_z)\ud\Omega = \\[6pt]
=\int_{\Gamma \setminus \Gamma_F} (f\vec{\Phi})\cdot \vec{n} \ud\Gamma-\int_\Omega f(\phi_x+\psi_y+\omega_z) \ud\Omega \quad \Rightarrow \\ \\
%
\int_\Omega (\tau_{11}\phi_x+\tau_{12}(\phi_y+\psi_x)+\tau_{22}\psi_y+\tau_{13}(\phi_z+\omega_x)+\tau_{23}(\psi_z+\omega_y)+\tau_{33}\omega_z)\ud\Omega- \\[6pt]
-\int_\Omega f(\phi_x+\psi_y+\omega_z) \ud\Omega +\Big[-\int_{\Gamma \setminus \Gamma_F} (\tau\vec{n})\cdot\vec{\Phi} \ud\Gamma + \int_{\Gamma \setminus \Gamma_F} (fI\vec{n})\cdot \vec{\Phi} \ud\Gamma\Big] = 0 
\end{gather*}
%
\\[-12pt]
Using (\ref{eq:uneq3d30}), (\ref{eq:uneq3d35}), and (\ref{eq:uneq3d41}),
\begin{gather*}
\int_\Omega (\tau_{11}\phi_x+\tau_{12}(\phi_y+\psi_x)+\tau_{22}\psi_y+\tau_{13}(\phi_z+\omega_x)+\tau_{23}(\psi_z+\omega_y)+\tau_{33}\omega_z)\ud\Omega- \\[6pt]
-\int_\Omega f(\phi_x+\psi_y+\omega_z) \ud\Omega -\int_{\Gamma \setminus \Gamma_F} (\hat{\sigma}\vec{n})\cdot \vec{\Phi} \ud\Gamma= 0 \quad \Rightarrow \\\\
%
\int_\Omega \tau_{11}\varepsilon_{11}(\vec{\Phi})+2\tau_{12}\varepsilon_{12}(\vec{\Phi})+\tau_{22}\varepsilon_{22}(\vec{\Phi})+2\tau_{13}\varepsilon_{13}(\vec{\Phi})+2\tau_{23}\varepsilon_{23}(\vec{\Phi})+\tau_{33}\varepsilon_{33}(\vec{\Phi})\ud\Omega \\[6pt]
-\int_\Omega f \text{tr}\varepsilon(\vec{\Phi}) \ud\Omega -\int_{\Gamma \setminus \Gamma_F} (\hat{\sigma}\vec{n})\cdot\vec{\Phi} \ud\Gamma= 0 \\[12pt]
%
\Rightarrow \qquad D\mathcal{V}(\vec{u})\vec{\Phi}=0, \qquad \forall ~\vec{\Phi} \in V_0
\end{gather*}
%
Thus,
\begin{equation}
D\mathcal{V}(\vec{u})\vec{\Phi}=0, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:wpes2}
\end{equation}
is a weak formulation of the elastic system (\ref{eq:uneq3d41}) and (\ref{eq:uneq3d43}). As before, $~D\mathcal{V}(\vec{u})\vec{\Phi}~$ is the G\^{a}teaux differential of $~\mathcal{V}$ with increment $~\vec{\Phi}$.\\
%
\indent
From (\ref{eq:uneq2d35}), (\ref{eq:uneq3d33}), (\ref{eq:wpes1}), and (\ref{eq:wpes2}), with the assumption that the applied boundary stress $~\hat{\sigma}$ is bounded on $\bar{\Omega} \times [0,t_f]$, ~the weak formulation of the two- and three-dimensional elastic problems has the following form: \\
Given $~f\in L^2\big(0,t_f;H^1(\Omega)\big)$, find $~\vec{u}\in L^2(0,t_f;V_0)$ such that $~\dot{\vec{u}}\in L^2(0,t_f;V_0)$ and
\begin{equation}
a_E(\vec{u},\vec{\Phi})-\int_\Omega f\big(\nabla \cdot \vec{\Phi}\big)\ud \Omega - \int_{\Gamma \setminus \Gamma_F}(\hat{\sigma}\vec{n})\cdot\vec{\Phi}\ud \Gamma = 0, \qquad \forall ~\vec{\Phi} \in V_0 \label{eq:wpes3}
\end{equation}
\indent
%
Next we prove the well-posedness of the elastic system (\ref{eq:wpes3}). Define a continuous linear functional $F$ on $V$ by a pairing
\begin{equation*}
\vec{\Phi}\mapsto <F, \vec{\Phi}>
\vspace{-0.1in}
\end{equation*}
where
\begin{equation}
<F,\vec{\Phi}> = \int_\Omega f (\nabla \cdot \vec{\Phi}) \ud \Omega + \int_{\Gamma \setminus \Gamma_F}(\hat{\sigma}\vec{n})\cdot\vec{\Phi}\ud \Gamma, \qquad \forall ~\vec{\Phi} \in V \label{eq:wpes4} 
\vspace{0.1in}
\end{equation}
%
Since $F$ is continuous on $V$, there exists a constant $c_F>0$ such that
\begin{equation}
|<F,\vec{\Phi}>|\leq c_F ||\vec{\Phi}||_V, \qquad \forall ~\vec{\Phi} \in V \label{eq:wpes5}
\end{equation}
%
From (\ref{eq:wpes3}) and (\ref{eq:wpes4}) in follows that 
\begin{equation}
a_E(\vec{u},\vec{\Phi})=<F,\vec{\Phi}>, \qquad \forall ~\vec{\Phi} \in V \label{eq:wpes6}
\end{equation}
%
\indent
The existence and uniqueness of a solution to the elastic problem is based on Korn's inequality ~\cite{Duvaut_Lions}:
\begin{theorem} \label{thm:Korn1}
Let $\Omega$ be a bounded open set with a regular boundary. Then there exists a constant $c>0$ that depends on $\Omega$ such that
\begin{equation*}
\int_\Omega \varepsilon_{ij}(\vec{v})\varepsilon_{ij}(\vec{v})\ud \Omega + \int_\Omega v_i^2 \ud \Omega \geq c||\vec{v}||_V^2, \qquad \forall ~\vec{v} \in V 
\end{equation*} 
\end{theorem}
\vspace{0.2in}
\noindent
The consequence of Theorem \ref{thm:Korn1} is the following result ~\cite{Duvaut_Lions}:
\begin{theorem} \label{thm:Korn2}
Let $\Omega$ be a bounded open set with a regular boundary. Let $\Gamma_F \subset \Gamma$ and $\Gamma_F$ have positive measure. Then there exists a constant $c_E>0$ such that
\begin{equation}
a_E(\vec{v},\vec{v}) \geq c_E||\vec{v}||_V^2, \qquad \forall ~\vec{v} \in V_0 \label{eq:wpes7}\\[12pt]
\end{equation}
\end{theorem}
%
For the sake of convenience, here and in the following we will take
$$\vec{u}=u_1\vec{i}+u_2\vec{j}, \quad \vec{v}=v_1\vec{i}+v_2\vec{j},~~ \text{and} \quad \bar{x}=x_1\vec{i}+x_2\vec{j} \in \Omega$$ for the 2D elastic problem and 
$$\vec{u}=u_1\vec{i}+u_2\vec{j}+u_3\vec{k}, \quad \vec{v}=v_1\vec{i}+v_2\vec{j}+v_3\vec{k},~~ \text{and} \quad\bar{x}=x_1\vec{i}+x_2\vec{j}+x_3\vec{k} \in \Omega$$ for the 3D elastic problem. \\
%
Then, from (\ref{eq:uneq2d23}), (\ref{eq:uneq2d33}), and (\ref{eq:uneq3d31}), for every $~\vec{u}, \vec{v}\in V$, 
\begin{gather*}
|a_E(\vec{u},\vec{v})|=\left|\int_\Omega a_{ijkl}\varepsilon_{ij}(\vec{u})\varepsilon_{kl}(\vec{v})\ud \Omega\right|\leq \underset{i,j,k,l}{max}\{a_{ijkl}\}\sum_{i,j,k,l=1}^n \left| \int_\Omega \varepsilon_{ij}(\vec{u})\varepsilon_{kl}(\vec{v})\ud \Omega\right|= \\[12pt]
%
=\frac{1}{4}\underset{i,j,k,l}{max}\{a_{ijkl}\}\sum_{i,j,k,l=1}^n \left| \int_\Omega \Big(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\Big) \Big(\frac{\partial v_k}{\partial x_l}+\frac{\partial v_l}{\partial x_k}\Big) \ud \Omega \right| \leq \\[12pt]
%
\leq \frac{1}{4}\underset{i,j,k,l}{max}\{a_{ijkl}\}\sum_{i,j,k,l=1}^n\Big\{\left\|\frac{\partial u_i}{\partial x_j}\right\|_{L^2(\Omega)} \left\|\frac{\partial v_k}{\partial x_l}\right\|_{L^2(\Omega)}+\left\|\frac{\partial u_j}{\partial x_i}\right\|_{L^2(\Omega)}\left\|\frac{\partial v_k}{\partial x_l}\right\|_{L^2(\Omega)}+ \\[6pt]
+\left\|\frac{\partial u_i}{\partial x_j}\right\|_{L^2(\Omega)} \left\|\frac{\partial v_l}{\partial x_k}\right\|_{L^2(\Omega)}+\left\|\frac{\partial u_j}{\partial x_i}\right\|_{L^2(\Omega)} \left\|\frac{\partial v_l}{\partial x_k}\right\|_{L^2(\Omega)} \Big\}=\\[12pt]
%
=\underset{i,j,k,l}{max}\{a_{ijkl}\}\sum_{i,j=1}^n \left\|\frac{\partial u_i}{\partial x_j}\right\|_{L^2(\Omega)}\sum_{k,l=1}^n \left\|\frac{\partial v_k}{\partial x_l}\right\|_{L^2(\Omega)} \leq \\[12pt]
%
\leq n\cdot \underset{i,j,k,l}{max}\{a_{ijkl}\} \bigg[\sum_{i,j=1}^n \left\|\frac{\partial u_i}{\partial x_j}\right\|^2_{L^2(\Omega)}\bigg]^{\frac{1}{2}} \bigg[\sum_{k,l=1}^n \left\|\frac{\partial v_k}{\partial x_l}\right\|^2_{L^2(\Omega)}\bigg]^{\frac{1}{2}} \leq \\[12pt]
%
\leq n\cdot \underset{i,j,k,l}{max}\{a_{ijkl}\} ||\vec{u}||_V ||\vec{v}||_V = \beta ||\vec{u}||_V ||\vec{v}||_V, \qquad \beta=n\cdot \underset{i,j,k,l}{max}\{a_{ijkl}\}, \quad n=2,3.
\end{gather*} 
%
Thus, there is a constant $\beta>0$ ($\beta=n\cdot \underset{i,j,k,l}{max}\{a_{ijkl}\}$) such that 
\begin{equation}
|a_E(\vec{u},\vec{v})|\leq \beta ||\vec{u}||_V ||\vec{v}||_V, \qquad \forall ~\vec{u},\vec{v} \in V \label{eq:wpes8}
\end{equation}
%
Next we show that $a_E(\cdot ,\cdot)$ is symmetric.
\vspace{1pt}
\begin{gather}
a_E(\vec{u},\vec{v})=\int_\Omega a_{ijkl}\varepsilon_{ij}(\vec{u})\varepsilon_{kl}(\vec{v})\ud \Omega = \int_\Omega a_{klij}\varepsilon_{ij}(\vec{u})\varepsilon_{kl}(\vec{v})\ud \Omega = \nonumber \\[6pt]
=\int_\Omega a_{klij}\varepsilon_{kl}(\vec{u})\varepsilon_{ij}(\vec{v})\ud \Omega = a_E(\vec{v},\vec{u}), \quad \forall ~\vec{u},\vec{v} \in V \label{eq:wpes9}
\end{gather}
%
From (\ref{eq:wpes7})-(\ref{eq:wpes9}) it follows that $a_E(\cdot ,\cdot)$ is symmetric and continuous on $V_0$ and there exist constants $c_E>0$ and $\beta>0$ such that
\begin{equation}
c_E||\vec{u}||^2_V \leq a_E(\vec{u},\vec{u}) \leq \beta ||\vec{u}||^2_V, \qquad \forall ~\vec{u} \in V_0 \label{eq:wpes10}
\end{equation}
%
Thus, $a_E(\cdot ,\cdot)$ is an inner product on $V_0$ that is associated with a topology equivalent to the usual topology on $V_0$. \\ 
\indent
From the Riesz representation theorem and (\ref{eq:wpes5}), (\ref{eq:wpes7}), we have the following existence and uniqueness result with an a priori estimate.
%
\begin{theorem}
There exists a unique function $\vec{u} \in V_0$ such that 
\begin{equation*}
a_E(\vec{u},\vec{\Phi})=<F,\vec{\Phi}>, \qquad \forall ~\vec{\Phi} \in V_0.
\end{equation*}
Furthermore,  
\begin{equation*}
||\vec{u}||_V \leq \frac{c_F}{c_E} 
\end{equation*} 
\end{theorem} 
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Well-posedness of the copuled TCPu system %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\vspace{0.65in}
\section{Coupled TCPu Problem}
%
In Section 3.2, for both the two- and three-dimensional elastic problems we obtained the following weak formulation:
\begin{equation*}
a_E(\vec{u},\vec{\Phi})-\int_\Omega f\big(\nabla \cdot \vec{\Phi}\big)\ud \Omega - \int_{\Gamma \setminus \Gamma_F}(\hat{\sigma}\vec{n})\cdot\vec{\Phi}\ud \Gamma = 0, \qquad \forall ~\vec{\Phi} \in V_0 \tag{\ref{eq:wpes3}}
\end{equation*}
%
Since $\dot{\vec{u}}\in V_0$, we have\\
\vspace{1pt}
\begin{align}
a_E(\dot{\vec{u}},\dot{\vec{u}})&=\int_{\Omega}\dot{f}(\nabla\cdot\dot{\vec{u}})\ud\Omega + \int_{\Gamma \setminus \Gamma_F}(\dot{\hat{\sigma}}\vec{n})\cdot \dot{\vec{u}}\ud\Gamma \nonumber \\[6pt]
&\leq ||\dot{f}||_{L^2(\Omega)}||\nabla\cdot\dot{\vec{u}}||_{L^2(\Omega)}+||\dot{\hat{\sigma}}\vec{n}||_{L^2(\Gamma;\reals^n)}||\dot{\vec{u}}||_{L^2(\Gamma;\reals^n)}, \quad n=2,3. \label{eq:wptcpu1}
\end{align}
%
Now
\begin{equation*}
||\nabla\cdot\dot{\vec{u}}||^2_{L^2(\Omega)}=\int_{\Omega}\bigg[\sum_{i=1}^n \frac{\partial\dot{u}_i}{\partial x_i}\bigg]^2 \ud\Omega  \leq n \int_{\Omega}\sum_{i=1}^n \left|\frac{\partial\dot{u}_i}{\partial x_i}\right|^2 \ud\Omega \leq n||\nabla\dot{\vec{u}}||_{H}^2
\end{equation*}
where 
\begin{equation*}
||\nabla\dot{\vec{u}}||_{H}^2 = \int_{\Omega}\sum_{i,j=1}^n \left|\frac{\partial\dot{u}_i}{\partial x_j}\right|^2\ud\Omega, \qquad n=2,3.
\end{equation*}
%
Thus,
\begin{equation}
||\nabla\cdot\dot{\vec{u}}||_{L^2(\Omega)} \leq \sqrt{n} ||\nabla\dot{\vec{u}}||_{H} \qquad n=2,3. \label{eq:wptcpu2}
\end{equation}
%
By the trace theorem ~\cite{Showalter}, there exists $~\gamma_1>0$ such that
\begin{equation}
||\vec{v}||_{L^2(\Gamma;\reals^n)} \leq \gamma_1||\vec{v}||_V, \qquad \forall ~\vec{v}\in V \label{eq:wptcpu3}
\end{equation}
%
By Poincare-Friedrichs inequality ~\cite{Dietrich}, there exists a constant $\gamma_2=\gamma_2(\Omega)>0$ such that
\begin{equation}
||\vec{v}||_H \leq \gamma_2||\nabla\vec{v}||_H, \qquad \forall ~\vec{v}\in V_0 \label{eq:wptcpu4}
\end{equation}
%
From (\ref{eq:wptcpu3}) and (\ref{eq:wptcpu4}) it follows that there exists $\gamma = \gamma(\Omega)>0$ such that
\begin{equation}
||\dot{\vec{u}}||_{L^2(\Gamma;\reals^n)} \leq \gamma||\nabla\dot{\vec{u}}||_H, \qquad n=2,3.  \label{eq:wptcpu5}
\end{equation}
%
Applying (\ref{eq:wptcpu2}) and (\ref{eq:wptcpu5}) to (\ref{eq:wptcpu1}), we have
\begin{equation}
a_E(\dot{\vec{u}},\dot{\vec{u}}) \leq \sqrt{n}||\dot{f}||_{L^2(\Omega)}||\nabla\dot{\vec{u}}||_{H} + \gamma ||\dot{\hat{\sigma}}\vec{n}||_{L^2(\Gamma;\reals^n)}||\nabla\dot{\vec{u}}||_H, \qquad n=2,3.  \label{eq:wptcpu6}
\end{equation}
%
From (\ref{eq:uneq2d30}) 
\begin{equation}
||\dot{f}||_{L^2(\Omega)} = ||\vec{b}_1^T\dot{\bar{V}}||_{L^2(\Omega)} \leq |\vec{b}_1| \cdot||\dot{\bar{V}}||_{\mathcal{H}} \label{eq:wptcpu7} 
\end{equation}
and from (\ref{eq:wpes10}), (\ref{eq:wptcpu6}), and (\ref{eq:wptcpu7}) it follows that
\begin{equation}
c_E||\nabla\dot{\vec{u}}||_{H} \leq \sqrt{n}|\vec{b}_1| \cdot||\dot{\bar{V}}||_{\mathcal{H}}+\gamma ||\dot{\hat{\sigma}}\vec{n}||_{L^2(\Gamma;\reals^n)}, \quad n=2,3. \label{eq:wptcpu8}
\end{equation}
%
The last inequality holds for all $t \in (0,t_f)$ and therefore
\begin{equation}
c_E||\nabla\dot{\vec{u}}||_{L^2(0,t_f;H)} \leq \sqrt{2n}|\vec{b}_1| \cdot||\dot{\bar{V}}||_{L^2(0,t_f;\mathcal{H})} +\sqrt{2}\gamma ||\dot{\hat{\sigma}}\vec{n}||_{L^2(0,t_f;L^2(\Gamma;\reals^n))}, \quad n=2,3. \label{eq:wptcpu9}
\end{equation}
%
\indent
Now we obtain a priori estimate $||\dot{\bar{V}}||_{L^2(0,t_f;\mathcal{H})}$. Applying (\ref{eq:dtcp2}) for the 2D problem or (\ref{eq:dtcp9}) for the 3D problem, (\ref{eq:dtcp16}), and (\ref{eq:dtcp20}) to (\ref{eq:wptcp20}), we have
\begin{gather*}
||\dot{\bar{V}}||_{L^2(0,t_f;\mathcal{H})} \leq ||P||\cdot ||\dot{U}||_{L^2(0,t_f;\mathcal{H})} + ||\dot{W}||_{L^2(0,t_f;\mathcal{H})} \leq\\\\
\leq ||P|| \Big[\sqrt{\lambda_1}||U_0||_{\mathcal{V}_0} + ||P||^{-1}||\dot{W}||_{L^2(0,t_f,\mathcal{H})} +||DP^{-1}||\cdot ||\nabla^2W||_{L^2(0,t_f,\mathcal{H})}+\\[6pt]
+|P^{-1}M^{-1}\vec{b}_0| \cdot||\nabla\cdot \dot{\vec{u}}||_{L^2(0,t_f,L^2(\Omega))}\Big] + ||\dot{W}||_{L^2(0,t_f,\mathcal{H})} \qquad \Rightarrow \\\\
%
||\dot{\bar{V}}||_{L^2(0,t_f;\mathcal{H})} \leq ||P||\cdot|P^{-1}M^{-1}\vec{b}_0|\cdot ||\nabla\cdot \dot{\vec{u}}||_{L^2(0,t_f,L^2(\Omega))} +\\[6pt]
+||P||\Big[\sqrt{\lambda_1}||U_0||_{\mathcal{V}_0} + ||P||^{-1}||\dot{W}||_{L^2(0,t_f,\mathcal{H})} +||DP^{-1}||\cdot ||\nabla^2W||_{L^2(0,t_f,\mathcal{H})}\Big] + \\[6pt]
+ ||\dot{W}||_{L^2(0,t_f,\mathcal{H})} 
\end{gather*}
%
Thus,
\begin{equation}
||\dot{\bar{V}}||_{L^2(0,t_f;\mathcal{H})} \leq ||P||\cdot|P^{-1}M^{-1}\vec{b}_0|\cdot ||\nabla\cdot \dot{\vec{u}}||_{L^2(0,t_f,L^2(\Omega))} + \theta  \label{eq:wptcpu10}
\end{equation}
where
\begin{gather*}
\theta = ||P||\Big[\sqrt{\lambda_1}||U_0||_{\mathcal{V}_0} + ||P||^{-1}||\dot{W}||_{L^2(0,t_f,\mathcal{H})} +||DP^{-1}||\cdot ||\nabla^2W||_{L^2(0,t_f,\mathcal{H})}\Big] + \\[6pt]
+ ||\dot{W}||_{L^2(0,t_f,\mathcal{H})} 
\end{gather*}
%
Applying (\ref{eq:wptcpu2}) and (\ref{eq:wptcpu10}) to (\ref{eq:wptcpu9}), we have
\begin{gather}
c_E||\nabla\dot{\vec{u}}||_{L^2(0,t_f;H)} \leq \sqrt{2}n|\vec{b}_1| \cdot ||P||\cdot|P^{-1}M^{-1}\vec{b}_0|\cdot ||\nabla\dot{\vec{u}}||_{L^2(0,t_f,H)} + \nonumber \\[6pt]
+\sqrt{2n}|\vec{b}_1|\theta + \sqrt{2}\gamma ||\dot{\hat{\sigma}}\vec{n}||_{L^2(0,t_f;L^2(\Gamma;\reals^n))}, \quad n=2,3.  \label{eq:wptcpu11}
\end{gather} 
%
From Corollaries \ref{cor:TCP1} and \ref{cor:TCP2}, Theorem \ref{thm:wptcp1}, Assumption \ref{assm:ch2assm1}, (\ref{eq:dtcp1}), (\ref{eq:dtcp2}), (\ref{eq:dtcp4})-(\ref{eq:dtcp9}), and (\ref{eq:wptcpu11}), we have the following result:
\begin{theorem} \label{thm:wptcpu6}
The coupled two- and three-dimensional TCPu systems (\ref{eq:uneq2d19})-(\ref{eq:uneq2d21}), (\ref{eq:uneq2d43}), (\ref{eq:uneq2d45}) and (\ref{eq:uneq3d17})-(\ref{eq:uneq3d22}), respectively, have unique solutions $~(\bar{V}, \vec{u}), ~\bar{V}\in L^2\big(0,t_f;H^1(\Omega,\reals^3)\big)$ and $~\vec{u}\in L^2(0,t_f;V_0)$ with $~\dot{\bar{V}}\in L^2(0,t_f;\mathcal{H})$ and $~\dot{\vec{u}}\in L^2(0,t_f;V_0)~$ if 
\begin{equation}
c_E - \sqrt{2}n||P||\cdot|P^{-1}M^{-1}\vec{b}_0|\cdot|\vec{b}_1| >0 \label{eq:wptcpu12}
\end{equation} 
where $~c_E>0$ is the Korn's constant, $~\vec{b}_0$ and $~\vec{b}_1$ are coupling vectors given by (\ref{eq:uneq2d17}) and (\ref{eq:uneq2d30}), respectively, and $M^{-1}A=PDP^{-1}.$
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  2D problem %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\vspace{0.1in}
\centerline{2D Special Case}
%
\begin{remark} 
For an annular domain $~\Omega \subset \reals^2$ of inner radius $~R_B$ and outer radius $~R_F$, with $~\delta=\dfrac{R_B}{R_F} \leq e^{-1}$, under the following side condition on the displacement $~\vec{u}$
\begin{equation}
\int_{\Omega} \sum_{i,j=1}^2 \Big( \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\Big) \ud \Omega =0 \label{eq:wptcpu13}
\vspace{0.1in}
\end{equation}
the following inequality holds ~\cite{Horgan}:\\
\vspace{-0.2in}
\begin{equation}
\frac{D(\dot{\vec{u}})}{S(\dot{\vec{u}})} \leq 4\bigg[1-\Big(\frac{3}{\delta^{-2}+1+\delta^2}\Big)^{1/2}\bigg]^{-1}  \label{eq:wptcpu14}
\end{equation}
where
\begin{align*}
 D(\dot{\vec{u}}) & = \int_{\Omega} \sum_{i,j=1}^2 \left|\frac{\partial\dot{u}_i}{\partial x_j}\right|^2\ud\Omega = ||\nabla\dot{\vec{u}}||_H^2\\[6pt]
S(\dot{\vec{u}}) &= \int_{\Omega} \varepsilon_{ij}(\dot{\vec{u}})^2 \ud\Omega 
\end{align*} 
\end{remark}
%
\noindent
It should be noted that the purpose of the condition (\ref{eq:wptcpu13}) is to eliminate pure rotation. From (\ref{eq:wptcpu14}) we have 
\begin{equation}
S(\dot{\vec{u}}) \geq \frac{1}{4}\bigg[1-\Big(\frac{3}{\delta^{-2}+1+\delta^2}\Big)^{1/2}\bigg] ||\nabla\dot{\vec{u}}||_H^2 \label{eq:wptcpu15}
\end{equation}
%
\indent
For homogeneous isotropic medium, the coefficients of elasticity $~a_{ijkl}, ~i,j,k,l=1,2$, in terms of Lam\'e parameters $\lambda$ and $\mu$ are 
\begin{equation}
a_{1111}=a_{2222}= 2\mu + \lambda, \quad a_{1122}=a_{2211}=\lambda, \quad a_{1212}=a_{2121}=a_{1221}=a_{2112}=\mu, \label{eq:wptcpu16}
\end{equation}
%
Then from (\ref{eq:uneq2d23}), (\ref{eq:uneq2d33}), and (\ref{eq:wptcpu16}) we have
\begin{align*}
a_E(\dot{\vec{u}},\dot{\vec{u}}) &= \int_{\Omega}(2\mu+\lambda)\varepsilon_{11}^2(\dot{\vec{u}}) + 2\mu \varepsilon_{12}^2(\dot{\vec{u}}) + 2\mu \varepsilon_{21}^2(\dot{\vec{u}}) \\[6pt]
&\quad +2\lambda \varepsilon_{11}(\dot{\vec{u}})\varepsilon_{22}(\dot{\vec{u}}) + (2\mu+\lambda)\varepsilon_{22}^2(\dot{\vec{u}})\ud\Omega \\[6pt]
& = \int_{\Omega} 2\mu(\varepsilon_{11}^2+\varepsilon_{12}^2+\varepsilon_{21}^2+\varepsilon_{22}^2) + \lambda (\varepsilon_{11}+\varepsilon_{22})^2 \ud\Omega \\[6pt]
& \geq 2\mu\int_{\Omega} \varepsilon_{11}^2+\varepsilon_{12}^2+\varepsilon_{21}^2+\varepsilon_{22}^2 \ud\Omega = 2\mu S(\dot{\vec{u}}) 
\end{align*}
%
Thus,
\begin{equation}
a_E(\dot{\vec{u}},\dot{\vec{u}}) \geq 2\mu S(\dot{\vec{u}}) \label{eq:wptcpu17}
\end{equation}
where $~\mu =G~$ is the shear modulus.\\
\indent
From (\ref{eq:wpes10}), (\ref{eq:wptcpu15}), and (\ref{eq:wptcpu17}),
\begin{equation}
c_E=\frac{G}{2}\bigg[1-\Big(\frac{3}{\delta^{-2}+1+\delta^2}\Big)^{1/2}\bigg]  \label{eq:wptcpu18}
\end{equation}
%
Applying (\ref{eq:wptcpu18}) to (\ref{eq:wptcpu12}), we obtain the following result.
\newpage
%
\begin{corollary}
For an annular domain $~\Omega \subset \reals^2$ of inner radius $~R_B$ and outer radius $~R_F$, with $~\delta=\dfrac{R_B}{R_F} \leq e^{-1}$ under side condition (\ref{eq:wptcpu13}), the coupled TCPu system (\ref{eq:uneq2d19})-(\ref{eq:uneq2d21}), (\ref{eq:uneq2d43}), and (\ref{eq:uneq2d45}) has a unique solution if 
\begin{equation}
\frac{G}{2}\bigg[1-\Big(\frac{3}{\delta^{-2}+1+\delta^2}\Big)^{1/2}\bigg] - 2\sqrt{2}||P||\cdot|P^{-1}M^{-1}\vec{b}_0|\cdot|\vec{b}_1| >0
\end{equation} \label{eq:wptcpu19}
where $~G$ is the shear modulus, $~\vec{b}_0$ and $~\vec{b}_1$ are coupling vectors given by (\ref{eq:uneq2d17}) and (\ref{eq:uneq2d30}), respectively, and $M^{-1}A=PDP^{-1}.$ 
\end{corollary}
%
\vspace{0.1in}
\indent
In this chapter, we obtained a sufficient condition for the well-posedness in the weak sense of the coupled parabolic-elliptic initial-boundary value problems for the two- and three-dimensional TCPu models. This condition depends on physical parameters of the system, coupling vectors, and the Korn's constant which in turn depends only on the shape of a domain. Evaluation of the Korn's constant becomes necessary for each specific domain and is essential for the proposed well-posedness theory. For a two-dimensional annular region that does not experience pure rotation, the Korn's constant can be expressed explicitly in terms of the ratio of the inner and outer radii of the domain. As a consequence, for such regions, the sufficient condition for the well-posedness of the problem was formulated in terms of physical parameters, coupling vectors, and the ratio of the inner and outer radii.   
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fourier-Finite Element Method for TCPu Problems %%%%%%%%%%%%%%%%%%%
%
\chapter {Fourier-Finite Element Method for TCPu Problems}
%
In this chapter, we will develop the Fourier-finite element method for the two- and three-dimensional radially nonsymmetric TCPu problems discussed in Chapters 2 and 3. The far-field radius can be specified arbitrarily and therefore we assume that far-field boundary conditions are time-independent and the non-zero displacement in the far-field is allowed with a penalty.\\ \\ 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2D  TCPu problem %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section {2D TCPu Problem}
%
As in Section 2.1, let $(X,Y)$ be a borehole Cartesian coordinate system with the $x$-axis in the direction of the maximum horizontal principal stress $\sigma_H$ and the $y$-axis in the direction of the minimum horizontal principal stress $\sigma_h$. For the sake of convenience, we will make the transformation from the coordinate system $(X,Y)$ to the borehole polar coordinate system $(r,\theta)$ with its origin in the center of the annular region $\Omega$. Then the boundary $~\Gamma$ of $~\Omega$ is $~\Gamma = \Gamma_B\cup\Gamma_F$ with
\begin{gather*}
\Gamma_B = \left\{ (R_B, \theta) \in \mathbb R^{2}: 0\leq \theta < 2 \pi \right\}\\
\Gamma_F = \left\{ (R_F, \theta) \in \mathbb R^{2}: 0\leq \theta < 2 \pi \right\} \\
\end{gather*} 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  TCP sytem  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection {Fourier-type approximation of the TCP sytem}
%
In the borehole polar coordinate system $(r,\theta)$, the TCP initial-boundary value problem (\ref{eq:uneq2d19})-(\ref{eq:uneq2d21}) is given by
%
\begin{equation} \label{eq:ffem2d1}
M \dot{\bar{V}} - A \nabla^{2} \bar{V}=-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f)
\end{equation}
%
with boundary conditions
\begin{equation} \label{eq:ffem2d2}
\bar{V}(r,\theta,t)=
\begin{cases}\\[-3em]
V_B(t) & \text{on } \Gamma_B \times [0,t_f)\\
V_F(\theta) & \text{on } \Gamma_F \times [0,t_f)
\end{cases}
\end{equation}
%
and initial conditions 
\begin{equation} \label{eq:ffem2d3}
\bar{V}(r,\theta,0)=\tilde{V}_I(r,\theta)
\end{equation}
where the displacement vector is
\begin{equation*}
\vec{u}(r,\theta,t)=u(r,\theta,t)\vec{i}_r +v(r,\theta,t)\vec{i}_\theta
\end{equation*}
\indent
The equivalent parabolic TCP-type system with homogeneous boundary conditions, (\ref{eq:dtcp11})-(\ref{eq:dtcp13}), in polar coordinates has the form
\begin{equation} \label{eq:ffem2d4}
M \dot{V} - A \nabla^{2} V =-M \dot{W}+A \nabla^{2}W-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f)
\end{equation}
%
with boundary conditions
\begin{equation}
V(r,\theta,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:eq:ffem2d5}
\end{equation}
%
and initial conditions
\begin{equation}
V(r,\theta,0)=\tilde{V}_I(r,\theta)-W(r,\theta,0) \label{eq:ffem2d6}
\end{equation}
where
\begin{gather}
W(r,\theta,t)=\frac{R_F-r}{R_F-R_B}V_B(t)+\frac{r-R_B}{R_F-R_B}V_F(\theta) \label{eq:ffem2d7}\\[6pt]
V=\bar{V}-W \label{eq:ffem2d8}
\end{gather}
\indent
We introduce the following $\theta$ - approximation: 
\begin{align}\label{eq:ffem2d9}
V(r,\theta,t) & =V_0(r,t)+V_1(r,t)\sin2\theta+V_2(r,t)\cos2\theta \nonumber \\
u(r,\theta,t) & =u_0(r,t)+u_1(r,t)\sin2\theta+u_2(r,t)\cos2\theta \nonumber\\
v(r,\theta,t) & =v_0(r,t)+v_1(r,t)\sin2\theta+v_2(r,t)\cos2\theta \nonumber\\
V_B(t) & =V_{B0}(t) \nonumber\\
V_F(\theta) & =V_{F0}+V_{F1}\sin2\theta+V_{F2}\cos2\theta \nonumber\\
\tilde{V}_I(r,\theta) & =\tilde{V}_{I0}(r)+\tilde{V}_{I1}(r)\sin2\theta+\tilde{V}_{I2}(r)\cos2\theta
\end{align}
%
Then
\begin{equation}
W(r,\theta,t) =W_0(r,t)+W_1(r,t)\sin2\theta+W_2(r,t)\cos2\theta \label{eq:ffem2d10}
\end{equation}
where
\begin{equation}
W_i(r,t)=\frac{R_F-r}{R_F-R_B}V_{Bi}(t)+\frac{r-R_B}{R_F-R_B}V_{Fi}, \qquad i=0,\ldots,2 \label{eq:ffem2d11}
\end{equation}
and $\qquad V_{Bi}(t)=0, \qquad i=1,2.$\\[6pt]
%
With this approximation, the system (\ref{eq:ffem2d4}) has the form:
\begin{gather} 
M\dot{V}-A\Big[ \frac{1}{r}(rV_r)_r +\frac{1}{r^2}V_{\theta\theta}\Big] = -M\dot{W}+A\Big( \frac{1}{r}(rW_r)_r+\frac{1}{r^2}W_{\theta\theta}\Big) \nonumber \\[6pt]
-\vec{b}_0\Big( \dot{u}_r+\frac{1}{r}(\dot{u}+\dot{v}_\theta)\Big) \label{eq:ffem2d12}
\end{gather}
%
where $\quad W$ is given by (\ref{eq:ffem2d10}) and (\ref{eq:ffem2d11}),
\begin{equation} 
W_r(r,\theta,t)=W_{0_r}(r,t)+W_{1_r}(r,t)\sin2\theta+W_{2_r}(r,t)\cos2\theta \label{eq:ffem2d13}
\end{equation}
with
\begin{equation}
W_{i_r}(r,t) =\frac{1}{R_F-R_B}\big(V_{Fi}-V_{Bi}(t)\big), \qquad i=0,\ldots,2 \label{eq:ffem2d14}
\end{equation}
and $\qquad V_{Bi}(t) =0, \qquad i=1,2.$\\[-6pt]
\begin{equation} 
W_{\theta\theta}(r,\theta,t) =-4W_1(r,t)\sin2\theta-4W_2(r,t)\cos2\theta \label{eq:ffem2d15}
\end{equation}
%
\indent
The $\theta$-approximation for the initial conditions $V(r,\theta,0)$ is
\begin{equation*}
V(r,\theta,0) =V_0(r,0)+V_1(r,0)\sin2\theta+V_2(r,0)\cos2\theta 
\end{equation*}
where
\begin{align}
&V_i(r,0) =\tilde{V}_{Ii}(r)-W_i(r,0) \nonumber \\[6pt]
& \qquad \quad =\tilde{V}_{Ii}(r)-\frac{R_F-r}{R_F-R_B}V_{Bi}(0)-\frac{r-R_B}{R_F-R_B}V_{Fi}, \qquad i=0,\ldots,2 \label{eq:ffem2d16}
\end{align}
and $\qquad V_{Bi}(0)=0, \qquad i=1,2.$\\[6pt]
%
Applying (\ref{eq:ffem2d13})-(\ref{eq:ffem2d16}) to (\ref{eq:ffem2d12}), multiplying by $1, ~\sin2\theta, ~\text{and} ~\cos2\theta$, and integrating over $(0,2\pi)$, we obtain the following three systems.\\
%
\textbf{System I}:
\begin{equation} 
M\dot{V}_0-A\frac{1}{r}(rV_{0_r})_r=-M\dot{W}_0+A\frac{1}{r}(rW_{0_r})_r-\vec{b}_0\Big(\dot{u}_{0_r}+\frac{1}{r}\dot{u}_0\Big) \label{eq:ffem2d17}
\end{equation}
with boundary conditions
\begin{equation}
V_0(r,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:ffem2d18}
\end{equation}
and initial conditions
\begin{equation}
V_0(r,0)=\tilde{V}_{I0}(r)-W_0(r,0) \label{eq:ffem2d19}
\end{equation} 
%
\textbf{System II}:
\begin{gather} 
M\dot{V}_1-A\Big[\frac{1}{r}(rV_{1_r})_r-\frac{4}{r^2}V_1\Big] = -M\dot{W}_1+A\Big(\frac{1}{r}(rW_{1_r})_r-\frac{4}{r^2}W_1\Big) \nonumber \\[6pt]
-\vec{b}_0\Big(\dot{u}_{1_r}+\frac{1}{r}(\dot{u}_1-2\dot{v}_2)\Big) \label{eq:ffem2d20}
\end{gather}
with boundary conditions
\begin{equation}
V_1(r,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:ffem2d21}
\end{equation}
and initial conditions
\begin{equation}
V_1(r,0)=\tilde{V}_{I1}(r)-W_1(r,0) \label{eq:ffem2d22}
\end{equation}
%
\textbf{System III}:
\begin{gather} 
M\dot{V}_2-A\Big[\frac{1}{r}(rV_{2_r})_r-\frac{4}{r^2}V_2\Big] = -M\dot{W}_2+A\Big(\frac{1}{r}(rW_{2_r})_r-\frac{4}{r^2}W_2\Big) \nonumber \\[6pt]
-\vec{b}_0\Big(\dot{u}_{2_r}+\frac{1}{r}(\dot{u}_2+2\dot{v}_1)\Big) \label{eq:ffem2d23}
\end{gather}
with boundary conditions
\begin{equation}
V_2(r,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:ffem2d24}
\end{equation}
and initial conditions
\begin{equation}
V_2(r,0)=\tilde{V}_{I2}(r)-W_2(r,0) \label{eq:ffem2d25} 
\end{equation} \\
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Elastic system  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection {Fourier-type approximation of the elastic system}
%
In the borehole polar coordinate system $(r, \theta)$, the displacement is
\begin{gather}
\vec{u}(r,\theta,t)=u(r,\theta,t)\vec{i}_r +v(r,\theta,t)\vec{i}_\theta \nonumber \\ 
\vec{u}\approx0 \quad \text{and} \quad \dot{\vec{u}}\approx0 \qquad \text{on  } \Gamma_F \label{eq:ffem2d26}
\end{gather}
and the strain tensor has the form
\begin{equation*}
\begin{bmatrix} \\[-2.5em]
\varepsilon_{rr}&\varepsilon_{r\theta} \\[-6pt]
\varepsilon_{\theta r}&\varepsilon_{\theta \theta}
\end{bmatrix}
\end{equation*}
where
\begin{equation}
\varepsilon_{rr}=u_r, \qquad \varepsilon_{r\theta}=\varepsilon_{\theta r}=\frac{1}{2}\big(v_r+\frac{u_\theta-v}{r}\big), \qquad \varepsilon_{\theta\theta}=\frac{u+v_\theta}{r} \label{eq:ffem2d27}
\end{equation}
\indent
The stress-strain relations for isotropic homogeneous medium in terms of the shear modulus $~G$ and the bulk modulus $~K$ are 
\begin{equation*}
\tau = 2G\varepsilon + \big(K-\frac{2}{3}G\big)(\text{tr}\varepsilon)I \tag{\ref{eq:uneq2d26}}
\end{equation*}
where
\begin{align}
\tau_{rr}&=2Gu_r+\big(K-\frac{2}{3}G\big)\big(u_r+\frac{u+v_\theta}{r}\big) \nonumber \\[6pt]
\tau_{\theta\theta}&=2G\frac{u+v_\theta}{r}+\big(K-\frac{2}{3}G\big)\big(u_r+\frac{u+v_\theta}{r}\big)\nonumber \\[6pt]
\tau_{r\theta}&=\tau_{\theta r}=G\big(v_r+\frac{u_\theta-v}{r}\big) \label{eq:ffem2d28}
\end{align}
%
\indent
The sum of the total potential energy and the penalty term is
\begin{align}
\Pi(\vec{u})=&\frac{1}{2}\int_{\Omega}\Big(\tau_{rr}(\vec{u})\varepsilon_{rr}(\vec{u})+2\tau_{r\theta}(\vec{u})\varepsilon_{r\theta}(\vec{u})+\tau_{\theta\theta}(\vec{u})\varepsilon_{\theta\theta}(\vec{u})\Big)\ud \Omega \nonumber \\[6pt]
&-\int_{\Omega}f\big(\text{tr}\varepsilon(\vec{u})\big)\ud \Omega-\int_{\Gamma}(\hat{\sigma}\vec{n})\cdot\vec{u}\ud\Gamma+K_F\int_{\Gamma_F}\vec{u}\cdot\vec{u}\ud \Gamma \label{eq:ffem2d29}
\end{align}
where $~f=\vec{b}_1^T\bar{V}=\vec{b}_1^T(V+W)$ is body force, \quad 
$\vec{b}_1^T= \begin{bmatrix} \\[-2.5em]
                     \tilde{\zeta} & -\lambda & \tilde{\alpha} \\[-6pt]
										\end{bmatrix},$\\
$K_F$ is a penalization constant,\\[12pt]
$\hat{\sigma}= \begin{bmatrix} \\[-2.5em]
\sigma_r & \sigma_{r\theta} \\[-6pt]
\sigma_{r\theta} & \sigma_\theta 
\end{bmatrix}$ is applied boundary stress in the system $(r, \theta)$.\\[12pt] 
%
\indent
Next, we describe stress around the borehole.\\[12pt]
%
$\sigma_{XY} =\begin{bmatrix} \\[-2.5em]
\sigma_x & \sigma_{xy} \\[-6pt]
\sigma_{xy} & \sigma_y 
\end{bmatrix} = \begin{bmatrix} \\[-2.5em]
\sigma_H & 0 \\[-6pt]
0 & \sigma_h 
\end{bmatrix}$ is the stress tensor in $(X,Y)$ coordinate system and \\[12pt]
% 
$\sigma(r,\theta) = \begin{bmatrix} \\[-2.5em]
\sigma_r & \sigma_{r\theta} \\[-6pt]
\sigma_{r\theta} & \sigma_\theta 
\end{bmatrix}(r,\theta)$ is the stress tensor in $(r,\theta)$ coordinate system. \\[12pt]
%
\indent
The transformation from $~\sigma_{XY}$ to $~\sigma(r,\theta)$ is given by ~\cite{Brad}:
\begin{flalign*}
&\sigma_r=\frac{\sigma_x+\sigma_y}{2}\Big(1-\frac{R_B^2}{r^2}\Big)+p_B\frac{R_B^2}{r^2}+\sigma_{xy}\Big(1-\frac{4R_B^2}{r^2}+\frac{3R_B^4}{r^4}\Big)\sin2\theta & \\[6pt]
&{}\quad \quad +\frac{\sigma_x-\sigma_y}{2}\Big(1-\frac{4R_B^2}{r^2}+\frac{3R_B^4}{r^4}\Big)\cos2\theta&\\ \\
&\sigma_\theta=\frac{\sigma_x+\sigma_y}{2}\Big(1+\frac{R_B^2}{r^2}\Big)-p_B\frac{R_B^2}{r^2}-\sigma_{xy}\Big(1+\frac{3R_B^4}{r^4}\Big)\sin2\theta & \\[6pt]
&{}\quad \quad -\frac{\sigma_x-\sigma_y}{2}\Big(1+\frac{3R_B^4}{r^4}\Big)\cos2\theta&\\ 
&\sigma_{r\theta}=\frac{\sigma_x-\sigma_y}{2}\Big(1+\frac{2R_B^2}{r^2}-\frac{3R_B^4}{r^4}\Big)\sin2\theta+\sigma_{xy}\Big(1+\frac{2R_B^2}{r^2}-\frac{3R_B^4}{r^4}\Big)\cos2\theta
\end{flalign*}
where $~p_B$ is the well bore pressure and $~\nu$ is Poisson's ratio of the formation.\\
%
From these equations we obtain applied stress on the boundaries.\\
On the borehole boundary $\Gamma_B$:\\
$\sigma_r(R_B,t) =p_B(t)$ \\
$\sigma_\theta(R_B,\theta,t) =\sigma_x+\sigma_y-p_B(t)-4\sigma_{xy}\sin2\theta-2\big(\sigma_x-\sigma_y\big)\cos2\theta$ \\
$\sigma_{r\theta}(R_B,\theta) =0$\\
%
On the far-field boundary $\Gamma_F$:\\
$\sigma_r(R_F,\theta)=\dfrac{\sigma_x+\sigma_y}{2}+\sigma_{xy}\sin2\theta+\dfrac{\sigma_x-\sigma_y}{2}\cos2\theta$ \\[6pt]
$\sigma_\theta(R_F,\theta)=\dfrac{\sigma_x+\sigma_y}{2}-\sigma_{xy}\sin2\theta-\dfrac{\sigma_x-\sigma_y}{2}\cos2\theta$ \\[6pt]
$\sigma_{r\theta}(R_F,\theta)=\dfrac{\sigma_x-\sigma_y}{2}\sin2\theta+\sigma_{xy}\cos2\theta$ \\[12pt]
%
\indent
Let $V=H^1\big(\Omega; \reals ^2\big).$ Our objective is to find the displacement $\vec{u}$ such that
$$\vec{u}\in L^2(0,t_f;V),\qquad \dot{\vec{u}}\in L^2(0,t_f;V)\qquad \text{and}$$
\begin{equation}
D\Pi(\vec{u})\vec{\Phi}=0, \qquad \forall ~\vec{\Phi} \in V \label{eq:ffem2d30}
\end{equation}
where $~\vec{u}(r,\theta,t)=u(r,\theta,t)\vec{i}_r +v(r,\theta,t)\vec{i}_\theta$, $~\vec{\Phi}(r,\theta)=\varphi(r,\theta)\vec{i}_r+\psi(r,\theta)\vec{i}_\theta~$, and $~D\Pi(\vec{u})\vec{\Phi}~$ is the G\^{a}teaux differential of $~\Pi$ with increment $~\vec{\Phi}$.\\
%
Applying (\ref{eq:ffem2d27}) and (\ref{eq:ffem2d28}) to (\ref{eq:ffem2d30}), we have
\noindent
\begin{align}
\int^{2\pi}_{0}\int^{R_F}_{R_B}&\Big[\Big(2Gu_r+\big(K-\frac{2}{3}G\big)\big(u_r+\frac{u+v_\theta}{r}\big)\Big)\varphi_r+G\big(v_r+\frac{u_\theta-v}{r}\big)\big(\psi_r+\frac{\varphi_\theta-\psi}{r}\big) \nonumber\\[6pt]
&+\Big(2G\frac{u+v_\theta}{r}+\big(K-\frac{2}{3}G\big)\big(u_r+\frac{u+v_\theta}{r}\big)\Big)\frac{\varphi+\psi_\theta}{r}\Big]r\ud r \ud \theta&\nonumber \\[6pt]
-\int^{2\pi}_{0}\int^{R_F}_{R_B}&f\big(\varphi_r+\frac{\varphi+\psi_\theta}{r}\big)r\ud r\ud \theta& \nonumber\\[6pt]
+\int^{2\pi}_{0}&\Big(\sigma_r(R_B,\theta)\varphi(R_B,\theta)+\sigma_{r\theta}(R_B,\theta)\psi(R_B,\theta)\Big)R_B\ud \theta- & \nonumber\\
-\int^{2\pi}_{0}&\Big(\sigma_r(R_F,\theta)\varphi(R_F,\theta)+\sigma_{r\theta}(R_F,\theta)\psi(R_F,\theta)\Big)R_F\ud \theta & \nonumber\\
+K_F\int^{2\pi}_{0}&\Big(u(R_F,\theta)\varphi(R_F,\theta)+v(R_F,\theta)\psi(R_F,\theta)\Big)R_F\ud \theta =0 \label{eq:ffem2d31}
\end{align} 
%
The first term of the left-hand side of (\ref{eq:ffem2d31}) is
\begin{multline}
\int^{2\pi}_{0}\int^{R_F}_{R_B}\Big[\big(K+\frac{4}{3}G\big)u_r\varphi_rr+\big(K-\frac{2}{3}G\big)u\varphi_r+\big(K-\frac{2}{3}G\big)v_\theta\varphi_r \\
+Gv_r\psi_rr+Gv_r\varphi_\theta-Gv_r\psi+Gu_\theta\psi_r-Gv\psi_r+Gu_\theta\varphi_\theta\frac{1}{r}-Gu_\theta\psi\frac{1}{r} \\
-Gv\varphi_\theta\frac{1}{r}+Gv\psi\frac{1}{r}+\big(K+\frac{4}{3}G\big)u\varphi\frac{1}{r}+\big(K+\frac{4}{3}G\big)u\psi_\theta\frac{1}{r} \\
+\big(K+\frac{4}{3}G\big)v_\theta\varphi\frac{1}{r}+\big(K+\frac{4}{3}G\big)v_\theta\psi_\theta\frac{1}{r}+\big(K-\frac{2}{3}G\big)u_r\varphi+\big(K-\frac{2}{3}G\big)u_r\psi_\theta \ud r \ud \theta \label{eq:ffem2d32}
\end{multline} 
%
Applying $\theta$-approximation (\ref{eq:ffem2d9}) to (\ref{eq:ffem2d32}) and integrating with respect to $\theta$ over ($0,2\pi$), we have
\begin{multline}
\pi\int^{R_F}_{R_B}\Big[\big(K+\frac{4}{3}G\big)\big(2u_{0_r}\varphi_{0_r}r+u_{1_r}\varphi_{1_r}r+u_{2_r}\varphi_{2_r}r\big) \\
+\big(K-\frac{2}{3}G\big)\big(2u_0\varphi_{0_r}+u_1\varphi_{1_r}+u_2\varphi_{2_r}-2v_2\varphi_{1_r}+2v_1\varphi_{2_r}\big) \\
+G\big(2v_{0_r}\psi_{0_r}r+v_{1_r}\psi_{1_r}r+v_{2_r}\psi_{2_r}r-2v_{1_r}\varphi_2+2v_{2_r}-2u_2\psi_{1_r}+2u_1\psi_{2_r}\varphi_1\big) \\
+G\big(-2v_{0_r}\psi_0-v_{1_r}\psi_1-v_{2_r}\psi_2-2v_0\psi_{0_r}-v_1\psi_{1_r}-v_2\psi_{2_r}\big) \\
+G\big(4u_2\varphi_2\frac{1}{r}+4u_1\varphi_1\frac{1}{r}+2u_2\psi_1\frac{1}{r}-2u_1\psi_2\frac{1}{r}\big) \\
+G\big(2v_1\varphi_2\frac{1}{r}-2v_2\varphi_1\frac{1}{r}+2v_0\psi_0\frac{1}{r}+v_1\psi_1\frac{1}{r}+v_2\psi_2\frac{1}{r}\big) \\
+\big(K+\frac{4}{3}G\big)\big(2u_0\varphi_0\frac{1}{r}+u_1\varphi_1\frac{1}{r}+u_2\varphi_2\frac{1}{r}-2u_1\psi_2\frac{1}{r}+2u_2\psi_1\frac{1}{r}\big) \\
+\big(K+\frac{4}{3}G\big)\big(-2v_2\varphi_1\frac{1}{r}+2v_1\varphi_2\frac{1}{r}+4v_2\psi_2\frac{1}{r}+4v_1\psi_1\frac{1}{r}\big) \\
+\big(K-\frac{2}{3}G\big)\big(2u_{0_r}\varphi_0+u_{1_r}\varphi_1+u_{2_r}\varphi_2-2u_{1_r}\psi_2+2u_{2_r}\psi_1\big)\Big]\ud r \label{eq:ffem2d33}
\end{multline}
%
Combining like terms with respect to the basis functions, (\ref{eq:ffem2d33}) has the following form:\\
%
$\varphi_0:$
\begin{multline}
\int^{R_F}_{R_B}\Big[2\big(K+\frac{4}{3}G\big)u_{0_r}\varphi_{0_r}r+2\big(K-\frac{2}{3}G\big)(u_0\varphi_{0_r}+u_{0_r}\varphi_0)+2\big(K+\frac{4}{3}G\big)u_0\varphi_0\frac{1}{r}\Big] \ud r \label{eq:ffem2d34}
\end{multline}
%
$\varphi_1:$
\begin{multline}
+\pi\int^{R_F}_{R_B}\Big[\big(K+\frac{4}{3}G\big)u_{1_r}\varphi_{1_r}r+\big(K-\frac{2}{3}G\big)u_1\varphi_{1_r}-2\big(K-\frac{2}{3}G\big)v_2\varphi_{1_r}+2Gv_{2_r}\varphi_1 \\
+\big(K+\frac{16}{3}G\big)u_1\varphi_1\frac{1}{r}-2\big(K+\frac{7}{3}G\big)v_2\varphi_1\frac{1}{r}+\big(K-\frac{2}{3}G\big)u_{1_r}\varphi_1\Big]\ud r \label{eq:ffem2d35}
\end{multline}
%
$\varphi_2:$
\begin{multline}
+\pi\int^{R_F}_{R_B}\Big[\big(K+\frac{4}{3}G\big)u_{2_r}\varphi_{2_r}r+\big(K-\frac{2}{3}G\big)u_2\varphi_{2_r}+2\big(K-\frac{2}{3}G\big)v_1\varphi_{2_r}-2Gv_{1_r}\varphi_2 \\
+\big(K+\frac{16}{3}G\big)u_2\varphi_2\frac{1}{r}+2\big(K+\frac{7}{3}G\big)v_1\varphi_2\frac{1}{r}+\big(K-\frac{2}{3}G\big)u_{2_r}\varphi_2\Big]\ud r \label{eq:ffem2d36}
\end{multline}
%
$\psi_0:$
\begin{equation}
+~\pi\int^{R_F}_{R_B}\Big[\underbrace{2Gv_{0_r}\psi_{0_r}r-2Gv_{0_r}\psi_0-2Gv_0\psi_{0_r}+2Gv_0\psi_0\frac{1}{r}}_{2G\Big[\big(v_{0_r}-\frac{v_0}{r}\big)\big(\psi_{0_r}-\frac{\psi_0}{r}\big)\Big]r}\Big] \ud r \label{eq:ffem2d37}
\end{equation}
%
$\psi_1:$
\begin{multline}
+\pi\int^{R_F}_{R_B}\Big[Gv_{1_r}\psi_{1_r}r-Gv_{1_r}\psi_1-2Gu_2\psi_{1_r}-Gv_1\psi_{1_r}+2\big(K+\frac{7}{3}G\big)u_2\psi_1\frac{1}{r} \\
+\big(4K+\frac{19}{3}G\big)v_1\psi_1\frac{1}{r}+2\big(K-\frac{2}{3}G\big)u_{2_r}\psi_1 \Big] \ud r \label{eq:ffem2d38}
\end{multline}
%
$\psi_2:$
\begin{multline}
+\pi\int^{R_F}_{R_B}\Big[Gv_{2_r}\psi_{2_r}r-Gv_{2_r}\psi_2+2Gu_1\psi_{2_r}-Gv_2\psi_{2_r}-2\big(K+\frac{7}{3}G\big)u_1\psi_2\frac{1}{r} \\
+\big(4K+\frac{19}{3}G\big)v_2\psi_2\frac{1}{r}-2\big(K-\frac{2}{3}G\big)u_{1_r}\psi_2\Big] \ud r \label{eq:ffem2d39}
\end{multline}
%
\noindent
The $\sigma$-approximation of body force $f$ has the form
\begin{equation}
f(r,\theta,t)=f_0(r,t)+f_1(r,t)\sin2\theta+f_2(r,t)\cos2\theta \label{eq:ffem2d40}
\end{equation}
%
The second term of the left-hand side of (\ref{eq:ffem2d31}) is
\begin{align}
-\pi\int^{R_F}_{R_B}&\big[2f_0\varphi_{0_r}r+f_1\varphi_{1_r}r+f_2\varphi_{2_r}r & \nonumber\\
&+2f_0\varphi_0+f_1\varphi_1+f_2\varphi_2-2f_1\psi_2+2f_2\psi_1 \big] \ud r \label{eq:ffem2d41}
\end{align}
%
The third term of the left-hand side of (\ref{eq:ffem2d31}) is
\begin{equation}
2\pi p_B\varphi_0(R_B)R_B \label{eq:ffem2d42}
\end{equation}
%
The fourth term of the left-hand side of (\ref{eq:ffem2d31}) is
\begin{align}
-\pi\Big[2\frac{\sigma_x+\sigma_y}{2}\varphi_0(R_F)&+\sigma_{xy}\varphi_1(R_F)+\frac{\sigma_x-\sigma_y}{2}\varphi_2(R_F) \nonumber\\
&+\frac{\sigma_x-\sigma_y}{2}\psi_1(R_F)+\sigma_{xy}\psi_2(R_F)\Big]R_F \label{eq:ffem2d43}
\end{align}
%
The fifth term of the left-hand side of (\ref{eq:ffem2d31}) is
\begin{multline}
\pi K_F\big[2u_0(R_F)\varphi_0(R_F)+u_1(R_F)\varphi_1(R_F)+u_2(R_F)\varphi_2(R_F) \\
+2v_0(R_F)\psi_0(R_F)+v_1(R_F)\psi_1(R_F)+v_2(R_F)\psi_2(R_F)\big]R_F \label{eq:ffem2d44}
\end{multline}
%
In (\ref{eq:ffem2d20})-(\ref{eq:ffem2d39}) and (\ref{eq:ffem2d41})-(\ref{eq:ffem2d44}), we combine terms corresponding to $\varphi_i$ and $\psi_i$, $i=0,\ldots,2$, and equal them to 0.\\ 
%
$\varphi_0: \quad (u_0)$
\begin{flalign}
&\int^{R_F}_{R_B}\Big[\big(K+\frac{4G}{3}\big)u_{0_r}\varphi_{0_r}r+\big(K-\frac{2G}{3}\big)(u_0\varphi_{0_r}+u_{0_r}\varphi_0)+\big(K+\frac{4G}{3}\big)u_0\varphi_0\frac{1}{r}& \nonumber\\[6pt]
-&\int^{R_F}_{R_B}f_0\big(\varphi_{0_r}r+\varphi_0\big)\ud r +p_B\varphi_0(R_B)R_B-\frac{\sigma_x+\sigma_y}{2}\varphi_0(R_F)R_F& \nonumber\\[6pt]
&\qquad \qquad +K_Fu_0(R_F)\varphi_0(R_F)R_F =0 \label{eq:ffem2d45}
\end{flalign}
%
$\varphi_1: \quad (u_1, v_2)$
\begin{align}
&\int^{R_F}_{R_B}\Big[\big(K+\frac{4G}{3}\big)u_{1_r}\varphi_{1_r}r+\big(K-\frac{2G}{3}\big)u_1\varphi_{1_r}+\big(K-\frac{2G}{3}\big)u_{1_r}\varphi_1 \nonumber\\[6pt]
&\qquad +\big(K+\frac{16G}{3}\big)u_1\varphi_1\frac{1}{r}-2\big(K-\frac{2G}{3}\big)v_2\varphi_{1_r}+2Gv_{2_r}\varphi_1-2\big(K+\frac{7G}{3}\big)v_2\varphi_1\frac{1}{r}\Big] \ud r \nonumber \\[6pt]
&-\int^{R_F}_{R_B}f_1\big(\varphi_{1_r}r+\varphi_1\big)\ud r -\sigma_{xy}\varphi_1(R_F)R_F+K_Fu_1(R_F)\varphi_1(R_F)R_F = 0 \label{eq:ffem2d46}
\end{align}\\
%
$\varphi_2: \quad (u_2, v_1)$
\begin{align}
&\int^{R_F}_{R_B}\Big[\big(K+\frac{4G}{3}\big)u_{2_r}\varphi_{2_r}r+\big(K-\frac{2G}{3}\big)u_2\varphi_{2_r}+\big(K-\frac{2G}{3}\big)u_{2_r}\varphi_2 \nonumber\\[6pt]
&\quad +\big(K+\frac{16G}{3}\big)u_2\varphi_2\frac{1}{r}+2\big(K-\frac{2G}{3}\big)v_1\varphi_{2_r}-2Gv_{1_r}\varphi_2+2\big(K+\frac{7G}{3}\big)v_1\varphi_2\frac{1}{r} \Big] \ud r \nonumber \\[6pt]
-&\int^{R_F}_{R_B}f_2\big(\varphi_{2_r}r+\varphi_2\big)\ud r -\frac{\sigma_x-\sigma_y}{2}\varphi_2(R_F)R_F+K_Fu_2(R_F)\varphi_2(R_F)R_F = 0 \label{eq:ffem2d47}
\end{align}
%
$\psi_0: \quad (v_0)$
\begin{equation*}
\int^{R_F}_{R_B}G\Big[\big(v_{0_r}-\frac{v_0}{r}\big)\big(\psi_{0_r}-\frac{\psi_0}{r}\big)\Big]r \ud r +K_Fv_0(R_F)\psi_0(R_F)R_F =0 
\vspace{0.2in}
\end{equation*}
%
Take $\psi_0=v_0$. Then
$$\int^{R_F}_{R_B}G\Big[\big(v_{0_r}-\frac{v_0}{r}\big)^2\Big]r \ud r +K_FR_Fv_0^2(R_F)=0$$
It follows that
\begin{equation*}
v_{0_r}-\frac{v_0}{r}=0, \quad R_B<r<R_F, \qquad \text{and} \qquad v_0(R_F)=0
\end{equation*}
and so,
\begin{equation}
v_0(r,t)=C_0(t)r, \quad R_B<r<R_F \label{eq:ffem2d48}
\end{equation}
%
$\psi_1: \quad(u_2, v_1)$
\begin{flalign}
&\int^{R_F}_{R_B}\Big[-2Gu_2\psi_{1_r}+2\big(K-\frac{2G}{3}\big)u_{2_r}\psi_1+2\big(K+\frac{7G}{3}\big)u_2\psi_1\frac{1}{r}&\nonumber\\[6pt]
&\qquad \quad +Gv_{1_r}\psi_{1_r}r-Gv_1\psi_{1_r}-Gv_{1_r}\psi_1+\big(4K+\frac{19G}{3}\big)v_1\psi_1\frac{1}{r} \Big]\ud r &\nonumber\\[6pt]
-&\int^{R_F}_{R_B}2f_2\psi_1\ud r -\frac{\sigma_x-\sigma_y}{2}\psi_1(R_F)R_F+K_Fv_1(R_F)\psi_1(R_F)R_F =0 \label{eq:ffem2d49}
\end{flalign}\\
%
$\psi_2: \quad(u_1, v_2)$
\begin{flalign}
&\int^{R_F}_{R_B}\Big[2Gu_1\psi_{2_r}-2\big(K-\frac{2G}{3}\big)u_{1_r}\psi_2-2\big(K+\frac{7G}{3}\big)u_1\psi_2\frac{1}{r}&\nonumber\\[6pt]
&\qquad \quad +Gv_{2_r}\psi_{2_r}r-Gv_2\psi_{2_r}-Gv_{2_r}\psi_2+\big(4K+\frac{19G}{3}\big)v_2\psi_2\frac{1}{r} \Big]\ud r & \nonumber\\[6pt]
+&\int^{R_F}_{R_B}2f_1\psi_2\ud r -\sigma_{xy}\psi_2(R_F)R_F+K_Fv_2(R_F)\psi_2(R_F)R_F =0 \label{eq:ffem2d50}
\end{flalign}
\\
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Finite Element Approximation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%% Basis Functions, TCP vector, and Boundary & Initial Conditions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection {Finite element approximation}
\centerline{Dirichlet boundary conditions for the TCP system}
$r$-approximation:\\
\noindent
\begin{figure}[ht]
\centering
\vspace{-0.4in}
\includegraphics[scale=0.65]{DR_r_2D.png}
\caption{$r$-approximation of 2D Dirichlet boundary conditions}
\end{figure}

\noindent
Here $N$ is the number of subintervals; $~M_{D}=N-1$ is the number of basis functions; $~\bar{b}(r)=\begin{bmatrix} \\[-2.7em]
\bar{b}_1(r) & \ldots & \bar{b}_{M_D}(r)
\end{bmatrix}^T$ is the vector of basis functions; and\\
%
$$B(r)=\begin{bmatrix} \\[-2.3em]
\bar{b}(r)^T & 0 & 0 \\[-6pt]
0 & \bar{b}(r)^T & 0 \\[-6pt]
0 & 0 & \bar{b}(r)^T
\end{bmatrix}_{3\times3M_{D}}
\vspace{0.1in}
$$
%
Using $r$-approximation, the coefficients of the $\theta$-approximation for $V(r,\theta,t)$ can be represented as
\begin{equation}
V_i(r,t)=B(r)\bar{c}_i(t) \label{eq:ffem2d51}
\end{equation}
where $~\bar{c}_i(t)~$ is a ($3M_{D}\times 1$) vector.\\
%
The coefficients of the $\theta$-approximation (\ref{eq:ffem2d9}) for boundary conditions (\ref{eq:ffem2d2}) and initial condition (\ref{eq:ffem2d3}) can be represented as
\begin{align}
V_{B0}(t)&=\bar{\nu}_{B0}(t) \nonumber\\
V_{Fi}&=\bar{\nu}_{Fi} \nonumber\\
\tilde{V}_{Ii}(r)&=B(r)\bar{\nu}_{Ii}, \qquad i=0, \ldots, 2 \label{eq:ffem2d52}
\end{align}
where $~\bar{\nu}_{B0}(t)$ and $~\bar{\nu}_{Fi}~$ are ($3\times 1$) vectors ~and $~\bar{\nu}_{Ii}~$ is a ($3M_{D}\times 1$) vector.\\
\\
%
\centerline{Neumann boundary conditions for the elastic system}

$r$-approximation:\\
\noindent
\begin{figure}[ht]
\centering
\vspace{-0.4in}
\includegraphics[scale=0.65]{Neu_r_2D.png}
\caption{$r$-approximation of 2D Neumann boundary conditions}
\end{figure}

\noindent
Here $~\uN$ is the number of subintervals; $~M=\uN + 1$ {} is the number of basis functions; $~\bar{\ub}(r)=\begin{bmatrix} \\[-2.7em]
\bar{\ub}_1(r) & \ldots & \bar{\ub}_{M}(r)
\end{bmatrix}^T$ is the vector of basis functions.\\[6pt]
%
Using $r$-approximation, the coefficients of the $\theta$-approximation (\ref{eq:ffem2d9}) for the displacement coordinates (\ref{eq:ffem2d51}) can be represented as
\begin{align}
u_i(r,t)&=\bar{\ub}(r)^T\bar{\mu}_i(t) \label{eq:ffem2d53} \\
v_i(r,t)&=\bar{\ub}(r)^T\bar{\eta}_i(t) \label{eq:ffem2d54}
\end{align}
where $~\bar{\mu}_i(t)~$ and $~\bar{\eta}_i(t)~$ are ($M\times 1$) vectors. \\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM approximation of the elastic system %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\\
\centerline{Finite element approximation of the elastic system}
\indent
Define the following $M\times M$ matrices:
\begin{align*}
g_2(r)&=\int^{R_F}_{R_B}\bar{\ub}_r(r)\bar{\ub}_r(r)^Tr\ud r \\[6pt]
g_1&=\int^{R_F}_{R_B}\bar{\ub}(r)\bar{\ub}_r(r)^T\ud r \\[6pt]
g_0\big(\frac{1}{r}\big)&=\int^{R_F}_{R_B}\bar{\ub}(r)\bar{\ub}(r)^T\frac{1}{r}\ud r \\[6pt]
g_F&=\bar{\ub}(R_F)\bar{\ub}(R_F)^T
\end{align*}
%
Let $P_3=\begin{bmatrix} \\[-2.6em]
0 & 0 & 1 \\[-0.3em]
\end{bmatrix}^T$. Then
\begin{equation}
p_B(t)=P_3^TV_B(t)=P_3^TV_{B0}(t)=P_3^T\bar{\nu}_{B0}(t) \label{eq:ffem2d55}
\end{equation}
%
Also, let
\begin{equation}
\sigma_R=\frac{\sigma_x+\sigma_y}{2}\qquad \text{and} \qquad \sigma_D=\frac{\sigma_x-\sigma_y}{2} \label{eq:ffem2d56}
\end{equation}
%
As before,
\begin{equation*}
f(r,\theta,t)=\vec{b}_1^T\bar{V}(r,\theta,t)=\vec{b}_1^T\big(V(r,\theta,t)+W(r,\theta,t)\big)
\end{equation*}
and from (\ref{eq:ffem2d9})-(\ref{eq:ffem2d11}), (\ref{eq:ffem2d13}), (\ref{eq:ffem2d51}), and (\ref{eq:ffem2d52}) it follows that
\begin{align}
f_i(r,t) & =\vec{b}_1^T\big(V_i(r,t)+W_i(r,t)\big) \nonumber \\[6pt]
& = \vec{b}_1^TB(r)\bar{c}_i(t)+\frac{R_F-r}{R_F-R_B}\vec{b}_1^T\bar{\nu}_{Bi}(t)+\frac{r-R_B}{R_F-R_B}\vec{b}_1^T\bar{\nu}_{Fi}, \quad i=0,1,2 \label{eq:ffem2d57}
\end{align} 
and $\bar{\nu}_{Bi}(t)=0,\quad i=1,2.$\\ 
%
%
From (\ref{eq:ffem2d45})-(\ref{eq:ffem2d50}) we obtain\\
%
$\varphi_0:$
\begin{align}
&\Big[\big(K+\frac{4G}{3}\big)g_2(r)+\big(K-\frac{2G}{3}\big)\big(g_1^T+g_1\big)+\big(K+\frac{4G}{3}\big)g_0\big(\frac{1}{r}\big)+K_FR_Fg_F\Big]\bar{\mu}_0(t) \nonumber \\[6pt]
=&\Big[\int_{R_B}^{R_F}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\vec{b}_1^TB(r)\ud r \Big] \bar{c}_0(t) \nonumber \\[6pt]
+&\bigg\{\Big[\int_{R_B}^{R_F}\frac{R_F-r}{R_F-R_B}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\ud r \Big]\vec{b}_1^T-R_B\bar{\ub}(R_B)P_3^T\bigg\} \bar{\nu}_{B0}(t) \nonumber \\[6pt]
+&\Big[\int_{R_B}^{R_F}\frac{r-R_B}{R_F-R_B}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\ud r \Big]\vec{b}_1^T \bar{\nu}_{F0}+R_F\sigma_R\bar{\ub}(R_F) \label{eq:ffem2d58}
\end{align}
%
$\varphi_1:$
\begin{align}
&\Big[\big(K+\frac{4G}{3}\big)g_2(r)+\big(K-\frac{2G}{3}\big)\big(g_1^T+g_1\big)+\big(K+\frac{16G}{3}\big)g_0\big(\frac{1}{r}\big)+K_FR_Fg_F\Big]\bar{\mu}_1(t) \nonumber \\[6pt]
-2&\Big[\big(K-\frac{2G}{3}\big)g_1^T-Gg_1+\big(K+\frac{7G}{3}\big)g_0\big(\frac{1}{r}\big)\Big]\bar{\eta}_2(t)\nonumber \\[6pt]
=&\Big[\int_{R_B}^{R_F}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\vec{b}_1^TB(r)\ud r \Big] \bar{c}_1(t) \nonumber \\[6pt]
+&\Big[\int_{R_B}^{R_F}\frac{r-R_B}{R_F-R_B}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\ud r \Big]\vec{b}_1^T \bar{\nu}_{F1}+R_F\sigma_{xy}\bar{\ub}(R_F) \label{eq:ffem2d59}
\end{align}
%
$\varphi_2:$
\begin{align}
&\Big[\big(K+\frac{4G}{3}\big)g_2(r)+\big(K-\frac{2G}{3}\big)\big(g_1^T+g_1\big)+\big(K+\frac{16G}{3}\big)g_0\big(\frac{1}{r}\big)+K_FR_Fg_F\Big]\bar{\mu}_2(t) \nonumber \\[6pt]
+2&\Big[\big(K-\frac{2G}{3}\big)g_1^T-Gg_1+\big(K+\frac{7G}{3}\big)g_0\big(\frac{1}{r}\big)\Big]\bar{\eta}_1(t)\nonumber \\[6pt]
=&\Big[\int_{R_B}^{R_F}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\vec{b}_1^TB(r)\ud r \Big] \bar{c}_2(t) \nonumber \\[6pt]
+&\Big[\int_{R_B}^{R_F}\frac{r-R_B}{R_F-R_B}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\ud r \Big]\vec{b}_1^T \bar{\nu}_{F2}+R_F\sigma_D\bar{\ub}(R_F) \label{eq:ffem2d60}
\end{align}
%
$\psi_0:$
\begin{equation*}
\Big[Gg_2(r)-G\big(g_1^T+g_1\big)+Gg_0\big(\frac{1}{r}\big)+K_FR_Fg_F\Big]\bar{\eta}_0(t)=0
\end{equation*}
It follows that
$$\bar{\eta}_0(t)=0$$
%
$\psi_1:$
\begin{align}
2&\Big[\big(K-\frac{2G}{3}\big)g_1-Gg_1^T+\big(K+\frac{7G}{3}\big)g_0\big(\frac{1}{r}\big)\Big]\bar{\mu}_2(t)\nonumber \\[6pt]
+&\Big[Gg_2(r)-G\big(g_1^T+g_1\big)+\big(4K+\frac{19G}{3}\big)g_0\big(\frac{1}{r}\big)+K_FR_Fg_F\Big]\bar{\eta}_1(t) \nonumber \\[6pt]
=2&\Big[\int_{R_B}^{R_F}\bar{\ub}(r)\vec{b}_1^TB(r)\ud r \Big] \bar{c}_2(t) \nonumber \\[6pt]
+2&\Big[\int_{R_B}^{R_F}\frac{r-R_B}{R_F-R_B}\bar{\ub}(r)\ud r \Big]\vec{b}_1^T \bar{\nu}_{F2}+R_F\sigma_D\bar{\ub}(R_F) \label{eq:ffem2d61}
\end{align}
%
$\psi_2:$
\begin{align}
-2&\Big[\big(K-\frac{2G}{3}\big)g_1-Gg_1^T+\big(K+\frac{7G}{3}\big)g_0\big(\frac{1}{r}\big)\Big]\bar{\mu}_1(t)\nonumber \\[6pt]
+&\Big[Gg_2(r)-G\big(g_1^T+g_1\big)+\big(4K+\frac{19G}{3}\big)g_0\big(\frac{1}{r}\big)+K_FR_Fg_F\Big]\bar{\eta}_2(t) \nonumber \\[6pt]
=-2&\Big[\int_{R_B}^{R_F}\bar{\ub}(r)\vec{b}_1^TB(r)\ud r \Big] \bar{c}_1(t) \nonumber \\[6pt]
-2&\Big[\int_{R_B}^{R_F}\frac{r-R_B}{R_F-R_B}\bar{\ub}(r)\ud r \Big]\vec{b}_1^T \bar{\nu}_{F1}+R_F\sigma_{xy}\bar{\ub}(R_F) \label{eq:ffem2d62} \\[-6pt] \nonumber
\end{align}
%
\indent
Define the following matrices and vectors.\\
\centerline{$M\times M$ matrices:}
\begin{align}
\mathcal{M}_0 &= \big(K+\frac{4G}{3}\big)g_2(r)+\big(K-\frac{2G}{3}\big)\big(g_1^T+g_1\big)+\big(K+\frac{4G}{3}\big)g_0\big(\frac{1}{r}\big)+K_FR_Fg_F \nonumber\\[6pt]
\mathcal{M}_1 & = \big(K+\frac{4G}{3}\big)g_2(r)+\big(K-\frac{2G}{3}\big)\big(g_1^T+g_1\big)+\big(K+\frac{16G}{3}\big)g_0\big(\frac{1}{r}\big)+K_FR_Fg_F\nonumber\\[6pt]
\mathcal{M}_2 & = Gg_2(r)-G\big(g_1^T+g_1\big)+\big(4K+\frac{19G}{3}\big)g_0\big(\frac{1}{r}\big)+K_FR_Fg_F \nonumber\\[6pt]
\mathcal{N} & = 2\Big[\big(K-\frac{2G}{3}\big)g_1^T-Gg_1+\big(K+\frac{7G}{3}\big)g_0\big(\frac{1}{r}\big)\Big] \label{eq:ffem2d63}
\end{align}
%
\centerline{$M\times 3M_D$ matrices:}
\begin{align}
\mathcal{Z}_0 &= \int_{R_B}^{R_F}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\vec{b}_1^TB(r)\ud r \label{eq:ffem2d64}\\[6pt]
\mathcal{Z}_1 &= 2\int_{R_B}^{R_F}\bar{\ub}(r)\vec{b}_1^TB(r)\ud r \label{eq:ffem2d65}
\end{align}\\[-6pt]
%
\centerline{$M\times 3$ matrices:}
\begin{align}
Y_{B0}& = \Big[\int_{R_B}^{R_F}\frac{R_F-r}{R_F-R_B}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\ud r\Big]\vec{b}_1^T-R_B\bar{\ub}(R_B)P_3^T \label{eq:ffem2d66}\\[6pt] 
Y_{F0}& = \Big[\int_{R_B}^{R_F}\frac{r-R_B}{R_F-R_B}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\ud r\Big]\vec{b}_1^T \label{eq:ffem2d67} \\[6pt]
Y_{F1}& = 2\Big[\int_{R_B}^{R_F}\frac{r-R_B}{R_F-R_B}\bar{\ub}(r)\ud r\Big]\vec{b}_1^T \label{eq:ffem2d68}
\end{align}\\[-6pt]
%
\centerline{$M\times 1$ vectors:}
\begin{align}
S_{R}& = R_F\sigma_R\bar{\ub}(R_F) \label{eq:ffem2d69}\\
S_{xy}& = R_F\sigma_{xy}\bar{\ub}(R_F) \label{eq:ffem2d70}\\
S_{D}& = R_F\sigma_D\bar{\ub}(R_F) \label{eq:ffem2d71}
\end{align} 
%
Applying (\ref{eq:ffem2d63})-(\ref{eq:ffem2d71}) to (\ref{eq:ffem2d58})-(\ref{eq:ffem2d62}), we obtain the following matrix equations:
%%% mu_0
\begin{equation}
\mathcal{M}_0\bar{\mu}_0(t)=\mathcal{Z}_0\bar{c}_0(t)+Y_{B0}\bar{\nu}_{B0}(t)+Y_{F0}\bar{\nu}_{F0}+S_R \label{eq:ffem2d72}
\end{equation}
%
%%%% mu_1, eta_2
\begin{gather}
\begin{bmatrix} \\[-2.5em]
\mathcal{M}_1 & -\mathcal{N} \\
-\mathcal{N}^T & \mathcal{M}_2  
\end{bmatrix} \begin{bmatrix} \\[-2.5em]
\bar{\mu}_1(t) \\
\bar{\eta}_2(t)  
\end{bmatrix} = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
-\mathcal{Z}_1 
\end{bmatrix}\bar{c}_1(t)+\begin{bmatrix} \\[-2.5em]
Y_{F0} \\
-Y_{F1} 
\end{bmatrix}\bar{\nu}_{F1}+\begin{bmatrix} \\[-2.5em]
1 \\
1
\end{bmatrix}S_{xy} \label{eq:ffem2d73}
\end{gather}
%
%%%% mu_2, eta_1
\begin{gather}
\begin{bmatrix} \\[-2.5em]
\mathcal{M}_1 & \mathcal{N} \\
\mathcal{N}^T & \mathcal{M}_2  
\end{bmatrix} \begin{bmatrix} \\[-2.5em]
\bar{\mu}_2(t) \\
\bar{\eta}_1(t) 
\end{bmatrix} = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
\mathcal{Z}_1  
\end{bmatrix}\bar{c}_2(t)+\begin{bmatrix} \\[-2.5em]
Y_{F0} \\
Y_{F1}  
\end{bmatrix}\bar{\nu}_{F2}+\begin{bmatrix} \\[-2.5em]
1 \\
1   
\end{bmatrix}S_D \label{eq:ffem2d74}
\end{gather}
%
\indent
Define the following matrices:
%
\begin{gather*}
J = \begin{bmatrix} \\[-2.5em]
I_{M \times M} & 0 \\
0 & -I_{M \times M}   
\end{bmatrix}
\\[9pt]
%
\mathcal{Q} = \begin{bmatrix} \\[-2.5em]
\mathcal{M}_1 & \mathcal{N} \\
\mathcal{N}^T & \mathcal{M}_2  
\end{bmatrix} \quad \Rightarrow \quad 
\begin{bmatrix} \\[-2.5em]
\mathcal{M}_1 & -\mathcal{N} \\
-\mathcal{N}^T & \mathcal{M}_2  
\end{bmatrix} = J\mathcal{Q} J
\\[9pt]
%
\bar{\mathcal{Z}} = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
\mathcal{Z}_1   
\end{bmatrix} \qquad \Rightarrow
%
\begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
-\mathcal{Z}_1 
\end{bmatrix} = J\bar{\mathcal{Z}}
\\[9pt]
%
\bar{Y}_F = \begin{bmatrix} \\[-2.5em]
Y_{F0} \\
Y_{F1}
\end{bmatrix} \qquad \Rightarrow
%
\begin{bmatrix} \\[-2.5em]
Y_{F0} \\
-Y_{F1}
\end{bmatrix} = J\bar{Y}_F
\\[9pt]
%
\bar{S}_{xy}=\begin{bmatrix} \\[-2.5em]
S_{xy} \\
S_{xy}   
\end{bmatrix}, \qquad 
%
\bar{S}_D=\begin{bmatrix} \\[-2.5em]
S_D \\
S_D   
\end{bmatrix}
\end{gather*}
%
\begin{align*}
P_{10} & = \begin{bmatrix} \\[-2.6em]
1 & 0 \\[-0.3em]
\end{bmatrix} \otimes I_{M\times M} \\
P_{01} & = \begin{bmatrix} \\[-2.6em]
0 & 1 \\[-0.3em]
\end{bmatrix} \otimes I_{M\times M} 
\end{align*}
%
From (\ref{eq:ffem2d72})-(\ref{eq:ffem2d74}) we have
%
\begin{align}
\bar{\mu}_0(t)& = \mathcal{M}_0^{-1}\Big[\mathcal{Z}_0\bar{c}_0(t)+Y_{B0}\bar{\nu}_{B0}(t)+Y_{F0}\bar{\nu}_{F0}+S_R\Big] \label{eq:ffem2d75} \\[6pt]
%
\bar{\mu}_1(t)& = P_{10}\mathcal{Q}^{-1}\Big[\bar{\mathcal{Z}}\bar{c}_1(t)+\bar{Y}_F\bar{\nu}_{F1}+J\bar{S}_{xy}\Big] \label{eq:ffem2d76}\\[6pt]
%
\bar{\eta}_2(t)& = -P_{01}\mathcal{Q}^{-1}\Big[\bar{\mathcal{Z}}\bar{c}_1(t)+\bar{Y}_F\bar{\nu}_{F1}+J\bar{S}_{xy}\Big] \label{eq:ffem2d77}\\[6pt]
%
\bar{\mu}_2(t)& = P_{10}\mathcal{Q}^{-1}\Big[\bar{\mathcal{Z}}\bar{c}_2(t)+\bar{Y}_F\bar{\nu}_{F2}+\bar{S}_D\Big] \label{eq:ffem2d78}\\[6pt]
%
\bar{\eta}_1(t)& = P_{01}\mathcal{Q}^{-1}\Big[\bar{\mathcal{Z}}\bar{c}_2(t)+\bar{Y}_F\bar{\nu}_{F2}+\bar{S}_D\Big] \label{eq:ffem2d79}
\end{align}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM Approximation of TCP System %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM Approximation of TCP System I (radially symmetric case) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\centerline{Finite element approximation of the TCP system}
%
\centerline{System I (radially symmetric case)}
%
System I is given by (\ref{eq:ffem2d17})-(\ref{eq:ffem2d19}):
\begin{equation} 
M\dot{V}_0-A\frac{1}{r}(rV_{0_r})_r=-M\dot{W}_0+A\frac{1}{r}(rW_{0_r})_r-\vec{b}_0\Big(\dot{u}_{0_r}+\frac{1}{r}\dot{u}_0\Big) \tag{\ref{eq:ffem2d17}}
\end{equation}
with boundary conditions
\begin{equation}
V_0(r,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \tag{\ref{eq:ffem2d18}}
\end{equation}
and initial conditions
\begin{equation}
V_0(r,0)=\tilde{V}_{I0}(r)-W_0(r,0) \tag{\ref{eq:ffem2d19}}
\end{equation}
%
\indent
Let
\begin{equation*}
\mathcal{A}=M^{-1}A \\[-6pt]
\end{equation*}
Then (\ref{eq:ffem2d17}) has the form
\begin{equation} 
\dot{V}_0-\mathcal{A}\frac{1}{r}(rV_{0_r})_r=-\dot{W}_0+\mathcal{A}\frac{1}{r}(rW_{0_r})_r-M^{-1}\vec{b}_0\Big( \dot{u}_{0_r}+\frac{1}{r}\dot{u}_0\Big) \label{eq:ffem2d80}
\end{equation}
%
Using (\ref{eq:ffem2d51}), the left-hand side of (\ref{eq:ffem2d80}) is
\begin{equation}
B(r)\dot{\bar{c}}_0(t)-\mathcal{A}\frac{1}{r}\Big(rB_r(r)\Big)_r\bar{c}_0(t)\label{eq:ffem2d81}
\end{equation}
%
Multiplying (\ref{eq:ffem2d81}) by $B(r)^T$ and integrating with respect to $r$, we have
\begin{align}
&\Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\dot{\bar{c}}_0(t)-\Big[\int_{R_B}^{R_F}B(r)^T\mathcal{A}\frac{1}{r}\Big(rB_r(r)\Big)_rr\ud r\Big]\bar{c}_0(t) \nonumber \\[6pt]
=&\Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\dot{\bar{c}}_0(t)+\Big[\int_{R_B}^{R_F}B_r(r)^T\mathcal{A}B_r(r)r\ud r\Big]\bar{c}_0(t) \nonumber \\[6pt]
=&\Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\dot{\bar{c}}_0(t)+\mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big]\bar{c}_0(t) \label{eq:ffem2d82}
\end{align}
%
Using (\ref{eq:ffem2d53}) and (\ref{eq:ffem2d54}), the third term of the right-hand side of (\ref{eq:ffem2d80}) is
\begin{equation}
-M^{-1}\vec{b}_0\Big( \dot{u}_{0_r}+\frac{1}{r}\dot{u}_0\Big)=-M^{-1}\vec{b}_0\Big[\bar{\ub}_r(r)^T+\frac{1}{r}\bar{\ub}(r)^T\Big]\dot{\bar{\mu}}_0(t) \label{eq:ffem2d83}
\end{equation}
%
Multiplying (\ref{eq:ffem2d83}) by $B(r)^T$ and integrating with respect to $r$, we have
%
\begin{equation}
-\Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\big(r\bar{\ub}_r(r)^T+\bar{\ub}(r)^T\big) \ud r \Big]\dot{\bar{\mu}}_0(t) \label{eq:ffem2d84}
\end{equation} 
%
From (\ref{eq:ffem2d11}) and (\ref{eq:ffem2d52})
\begin{equation}
W_i(r,t) = \frac{R_F-r}{R_F-R_B}\bar{\nu}_{Bi}(t)+\frac{r-R_B}{R_F-R_B}\bar{\nu}_{Fi}, \quad i=0,\ldots,2 \label{eq:ffem2d85}
\end{equation}
and $\bar{\nu}_{Bi}(t)=0,\quad i=1,2.$\\ 
It follows that
% 
\begin{equation}
\dot{W}_i(r,t) = \frac{R_F-r}{R_F-R_B}\dot{\bar{\nu}}_{Bi}(t), \quad i=0,\ldots,2 \label{eq:ffem2d86}
\end{equation}
and $\dot{\bar{\nu}}_{Bi}(t)=0,\quad i=1,2.$\\ 
%
Multiplying (\ref{eq:ffem2d86}) by $B(r)^T$ and integrating with respect to $r$, we have
%
\begin{equation}
\Big[\int_{R_B}^{R_F}B(r)^T\frac{R_F-r}{R_F-R_B}r\ud r\Big] \dot{\bar{\nu}}_{Bi}(t) \label{eq:ffem2d87}
\end{equation}
%
From (\ref{eq:ffem2d14}) and (\ref{eq:ffem2d52})
\begin{equation}
W_{i_r}(r,t) = -\frac{1}{R_F-R_B}\bar{\nu}_{Bi}(t)+\frac{1}{R_F-R_B}\bar{\nu}_{Fi}, \quad i=0,\ldots,2 \label{eq:ffem2d88}
\end{equation}
and $\bar{\nu}_{Bi}(t)=0,\quad i=1,2.$\\[12pt]
%
%
Multiplying $\mathcal{A}\dfrac{1}{r}(rW_{i_r})_r$ by $B(r)^T$ and integrating with respect to $r$, we have
%
\begin{equation}
\Big[\int_{R_B}^{R_F}B(r)^T\mathcal{A}\frac{1}{R_F-R_B} \ud r\Big] \Big(-\bar{\nu}_{Bi}(t)+\bar{\nu}_{Fi}\Big),\quad i=0,\ldots,2 \label{eq:ffem2d89}
\end{equation}
and $\bar{\nu}_{Bi}(t)=0,\quad i=1,2.$\\[12pt]
\indent
%
Define the following matrices.\\
\centerline{$3M_D\times 3M_D$ matrices:}
\begin{align*}
G_0&=\int_{R_B}^{R_F}B(r)^TB(r)r\ud r \\[6pt]
G_2&=\mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big] 
\end{align*}\\[-12pt]
%
\centerline{$3M_D\times M$ matrices:}
\begin{equation*}
H_0=\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0(r\bar{\ub}_r(r)+\bar{\ub}(r))^T \ud r 
\end{equation*}\\[-12pt]
%
\centerline{$3M_D\times 3$ matrices:}
\begin{align*}
\mathcal{R}_B&=\int_{R_B}^{R_F}B(r)^T\frac{R_F-r}{R_F-R_B}r\ud r \\[6pt]
\mathcal{R}_F&=\int_{R_B}^{R_F}B(r)^T\frac{r-R_B}{R_F-R_B}r\ud r \\[6pt]
\mathcal{R}&=\Big[\int_{R_B}^{R_F}\frac{1}{R_F-R_B}B(r)^T\ud r\Big]\mathcal{A} 
\end{align*}
%
Using these matrices, (\ref{eq:ffem2d80}) multiplied by $B(r)^T$ and integrated with respect to $r$, has the form 
\begin{equation}
G_0\dot{\bar{c}}_0(t)+G_2\bar{c}_0(t)=-H_0\dot{\bar{\mu}}_0(t)-\mathcal{R}_B\dot{\bar{\nu}}_{B0}(t)-\mathcal{R}\bar{\nu}_{B0}(t)+\mathcal{R}\bar{\nu}_{F0}\label{eq:ffem2d90}
\end{equation}
%
From (\ref{eq:ffem2d75})
\begin{equation}
\dot{\bar{\mu}}_0(t) = \mathcal{M}_0^{-1}\Big[\mathcal{Z}_0\dot{\bar{c}}_0(t)+Y_{B0}\dot{\bar{\nu}}_{B0}(t)\Big] \label{eq:ffem2d91} 
\end{equation}
%
Applying (\ref{eq:ffem2d91}) to (\ref{eq:ffem2d90}), we have
\begin{equation}
\mathcal{H}_0\dot{\bar{c}}_0(t)+G_2\bar{c}_0(t)=\mathcal{D}_{B0}\dot{\bar{\nu}}_{B0}(t)-\mathcal{R}\bar{\nu}_{B0}(t)+\mathcal{R}\bar{\nu}_{F0} \label{eq:ffem2d92}
\end{equation}
where $\quad \mathcal{H}_0=G_0+H_0\mathcal{M}_0^{-1}\mathcal{Z}_0$\\
${}\qquad \qquad \mathcal{D}_{B0}=-H_0\mathcal{M}_0^{-1}Y_{B0}-\mathcal{R}_B$\\[12pt]
%
\indent
Now we approximate initial conditions. ~From (\ref{eq:ffem2d11}), (\ref{eq:ffem2d19}) and (\ref{eq:ffem2d52}), the coefficients $V_i(r,0)$ of $\theta$-approximation of initial conditions have the form
\begin{equation*}
V_i(r,0)=\tilde{V}_{Ii}(r)-W_i(r,0)
\end{equation*} 
where $\quad \tilde{V}_{Ii}(r)=B(r)\bar{\nu}_{Ii}$
\begin{equation}
W_i(r,0)=\frac{R_F-r}{R_F-R_B}\bar{\nu}_{Bi}(0)+ \frac{r-R_B}{R_F-R_B}\bar{\nu}_{Fi}, \quad i=0,\ldots,2 \label{eq:ffem2d93}
\end{equation}
and $\bar{\nu}_{Bi}(0)=0,\quad i=1,2.$\\ 
%
Multiplying $V_i(r,0)$ by $B(r)^T$ and integrating with respect to $r$, we have
\begin{gather}
\Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bar{c}_i(0)=\Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bar{\nu}_{Ii} \nonumber \\[6pt]
-\Big[\int_{R_B}^{R_F}B(r)^T\frac{R_F-r}{R_F-R_B}r\ud r\Big]\bar{\nu}_{Bi}(0)-\Big[\int_{R_B}^{R_F}B(r)^T\frac{r-R_B}{R_F-R_B}r\ud r\Big]\bar{\nu}_{Fi} \label{eq:ffem2d94}
\end{gather}
%
Using matrices defined above, (\ref{eq:ffem2d94}) has the form
\begin{equation}
G_0\bar{c}_i(0)=G_0\bar{\nu}_{Ii}-\mathcal{R}_B\bar{\nu}_{Bi}(0)-\mathcal{R}_F\bar{\nu}_{Fi}\label{eq:ffem2d95}
\end{equation}
%
From (\ref{eq:ffem2d92}) and (\ref{eq:ffem2d95}) with $i=0$ we obtain the following system of ordinary differential equations (ODEs).\\ 
\centerline{System of ODEs for TCP I (radially symmetric case):}
\begin{equation*}
\mathcal{H}_0\dot{\bar{c}}_0(t)+G_2\bar{c}_0(t)=\mathcal{D}_{B0}\dot{\bar{\nu}}_{B0}(t)-\mathcal{R}\bar{\nu}_{B0}(t)+\mathcal{R}\bar{\nu}_{F0}\tag{\ref{eq:ffem2d92}}
\end{equation*}
with initial conditions
\begin{equation}
\bar{c}_0(0)=\bar{\nu}_{I0}-G_0^{-1}\big[\mathcal{R}_B\bar{\nu}_{B0}(0)+\mathcal{R}_F\bar{\nu}_{F0}\big] \label{eq:ffem2d96}
\end{equation}
\\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM Approximation of TCP System II (sin 2\theta) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\noindent
\centerline{System II (corresponds to $\sin2\theta$ approximation)}
%
System II is given by (\ref{eq:ffem2d20})-(\ref{eq:ffem2d22}):
\begin{gather*} 
M\dot{V}_1-A\Big[\frac{1}{r}(rV_{1_r})_r-\frac{4}{r^2}V_1\Big] = -M\dot{W}_1+A\Big(\frac{1}{r}(rW_{1_r})_r-\frac{4}{r^2}W_1\Big)\nonumber \\[6pt]
-\vec{b}_0\Big(\dot{u}_{1_r}+\frac{1}{r}(\dot{u}_1-2\dot{v}_2)\Big) \tag{\ref{eq:ffem2d20}}
\end{gather*}
with boundary conditions\\
\vspace{-0.2in}
\begin{equation*}
V_1(r,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \tag{\ref{eq:ffem2d21}}
\end{equation*}
and initial conditions
\begin{equation*}
V_1(r,0)=\tilde{V}_{I1}(r)-W_1(r,0) \tag{\ref{eq:ffem2d22}}
\end{equation*}
%
Using $\mathcal{A}=M^{-1}A$, (\ref{eq:ffem2d20}) has the form
\begin{gather} 
\dot{V}_1-\mathcal{A}\Big[\frac{1}{r}(rV_{1_r})_r-\frac{4}{r^2}V_1\Big] =-M^{-1}\vec{b}_0\Big( \dot{u}_{1_r}+\frac{1}{r}\dot{u}_1-\frac{2}{r}\dot{v}_2\Big) \nonumber \\[6pt]
-\dot{W}_1+\mathcal{A}\Big[\frac{1}{r}(rW_{1_r})_r-\frac{4}{r^2}W_1\Big] \label{eq:ffem2d97}
\end{gather}
%
Using (\ref{eq:ffem2d51}), the left-hand side of (\ref{eq:ffem2d97}) is
\begin{align}
B(r)\dot{\bar{c}}_1(t)-\mathcal{A}\frac{1}{r}\Big(rB_r(r)\Big)_r\bar{c}_1(t)+\mathcal{A}\frac{4}{r^2}B(r)\bar{c}_1(t)\label{eq:ffem2d98}
\end{align}
%
Multiplying (\ref{eq:ffem2d98}) by $B(r)^T$ and integrating with respect to $r$, we have
\begin{gather}
\Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\dot{\bar{c}}_1(t) \nonumber \\[6pt]
+\mathcal{A}\otimes \bigg\{\Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big]+4\Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r} \ud r\Big]\bigg\} \bar{c}_1(t) \label{eq:ffem2d99}
\end{gather}
%
Using (\ref{eq:ffem2d53}) and (\ref{eq:ffem2d54}), the third term of the right-hand side of (\ref{eq:ffem2d97}) is
\begin{align*}
&-M^{-1}\vec{b}_0\Big( \dot{u}_{1_r}+\frac{1}{r}\dot{u}_1-\frac{2}{r}\dot{v}_2\Big) \\[6pt]
=&-M^{-1}\vec{b}_0\Big[\bar{\ub}_r(r)^T+\frac{1}{r}\bar{\ub}(r)^T\Big]\dot{\bar{\mu}}_1(t)+M^{-1}\vec{b}_0\frac{2}{r}\bar{\ub}(r)^T\dot{\bar{\eta}}_2(t)
\end{align*}
%
Multiplying by $B(r)^T$ and integrating with respect to $r$, we have
%
\begin{gather}
-\Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0(r\bar{\ub}_r(r)+\bar{\ub}(r))^T \ud r\Big]\dot{\bar{\mu}}_1(t) \nonumber \\[6pt]
+2\Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^T \ud r\Big]\dot{\bar{\eta}}_2(t) \label{eq:ffem2d100}
\end{gather} 
%
Multiplying the corresponding term in (\ref{eq:ffem2d97}) by $B(r)^T$ and integrating with respect to $r$, we have the following results\\
for $\dot{W}_1$: (\ref{eq:ffem2d87}) with $\dot{\bar{\nu}}_{B1}(t)=0$ and\\
for $\mathcal{A}\dfrac{1}{r}(rW_{1_r})_r$: (\ref{eq:ffem2d89}) with $\bar{\nu}_{B1}(t)=0.$\\ %
$W_1(r,t)$ is given by (\ref{eq:ffem2d85}) with $\bar{\nu}_{B1}(t)=0.$\\[6pt]
%
Multiplying $\mathcal{A}\dfrac{4}{r^2}W_i$ by $B(r)^T$ and integrating with respect to $r$, we have
%
\begin{align}
&\Big[\int_{R_B}^{R_F}B(r)^T\mathcal{A}\frac{4}{r^2}\frac{r-R_B}{R_F-R_B}r \ud r\Big] \bar{\nu}_{Fi} \nonumber \\[6pt]
=&4\Big[\int_{R_B}^{R_F}B(r)^T\frac{r-R_B}{R_F-R_B}\frac{1}{r}\ud r\Big]\mathcal{A}\bar{\nu}_{Fi}, \qquad i=1,2. \label{eq:ffem2d101}
\end{align}
%
\\[-12pt]
\indent
Define the following matrices.\\
\centerline{$3M_D\times 3M_D$ matrix:}
\begin{equation*}
G_0\big(\frac{1}{r}\big)=\mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r} \ud r\Big] 
\end{equation*}
%
\centerline{$3M_D\times M$ matrix:}
\begin{equation*}
H_0(r)=\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^T \ud r 
\end{equation*}
%
\centerline{$3M_D\times 3$ matrix:}
\begin{equation*}
\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)=\Big[\int_{R_B}^{R_F}B(r)^T\frac{r-R_B}{R_F-R_B}\frac{1}{r}\ud r\Big]\mathcal{A}
\end{equation*}
%
Using these matrices together with matrices defined for System I, (\ref{eq:ffem2d97}) multiplied by $B(r)^T$ and integrated with respect to $r$, has the form 
\begin{equation}
G_0\dot{\bar{c}}_1(t)+\Big[G_2+4G_0\big(\frac{1}{r}\big)\Big]\bar{c}_1(t)=-H_0\dot{\bar{\mu}}_1(t)+2H_0(r)\dot{\bar{\eta}}_2(t)+\Big[\mathcal{R}-4\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)\Big]\bar{\nu}_{F1} \label{eq:ffem2d102}
\end{equation}
%
From (\ref{eq:ffem2d76}) and (\ref{eq:ffem2d77})
\begin{align}
\dot{\bar{\mu}}_1(t)&=P_{10}\mathcal{Q}^{-1}\bar{\mathcal{Z}}\dot{\bar{c}}_1(t) \label{eq:ffem2d103} \\
%
\dot{\bar{\eta}}_2(t)&=-P_{01}\mathcal{Q}^{-1}\bar{\mathcal{Z}}\dot{\bar{c}}_1(t)\label{eq:ffem2d104} 
\end{align}
%
Applying (\ref{eq:ffem2d103}) and (\ref{eq:ffem2d104}) to (\ref{eq:ffem2d102}), we have
%
\begin{equation}
\mathcal{H}_1\dot{\bar{c}}_1(t)+\mathcal{G}_1\bar{c}_1(t)=\mathcal{F}_{F1}\bar{\nu}_{F1} \label{eq:ffem2d105}
\end{equation}
where 
\begin{align}
\mathcal{H}_1&=G_0+\big(H_0P_{10}+2H_0(r)P_{01}\big)\mathcal{Q}^{-1}\bar{\mathcal{Z}} \nonumber \\
\mathcal{G}_1&=G_2+4G_0\big(\frac{1}{r}\big) \nonumber \\
\mathcal{F}_{F1}&=\Big(\mathcal{R}-4\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)\Big) \label{eq:ffem2d106}
\end{align}
\indent
Initial conditions are given by (\ref{eq:ffem2d95}) with $i=1$ and $\bar{\nu}_{B1}(0)=0$. ~From (\ref{eq:ffem2d106}) and (\ref{eq:ffem2d95}) with $i=1$ and $\bar{\nu}_{B1}(0)=0$ we obtain the following system.\\ 
%
\centerline{System of ODEs for TCP II (corresponds to $\sin2\theta$ approximation):}
\begin{equation*}
\mathcal{H}_1\dot{\bar{c}}_1(t)+\mathcal{G}_1\bar{c}_1(t)=\mathcal{F}_{F1}\bar{\nu}_{F1} \tag{\ref{eq:ffem2d105}}
\end{equation*}
with initial conditions
\begin{equation}
\bar{c}_1(0)=\bar{\nu}_{I1}-G_0^{-1}\mathcal{R}_F\bar{\nu}_{F1} \label{eq:ffem2d107}
\end{equation}
\\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM Approximation of TCP System III (cos 2\theta) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\noindent
\centerline{System III (corresponds to $\cos2\theta$ approximation)}
%
System III is given by (\ref{eq:ffem2d23})-(\ref{eq:ffem2d25}):
\begin{gather*} 
M\dot{V}_2-A\Big[\frac{1}{r}(rV_{2_r})_r-\frac{4}{r^2}V_2\Big] = -M\dot{W}_2+A\Big(\frac{1}{r}(rW_{2_r})_r-\frac{4}{r^2}W_2\Big)\nonumber \\[6pt]
-\vec{b}_0\Big(\dot{u}_{2_r}+\frac{1}{r}(\dot{u}_2+2\dot{v}_1)\Big) \tag{\ref{eq:ffem2d23}}
\end{gather*}
with boundary conditions
\begin{equation*}
V_2(r,t)=0 \qquad \text{on } \Gamma \times(0,t_f) \tag{\ref{eq:ffem2d24}}
\end{equation*}
and initial conditions\\
\vspace{-0.2in}
\begin{equation*}
V_2(r,0)=\tilde{V}_{I2}(r)-W_2(r,0) \tag{\ref{eq:ffem2d25}}
\end{equation*}
%
%
Using $\mathcal{A}=M^{-1}A$, (\ref{eq:ffem2d23}) has the form
\begin{gather} 
\dot{V}_2-\mathcal{A}\Big[\frac{1}{r}(rV_{2_r})_r-\frac{4}{r^2}V_2\Big] =-M^{-1}\vec{b}_0\Big(\dot{u}_{2_r}+\frac{1}{r}\dot{u}_2+\frac{2}{r}\dot{v}_1\Big) \nonumber \\[6pt]
-\dot{W}_2+\mathcal{A}\Big[\frac{1}{r}(rW_{2_r})_r-\frac{4}{r^2}W_2\Big] \label{eq:ffem2d108}
\end{gather}
%
Using (\ref{eq:ffem2d51}), the left-hand side of (\ref{eq:ffem2d108}) is
\begin{equation}
B(r)\dot{\bar{c}}_2(t)-\mathcal{A}\frac{1}{r}\Big(rB_r(r)\Big)_r\bar{c}_2(t)+\mathcal{A}\frac{4}{r^2}B(r)\bar{c}_2(t)\label{eq:ffem2d109}
\end{equation}
%
Multiplying (\ref{eq:ffem2d109}) by $B(r)^T$ and integrating with respect to $r$, we have
\begin{gather}
\Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\dot{\bar{c}}_2(t) \nonumber \\[6pt]
+\mathcal{A}\otimes \bigg\{\Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big]+4\Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r} \ud r\Big]\bigg\} \bar{c}_2(t) \label{eq:ffem2d110}
\end{gather}
%
Using (\ref{eq:ffem2d53}) and (\ref{eq:ffem2d54}), the first term of the right-hand side of (\ref{eq:ffem2d108}) is
\begin{align*}
&-M^{-1}\vec{b}_0\Big( \dot{u}_{2_r}+\frac{1}{r}\dot{u}_2+\frac{2}{r}\dot{v}_1\Big) \\[6pt]
=&-M^{-1}\vec{b}_0\Big[\bar{\ub}_r(r)^T+\frac{1}{r}\bar{\ub}(r)^T\Big]\dot{\bar{\mu}}_2(t)-M^{-1}\vec{b}_0\frac{2}{r}\bar{\ub}(r)^T\dot{\bar{\eta}}_1(t)
\end{align*}
%
Multiplying by $B(r)^T$ and integrating with respect to $r$, we have
%
\begin{gather}
-\Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0(r\bar{\ub}_r(r)+\bar{\ub}(r))^T \ud r\Big]\dot{\bar{\mu}}_2(t) \nonumber \\[6pt]
-2\Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^T \ud r\Big]\dot{\bar{\eta}}_1(t) \label{eq:ffem2d111}
\end{gather} 
%
Multiplying the corresponding term in (\ref{eq:ffem2d108}) by $B(r)^T$ and integrating with respect to $r$, we have the following results\\
for $\dot{W}_2$: (\ref{eq:ffem2d87}) with $\dot{\bar{\nu}}_{B2}(t)=0$\\
for $\mathcal{A}\dfrac{1}{r}(rW_{2_r})_r$: (\ref{eq:ffem2d89}) with $\bar{\nu}_{B2}(t)=0.$\\
for $\mathcal{A}\dfrac{4}{r^2}W_2$: (\ref{eq:ffem2d101})\\[6pt]
%
Using the matrices defined for Systems I and II, (\ref{eq:ffem2d108}) multiplied by $B(r)^T$ and integrated with respect to $r$, has the form 
\begin{equation}
G_0\dot{\bar{c}}_2(t)+\Big[G_2+4G_0\big(\frac{1}{r}\big)\Big]\bar{c}_2(t)=-H_0\dot{\bar{\mu}}_2(t)-2H_0(r)\dot{\bar{\eta}}_1(t)+\Big[\mathcal{R}-4\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)\Big]\bar{\nu}_{F2} \label{eq:ffem2d112}
\end{equation}
%
From (\ref{eq:ffem2d78}) and (\ref{eq:ffem2d79})
\begin{align}
\dot{\bar{\mu}}_2(t)&=P_{10}\mathcal{Q}^{-1}\bar{\mathcal{Z}}\dot{\bar{c}}_2(t) \label{eq:ffem2d113} \\
%
\dot{\bar{\eta}}_1(t)&=P_{01}\mathcal{Q}^{-1}\bar{\mathcal{Z}}\dot{\bar{c}}_2(t) \label{eq:ffem2d114} 
\end{align}
%
Applying (\ref{eq:ffem2d113}) and (\ref{eq:ffem2d114}) to (\ref{eq:ffem2d112}), we have
%
\begin{equation}
\mathcal{H}_1\dot{\bar{c}}_2(t)+\mathcal{G}_1\bar{c}_2(t)=\mathcal{F}_{F1}\bar{\nu}_{F2} \label{eq:ffem2d115}
\end{equation}
where the matrices $\mathcal{H}_1,$ $\mathcal{G}_1,$ and $\mathcal{F}_{F1}$ are given by (\ref{eq:ffem2d106})\\ 
%
Initial conditions are given by (\ref{eq:ffem2d95}) with $i=2$ and $\bar{\nu}_{B2}(0)=0$. ~From (\ref{eq:ffem2d115}) and (\ref{eq:ffem2d95}) with $i=2$ and $\bar{\nu}_{B2}(0)=0$ we obtain the following system.\\ 
%
\centerline{System of ODEs for TCP III (corresponds to $\cos2\theta$ approximation):}
\begin{equation*}
\mathcal{H}_1\dot{\bar{c}}_2(t)+\mathcal{G}_1\bar{c}_2(t)=\mathcal{F}_{F1}\bar{\nu}_{F2} \tag{\ref{eq:ffem2d115}}
\end{equation*}
with initial conditions
\begin{equation}
\bar{c}_2(0)=\bar{\nu}_{I2}-G_0^{-1}\mathcal{R}_F\bar{\nu}_{F2} \label{eq:ffem2d116}
\vspace{0.2in}
\end{equation}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Three Systems of ODEs %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\indent
We have the following three systems of ODEs:\\[6pt] 
%
\centerline{System of ODEs for TCP I (radially symmetric case):}
\begin{equation*}
\mathcal{H}_0\dot{\bar{c}}_0(t)+G_2\bar{c}_0(t)=\mathcal{D}_{B0}\dot{\bar{\nu}}_{B0}(t)-\mathcal{R}\bar{\nu}_{B0}(t)+\mathcal{R}\bar{\nu}_{F0} \tag{\ref{eq:ffem2d92}}
\end{equation*}
with initial conditions
\begin{equation*}
\bar{c}_0(0)=\bar{\nu}_{I0}-G_0^{-1}\big[\mathcal{R}_B\bar{\nu}_{B0}(0)+\mathcal{R}_F\bar{\nu}_{F0}\big] \tag{\ref{eq:ffem2d96}}
\end{equation*}
%
System I gives coefficients $~V_0~$ and $~u_0~$ of the $~\theta$-approximation (\ref{eq:ffem2d9}).\\[12pt]
%
\centerline{System of ODEs for TCP II (corresponds to $\sin2\theta$ approximation):}
\begin{equation*}
\mathcal{H}_1\dot{\bar{c}}_1(t)+\mathcal{G}_1\bar{c}_1(t)=\mathcal{F}_{F1}\bar{\nu}_{F1} \tag{\ref{eq:ffem2d105}}
\end{equation*}
with initial conditions
\begin{equation*}
\bar{c}_1(0)=\bar{\nu}_{I1}-G_0^{-1}\mathcal{R}_F\bar{\nu}_{F1} \tag{\ref{eq:ffem2d107}}
\end{equation*}
%
System II gives $~\sin2\theta$-coefficients $~V_1,~u_1~$ and $~\cos2\theta$-coefficient $~v_2~$ of the $~\theta$-approximation (\ref{eq:ffem2d9}).\\[12pt]
%
\centerline{System of ODEs for TCP III (corresponds to $\cos2\theta$ approximation):}
\begin{equation*}
\mathcal{H}_1\dot{\bar{c}}_2(t)+\mathcal{G}_1\bar{c}_2(t)=\mathcal{F}_{F1}\bar{\nu}_{F2} \tag{\ref{eq:ffem2d115}}
\end{equation*}
with initial conditions
\begin{equation*}
\bar{c}_2(0)=\bar{\nu}_{I2}-G_0^{-1}\mathcal{R}_F\bar{\nu}_{F2} \tag{\ref{eq:ffem2d116}}
\end{equation*}
%
System III gives $~\sin2\theta$-coefficient $~v_1~$ and $~\cos2\theta$-coefficients $~V_2,~u_2~$ of the $~\theta$-approximation (\ref{eq:ffem2d9}).\\ \\
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM for 3D TCPu problem  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section {3D TCPu Problem for an Inclined Borehole}
%
As in Section 2.2, let $~(X,Y,Z)$ be a borehole Cartesian coordinate system with the $~x$ -axis in the direction of the maximum horizontal principal stress $~\sigma_H~$, the $~y$ -axis in the direction of the minimum horizontal principal stress $\sigma_h$, and the $~z$ -axis in the direction of overburden stress $\sigma_V$ and let $~(X_{b}, Y_{b}, Z_{b})$ be the borehole Cartesian coordinate system.  For the sake of convenience, we will make the transformation from the coordinate system $~(X_{b}, Y_{b}, Z_{b})$ to the borehole cylindrical coordinate system $~(r,\theta,z)$ with the $z$ -axis passing through the central axis of the cylindrical region $~\Omega$. Then the boundary $\Gamma$ of $\Omega$ is $\Gamma = \Gamma_B\cup\Gamma_F\cup\tilde{\Gamma}_T\cup\tilde{\Gamma}_B$:
%
\begin{gather*}
\Gamma_B = \left\{ (R_B, \theta,z) \in \mathbb R^{3}: 0\leq \theta < 2 \pi, z_T \leq z \leq z_B \right\} \\
\Gamma_F = \left\{ (R_F, \theta,z) \in \mathbb R^{3}: 0\leq \theta < 2 \pi, z_T \leq z \leq z_B \right\} \\
\tilde{\Gamma}_T = \left\{ (r,\theta,z_T) \in \mathbb R^{3}: R_B \leq r \leq R_F, 0\leq \theta < 2 \pi \right\} \\
\tilde{\Gamma}_B = \left\{ (r,\theta,z_B) \in \mathbb R^{3}: R_B \leq r \leq R_F, 0\leq \theta < 2 \pi \right\} \\[6pt]
\end{gather*}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  TCP sytem  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection {Fourier-type approximation of the TCP sytem}
%
In the borehole cylindrical coordinate system $~(r,\theta,z)$ the TCP initial-boundary value problem (\ref{eq:uneq3d17})-(\ref{eq:uneq3d20}) is given by
\begin{equation} \label{eq:ffem3d1}
M \dot{\bar{V}} - A \nabla^{2} \bar{V}=-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f)
\end{equation}
%
with boundary conditions
\begin{equation} \label{eq:ffem3d2}
\bar{V}(r,\theta,z,t)=
\begin{cases}
V_B(z,t) & \text{on } \Gamma_B \times [0,t_f)\\
V_F(\theta,z) & \text{on } \Gamma_F \times [0,t_f)\\
\tilde{V}_T(r,\theta,t) & \text{on } \tilde{\Gamma}_T \times [0,t_f)\\
\tilde{V}_B(r,\theta,t) & \text{on } \tilde{\Gamma}_B \times [0,t_f)\\
\end{cases}
\end{equation}
%
initial conditions 
\begin{equation} 
\bar{V}(r,\theta,z,0)=\tilde{V}_I(r,\theta,z) \label{eq:ffem3d3}
\end{equation}
%
and compatibility conditions: \label{eq:ffem3d4}
\begin{subequations}
\begin{align}
\tilde{V}_T(R_B,\theta,t) & =V_B(z_T,t) \\
\tilde{V}_T(R_F,\theta,t) & =V_F(\theta,z_T) \\
\tilde{V}_B(R_B,\theta,t) & =V_B(z_B,t) \\
\tilde{V}_B(R_F,\theta,t) & =V_F(\theta,z_B) 
\end{align}
\end{subequations}
%
where the displacement vector is
\begin{equation*}
\vec{u}(r,\theta,z,t)=u(r,\theta,z,t)\vec{i}_r +v(r,\theta,z,t)\vec{i}_\theta +w(r,\theta,z,t)\vec{i}_z
\end{equation*} 
%
\indent
To obtain the equivalent parabolic TCP-type system with homogeneous boundary conditions (\ref{eq:dtcp11})-(\ref{eq:dtcp13}), we first define
\begin{gather} 
W^r(r,\theta,z,t)=\frac{R_F-r}{R_F-R_B}V_B(z,t)+\frac{r-R_B}{R_F-R_B}V_F(\theta,z) \label{eq:ffem3d5} \\[12pt]
\bar{V}^0=\bar{V}-W^r \label{eq:ffem3d6}
\end{gather}
%
Then (\ref{eq:ffem3d1})-(\ref{eq:ffem3d3}) has the form 
\begin{equation} 
M \dot{\bar{V}}^0 - A \nabla^{2} \bar{V}^0=-M \dot{W}^r+A \nabla^{2}W^r-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f) \label{eq:ffem3d7}
\end{equation}
%
with boundary conditions
\begin{equation} \label{eq:ffem3d8}
\bar{V}^0(r,\theta,z,t)=
\begin{cases}
\qquad 0 & \text{on } \Gamma_B \times [0,t_f)\\
\qquad 0 & \text{on } \Gamma_F \times [0,t_f)\\
\tilde{V}_T(r,\theta,t)-W^r(r,\theta,z_T,t) & \text{on } \tilde{\Gamma}_T \times [0,t_f)\\
\tilde{V}_B(r,\theta,t)-W^r(r,\theta,z_B,t) & \text{on } \tilde{\Gamma}_B \times [0,t_f)\\
\end{cases}
\end{equation}
%
and initial conditions 
\begin{equation} 
\bar{V}^0(r,\theta,z,0)=\tilde{V}_I(r,\theta,z)-W^r(r,\theta,z,0) \label{eq:ffem3d9}
\end{equation}
%
Now define
\begin{align} 
\hat{V}_T(r,\theta,t) & =\tilde{V}_T(r,\theta,t)-W^r(r,\theta,z_T,t)  \label{eq:ffem3d10} \\
\hat{V}_B(r,\theta,t) & =\tilde{V}_B(r,\theta,t)-W^r(r,\theta,z_B,t) \label{eq:ffem3d11}
\end{align}
%
and 
\begin{equation} 
W^z(r,\theta,z,t) =\frac{z-z_T}{z_B-z_T}\hat{V}_B(r,\theta,t)+\frac{z_B-z}{z_B-z_T}\hat{V}_T(r,\theta,t) \label{eq:ffem3d12}
\end{equation}
\begin{align}
V & =\bar{V}^0-W^z=\bar{V}-W^r-W^z \label{eq:ffem3d13} \\
V & =\bar{V}-W(r,\theta,z,t), \qquad \text{ where } W=W^r+W^z  \label{eq:ffem3d14}
\end{align}
%
Then (\ref{eq:ffem3d1})-(\ref{eq:ffem3d3}) has the form
\begin{equation} 
M \dot{V} - A \nabla^{2} V =-M \dot{W}+A \nabla^{2}W-\vec{b}_0 (\nabla \cdot \dot{\vec{u}}) \qquad \text{in }  \Omega \times (0, t_f) \label{eq:ffem3d15}
\end{equation}
%
with boundary conditions
\begin{equation}
V(r,\theta,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:ffem3d16}
\end{equation}
%
and initial conditions
\begin{equation}
V(r,\theta,z,0)=\tilde{V}_I(r,\theta,z)-W(r,\theta,z,0) \label{eq:ffem3d17}
\end{equation} 
%
\indent
We introduce the following $\theta$ - approximation: 
\begin{align}
V(r,\theta,z,t) & =V_0(r,z,t)+V_1(r,z,t)\sin\theta+V_2(r,z,t)\cos\theta \nonumber \\
& \quad +V_3(r,z,t)\sin2\theta+V_4(r,z,t)\cos2\theta\nonumber \\[6pt]
u(r,\theta,z,t) & =u_0(r,z,t)+u_1(r,z,t)\sin\theta+u_2(r,z,t)\cos\theta \nonumber\\
& \quad +u_3(r,z,t)\sin2\theta+u_4(r,z,t)\cos2\theta \nonumber\\[6pt]
v(r,\theta,z,t) & =v_0(r,z,t)+v_1(r,z,t)\sin\theta+v_2(r,z,t)\cos\theta \nonumber\\
& \quad +v_3(r,z,t)\sin2\theta+v_4(r,z,t)\cos2\theta \nonumber\\[6pt]
w(r,\theta,z,t) & =w_0(r,z,t)+w_1(r,z,t)\sin\theta+w_2(r,z,t)\cos\theta \nonumber\\
& \quad +w_3(r,z,t)\sin2\theta+w_4(r,z,t)\cos2\theta \nonumber\\[6pt]
V_B(z,t) & =V_{B0}(z,t) \nonumber\\[6pt]
V_F(\theta,z) & =V_{F0}(z)+V_{F1}(z)\sin\theta+V_{F2}(z)\cos\theta \nonumber\\
& \quad +V_{F3}(z)\sin2\theta+V_{F4}(z)\cos2\theta \nonumber\\[6pt]
\tilde{V}_T(r,\theta,t) & =\tilde{V}_{T0}(r,t)+\tilde{V}_{T1}(r,t)\sin\theta+\tilde{V}_{T2}(r,t)\cos\theta \nonumber\\
& \quad +\tilde{V}_{T3}(r,t)\sin2\theta+\tilde{V}_{T4}(r,t)\cos2\theta \nonumber\\[6pt]
\tilde{V}_B(r,\theta,t) & =\tilde{V}_{B0}(r,t)+\tilde{V}_{B1}(r,t)\sin\theta+\tilde{V}_{B2}(r,t)\cos\theta \nonumber\\
& \quad +\tilde{V}_{B3}(r,t)\sin2\theta+\tilde{V}_{B4}(r,t)\cos2\theta \nonumber\\[6pt]
\tilde{V}_I(r,\theta,z) & =\tilde{V}_{I0}(r,z)+\tilde{V}_{I1}(r,z)\sin\theta+\tilde{V}_{I2}(r,z)\cos\theta \nonumber\\
& \quad +\tilde{V}_{I3}(r,z)\sin2\theta+\tilde{V}_{I4}(r,z)\cos2\theta \label{eq:ffem3d18}
\end{align}
%
Then
\vspace{-0.1in}
\begin{align}
W(r,\theta,z,t) & =W_0(r,z,t)+W_1(r,z,t)\sin\theta+W_2(r,z,t)\cos\theta \nonumber \\
& \quad +W_3(r,z,t)\sin2\theta+W_4(r,z,t)\cos2\theta  \label{eq:ffem3d19}
\end{align}
where
\vspace{-0.1in}
\begin{gather}
W_i(r,z,t)=\frac{R_F-r}{R_F-R_B}V_{Bi}(z,t)+\frac{r-R_B}{R_F-R_B}V_{Fi}(z)+ \nonumber \\[6pt]
+\frac{z-z_T}{z_B-z_T}\Big[\tilde{V}_{Bi}(r,t)-\frac{R_F-r}{R_F-R_B}V_{Bi}(z_B,t)-\frac{r-R_B}{R_F-R_B}V_{Fi}(z_B)\Big] \nonumber \\[6pt]
+\frac{z_B-z}{z_B-z_T}\Big[\tilde{V}_{Ti}(r,t)-\frac{R_F-r}{R_F-R_B}V_{Bi}(z_T,t)-\frac{r-R_B}{R_F-R_B}V_{Fi}(z_T)\Big], \qquad i=0,\ldots,4 \label{eq:ffem3d20}
\end{gather}
and $\qquad V_{Bi}(z,t)=0, \qquad z_T\leq z\leq z_B, \qquad i=1,\ldots,4.$\\[3pt]
%
With this approximation, the system (\ref{eq:ffem3d15}) has the form:
\begin{gather} 
M\dot{V}-A\Big[ \frac{1}{r}(rV_r)_r +\frac{1}{r^2}V_{\theta\theta}+V_{zz}\Big] = -M\dot{W}+A\Big( \frac{1}{r}(rW_r)_r+\frac{1}{r^2}W_{\theta\theta}+W_{zz}\Big)  \nonumber \\[6pt]
-\vec{b}_0\Big( \dot{u}_r+\frac{1}{r}(\dot{u}+\dot{v}_\theta)+\dot{w}_z\Big)  \label{eq:ffem3d21}
\end{gather}
where $~W$ is given by (\ref{eq:ffem3d19}) and (\ref{eq:ffem3d20}),
%
\begin{align} 
W_r(r,\theta,z,t)& =W_{0_r}(r,z,t)+W_{1_r}(r,z,t)\sin\theta+W_{2_r}(r,z,t)\cos\theta \nonumber \\
& \quad +W_{3_r}(r,z,t)\sin2\theta+W_{4_r}(r,z,t)\cos2\theta  \label{eq:ffem3d22}
\end{align}
with
\vspace{-0.1in}
\begin{gather}
W_{i_r}(r,z,t) =\frac{1}{R_F-R_B}\big(V_{Fi}(z)-V_{Bi}(z,t)\big) \nonumber \\[6pt] 
+\frac{z-z_T}{z_B-z_T}\Big[\tilde{V}_{Bi_r}(r,t)+\frac{1}{R_F-R_B}\big(V_{Bi}(z_B,t)-V_{Fi}(z_B)\big)\Big] \nonumber \\[6pt] 
+\frac{z_B-z}{z_B-z_T}\Big[\tilde{V}_{Ti_r}(r,t)+\frac{1}{R_F-R_B}\big(V_{Bi}(z_T,t)-V_{Fi}(z_T)\big)\Big], \qquad i=0,\ldots,4 \label{eq:ffem3d23}
\end{gather}
and $\qquad V_{Bi}(z,t) =0, \qquad z_T\leq z\leq z_B, \qquad i=1,\ldots,4$;\\[-12pt]
%
\begin{align} 
W_{\theta\theta}(r,\theta,z,t) & =-W_1(r,z,t)\sin\theta-W_2(r,z,t)\cos\theta \nonumber \\
& \quad -4W_3(r,z,t)\sin2\theta-4W_4(r,z,t)\cos2\theta \label{eq:ffem3d24} \\[6pt]
W_z(r,\theta,z,t)& =W_{0_z}(r,z,t)+W_{1_z}(r,z,t)\sin\theta+W_{2_z}(r,z,t)\cos\theta \nonumber \\
& \quad +W_{3_z}(r,z,t)\sin2\theta+W_{4_z}(r,z,t)\cos2\theta  \label{eq:ffem3d25}
\end{align}
with
\begin{gather}
W_{i_z}(r,z,t) =\frac{R_F-r}{R_F-R_B}V_{Bi_z}(z,t)+\frac{r-R_B}{R_F-R_B}V_{Fi_z}(z) \nonumber \\[6pt]
+\frac{1}{z_B-z_T}\Big[ \tilde{V}_{Bi}(r,t)-\tilde{V}_{Ti}(r,t)-\frac{R_F-r}{R_F-R_B}\big(V_{Bi}(z_B,t)-V_{Bi}(z_T,t)\big) \nonumber\\[6pt]
-\frac{r-R_B}{R_F-R_B}\big(V_{Fi}(z_B)-V_{Fi}(z_T)\Big)\Big],\qquad i=0,\ldots,4 \label{eq:ffem3d26}
\end{gather}
and $\qquad V_{Bi_z}(z,t) =0, \qquad z_T\leq z\leq z_B, \qquad i=1,\ldots,4.$\\[12pt] 
%
\indent
The $\theta$-approximation for the initial conditions $~V(r,\theta,z,0)$ (\ref{eq:ffem3d17}) is
\begin{align*}
V(r,\theta,z,0) & =V_0(r,z,0)+V_1(r,z,0)\sin\theta+V_2(r,z,0)\cos\theta \\
& \quad +V_3(r,z,0)\sin2\theta+V_4(r,z,0)\cos2\theta
\end{align*}
where
\begin{align}
&V_i(r,z,0) =\tilde{V}_{Ii}(r,z)-W_i(r,z,0)= \nonumber \\[6pt]
& \qquad \qquad =\tilde{V}_{Ii}(r,z)-\frac{R_F-r}{R_F-R_B}V_{Bi}(z,0)-\frac{r-R_B}{R_F-R_B}V_{Fi}(z) \nonumber \\[6pt]
&-\frac{z-z_T}{z_B-z_T}\Big(\tilde{V}_{Bi}(r,0)-\frac{R_F-r}{R_F-R_B}V_{Bi}(z_B,0)-\frac{r-R_B}{R_F-R_B}V_{Fi}(z_B)\Big)- \nonumber \\[6pt]
&-\frac{z_B-z}{z_B-z_T}\Big(\tilde{V}_{Ti}(r,0)-\frac{R_F-r}{R_F-R_B}V_{Bi}(z_T,0)-\frac{r-R_B}{R_F-R_B}V_{Fi}(z_T)\Big), \qquad i=0,\ldots,4 \label{eq:ffem3d27}
\end{align}
and $\qquad V_{Bi}(z,0)=0, \qquad z_T\leq z\leq z_B, \qquad i=1,\ldots,4.$\\
%
Applying (\ref{eq:ffem3d22})-(\ref{eq:ffem3d27}) to (\ref{eq:ffem3d21}), multiplying by $1, \sin\theta, \cos\theta, \sin2\theta, \text{and}\cos2\theta$, and integrating over $(0,2\pi)$, we obtain five systems.\\
%
\textbf{System I}:
\begin{gather} 
M\dot{V}_0-A\Big[\frac{1}{r}(rV_{0_r})_r+(V_{0_z})_z\Big] = -M\dot{W}_0+A\Big(\frac{1}{r}(rW_{0_r})_r+(W_{0_z})_z\Big) \nonumber \\[6pt]
-\vec{b}_0\Big( \dot{u}_{0_r}+\frac{1}{r}\dot{u}_0+\dot{w}_{0_z}\Big)  \label{eq:ffem3d28}
\end{gather}
with boundary conditions
\begin{equation}
V_0(r,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:ffem3d29}
\end{equation}
and initial conditions
\begin{equation}
V_0(r,z,0)=\tilde{V}_{I0}(r,z)-W_0(r,z,0) \label{eq:ffem3d30}
\end{equation}
%
\textbf{System II}:
\begin{gather} 
M\dot{V}_1-A\Big[\frac{1}{r}(rV_{1_r})_r-\frac{1}{r^2}V_1+(V_{1_z})_z\Big] = -M\dot{W}_1+A\Big(\frac{1}{r}(rW_{1_r})_r-\frac{1}{r^2}W_1+(W_{1_z})_z\Big)  \nonumber \\[6pt]
-\vec{b}_0\Big(\dot{u}_{1_r}+\frac{1}{r}(\dot{u}_1-\dot{v}_2)+\dot{w}_{1_z}\Big) \label{eq:ffem3d31}
\end{gather}
with boundary conditions
\begin{equation}
V_1(r,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:ffem3d32}
\end{equation}
and initial conditions
\vspace{1pt}
\begin{equation}
V_1(r,z,0)=\tilde{V}_{I1}(r,z)-W_1(r,z,0) \label{eq:ffem3d33}
\end{equation}
%
\textbf{System III}:
\begin{gather} 
M\dot{V}_2-A\Big[\frac{1}{r}(rV_{2_r})_r-\frac{1}{r^2}V_2+(V_{2_z})_z\Big] = -M\dot{W}_2+A\Big(\frac{1}{r}(rW_{2_r})_r-\frac{1}{r^2}W_2+(W_{2_z})_z\Big)  \nonumber \\[6pt]
-\vec{b}_0\Big(\dot{u}_{2_r}+\frac{1}{r}(\dot{u}_2+\dot{v}_1)+\dot{w}_{2_z}\Big) \label{eq:ffem3d34}
\end{gather}
with boundary conditions
\begin{equation}
V_2(r,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:ffem3d35}
\end{equation}
and initial conditions
\begin{equation}
V_2(r,z,0)=\tilde{V}_{I2}(r,z)-W_2(r,z,0) \label{eq:ffem3d36}
\end{equation}
%
\textbf{System IV}:
\begin{gather} 
M\dot{V}_3-A\Big[\frac{1}{r}(rV_{3_r})_r-\frac{4}{r^2}V_3+(V_{3_z})_z\Big] = -M\dot{W}_3+A\Big(\frac{1}{r}(rW_{3_r})_r-\frac{4}{r^2}W_3+(W_{3_z})_z\Big) \nonumber \\[6pt]
-\vec{b}_0\Big(\dot{u}_{3_r}+\frac{1}{r}(\dot{u}_3-2\dot{v}_4)+\dot{w}_{3_z}\Big)  \label{eq:ffem3d37}
\end{gather}
with boundary conditions\\
\vspace{-0.2in}
\begin{equation}
V_3(r,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:ffem3d38}
\end{equation}
and initial conditions
\begin{equation}
V_3(r,z,0)=\tilde{V}_{I3}(r,z)-W_3(r,z,0) \label{eq:ffem3d39}
\end{equation}
%
\textbf{System V}:
\begin{gather} 
M\dot{V}_4-A\Big[\frac{1}{r}(rV_{4_r})_r-\frac{4}{r^2}V_4+(V_{4_z})_z\Big] = -M\dot{W}_4+A\Big(\frac{1}{r}(rW_{4_r})_r-\frac{4}{r^2}W_4+(W_{4_z})_z\Big) \nonumber \\[6pt]
-\vec{b}_0\Big(\dot{u}_{4_r}+\frac{1}{r}(\dot{u}_4+2\dot{v}_3)+\dot{w}_{4_z}\Big) \label{eq:ffem3d40}
\end{gather}
with boundary conditions
\begin{equation}
V_4(r,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \label{eq:ffem3d41}
\end{equation}
and initial conditions
\begin{equation}
V_4(r,z,0)=\tilde{V}_{I4}(r,z)-W_4(r,z,0) \label{eq:ffem3d42}
\end{equation}
\\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Elastic System %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection {Fourier-type approximation of the elastic system}
%
In the borehole cylindrical coordinate system $(r, \theta, z)$, the displacement is
\begin{gather*}
\vec{u}(r,\theta,z,t)=u(r,\theta,z,t)\vec{i}_r +v(r,\theta,z,t)\vec{i}_\theta + w(r,\theta,z,t)\vec{i}_z \nonumber \\ 
\vec{u}\approx0 \quad \text{and} \quad \dot{\vec{u}}\approx0 \qquad \text{on  } \Gamma_F 
\end{gather*}
and the strain tensor has the form
\begin{equation*}
\begin{bmatrix} \\[-2.5em]
\varepsilon_{rr}&\varepsilon_{r\theta}&\varepsilon_{rz} \\[-6pt]
\varepsilon_{\theta r}&\varepsilon_{\theta \theta}&\varepsilon_{\theta z} \\[-6pt]
\varepsilon_{zr}&\varepsilon_{z\theta}&\varepsilon_{zz} 
\end{bmatrix}
\end{equation*}
where
\vspace{-0.2in}
\begin{gather}
\varepsilon_{rr}=u_r, \qquad \varepsilon_{r\theta}=\varepsilon_{\theta r}=\frac{1}{2}\big(v_r+\frac{u_\theta-v}{r}\big), \qquad \varepsilon_{\theta\theta}=\frac{u+v_\theta}{r} \nonumber \\[12pt]
\varepsilon_{rz}=\varepsilon_{zr}=\frac{1}{2}\big(u_z+w_r\big), \qquad \varepsilon_{\theta z}=\varepsilon_{z\theta}=\frac{1}{2}\big(\frac{w_\theta}{r}+v_z\big), \qquad \varepsilon_{zz}=w_z \label{eq:ffem3d43}
\end{gather}
%
\indent
The stress-strain relations for isotropic homogeneous medium in terms of the shear modulus $~G$ and the bulk modulus $~K$ are 
\begin{equation*}
\tau = 2G\varepsilon + \big(K-\frac{2}{3}G\big)(\text{tr}\varepsilon)I \tag{\ref{eq:uneq3d23}}
\end{equation*}
where
\vspace{-0.1in}
\begin{align}
\tau_{rr}&=2Gu_r+\big(K-\frac{2}{3}G\big)\big(u_r+\frac{u+v_\theta}{r}+w_z\big) \nonumber \\[6pt]
\tau_{\theta\theta}&=2G\frac{u+v_\theta}{r}+\big(K-\frac{2}{3}G\big)\big(u_r+\frac{u+v_\theta}{r}+w_z\big) \nonumber \\[6pt]
\tau_{zz}&=2Gw_z+\big(K-\frac{2}{3}G\big)\big(u_r+\frac{u+v_\theta}{r}+w_z\big) \nonumber \\[6pt]
\tau_{r\theta}&=\tau_{\theta r}=G\big(v_r+\frac{u_\theta-v}{r}\big) \nonumber \\[6pt]
\tau_{rz}&=\tau_{zr}=G\big(u_z+w_r\big) \nonumber \\[6pt]
\tau_{\theta z}&=\tau_{z\theta}=G\big(\frac{w_\theta}{r}+v_z\big) \label{eq:ffem3d44}
\end{align}
%
\indent
The sum of total potential energy and penalty term is
\begin{align}
\Pi(\vec{u})=&\frac{1}{2}\int_{\Omega}\Big(\tau_{rr}(\vec{u})\varepsilon_{rr}(\vec{u})+2\tau_{r\theta}(\vec{u})\varepsilon_{r\theta}(\vec{u})+2\tau_{rz}(\vec{u})\varepsilon_{rz}(\vec{u})\nonumber \\[6pt]
&+\tau_{\theta\theta}(\vec{u})\varepsilon_{\theta\theta}(\vec{u})+2\tau_{\theta z}(\vec{u})\varepsilon_{\theta z}(\vec{u})+2\tau_{zz}(\vec{u})\varepsilon_{zz}(\vec{u})\Big)\ud \Omega \nonumber \\[6pt]
&-\int_{\Omega}f\big(\text{tr}\varepsilon(\vec{u})\big)\ud \Omega-\int_{\Gamma}(\hat{\sigma}\vec{n})\cdot\vec{u}\ud\Gamma+K_F\int_{\Gamma_F}\vec{u}\cdot\vec{u}\ud \Gamma \label{eq:ffem3d45}
\end{align}
where $~f=\vec{b}_1^T\bar{V}=\vec{b}_1^T(V+W)$ is body force, \quad 
$\vec{b}_1^T= \begin{bmatrix} \\[-2.5em]
                     \tilde{\zeta} & -\lambda & \tilde{\alpha} \\[-6pt]
										\end{bmatrix},$\\[6pt]
$\hat{\sigma}= \begin{bmatrix} \\[-2.5em]
\sigma_r & \sigma_{r\theta} & \sigma_{rz} \\[-6pt]
\sigma_{r\theta} & \sigma_\theta & \sigma_{\theta z} \\[-6pt]
\sigma_{rz} & \sigma_{\theta z} & \sigma_z 
\end{bmatrix}$ is applied boundary stress in the system $(r, \theta, Z_{b})$, and\\[6pt]
$K_F$ is a penalization constant.\\
%
\indent
Next, we describe stress around the borehole. Let
%
\begin{gather*}
\sigma_{XYZ}(z) = \begin{bmatrix} \\[-2.5em]
\sigma_H(z) & 0 & 0 \\[-6pt]
0 & \sigma_h(z) & 0 \\[-6pt]
0 & 0 & \sigma_V
\end{bmatrix}, \qquad 
%
\sigma_{XYZ}^b(z) = \begin{bmatrix} \\[-2.5em]
\sigma_x & \sigma_{xy} & \sigma_{xz} \\[-6pt]
\sigma_{xy} & \sigma_y & \sigma_{yz} \\[-6pt]
\sigma_{xz} & \sigma_{yz} & \sigma_{zz}
\end{bmatrix}(z) \\[6pt]
%
\sigma(r,\theta,z) = \begin{bmatrix} \\[-2.5em]
\sigma_r & \sigma_{r\theta} & \sigma_{rz} \\[-6pt]
\sigma_{r\theta} & \sigma_\theta & \sigma_{\theta z} \\[-6pt]
\sigma_{rz} & \sigma_{\theta z} & \sigma_{z}
\end{bmatrix}(r,\theta,z)
\end{gather*}
be the stress tensors in the $~(X,Y,Z)$, $~(X_{b}, Y_{b}, Z_{b})$, and $~(r, \theta, z)$ coordinate systems, respectively. \\
%
The transformation from $~\sigma_{XYZ}(z)$ to $~\sigma_{XYZ}^b(z)$ is given by ~\cite{Jae}:
%
\begin{equation*}
\sigma_{XYZ}^b(z) = L_\sigma\cdot \sigma_{XYZ}(z) \cdot L_\sigma^T
\end{equation*}
%
where
\begin{equation*}
L_\sigma = \begin{bmatrix} \\[-2.5em]
l_{11} & l_{12} & l_{13} \\[-6pt]
l_{21} & l_{22} & l_{23} \\[-6pt]
l_{31} & l_{32} & l_{33}
\end{bmatrix} = \begin{bmatrix} \\[-2.5em]
\cos\varphi_x\cos\varphi_z & \sin\varphi_x\cos\varphi_z & -\sin\varphi_z \\[-6pt]
-\sin\varphi_x & \cos\varphi_x & 0 \\[-6pt]
\cos\varphi_x\sin\varphi_z & \sin\varphi_x\sin\varphi_z & \cos\varphi_z
\end{bmatrix}
\end{equation*}
is the transformation matrix, $~\varphi_x~$ is the horizontal angle between the $x$-axis and the borehole axis measures counterclockwise, and
$~\varphi_z~$ is the vertical angle between the $z$-axis and the borehole axis measures clockwise.\\
% 
Equivalently,
\begin{equation*}
\begin{bmatrix} \\[-2.5em]
\sigma_x \\[-6pt]
\sigma_y \\[-6pt] 
\sigma_{zz} \\[-6pt]
\sigma_{xy} \\[-6pt] 
\sigma_{yz} \\[-6pt]
\sigma_{xz} 
\end{bmatrix}(z) = T_\sigma\begin{bmatrix} \\[-2.5em]
\sigma_H(z) \\[-6pt]
\sigma_h(z) \\[-6pt] 
\sigma_V 
\end{bmatrix}(z), \qquad \text{where } T_\sigma = \begin{bmatrix} \\[-2.5em]
l_{11}^2 & l_{12}^2 & l_{13}^2 \\[-6pt]
l_{21}^2 & l_{22}^2 & l_{23}^2 \\[-6pt]
l_{31}^2 & l_{32}^2 & l_{33}^2 \\[-6pt]
l_{11}l_{12} & l_{12}l_{22} & l_{13}l_{23} \\[-6pt]
l_{21}l_{31} & l_{22}l_{32} & l_{23}l_{33} \\[-6pt]
l_{11}l_{31} & l_{12}l_{32} & l_{13}l_{33}
\end{bmatrix}
\end{equation*}
%
\indent
The transformation from $~\sigma_{XYZ}^b(z)$ to $~\sigma(r,\theta,z)$ is given by ~\cite{Brad}:
%
\begin{flalign*}
&\sigma_r=\frac{\sigma_x+\sigma_y}{2}\Big(1-\frac{R_B^2}{r^2}\Big)+p_B\frac{R_B^2}{r^2}+\sigma_{xy}\Big(1-\frac{4R_B^2}{r^2}+\frac{3R_B^4}{r^4}\Big)\sin2\theta & \\[6pt]
&{}\quad \quad +\frac{\sigma_x-\sigma_y}{2}\Big(1-\frac{4R_B^2}{r^2}+\frac{3R_B^4}{r^4}\Big)\cos2\theta&\\[12pt]
&\sigma_\theta=\frac{\sigma_x+\sigma_y}{2}\Big(1+\frac{R_B^2}{r^2}\Big)-p_B\frac{R_B^2}{r^2}-\sigma_{xy}\Big(1+\frac{3R_B^4}{r^4}\Big)\sin2\theta & \\[6pt]
&{}\quad \quad -\frac{\sigma_x-\sigma_y}{2}\Big(1+\frac{3R_B^4}{r^4}\Big)\cos2\theta&\\[12pt]
&\sigma_z=\sigma_{zz}-4\nu\sigma_{xy}\frac{R_B^2}{r^2}\sin2\theta-2\nu(\sigma_x-\sigma_y)\frac{R_B^2}{r^2}\cos2\theta&\\[12pt]
&\sigma_{r\theta}=\frac{\sigma_x-\sigma_y}{2}\Big(1+\frac{2R_B^2}{r^2}-\frac{3R_B^4}{r^4}\Big)\sin2\theta+\sigma_{xy}\Big(1+\frac{2R_B^2}{r^2}-\frac{3R_B^4}{r^4}\Big)\cos2\theta&\\[12pt]
&\sigma_{\theta z}=(-\sigma_{xz}\sin\theta+\sigma_{yz}\cos\theta)\Big(1+\frac{R_B^2}{r^2}\Big)&\\[12pt]
&\sigma_{rz}=(\sigma_{yz}\sin\theta+\sigma_{xz}\cos\theta)\Big(1-\frac{R_B^2}{r^2}\Big)
\end{flalign*}
where $p_B$ is well bore pressure and $\nu$ is Poisson's ratio of the formation.\\
%
From these equations we obtain applied stress on the boundaries.\\
On the borehole boundary $\Gamma_B$:\\
$\sigma_r(R_B,z,t) =p_B(z,t)$ \\
$\sigma_\theta(R_B,\theta,z,t) =\sigma_x(z)+\sigma_y(z)-p_B(z,t)-4\sigma_{xy}(z)\sin2\theta-2\big(\sigma_x(z)-\sigma_y(z)\big)\cos2\theta$ \\
$\sigma_z(R_B,\theta,z) =\sigma_{zz}(z)-4\nu\sigma_{xy}(z)\sin2\theta-2\nu\big(\sigma_x(z)-\sigma_y(z)\big)\cos2\theta$\\
$\sigma_{r\theta}(R_B,\theta,z) =0$\\
$\sigma_{rz}(R_B,\theta,z) =0$\\
$\sigma_{\theta z}(R_B,\theta,z) =2\big(-\sigma_{xz}(z)\sin\theta+\sigma_{yz}(z)\cos\theta\big)$\\[12pt]
%
On the far-field boundary $\Gamma_F$:
\begin{flalign*}
&\sigma_r(R_F,\theta,z)=\frac{\sigma_x(z)+\sigma_y(z)}{2}+\sigma_{xy}(z)\sin2\theta+\frac{\sigma_x(z)-\sigma_y(z)}{2}\cos2\theta& \\
&\sigma_\theta(R_F,\theta,z)=\frac{\sigma_x(z)+\sigma_y(z)}{2}-\sigma_{xy}(z)\sin2\theta-\frac{\sigma_x(z)-\sigma_y(z)}{2}\cos2\theta &\\
&\sigma_z(R_F,\theta,z)=\sigma_{zz}(z) &\\
&\sigma_{r\theta}(R_F,\theta,z)=\frac{\sigma_x(z)-\sigma_y(z)}{2}\sin2\theta+\sigma_{xy}(z)\cos2\theta &\\
&\sigma_{rz}(R_F,\theta,z)=\sigma_{yz}(z)\sin\theta+\sigma_{xz}(z)\cos\theta &\\
&\sigma_{\theta z}(R_F,\theta,z)=-\sigma_{xz}(z)\sin\theta+\sigma_{yz}(z)\cos\theta
\end{flalign*}
%
On the top boundary $\tilde{\Gamma}_T$ and the bottom boundary $\tilde{\Gamma}_B$:
\begin{flalign}
&\sigma_r(r,\theta,z_i,t)=\frac{\sigma_x(z_i)+\sigma_y(z_i)}{2}\Big(1-\frac{R^2_B}{r^2}\Big)+p_B(z_i,t)\frac{R_B^2}{r^2}& \nonumber \\[6pt]
&{}\qquad \qquad\qquad +\Big(1-\frac{4R_B^2}{r^2}+\frac{3R_B^4}{r^4}\Big)\Big(\sigma_{xy}(z_i)\sin2\theta+\frac{\sigma_x(z_i)-\sigma_y(z_i)}{2}\cos2\theta\Big)& \nonumber \\[12pt] 
&\sigma_\theta(r,\theta,z_i,t) =\frac{\sigma_x(z_i)+\sigma_y(z_i)}{2}\Big(1+\frac{R_B^2}{r^2}\Big)-p_B(z_i,t)\frac{R_B^2}{r^2}& \nonumber\\[6pt]
&{}\qquad \qquad\qquad-\Big(1+\frac{3R_B^4}{r^4}\Big)\Big(\sigma_{xy}(z_i)\sin2\theta+\frac{\sigma_x(z_i)-\sigma_y(z_i)}{2}\cos2\theta\Big)& \nonumber\\[12pt]
&\sigma_z(r,\theta,z_i)=\sigma_{zz}(z_i)-4\nu\frac{R_B^2}{r^2}\Big(\sigma_{xy}(z_i)\sin2\theta+\frac{\sigma_x(z_i)-\sigma_y(z_i)}{2}\cos2\theta\Big) &\nonumber\\[12pt]
&\sigma_{r\theta}(r,\theta,z_i)=\Big(1+\frac{2R_B^2}{r^2}-\frac{3R_B^4}{r^4}\Big)\Big(\frac{\sigma_x(z)-\sigma_y(z)}{2}\sin2\theta+\sigma_{xy}(z)\cos2\theta\Big)& \nonumber\\[12pt]
&\sigma_{rz}(r,\theta,z_i)=\Big(1-\frac{R_B^2}{r^2}\Big)\big(\sigma_{yz}(z_i)\sin\theta+\sigma_{xz}(z_i)\cos\theta\big)& \nonumber\\[12pt]
&\sigma_{\theta z}(r,\theta,z_i)=\Big(1+\frac{R_B^2}{r^2}\Big)\big(-\sigma_{xz}(z_i)\sin\theta+\sigma_{yz}(z_i)\cos\theta\big)& \label{eq:ffem3d46}
\end{flalign}
where $\qquad i=T$ (top) or $i=B$ (bottom).\\[12pt] 
%
\indent
Let $V=H^1(\Omega;\reals^3).$ Our objective is to find the displacement $\vec{u}$ such that
$$\vec{u}\in L^2(0,t_f;V),\qquad \dot{\vec{u}}\in L^2(0,t_f;V)\qquad \text{and}$$
\begin{equation}
D\Pi(\vec{u})\vec{\Phi}=0, \qquad \forall ~\vec{\Phi} \in V \label{eq:ffem3d47}
\end{equation}
where $~\vec{u}(r,\theta,z,t)=u(r,\theta,z,t)\vec{i}_r +v(r,\theta,z,t)\vec{i}_\theta+w(r,\theta,z,t)\vec{i}_z$,\\
${}\qquad \quad ~\vec{\Phi}(r,\theta,z)=\varphi(r,\theta,z)\vec{i}_r+\psi(r,\theta,z)\vec{i}_\theta+\omega(r,\theta,z)\vec{i}_z$,\\
${}\qquad \quad ~D\Pi(\vec{u})\vec{\Phi}~$ is the G\^{a}teaux differential of $~\Pi$ with increment $~\vec{\Phi}$.\\[12pt]
%
Then (\ref{eq:ffem3d47}) has the form
\noindent
\begin{multline}
\int^{z_B}_{z_T}\int^{2\pi}_{0}\int^{R_F}_{R_B}\Big[\Big(2Gu_r+\big(K-\frac{2G}{3}\big)
\big(u_r+\frac{u+v_\theta}{r}+w_z\big)\Big)\varphi_r \\[6pt]
+G\big(v_r+\frac{u_\theta-v}{r}\big)\big(\psi_r+\frac{\varphi_\theta-\psi}{r}\big)+G(u_z+w_r)(\varphi_z+\omega_r) \\[6pt]
+\Big(2G\frac{u+v_\theta}{r}+\big(K-\frac{2G}{3}\big)\big(u_r+\frac{u+v_\theta}{r}+w_z\big)\Big)\frac{\varphi+\psi_\theta}{r} \\[6pt]
+G\big(\frac{w_\theta}{r}+v_z\big)\big(\frac{\omega_\theta}{r}+\psi_z\big)+\Big(2Gw_z+\big(K-\frac{2G}{3}\big)\big(u_r+\frac{u+v_\theta}{r}+w_z\big)\Big)\omega_z\Big]r\ud r \ud \theta \ud z \\[6pt]
-\int^{z_B}_{z_T}\int^{2\pi}_{0}\int^{R_F}_{R_B}f\big(\varphi_r+\frac{\varphi+\psi_\theta}{r}+\omega_z\big)r\ud r\ud \theta \ud z\\[6pt]
+\int^{z_B}_{z_T}\int^{2\pi}_{0}\Big(\sigma_r(R_B,\theta,z)\varphi(R_B,\theta,z)+\sigma_{r\theta}(R_B,\theta,z)\psi(R_B,\theta,z)\\
+\sigma_{rz}(R_B,\theta,z)\omega(R_B,\theta,z)\Big)R_B\ud \theta \ud z\\
-\int^{z_B}_{z_T}\int^{2\pi}_{0}\Big(\sigma_r(R_F,\theta,z)\varphi(R_F,\theta,z)+\sigma_{r\theta}(R_F,\theta,z)\psi(R_F,\theta,z)\\
+\sigma_{rz}(R_F,\theta,z)\omega(R_F,\theta,z)\Big)R_F\ud \theta \ud z\\
+\int^{2\pi}_{0}\int^{R_F}_{R_B}\Big(\sigma_{rz}(r,\theta,z_T)\varphi(r,\theta,z_T)+\sigma_{\theta z}(r,\theta,z_T)\psi(r,\theta,z_T)\\
+\sigma_{z}(r,\theta,z_T)\omega(r,\theta,z_T)\Big)r\ud r \ud \theta \\
-\int^{2\pi}_{0}\int^{R_F}_{R_B}\Big(\sigma_{rz}(r,\theta,z_B)\varphi(r,\theta,z_B)+\sigma_{\theta z}(r,\theta,z_B)\psi(r,\theta,z_B)\\
+\sigma_{z}(r,\theta,z_B)\omega(r,\theta,z_B)\Big)r\ud r \ud \theta \\
+K_F\int^{z_B}_{z_T}\int^{2\pi}_{0}\Big(u(R_F,\theta,z)\varphi(R_F,\theta,z)+v(R_F,\theta,z)\psi(R_F,\theta,z)\\
+w(R_F,\theta,z)\omega(R_F,\theta,z)\Big)R_F\ud \theta \ud z =0 \label{eq:ffem3d48}
\end{multline}
\\
The first term of the left-hand side of (\ref{eq:ffem3d48}) is
\begin{multline}
\int^{z_B}_{z_T}\int^{2\pi}_{0}\int^{R_F}_{R_B}\Big[\big(K+\frac{4G}{3}\big)u_r\varphi_rr+\big(K-\frac{2G}{3}\big)u\varphi_r+\big(K-\frac{2G}{3}\big)v_\theta\varphi_r+\big(K-\frac{2G}{3}\big)w_z\varphi_rr \\
+Gv_r\psi_rr+Gv_r\varphi_\theta-Gv_r\psi+Gu_\theta\psi_r-Gv\psi_r+Gu_\theta\varphi_\theta\frac{1}{r}-Gu_\theta\psi\frac{1}{r} \\
-Gv\varphi_\theta\frac{1}{r}+Gv\psi\frac{1}{r}+Gu_z\varphi_zr+Gu_z\omega_rr+Gw_r\varphi_zr+Gw_r\omega_rr \\
+\big(K+\frac{4G}{3}\big)u\varphi\frac{1}{r}+\big(K+\frac{4G}{3}\big)u\psi_\theta\frac{1}{r}+\big(K+\frac{4G}{3}\big)v_\theta\varphi\frac{1}{r}+\big(K+\frac{4G}{3}\big)v_\theta\psi_\theta\frac{1}{r} \\
+\big(K-\frac{2G}{3}\big)u_r\varphi+\big(K-\frac{2G}{3}\big)u_r\psi_\theta+\big(K-\frac{2G}{3}\big)w_z\varphi+\big(K-\frac{2G}{3}\big)w_z\psi_\theta \\
+Gw_\theta\omega_\theta\frac{1}{r}+Gw_\theta\psi_z+Gv_z\omega_\theta+Gv_z\psi_zr+\big(K+\frac{4G}{3}\big)w_z\omega_zr \\
+\big(K-\frac{2G}{3}\big)u_r\omega_zr+\big(K-\frac{2G}{3}\big)u\omega_z+\big(K-\frac{2G}{3}\big)v_\theta\omega_z\Big]\ud r \ud \theta \ud z \label{eq:ffem3d49}
\end{multline} 
%
%
Applying $\theta$-approximation (\ref{eq:ffem3d18}) to (\ref{eq:ffem3d49}), integrating with respect to $\theta$ over ($0,2\pi$), and combining like terms with respect to the basis functions, we have\\
$\varphi_0:$
\begin{multline}
\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[2\big(K+\frac{4}{3}G\big)u_{0_r}\varphi_{0_r}r+2\big(K-\frac{2}{3}G\big)u_0\varphi_{0_r}+2\big(K-\frac{2}{3}G\big)w_{0_z}\varphi_{0_r}r \\[6pt]
+2Gu_{0_z}\varphi_{0_z}r+2Gw_{0_r}\varphi_{0_z}r+2\big(K+\frac{4}{3}G\big)u_0\varphi_0\frac{1}{r} \\[6pt]
+2\big(K-\frac{2}{3}G\big)u_{0_r}\varphi_0+2\big(K-\frac{2}{3}G\big)w_{0_z}\varphi_0\Big] \ud r \ud z \label{eq:ffem3d51}
\end{multline}
%
$\varphi_1:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[\big(K+\frac{4}{3}G\big)u_{1_r}\varphi_{1_r}r+\big(K-\frac{2}{3}G\big)u_1\varphi_{1_r}-\big(K-\frac{2}{3}G\big)v_2\varphi_{1_r} \\[6pt]
+\big(K-\frac{2}{3}G\big)w_{1_z}\varphi_{1_r}r+Gv_{2_r}\varphi_1+\big(K+\frac{7}{3}G\big)u_1\varphi_1\frac{1}{r}+Gu_{1_z}\varphi_{1_z}r+Gw_{1_r}\varphi_{1_z}r \\[6pt]
-\big(K+\frac{7}{3}G\big)v_2\varphi_1\frac{1}{r}+\big(K-\frac{2}{3}G\big)u_{1_r}\varphi_1+\big(K-\frac{2}{3}G\big)w_{1_z}\varphi_1\Big] \ud r \ud z \label{eq:ffem3d52}
\end{multline}
%
$\varphi_2:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[\big(K+\frac{4}{3}G\big)u_{2_r}\varphi_{2_r}r+\big(K-\frac{2}{3}G\big)u_2\varphi_{2_r}+\big(K-\frac{2}{3}G\big)v_1\varphi_{2_r} \\[6pt]
+\big(K-\frac{2}{3}G\big)w_{2_z}\varphi_{2_r}r-Gv_{1_r}\varphi_2+\big(K+\frac{7}{3}G\big)u_2\varphi_2\frac{1}{r}+Gu_{2_z}\varphi_{2_z}r+Gw_{2_r}\varphi_{2_z}r \\[6pt]
+\big(K+\frac{7}{3}G\big)v_1\varphi_2\frac{1}{r}+\big(K-\frac{2}{3}G\big)u_{2_r}\varphi_2+\big(K-\frac{2}{3}G\big)w_{2_z}\varphi_2\Big] \ud r \ud z \label{eq:ffem3d53}
\end{multline}
%
$\varphi_3:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[\big(K+\frac{4}{3}G\big)u_{3_r}\varphi_{3_r}r+\big(K-\frac{2}{3}G\big)u_3\varphi_{3_r}-2\big(K-\frac{2}{3}G\big)v_4\varphi_{3_r} \\[6pt]
+\big(K-\frac{2}{3}G\big)w_{3_z}\varphi_{3_r}r+2Gv_{4_r}\varphi_3+\big(K+\frac{16}{3}G\big)u_3\varphi_3\frac{1}{r}+Gu_{3_z}\varphi_{3_z}r+Gw_{3_r}\varphi_{3_z}r \\[6pt]
-2\big(K+\frac{7}{3}G\big)v_4\varphi_3\frac{1}{r}+\big(K-\frac{2}{3}G\big)u_{3_r}\varphi_3+\big(K-\frac{2}{3}G\big)w_{3_z}\varphi_3\Big]\ud r \ud z \label{eq:ffem3d54}
\end{multline}
%
\newpage
\noindent
$\varphi_4:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[\big(K+\frac{4}{3}G\big)u_{4_r}\varphi_{4_r}r+\big(K-\frac{2}{3}G\big)u_4\varphi_{4_r}+2\big(K-\frac{2}{3}G\big)v_3\varphi_{4_r} \\[6pt]
+\big(K-\frac{2}{3}G\big)w_{4_z}\varphi_{4_r}r-2Gv_{3_r}\varphi_4+\big(K+\frac{16}{3}G\big)u_4\varphi_4\frac{1}{r}+Gu_{4_z}\varphi_{4_z}r+Gw_{4_r}\varphi_{4_z}r \\[6pt]
+2\big(K+\frac{7}{3}G\big)v_3\varphi_4\frac{1}{r}+\big(K-\frac{2}{3}G\big)u_{4_r}\varphi_4+\big(K-\frac{2}{3}G\big)w_{4_z}\varphi_4\Big]\ud r \ud z \label{eq:ffem3d55}
\end{multline}
%
$\psi_0:$
\begin{equation}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[2G\Big(\big(v_{0_r}-\frac{v_0}{r}\big)\big(\psi_{0_r}-\frac{\psi_0}{r}\big)+v_{0_z}\psi_{0_z}\Big)r\Big] \ud r \ud z \label{eq:ffem3d56}
\vspace{0.2in}
\end{equation}
%
$\psi_1:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gv_{1_r}\psi_{1_r}r-Gv_{1_r}\psi_1-Gu_2\psi_{1_r}-Gv_1\psi_{1_r}+\big(K+\frac{7}{3}G\big)u_2\psi_1\frac{1}{r} \\[6pt]
+\big(K+\frac{7}{3}G\big)v_1\psi_1\frac{1}{r}+\big(K-\frac{2}{3}G\big)u_{2_r}\psi_1+\big(K-\frac{2}{3}G\big)w_{2_z}\psi_1 \\[6pt]
-Gw_2\psi_{1_z}+Gv_{1_z}\psi_{1_z}r\Big] \ud r \ud z \label{eq:ffem3d57}
\end{multline}
%
$\psi_2:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gv_{2_r}\psi_{2_r}r-Gv_{2_r}\psi_2+Gu_1\psi_{2_r}-Gv_2\psi_{2_r}-\big(K+\frac{7}{3}G\big)u_1\psi_2\frac{1}{r} \\[6pt]
+\big(K+\frac{7}{3}G\big)v_2\psi_2\frac{1}{r}-\big(K-\frac{2}{3}G\big)u_{1_r}\psi_2-\big(K-\frac{2}{3}G\big)w_{1_z}\psi_2 \\[6pt]
+Gw_1\psi_{2_z}+Gv_{2_z}\psi_{2_z}r\Big] \ud r \ud z \label{eq:ffem3d58}
\end{multline}
%
$\psi_3:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gv_{3_r}\psi_{3_r}r-Gv_{3_r}\psi_3-2Gu_4\psi_{3_r}-Gv_3\psi_{3_r}+2\big(K+\frac{7}{3}G\big)u_4\psi_3\frac{1}{r} \\[6pt]
+\big(4K+\frac{19}{3}G\big)v_3\psi_3\frac{1}{r}+2\big(K-\frac{2}{3}G\big)u_{4_r}\psi_3+2\big(K-\frac{2}{3}G\big)w_{4_z}\psi_3 \\[6pt]
-2Gw_4\psi_{3_z}+Gv_{3_z}\psi_{3_z}r\Big] \ud r \ud z \label{eq:ffem3d59}
\end{multline}
%
\newpage
\noindent
$\psi_4:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gv_{4_r}\psi_{4_r}r-Gv_{4_r}\psi_4+2Gu_3\psi_{4_r}-Gv_4\psi_{4_r}-2\big(K+\frac{7}{3}G\big)u_3\psi_4\frac{1}{r} \\[6pt]
+\big(4K+\frac{19}{3}G\big)v_4\psi_4\frac{1}{r}-2\big(K-\frac{2}{3}G\big)u_{3_r}\psi_4-2\big(K-\frac{2}{3}G\big)w_{3_z}\psi_4 \\[6pt]
+2Gw_3\psi_{4_z}+Gv_{4_z}\psi_{4_z}r\Big] \ud r \ud z \label{eq:ffem3d60}
\end{multline}
%
$\omega_0:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[2Gu_{0_z}\omega_{0_r}r+2Gw_{0_r}\omega_{0_r}r+2\big(K+\frac{4}{3}G\big)w_{0_z}\omega_{0_z}r \\[6pt]
+2\big(K-\frac{2}{3}G\big)u_{0_r}\omega_{0_z}r+2\big(K-\frac{2}{3}G\big)u_0\omega_{0_z}\Big]\ud r \ud z \label{eq:ffem3d61}
\end{multline}
%
$\omega_1:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gu_{1_z}\omega_{1_r}r+Gw_{1_r}\omega_{1_r}r+Gw_1\omega_1\frac{1}{r}+Gv_{2_z}\omega_1+\big(K+\frac{4}{3}G\big)w_{1_z}\omega_{1_z}r \\[6pt]
+\big(K-\frac{2}{3}G\big)u_{1_r}\omega_{1_z}r+\big(K-\frac{2}{3}G\big)u_1\omega_{1_z}-\big(K-\frac{2}{3}G\big)v_2\omega_{1_z}\Big]\ud r \ud z \label{eq:ffem3d62}
\end{multline}
%
$\omega_2:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gu_{2_z}\omega_{2_r}r+Gw_{2_r}\omega_{2_r}r+Gw_2\omega_2\frac{1}{r}-Gv_{1_z}\omega_2+\big(K+\frac{4}{3}G\big)w_{2_z}\omega_{2_z}r \\[6pt]
+\big(K-\frac{2}{3}G\big)u_{2_r}\omega_{2_z}r+\big(K-\frac{2}{3}G\big)u_2\omega_{2_z} +\big(K-\frac{2}{3}G\big)v_1\omega_{2_z}\Big]\ud r \ud z \label{eq:ffem3d63}
\end{multline}
%
$\omega_3:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gu_{3_z}\omega_{3_r}r+Gw_{3_r}\omega_{3_r}r+4Gw_3\omega_3\frac{1}{r}+2Gv_{4_z}\omega_3 +\big(K+\frac{4}{3}G\big)w_{3_z}\omega_{3_z}r\\[6pt]
+\big(K-\frac{2}{3}G\big)u_{3_r}\omega_{3_z}r+\big(K-\frac{2}{3}G\big)u_3\omega_{3_z}-2\big(K-\frac{2}{3}G\big)v_4\omega_{3_z}\Big]\ud r \ud z \label{eq:ffem3d64}
\end{multline}
%
\newpage
\noindent
$\omega_4:$
\begin{multline}
+\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gu_{4_z}\omega_{4_r}r+Gw_{4_r}\omega_{4_r}r+4Gw_4\omega_4\frac{1}{r}-2Gv_{3_z}\omega_4+\big(K+\frac{4}{3}G\big)w_{4_z}\omega_{4_z}r \\[6pt]
+\big(K-\frac{2}{3}G\big)u_{4_r}\omega_{4_z}r+\big(K-\frac{2}{3}G\big)u_4\omega_{4_z} +2\big(K-\frac{2}{3}G\big)v_3\omega_{4_z}\Big]\ud r \ud z \label{eq:ffem3d65}
\end{multline}
%
\noindent
The $\sigma$-approximation of body force {f} has the form
\begin{align}
f(r,\theta,z,t)&=f_0(r,z,t)+f_1(r,z,t)\sin\theta+f_2(r,z,t)\cos\theta \nonumber \\
&=f_3(r,z,t)\sin2\theta+f_4(r,z,t)\cos2\theta \label{eq:ffem3d66}
\end{align}
%
The second term of the left-hand side of (\ref{eq:ffem3d48}) is
\begin{multline}
-\pi\int^{z_B}_{z_T}\int^{R_F}_{R_B}\big[2f_0\varphi_{0_r}r+f_1\varphi_{1_r}r+f_2\varphi_{2_r}r+f_3\varphi_{3_r}r+f_4\varphi_{4_r}r \\
+2f_0\varphi_0+f_1\varphi_1+f_2\varphi_2+f_3\varphi_3+f_4\varphi_4-f_1\psi_2+f_2\psi_1-2f_3\psi_4+2f_4\psi_3 \\
+2f_0\omega_{0_z}r+f_1\omega_{1_z}r+f_2\omega_{2_z}r+f_3\omega_{3_z}r+f_4\omega_{4_z}r\big] \ud r \ud z \label{eq:ffem3d67}
\end{multline}
%
The third term of the left-hand side of (\ref{eq:ffem3d48}) is
\begin{equation}
\pi\int^{z_B}_{z_T}2p_B(z)\varphi_0(R_B,z)R_B\ud z \label{eq:ffem3d68}
\end{equation}
%
The fourth term of the left-hand side of (\ref{eq:ffem3d48}) is
\begin{multline}
-\pi\int^{z_B}_{z_T}\Big[2\frac{\sigma_x(z)+\sigma_y(z)}{2}\varphi_0(R_F,z)+\sigma_{xy}(z)\varphi_3(R_F,z)+\frac{\sigma_x(z)-\sigma_y(z)}{2}\varphi_4(R_F,z) \\
+\frac{\sigma_x(z)-\sigma_y(z)}{2}\psi_3(R_F,z)+\sigma_{xy}(z)\psi_4(R_F,z) \\
+\sigma_{yz}(z)\omega_1(R_F,z)+\sigma_{xz}(z)\omega_2(R_F,z)\Big]R_F \ud z \label{eq:ffem3d69}
\end{multline}
%
The fifth term of the left-hand side of (\ref{eq:ffem3d48}) is
\begin{multline}
\pi\int^{R_F}_{R_B}\Big[\big(1-\frac{R^{2}_{B}}{r^2}\big)r\big(\sigma_{yz}(z_T)\varphi_1(r,z_T)+\sigma_{xz}(z_T)\varphi_2(r,z_T)\big) \\[6pt]
+\big(1+\frac{R^{2}_{B}}{r^2}\big)r\big(-\sigma_{xz}(z_T)\psi_1(r,z_T)+\sigma_{yz}(z_T)\psi_2(r,z_T)\big)+2\sigma_{zz}(z_T)\omega_0(r,z_T)r- \\[6pt]
-4\nu\frac{R^{2}_{B}}{r}\sigma_{xy}(z_T)\omega_3(r,z_T)-4\nu\frac{R^{2}_{B}}{r}\frac{\sigma_x(z_T)-\sigma_y(z_T)}{2}\omega_4(r,z_T)\Big] \ud r \label{eq:ffem3d70}
\end{multline}
%
The sixth term of the left-hand side of (\ref{eq:ffem3d48}) is
\begin{multline}
-\pi\int^{R_F}_{R_B}\Big[\big(1-\frac{R^{2}_{B}}{r^2}\big)r\big(\sigma_{yz}(z_B)\varphi_1(r,z_B)+\sigma_{xz}(z_B)\varphi_2(r,z_B)\big) \\[6pt]
+\big(1+\frac{R^{2}_{B}}{r^2}\big)r\big(-\sigma_{xz}(z_B)\psi_1(r,z_B)+\sigma_{yz}(z_B)\psi_2(r,z_B)\big)+2\sigma_{zz}(z_B)\omega_0(r,z_B)r \\[6pt]
-4\nu\frac{R^{2}_{B}}{r}\sigma_{xy}(z_B)\omega_3(r,z_B)-4\nu\frac{R^{2}_{B}}{r}\frac{\sigma_x(z_B)-\sigma_y(z_B)}{2}\omega_4(r,z_B)\Big] \ud r \label{eq:ffem3d71}
\end{multline}
%
The seventh term of the left-hand side of (\ref{eq:ffem3d48}) is
\begin{multline}
K_F\pi\int^{z_B}_{z_T}\big[2u_0(R_F,z)\varphi_0(R_F,z)+u_1(R_F,z)\varphi_1(R_F,z)+u_2(R_F,z)\varphi_2(R_F,z) \\
+u_3(R_F,z)\varphi_3(R_F,z)+u_4(R_F,z)\varphi_4(R_F,z) +2v_0(R_F,z)\psi_0(R_F,z)\\
+v_1(R_F,z)\psi_1(R_F,z)+v_2(R_F,z)\psi_2(R_F,z) +v_3(R_F,z)\psi_3(R_F,z)\\
+v_4(R_F,z)\psi_4(R_F,z) +2w_0(R_F,z)\omega_0(R_F,z)+w_1(R_F,z)\omega_1(R_F,z)\\
+w_2(R_F,z)\omega_2(R_F,z) +w_3(R_F,z)\omega_3(R_F,z)+w_4(R_F,z)\omega_4(R_F,z)\big]R_F \ud z \label{eq:ffem3d72}
\end{multline}
%
\indent
In (\ref{eq:ffem3d51})-(\ref{eq:ffem3d65}) and (\ref{eq:ffem3d67})-(\ref{eq:ffem3d72}), we combine terms corresponding to $\varphi_i$, $\psi_i$, $\omega_i$, $i=0,\ldots,4$, and equal them to 0.\\ 
%
$\varphi_0: \quad (u_0, w_0)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[\big(K+\frac{4G}{3}\big)u_{0_r}\varphi_{0_r}r+\big(K-\frac{2G}{3}\big)u_0\varphi_{0_r}+\big(K-\frac{2G}{3}\big)u_{0_r}\varphi_0 \nonumber\\[6pt]
&\qquad \quad +\big(K+\frac{4G}{3}\big)u_0\varphi_0\frac{1}{r}+Gu_{0_z}\varphi_{0_z}r+\big(K-\frac{2G}{3}\big)w_{0_z}\varphi_{0_r}r +Gw_{0_r}\varphi_{0_z}r \nonumber\\[6pt]
&\qquad \quad+\big(K-\frac{2G}{3}\big)w_{0_z}\varphi_0\Big] \ud r \ud z -\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_0\big(\varphi_{0_r}r+\varphi_0\big)\ud r \ud z \nonumber\\[6pt]
+&\int^{z_B}_{z_T}p_B(z)\varphi_0(R_B,z)R_B\ud z-\int^{z_B}_{z_T}\frac{\sigma_x(z)+\sigma_y(z)}{2}\varphi_0(R_F,z)R_F \ud z\nonumber \\[6pt]
+&K_F\int^{z_B}_{z_T}u_0(R_F,z)\varphi_0(R_F,z)R_F\ud z =0 \label{eq:ffem3d73}
\end{align}
%
$\varphi_1: \quad (u_1, v_2, w_1)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[\big(K+\frac{4G}{3}\big)u_{1_r}\varphi_{1_r}r+\big(K-\frac{2G}{3}\big)u_1\varphi_{1_r}+\big(K-\frac{2G}{3}\big)u_{1_r}\varphi_1 +Gw_{1_r}\varphi_{1_z}r\nonumber\\[6pt]
&\qquad \quad +\big(K+\frac{7G}{3}\big)u_1\varphi_1\frac{1}{r}+Gu_{1_z}\varphi_{1_z}r-\big(K-\frac{2G}{3}\big)v_2\varphi_{1_r}+Gv_{2_r}\varphi_1 \nonumber\\[6pt]
&\qquad \quad -\big(K+\frac{7G}{3}\big)v_2\varphi_1\frac{1}{r} +\big(K-\frac{2G}{3}\big)w_{1_z}\varphi_{1_r}r+\big(K-\frac{2G}{3}\big)w_{1_z}\varphi_1\Big] \ud r \ud z \nonumber\\[6pt]
-&\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_1\big(\varphi_{1_r}r+\varphi_1\big)\ud r \ud z+\int^{R_F}_{R_B}\big(1-\frac{R^{2}_{B}}{r^2}\big)r\sigma_{yz}(z_T)\varphi_1(r,z_T) \ud r \nonumber\\[6pt]
-&\int^{R_F}_{R_B}\big(1-\frac{R^{2}_{B}}{r^2}\big)r\sigma_{yz}(z_B)\varphi_1(r,z_B) \ud r+K_F\int^{z_B}_{z_T}u_1(R_F,z)\varphi_1(R_F,z)R_F \ud z = 0 \label{eq:ffem3d74}
\end{align}
%
$\varphi_2: \quad (u_2, v_1, w_2)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[\big(K+\frac{4G}{3}\big)u_{2_r}\varphi_{2_r}r+\big(K-\frac{2G}{3}\big)u_2\varphi_{2_r}+\big(K-\frac{2G}{3}\big)u_{2_r}\varphi_2 +Gu_{2_z}\varphi_{2_z}r\nonumber\\[6pt]
&\qquad \quad +\big(K+\frac{7G}{3}\big)u_2\varphi_2\frac{1}{r} +\big(K-\frac{2G}{3}\big)v_1\varphi_{2_r}-Gv_{1_r}\varphi_2+\big(K+\frac{7G}{3}\big)v_1\varphi_2\frac{1}{r} \nonumber\\[6pt]
&\qquad \quad +Gw_{2_r}\varphi_{2_z}r+\big(K-\frac{2G}{3}\big)w_{2_z}\varphi_{2_r}r+\big(K-\frac{2G}{3}\big)w_{2_z}\varphi_2\Big] \ud r \ud z \nonumber\\[6pt]
-&\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_2\big(\varphi_{2_r}r+\varphi_2\big)\ud r \ud z+\int^{R_F}_{R_B}\big(1-\frac{R^{2}_{B}}{r^2}\big)r\sigma_{xz}(z_T)\varphi_2(r,z_T) \ud r \nonumber\\[6pt]
-&\int^{R_F}_{R_B}\big(1-\frac{R^{2}_{B}}{r^2}\big)r\sigma_{xz}(z_B)\varphi_2(r,z_B) \ud r+K_F\int^{z_B}_{z_T}u_2(R_F,z)\varphi_2(R_F,z)R_F \ud z = 0 \label{eq:ffem3d75}
\end{align}
%
$\varphi_3: \quad (u_3, v_4, w_3)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[\big(K+\frac{4G}{3}\big)u_{3_r}\varphi_{3_r}r+\big(K-\frac{2G}{3}\big)u_3\varphi_{3_r}+\big(K-\frac{2G}{3}\big)u_{3_r}\varphi_3 +Gu_{3_z}\varphi_{3_z}r\nonumber\\[6pt]
&\qquad\quad +\big(K+\frac{16G}{3}\big)u_3\varphi_3\frac{1}{r} -2\big(K-\frac{2G}{3}\big)v_4\varphi_{3_r}-2\big(K+\frac{7G}{3}\big)v_4\varphi_3\frac{1}{r} \nonumber\\[6pt]
&\qquad \quad +2Gv_{4_r}\varphi_3+Gw_{3_r}\varphi_{3_z}r+\big(K-\frac{2G}{3}\big)w_{3_z}\varphi_{3_r}r+\big(K-\frac{2G}{3}\big)w_{3_z}\varphi_3\Big] \ud r \ud z\nonumber \\[6pt]
-&\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_3\big(\varphi_{3_r}r+\varphi_3\big)\ud r \ud z-\int^{z_T}_{z_B}\sigma_{xy}(z)\varphi_3(R_F,z)R_F \ud z\nonumber\\[6pt]
+&K_F\int^{z_B}_{z_T}u_3(R_F,z)\varphi_3(R_F,z)R_F \ud z = 0 \label{eq:ffem3d76}
\end{align}\\[-6pt]
%
$\varphi_4: \quad (u_4, v_3, w_4)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[\big(K+\frac{4G}{3}\big)u_{4_r}\varphi_{4_r}r+\big(K-\frac{2G}{3}\big)u_4\varphi_{4_r}+\big(K-\frac{2G}{3}\big)u_{4_r}\varphi_4 +Gu_{4_z}\varphi_{4_z}r \nonumber\\[6pt]
&\qquad \quad +\big(K+\frac{16G}{3}\big)u_4\varphi_4\frac{1}{r}+2\big(K-\frac{2G}{3}\big)v_3\varphi_{4_r}+2\big(K+\frac{7G}{3}\big)v_3\varphi_4\frac{1}{r} \nonumber\\[6pt]
&\qquad \quad -2Gv_{3_r}\varphi_4+Gw_{4_r}\varphi_{4_z}r+\big(K-\frac{2G}{3}\big)w_{4_z}\varphi_{4_r}r+\big(K-\frac{2G}{3}\big)w_{4_z}\varphi_4\Big] \ud r \ud z \nonumber\\[6pt]
-&\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_4\big(\varphi_{4_r}r+\varphi_4\big)\ud r \ud z-\int^{z_T}_{z_B}\frac{\sigma_x(z)-\sigma_y(z)}{2}\varphi_4(R_F,z)R_F \ud z \nonumber\\[6pt]
+&K_F\int^{z_B}_{z_T}u_4(R_F,z)\varphi_4(R_F,z)R_F \ud z = 0 \label{eq:ffem3d77}
\end{align}\\[-6pt]
%
$\psi_0: \quad (v_0)$
\begin{multline*}
\int^{z_B}_{z_T}\int^{R_F}_{R_B}G\Big[\big(v_{0_r}-\frac{v_0}{r}\big)\big(\psi_{0_r}-\frac{\psi_0}{r}\big)+v_{0_z}\psi_{0_z}\Big]r \ud r \ud z\\
+K_F\int^{z_B}_{z_T}v_0(R_F,z)\psi_0(R_F,z)R_F \ud z=0 
\end{multline*}
%
Take $\psi_0=v_0$. Then
$$\int^{z_B}_{z_T}\int^{R_F}_{R_B}G\Big[\big(v_{0_r}-\frac{v_0}{r}\big)^2+v_{0_z}^2\Big]r \ud r \ud z+K_FR_F\int^{z_B}_{z_T}v_0^2(R_F,z)\ud z=0$$
It follows that
\begin{gather*}
v_{0_r}-\frac{v_0}{r}=0\quad \text{and} \quad v_{0_z}=0,   \qquad z_T<z<z_B, \quad R_B<r<R_F \\
v_0(R_F,z)=0, \quad z_T<z<z_B
\end{gather*}
and so,
\vspace{-0.1in}
\begin{equation}
v_0(r,z,t)=C_0(t)r, \quad R_B<r<R_F \label{eq:ffem3d78}
\vspace{0.2in}
\end{equation}
%
$\psi_1: \quad(u_2, v_1, w_2)$
\begin{gather}
\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[-Gu_2\psi_{1_r}+\big(K-\frac{2G}{3}\big)u_{2_r}\psi_1+\big(K+\frac{7G}{3}\big)u_2\psi_1\frac{1}{r}+Gv_{1_r}\psi_{1_r}r-Gv_1\psi_{1_r}\nonumber\\[6pt]
-Gv_{1_r}\psi_1+\big(K+\frac{7G}{3}\big)v_1\psi_1\frac{1}{r}+Gv_{1_z}\psi_{1_z}r -Gw_2\psi_{1_z}+\big(K-\frac{2G}{3}\big)w_{2_z}\psi_1\Big]\ud r \ud z \nonumber\\[6pt]
-\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_2\psi_1\ud r \ud z-\int^{R_F}_{R_B}\big(1+\frac{R^{2}_{B}}{r^2}\big)r\sigma_{xz}(z_T)\psi_1(r,z_T) \ud r  \nonumber\\[6pt]
+\int^{R_F}_{R_B}\big(1+\frac{R^{2}_{B}}{r^2}\big)r\sigma_{xz}(z_B)\psi_1(r,z_B) \ud r+K_F\int^{z_B}_{z_T}v_1(R_F,z)\psi_1(R_F,z)R_F \ud z=0 \label{eq:ffem3d79}
\end{gather}
%
$\psi_2: \quad(u_1, v_2, w_1)$
\begin{gather}
\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gu_1\psi_{2_r}-\big(K-\frac{2G}{3}\big)u_{1_r}\psi_2-\big(K+\frac{7G}{3}\big)u_1\psi_2\frac{1}{r} +Gv_{2_r}\psi_{2_r}r-Gv_2\psi_{2_r}\nonumber\\[6pt]
-Gv_{2_r}\psi_2+\big(K+\frac{7G}{3}\big)v_2\psi_2\frac{1}{r}+Gv_{2_z}\psi_{2_z}r+Gw_1\psi_{2_z}-\big(K-\frac{2G}{3}\big)w_{1_z}\psi_2\Big]\ud r \ud z+ \nonumber\\[6pt]
+\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_1\psi_2\ud r \ud z+\int^{R_F}_{R_B}\big(1+\frac{R^{2}_{B}}{r^2}\big)r\sigma_{yz}(z_T)\psi_2(r,z_T) \ud r  \nonumber\\[6pt]
-\int^{R_F}_{R_B}\big(1+\frac{R^{2}_{B}}{r^2}\big)r\sigma_{yz}(z_B)\psi_2(r,z_B) \ud r+K_F\int^{z_B}_{z_T}v_2(R_F,z)\psi_2(R_F,z)R_F \ud z=0 \label{eq:ffem3d80}
\end{gather}
%
$\psi_3: \quad(u_4, v_3, w_4)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[-2Gu_4\psi_{3_r}+2\big(K-\frac{2G}{3}\big)u_{4_r}\psi_3+2\big(K+\frac{7G}{3}\big)u_4\psi_3\frac{1}{r}+Gv_{3_r}\psi_{3_r}r\nonumber\\[6pt]
&\qquad \qquad -Gv_3\psi_{3_r}-Gv_{3_r}\psi_3+\big(4K+\frac{19G}{3}\big)v_3\psi_3\frac{1}{r}+Gv_{3_z}\psi_{3_z}r-2Gw_4\psi_{3_z}\nonumber\\[6pt]
&\qquad \qquad +2\big(K-\frac{2G}{3}\big)w_{4_z}\psi_3\Big]\ud r \ud z-\int^{z_B}_{z_T}\int^{R_F}_{R_B}2f_4\psi_3\ud r \ud z  \nonumber\\[6pt]
-&\int^{z_B}_{z_T}\frac{\sigma_x(z)-\sigma_y(z)}{2}\psi_3(R_F,z)R_F \ud z+K_F\int^{z_B}_{z_T}v_3(R_F,z)\psi_3(R_F,z)R_F \ud z=0 \label{eq:ffem3d81}
\end{align}
%
$\psi_4: \quad(u_3, v_4, w_3)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[2Gu_3\psi_{4_r}-2\big(K-\frac{2G}{3}\big)u_{3_r}\psi_4-2\big(K+\frac{7G}{3}\big)u_3\psi_4\frac{1}{r}+Gv_{4_r}\psi_{4_r}r\nonumber\\[6pt]
&\qquad \qquad -Gv_4\psi_{4_r}-Gv_{4_r}\psi_4+\big(4K+\frac{19G}{3}\big)v_4\psi_4\frac{1}{r}+Gv_{4_z}\psi_{4_z}r+2Gw_3\psi_{4_z}\nonumber\\[6pt]
&\qquad \qquad -2\big(K-\frac{2G}{3}\big)w_{3_z}\psi_4\Big]\ud r \ud z+\int^{z_B}_{z_T}\int^{R_F}_{R_B}2f_3\psi_4\ud r \ud z \nonumber\\[6pt]
-&\int^{z_B}_{z_T}\sigma_{xy}(z)\psi_4(R_F,z)R_F \ud z+K_F\int^{z_B}_{z_T}v_4(R_F,z)\psi_4(R_F,z)R_F \ud z=0 \label{eq:ffem3d82}
\end{align}\\[-6pt]
%
$\omega_0: \quad(u_0, w_0)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gu_{0_z}\omega_{0_r}r+\big(K-\frac{2G}{3}\big)u_{0_r}\omega_{0_z}r+\big(K-\frac{2G}{3}\big)u_0\omega_{0_z}\nonumber\\[6pt]
&\qquad \qquad +\big(K+\frac{4G}{3}\big)w_{0_z}\omega_{0_z}r \Big]\ud r \ud z \nonumber\\[6pt]
-&\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_0\omega_{0_z}r\ud r \ud z+\int^{R_F}_{F_B}\sigma_{zz}(z_T)\omega_0(r,z_T)r \ud r-\int^{R_F}_{F_B}\sigma_{zz}(z_B)\omega_0(r,z_B)r \ud r\nonumber\\[6pt]
+&K_F\int^{z_B}_{z_T}w_0(R_F,z)\omega_0(R_F,z)R_F \ud z=0 \label{eq:ffem3d83}
\end{align}\\[-6pt]
%
$\omega_1: \quad(u_1, v_2, w_1)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gu_{1_z}\omega_{1_r}r+\big(K-\frac{2G}{3}\big)u_{1_r}\omega_{1_z}r+\big(K-\frac{2G}{3}\big)u_1\omega_{1_z}+Gv_{2_z}\omega_1\nonumber\\[6pt]
&\qquad \quad -\big(K-\frac{2G}{3}\big)v_2\omega_{1_z}+Gw_{1_r}\omega_{1_r}r+Gw_1\omega_1\frac{1}{r}+\big(K+\frac{4G}{3}\big)w_{1_z}\omega_{1_z}r \Big]\ud r \ud z \nonumber\\[6pt]
-&\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_1\omega_{1_z}r\ud r \ud z-\int^{z_B}_{z_T}\sigma_{yz}(z)\omega_1(R_F,z)R_F \ud z\nonumber\\[6pt]
+&K_F\int^{z_B}_{z_T}w_1(R_F,z)\omega_1(R_F,z)R_F \ud z=0 \label{eq:ffem3d84}
\end{align}
%
$\omega_2: \quad(u_2, v_1, w_2)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gu_{2_z}\omega_{2_r}r+\big(K-\frac{2G}{3}\big)u_{2_r}\omega_{2_z}r+\big(K-\frac{2G}{3}\big)u_2\omega_{2_z}-Gv_{1_z}\omega_2\nonumber\\[6pt]
&\qquad \quad +\big(K-\frac{2G}{3}\big)v_1\omega_{2_z}+Gw_{2_r}\omega_{2_r}r+Gw_2\omega_2\frac{1}{r}+\big(K+\frac{4G}{3}\big)w_{2_z}\omega_{2_z}r \Big]\ud r \ud z \nonumber\\[6pt]
-&\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_2\omega_{2_z}r\ud r \ud z-\int^{z_B}_{z_T}\sigma_{xz}(z)\omega_2(R_F,z)R_F \ud z\nonumber\\[6pt]
+&K_F\int^{z_B}_{z_T}w_2(R_F,z)\omega_2(R_F,z)R_F \ud z=0 \label{eq:ffem3d85}
\end{align}\\[-6pt]
%
$\omega_3: \quad(u_3, v_4, w_3)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gu_{3_z}\omega_{3_r}r+\big(K-\frac{2G}{3}\big)u_{3_r}\omega_{3_z}r+\big(K-\frac{2G}{3}\big)u_3\omega_{3_z}+2Gv_{4_z}\omega_3\nonumber\\[6pt]
&\qquad \quad -2\big(K-\frac{2G}{3}\big)v_4\omega_{3_z}+Gw_{3_r}\omega_{3_r}r+4Gw_3\omega_3\frac{1}{r}+\big(K+\frac{4G}{3}\big)w_{3_z}\omega_{3_z}r \Big]\ud r \ud z \nonumber\\[6pt]
-&\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_3\omega_{3_z}r\ud r \ud z-\int^{R_F}_{R_B}4\nu\frac{R_B^2}{r}\sigma_{xy}(z_T)\omega_3(r,z_T) \ud r\nonumber\\[6pt]
+&\int^{R_F}_{R_B}4\nu\frac{R_B^2}{r}\sigma_{xy}(z_B)\omega_3(r,z_B) \ud r+K_F\int^{z_B}_{z_T}w_3(R_F,z)\omega_3(R_F,z)R_F \ud z=0  \label{eq:ffem3d86}
\end{align}\\[-6pt]
%
$\omega_4: \quad(u_4, v_3, w_4)$
\begin{align}
&\int^{z_B}_{z_T}\int^{R_F}_{R_B}\Big[Gu_{4_z}\omega_{4_r}r+\big(K-\frac{2G}{3}\big)u_{4_r}\omega_{4_z}r+\big(K-\frac{2G}{3}\big)u_4\omega_{4_z}-2Gv_{3_z}\omega_4\nonumber\\[6pt]
&\qquad \quad +2\big(K-\frac{2G}{3}\big)v_3\omega_{4_z}+Gw_{4_r}\omega_{4_r}r+4Gw_4\omega_4\frac{1}{r}+\big(K+\frac{4G}{3}\big)w_{4_z}\omega_{4_z}r \Big]\ud r \ud z \nonumber\\[6pt]
-&\int^{z_B}_{z_T}\int^{R_F}_{R_B}f_4\omega_{4_z}r\ud r \ud z-\int^{R_F}_{R_B}4\nu\frac{R_B^2}{r}\frac{\sigma_x(z_T)-\sigma_y(z_T)}{2}\omega_4(r,z_T) \ud r\nonumber\\[6pt]
+&\int^{R_F}_{R_B}4\nu\frac{R_B^2}{r}\frac{\sigma_x(z_B)-\sigma_y(z_B)}{2}\omega_4(r,z_B) \ud r+K_F\int^{z_B}_{z_T}w_4(R_F,z)\omega_4(R_F,z)R_F \ud z=0  \label{eq:ffem3d87}
\end{align}\\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Finite Element Approximation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%% Basis Functions, TCP vector, and Boundary & Initial Conditions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection {Finite element approximation}
%
\centerline{Dirichlet boundary conditions for the TCP system}

$r$-approximation:\\
\noindent
\begin{figure}[ht]
\centering
\vspace{-0.4in}
\includegraphics[scale=0.65]{DR_r.png}
\caption{$r$-approximation of 3D Dirichlet boundary conditions}
\end{figure}

\noindent
Here $~N_r$ is the number of subintervals; $~M_{Dr}=N_r-1$ is the number of basis functions $~\bar{b}(r)=\begin{bmatrix} \\[-2.7em]
\bar{b}_1(r) & \ldots & \bar{b}_{M_Dr}(r)
\end{bmatrix}^T$ is the vector of basis functions; and\\ 
%
$$B(r)=\begin{bmatrix} \\[-2.3em]
\bar{b}(r)^T & 0 & 0 \\[-6pt]
0 & \bar{b}(r)^T & 0 \\[-6pt]
0 & 0 & \bar{b}(r)^T
\end{bmatrix}_{3\times3M_{Dr}}$$ 
%
\\$z$-approximation:\\
\noindent
\begin{figure}[ht]
\centering
\vspace{-0.4in}
\includegraphics[scale=0.65]{DR_z.png}
\caption{$z$-approximation of 3D Dirichlet boundary conditions}
\end{figure}

\noindent
Here $~N_z$ is the number of subintervals; $~M_{Dz}=N_z-1$ is the number of basis functions $~\hat{b}(z)=\begin{bmatrix} \\[-2.7em]
\hat{b}_1(z) & \ldots & \hat{b}_{M_Dz}(z)
\end{bmatrix}^T$ is the vector of basis functions; and\\ 
%
$$\hat{B}(z)=\begin{bmatrix} \\[-2.3em]
\hat{b}^T(z) & 0 & 0 \\[-6pt]
0 & \hat{b}(z)^T & 0 \\[-6pt]
0 & 0 & \hat{b}(z)^T
\end{bmatrix}_{3\times3M_{Dz}}=I_{3\times3}\otimes\hat{b}(z)^T$$\\ 
%
Using $r$-and $z$-approximations, the coefficients of the $\theta$-approximation for $V(r,\theta,z,t)$ can be represented as
\begin{multline}
V_i(r,z,t)=\hat{B}(z)\cdot\begin{bmatrix} \\[-2.3em]
\bar{b}(r)^T & \ldots & 0 \\[-6pt]
 & \ldots & \\[-6pt]
0 & \ldots & \hat{b}(r)^T
\end{bmatrix}_{3M_{Dz}\times(M_{Dr}\cdot3M_{Dz})}\cdot\bar{c}_i(t)_{(M_{Dr}\cdot3M_{Dz})\times 1}\\
i=0,\ldots,4 \label{eq:ffem3d88}
\end{multline}
%
We can rewrite (\ref{eq:ffem3d88}) as
\begin{equation}
V_i(r,z,t)=\Big(\hat{b}(z)^T\otimes B(r)\Big)\bar{c}_i(t) \label{eq:ffem3d89}
\end{equation}
where $\bar{c}_i(t)$ is a ($3M_{Dr}M_{Dz}\times 1$) vector and $\hat{b}(z)^T\otimes B(r)$ is a ($3\times3M_{Dr}M_{Dz}$) matrix.\\
%
The coefficients of the $\theta$-approximation for boundary conditions (\ref{eq:ffem3d2}) and initial condition (\ref{eq:ffem3d3}) can be represented as
\begin{align}
V_{B0}(z,t)&=\hat{B}(z)\bar{\nu}_{B0}(t) \nonumber\\
V_{Fi}(z)&=\hat{B}(z)\bar{\nu}_{Fi} \nonumber\\
\tilde{V}_{Ti}(r,t)&=B(r)\bar{\nu}_{\tilde{T}i}(t) \label{eq:ffem3d90}\\
\tilde{V}_{Bi}(r,t)&=B(r)\bar{\nu}_{\tilde{B}i}(t) \nonumber\\
\tilde{V}_{Ii}(r,z)&=\Big(\hat{b}(z)^T\otimes B(r)\Big)\bar{\nu}_{Ii}, \qquad i=0, \ldots, 4 \nonumber
\end{align}
where $~\bar{\nu}_{B0}(t)~$ and $~\bar{\nu}_{Fi}~$ are ($3M_{Dz}\times 1$) vectors; $~~\bar{\nu}_{\tilde{B}i}(t)~$ and $~\bar{\nu}_{\tilde{T}i}(t)~$ are ($3M_{Dr}\times 1$) vectors; ~and $~\bar{\nu}_{Ii}~$ is a ($3M_{Dr}M_{Dz}\times 1$) vector.
%
\newpage
\centerline{Neumann boundary conditions for the elastic system}

$r$-approximation:\\
\noindent
\begin{figure}[ht]
\centering
\vspace{-0.4in}
\includegraphics[scale=0.65]{Neu_r.png}
\caption{$r$-approximation of 3D Neumann boundary conditions}
\end{figure}

\noindent
$~\uN_r$ is the number of subintervals; $~M_r=\uN_r + 1$ {} is the number of basis functions $~\bar{\ub}(r)=\begin{bmatrix} \\[-2.7em]
\bar{\ub}_1(r) & \ldots & \bar{\ub}_{M_r}(r)
\end{bmatrix}^T$ is the vector of basis functions.\\[12pt]
\noindent
%
$z$-approximation:\\
\noindent
\begin{figure}[ht]
\centering
\vspace{-0.4in}
\includegraphics[scale=0.65]{Neu_z.png}
\caption{$z$-approximation of 3D Neumann boundary conditions}
\end{figure}

\noindent
$~\uN_z$ is the number of subintervals; $~M_z=\uN_z + 1$ {} is the number of basis functions $~\hat{\ub}(z)=\begin{bmatrix} \\[-2.7em]
\hat{\ub}_1(z) & \ldots & \hat{\ub}_{M_z}(z)
\end{bmatrix}^T$ is the vector of basis functions.\\[6pt]
%
Using $r$-and $z$-approximations, the coefficients of the $\theta$-approximation for the displacement coordinates can be represented as
\begin{align}
u_i(r,z,t)&=\Big(\hat{\ub}(z)^T\otimes\bar{\ub}(r)^T\Big)\bar{\mu}_i(t) \label{eq:ffem3d91} \\
v_i(r,z,t)&=\Big(\hat{\ub}(z)^T\otimes\bar{\ub}(r)^T\Big)\bar{\eta}_i(t) \label{eq:ffem3d92} \\
w_i(r,z,t)&=\Big(\hat{\ub}(z)^T\otimes\bar{\ub}(r)^T\Big)\bar{\lambda}_i(t) \label{eq:ffem3d93}
\end{align}
where $~\hat{\ub}(z)^T\otimes\bar{\ub}(r)^T~$ is a ($1\times M_rM_z$) vector ~and $~\bar{\mu}_i(t),{}\bar{\eta}_i(t),{}\bar{\lambda}_i(t)~$ are ($M_rM_z\times 1$) vectors. 
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM approximation of the elastic system %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\centerline{Finite element approximation of the elastic system}
\indent
Define the following matrices.\\
\centerline{$M_r\times M_r$ matrices:}
\begin{align*}
g_2(r)&=\int^{R_F}_{R_B}\bar{\ub}_r(r)\bar{\ub}_r(r)^Tr\ud r \\[6pt]
g_1^{(r)}&=\int^{R_F}_{R_B}\bar{\ub}(r)\bar{\ub}_r(r)^T\ud r \\[6pt]
g_1(r)&=\int^{R_F}_{R_B}\bar{\ub}(r)\bar{\ub}_r(r)^Tr\ud r \\[6pt]
g_0\big(\frac{1}{r}\big)&=\int^{R_F}_{R_B}\bar{\ub}(r)\bar{\ub}(r)^T\frac{1}{r}\ud r \\[6pt]
g_0(r)&=\int^{R_F}_{R_B}\bar{\ub}(r)\bar{\ub}(r)^Tr\ud r \\[6pt]
g_0^{(r)}&=\int^{R_F}_{R_B}\bar{\ub}(r)\bar{\ub}(r)^T\ud r
\end{align*}
%
\centerline{$M_z\times M_z$ matrices:}
\begin{align*}
g_2^{(z)}&=\int^{z_B}_{z_T}\hat{\ub}_z(z)\hat{\ub}_z(z)^T\ud z \\[6pt]
g_1^{(z)}&=\int^{z_B}_{z_T}\hat{\ub}(z)\hat{\ub}_z(z)^T\ud z \\[6pt]
g_0^{(z)}&=\int^{z_B}_{z_T}\hat{\ub}(z)\hat{\ub}(z)^T\ud z 
\end{align*}
%
\centerline{$M_rM_z\times M_rM_z$ matrix:}
\begin{equation*}
g_F=\int^{z_B}_{z_T}\Big(\hat{\ub}(z)\otimes\bar{\ub}(R_F)\Big)\Big(\hat{\ub}(z)\otimes\bar{\ub}(R_F)\Big)^T \ud z
\end{equation*}
%
\indent
Let $P_3=\begin{bmatrix} \\[-2.6em]
0 & 0 & 1 \\[-0.3em]
\end{bmatrix}^T$. Then
\begin{align}
p_B(z,t)&=P_3^TV_B(z,t)=P_3^TV_{B0}(z,t) \nonumber \\
&=P_3^T\hat{B}(z)\bar{\nu}_{B0}(t) \label{eq:ffem3d94}
\end{align}
Also, let
\begin{align}
\sigma_R(z)&=\frac{\sigma_x(z)+\sigma_y(z)}{2}=\hat{\ub}(z)^T\bar{\sigma}_R \nonumber \\
\sigma_D(z)&=\frac{\sigma_x(z)-\sigma_y(z)}{2}=\hat{\ub}(z)^T\bar{\sigma}_D \nonumber \\
\sigma_{xy}(z)&=\hat{\ub}(z)^T\bar{\sigma}_{xy} \label{eq:ffem3d95}
\end{align}
%
As before,
\begin{equation*}
f(r,\theta,z,t)=\vec{b}_1^T\bar{V}(r,\theta,z,t)=\vec{b}_1^T(V(r,\theta,z,t)+W(r,\theta,z,t))
\end{equation*} 
and from (\ref{eq:ffem3d18})-(\ref{eq:ffem3d20}), (\ref{eq:ffem3d66}), (\ref{eq:ffem3d89}), and (\ref{eq:ffem3d90}) it follows that
\begin{align}
f_i(r,z,t) & =\vec{b}_1^T\big(V_i(r,z,t)+W_i(r,z,t)\big) \nonumber \\[6pt]
& = \vec{b}_1^T\Big(\hat{b}(z)^T\otimes B(r)\Big)\bar{c}_i(t) \nonumber \\[6pt]
&+\frac{R_F-r}{R_F-R_B}\vec{b}_1^T\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\bar{\nu}_{Bi}(t) \nonumber \\[6pt]
&+\frac{r-R_B}{R_F-R_B}\vec{b}_1^T\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\bar{\nu}_{Fi} \nonumber \\[6pt]
&+\frac{z-z_T}{z_B-z_T}\vec{b}_1^TB(r)\bar{\nu}_{\tilde{B}i}(t)+\frac{z_B-z}{z_B-z_T}\vec{b}_1^TB(r)\bar{\nu}_{\tilde{T}i}(t), \quad i=0,\ldots,4 \label{eq:ffem3d96}
\end{align} 
and $\bar{\nu}_{Bi}(t)=0,\quad i=1,\ldots,4.$\\ 
%
From (\ref{eq:ffem3d73})-(\ref{eq:ffem3d87}) we obtain\\
%
$\varphi_0:$
\begin{align}
&\Big[\big(K+\frac{4G}{3}\big)g_0^{(z)}\otimes g_2(r)+\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big) \nonumber \\[6pt]
&{}+\big(K+\frac{4G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\mu}_0(t) \nonumber \\[6pt]
+&\Big[\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_1(r)^T+Gg_1^{(z)T}\otimes g_1(r)+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)}\Big]\bar{\lambda}_0(t)\nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_0\big[\hat{\ub}(z)\otimes (r\bar{\ub}_r(r)+\bar{\ub}(r))\big]\ud r \ud z- \nonumber \\[6pt]
-&R_B\Big[\int_{z_T}^{z_B}(\hat{\ub}(z)\otimes \bar{\ub}(R_B))P_3^T\hat{B}(z)\ud z\Big]\bar{\nu}_{B0}(t)+R_F\Big(g_0^{(z)}\otimes \bar{\ub}(R_F)\Big)\bar{\sigma}_R \label{eq:ffem3d97}
\end{align}
%
\\
$\varphi_1:$
\begin{align}
&\Big[\big(K+\frac{4G}{3}\big)g_0^{(z)}\otimes g_2(r)+\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big) \nonumber \\[6pt]
&{}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\mu}_1(t) \nonumber \\[6pt]
-&\Big[\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes g_1^{(r)T}-Gg_0^{(z)}\otimes g_1^{(r)}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)\Big]\bar{\eta}_2(t)\nonumber \\[6pt]
+&\Big[\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_1(r)^T+Gg_1^{(z)T}\otimes g_1(r)+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)}\Big]\bar{\lambda}_1(t)\nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_1\big[\hat{\ub}(z)\otimes (r\bar{\ub}_r(r)+\bar{\ub}(r))\big]\ud r \ud z \nonumber \\[6pt]
+&\bigg\{\Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \Big[\int_{R_B}^{R_F}\big(1-\frac{R_B^2}{r^2}\big)r\bar{\ub}(r)\ud r\Big]\bigg\}\bar{\sigma}_{yz}\label{eq:ffem3d98}
\end{align} 
%
\\
$\varphi_2:$
\begin{align}
&\Big[\big(K+\frac{4G}{3}\big)g_0^{(z)}\otimes g_2(r)+\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big) \nonumber \\[6pt]
&{}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\mu}_2(t) \nonumber \\[6pt]
+&\Big[\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes g_1^{(r)T}-Gg_0^{(z)}\otimes g_1^{(r)}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)\Big]\bar{\eta}_1(t)\nonumber \\[6pt]
+&\Big[\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_1(r)^T+Gg_1^{(z)T}\otimes g_1(r)+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)}\Big]\bar{\lambda}_2(t)\nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_2\big[\hat{\ub}(z)\otimes (r\bar{\ub}_r(r)+\bar{\ub}(r))\big]\ud r \ud z \nonumber \\[6pt]
+&\bigg\{\Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \Big[\int_{R_B}^{R_F}\big(1-\frac{R_B^2}{r^2}\big)r\bar{\ub}(r)\ud r\Big]\bigg\}\bar{\sigma}_{xz} \label{eq:ffem3d99}
\end{align} 
\\
$\varphi_3:$
\begin{align}
&\Big[\big(K+\frac{4G}{3}\big)g_0^{(z)}\otimes g_2(r)+\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big) \nonumber \\[6pt]
&{}+\big(K+\frac{16G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\mu}_3(t) \nonumber \\[6pt]
-2&\Big[\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes g_1^{(r)T}-Gg_0^{(z)}\otimes g_1^{(r)}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)\Big]\bar{\eta}_4(t)\nonumber \\[6pt]
+&\Big[\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_1(r)^T+Gg_1^{(z)T}\otimes g_1(r)+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)}\Big]\bar{\lambda}_3(t)\nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_3\big[\hat{\ub}(z)\otimes (r\bar{\ub}_r(r)+\bar{\ub}(r))\big]\ud r \ud z+R_F\Big(g_0^{(z)}\otimes \bar{\ub}(R_F)\Big)\bar{\sigma}_{xy} \label{eq:ffem3d100}
\end{align}
\\
$\varphi_4:$
\begin{align}
&\Big[\big(K+\frac{4G}{3}\big)g_0^{(z)}\otimes g_2(r)+\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big) \nonumber \\[6pt]
&{}+\big(K+\frac{16G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\mu}_4(t) \nonumber \\[6pt]
+2&\Big[\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes g_1^{(r)T}-Gg_0^{(z)}\otimes g_1^{(r)}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)\Big]\bar{\eta}_3(t)\nonumber \\[6pt]
+&\Big[\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_1(r)^T+Gg_1^{(z)T}\otimes g_1(r)+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)}\Big]\bar{\lambda}_4(t)\nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_4\big[\hat{\ub}(z)\otimes (r\bar{\ub}_r(r)+\bar{\ub}(r))\big]\ud r \ud z+R_F\Big(g_0^{(z)}\otimes \bar{\ub}(R_F)\Big)\bar{\sigma}_D \label{eq:ffem3d101}
\end{align}
\\
$\psi_0:$
\begin{align*}
&\Big[Gg_0^{(z)}\otimes g_2(r)-Gg_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big)+Gg_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+Gg_2^{(z)}\otimes g_0(r) \\[6pt]
&{}+K_FR_Fg_F\Big]\bar{\eta}_0(t)=0
\end{align*}
It follows that
$$\bar{\eta}_0(t)=0$$
%
$\psi_1:$
\begin{align}
&\Big[\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes g_1^{(r)}-Gg_0^{(z)}\otimes g_1^{(r)T}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)\Big]\bar{\mu}_2(t)\nonumber \\[6pt]
+&\Big[Gg_0^{(z)}\otimes g_2(r)-Gg_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big)+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big) \nonumber \\[6pt]
&{} +Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\eta}_1(t) \nonumber \\[6pt]
+&\Big[-Gg_1^{(z)T}\otimes g_0^{(r)}+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)}\Big]\bar{\lambda}_2(t) \nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_2\big[\hat{\ub}(z)\otimes \bar{\ub}(r)\big]\ud r \ud z \nonumber \\[6pt]
-&\bigg\{\Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \Big[\int_{R_B}^{R_F}\big(1+\frac{R_B^2}{r^2}\big)r\bar{\ub}(r)\ud r\Big]\bigg\}\bar{\sigma}_{xz} \label{eq:ffem3d102}
\end{align}
%
\\
$\psi_2:$
\begin{align}
-&\Big[\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes g_1^{(r)}-Gg_0^{(z)}\otimes g_1^{(r)T}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)\Big]\bar{\mu}_1(t)\nonumber \\[6pt]
+&\Big[Gg_0^{(z)}\otimes g_2(r)-Gg_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big)+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big) \nonumber \\[6pt]
&{} +Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\eta}_2(t) \nonumber \\[6pt]
-&\Big[-Gg_1^{(z)T}\otimes g_0^{(r)}+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)}\Big]\bar{\lambda}_1(t) \nonumber \\[6pt]
=-&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_1\big[\hat{\ub}(z)\otimes \bar{\ub}(r)\big]\ud r \ud z \nonumber \\[6pt]
+&\bigg\{\Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \Big[\int_{R_B}^{R_F}\big(1+\frac{R_B^2}{r^2}\big)r\bar{\ub}(r)\ud r\Big]\bigg\}\bar{\sigma}_{yz} \label{eq:ffem3d103}
\end{align}
\\
$\psi_3:$
\begin{align}
2&\Big[\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes g_1^{(r)}-Gg_0^{(z)}\otimes g_1^{(r)T}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)\Big]\bar{\mu}_4(t)\nonumber \\[6pt]
+&\Big[Gg_0^{(z)}\otimes g_2(r)-Gg_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big)+\big(4K+\frac{19G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big) +\nonumber \\[6pt]
&{}+Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\eta}_3(t) \nonumber \\[6pt]
+2&\Big[-Gg_1^{(z)T}\otimes g_0^{(r)}+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)}\Big]\bar{\lambda}_4(t) \nonumber \\[6pt]
=2&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_4\big[\hat{\ub}(z)\otimes \bar{\ub}(r)\big]\ud r \ud z+R_F\Big(g_0^{(z)}\otimes \bar{\ub}(R_F)\Big)\bar{\sigma}_D \label{eq:ffem3d104}
\end{align}
\\
$\psi_4:$
\begin{align}
-2&\Big[\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes g_1^{(r)}-Gg_0^{(z)}\otimes g_1^{(r)T}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)\Big]\bar{\mu}_3(t)\nonumber \\[6pt]
+&\Big[Gg_0^{(z)}\otimes g_2(r)-Gg_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big)+\big(4K+\frac{19G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big) \nonumber \\[6pt]
&{}+Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\eta}_4(t) \nonumber \\[6pt]
-2&\Big[-Gg_1^{(z)T}\otimes g_0^{(r)}+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)}\Big]\bar{\lambda}_3(t) \nonumber \\[6pt]
=-2&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_3\big[\hat{\ub}(z)\otimes \bar{\ub}(r)\big]\ud r \ud z+R_F\Big(g_0^{(z)}\otimes \bar{\ub}(R_F)\Big)\bar{\sigma}_{xy} \label{eq:ffem3d105}
\end{align}
\\
$\omega_0:$
\begin{align}
&\Big[\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_1(r)+Gg_1^{(z)}\otimes g_1(r)^T+\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_0^{(r)}\Big]\bar{\mu}_0(t)\nonumber \\[6pt]
+&\Big[Gg_0^{(z)}\otimes g_2(r)+\big(K+\frac{4G}{3}\big)g_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\lambda}_0(t) \nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_0\big[\hat{\ub}_z(z)\otimes r\bar{\ub}(r)\big]\ud r \ud z \nonumber \\[6pt]
+&\bigg\{\Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \Big[\int_{R_B}^{R_F}r\bar{\ub}(r)\ud r\Big]\bigg\}\bar{\sigma}_{zz} \label{eq:ffem3d106}
\end{align}
\\
%
$\omega_1:$
\begin{align}
&\Big[\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_1(r)+Gg_1^{(z)}\otimes g_1(r)^T+\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_0^{(r)}\Big]\bar{\mu}_1(t)-\nonumber \\[6pt]
-&\Big[-Gg_1^{(z)}\otimes g_0^{(r)}+\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_0^{(r)}\Big]\bar{\eta}_2(t) \nonumber \\[6pt]
+&\Big[Gg_0^{(z)}\otimes g_2(r)+Gg_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+\big(K+\frac{4G}{3}\big)g_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\lambda}_1(t) \nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_1\big[\hat{\ub}_z(z)\otimes r\bar{\ub}(r)\big]\ud r \ud z+R_F\Big(g_0^{(z)}\otimes \bar{\ub}(R_F)\Big)\bar{\sigma}_{yz} \label{eq:ffem3d107}
\end{align}
\\
$\omega_2:$
\begin{align}
&\Big[\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_1(r)+Gg_1^{(z)}\otimes g_1(r)^T+\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_0^{(r)}\Big]\bar{\mu}_2(t)\nonumber \\[6pt]
+&\Big[-Gg_1^{(z)}\otimes g_0^{(r)}+\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_0^{(r)}\Big]\bar{\eta}_1(t) \nonumber \\[6pt]
+&\Big[Gg_0^{(z)}\otimes g_2(r)+Gg_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+\big(K+\frac{4G}{3}\big)g_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\lambda}_2(t) \nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_2\big[\hat{\ub}_z(z)\otimes r\bar{\ub}(r)\big]\ud r \ud z+R_F\Big(g_0^{(z)}\otimes \bar{\ub}(R_F)\Big)\bar{\sigma}_{xz} \label{eq:ffem3d108}
\end{align}
\\
$\omega_3:$
\begin{align}
&\Big[\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_1(r)+Gg_1^{(z)}\otimes g_1(r)^T+\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_0^{(r)}\Big]\bar{\mu}_3(t)\nonumber \\[6pt]
-2&\Big[-Gg_1^{(z)}\otimes g_0^{(r)}+\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_0^{(r)}\Big]\bar{\eta}_4(t) \nonumber \\[6pt]
+&\Big[Gg_0^{(z)}\otimes g_2(r)+4Gg_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+\big(K+\frac{4G}{3}\big)g_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\lambda}_3(t) \nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_3\big[\hat{\ub}_z(z)\otimes r\bar{\ub}(r)\big]\ud r \ud z \nonumber \\[6pt]
-&4\nu R_B^2\bigg\{\Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \Big[\int_{R_B}^{R_F}\frac{1}{r}\bar{\ub}(r)\ud r\Big]\bigg\}\bar{\sigma}_{xy} \label{eq:ffem3d109}
\end{align}
\\
$\omega_4:$
\begin{align}
&\Big[\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_1(r)+Gg_1^{(z)}\otimes g_1(r)^T+\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_0^{(r)}\Big]\bar{\mu}_4(t)+\nonumber \\[6pt]
+2&\Big[-Gg_1^{(z)}\otimes g_0^{(r)}+\big(K-\frac{2G}{3}\big)g_1^{(z)T}\otimes g_0^{(r)}\Big]\bar{\eta}_3(t) \nonumber \\[6pt]
+&\Big[Gg_0^{(z)}\otimes g_2(r)+4Gg_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+\big(K+\frac{4G}{3}\big)g_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\Big]\bar{\lambda}_4(t) \nonumber \\[6pt]
=&\int_{z_T}^{z_B}\int_{R_B}^{R_F}f_4\big[\hat{\ub}_z(z)\otimes r\bar{\ub}(r)\big]\ud r \ud z \nonumber \\[6pt]
-&4\nu R_B^2\bigg\{\Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \Big[\int_{R_B}^{R_F}\frac{1}{r}\bar{\ub}(r)\ud r\Big]\bigg\}\bar{\sigma}_D \label{eq:ffem3d110}
\end{align}\\
%
\indent
Define the following matrices.\\
\centerline{$M_rM_z\times M_rM_z$ matrices:}
\begin{align}
\mathcal{M}_0 &= \big(K+\frac{4G}{3}\big)g_0^{(z)}\otimes g_2(r)+\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big)& \nonumber\\[6pt]
&\quad +\big(K+\frac{4G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F \nonumber\\[18pt]
%
\mathcal{M}_1 & = \big(K+\frac{4G}{3}\big)g_0^{(z)}\otimes g_2(r)+\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big) \nonumber\\[6pt]
&\quad +\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\nonumber\\[18pt]
%
\mathcal{M}_2 & = \big(K+\frac{4G}{3}\big)g_0^{(z)}\otimes g_2(r)+\big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big) \nonumber\\[6pt]
&\quad +\big(K+\frac{16G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F \nonumber\\[18pt]
%
\mathcal{M}_3 & = \big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_1(r)^T+Gg_1^{(z)T}\otimes g_1(r)+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)}\nonumber\\[18pt]
%
\mathcal{N}_1 & = \big(K-\frac{2G}{3}\big)g_0^{(z)}\otimes g_1^{(r)T}-Gg_0^{(z)}\otimes g_1^{(r)}+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)\nonumber\\[18pt]
%
\mathcal{N}_2 & = Gg_0^{(z)}\otimes g_2(r)-Gg_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big)+\big(K+\frac{7G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big) \nonumber\\[6pt]
&\quad +Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F\nonumber\\[18pt]
%
\mathcal{N}_3 & = Gg_0^{(z)}\otimes g_2(r)-Gg_0^{(z)}\otimes \big(g_1^{(r)T}+g_1^{(r)}\big)+\big(4K+\frac{19G}{3}\big)g_0^{(z)}\otimes g_0\big(\frac{1}{r}\big) \nonumber\\[6pt]
&\quad +Gg_2^{(z)}\otimes g_0(r)+K_FR_Fg_F \nonumber\\[18pt]
%
\mathcal{L}_0 & = Gg_0^{(z)}\otimes g_2(r)+\big(K+\frac{4G}{3}\big)g_2^{(z)}\otimes g_0(r)+K_FR_Fg_F \nonumber\\[18pt]
%
\mathcal{L}_1 & = -Gg_1^{(z)T}\otimes g_0^{(r)}+\big(K-\frac{2G}{3}\big)g_1^{(z)}\otimes g_0^{(r)} \nonumber\\[18pt]
%
\mathcal{L}_2 & = Gg_0^{(z)}\otimes g_2(r)+Gg_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+\big(K+\frac{4G}{3}\big)g_2^{(z)}\otimes g_0(r)+K_FR_Fg_F \nonumber\\[18pt]
%
\mathcal{L}_3 & = Gg_0^{(z)}\otimes g_2(r)+4Gg_0^{(z)}\otimes g_0\big(\frac{1}{r}\big)+\big(K+\frac{4G}{3}\big)g_2^{(z)}\otimes g_0(r)+K_FR_Fg_F \label{eq:ffem3d111}
\end{align} 
%
%
\centerline{$M_rM_z\times 3M_{Dr}M_{Dz}$ matrices:}
\begin{align}
\mathcal{Z}_0 &= \int_{z_T}^{z_B}\int_{R_B}^{R_F}\big[\hat{\ub}(z)\otimes \big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\big]\vec{b}_1^T\big(\hat{b}(z)^T\otimes B(r)\big) \ud r \ud z \nonumber \\[6pt]
&= \bigg[\int_{z_T}^{z_B}\hat{\ub}(z)\hat{b}(z)^T \ud z\bigg]\otimes \bigg[\int_{R_B}^{R_F}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\vec{b}_1^TB(r)\ud r\bigg] \label{eq:ffem3d112}\\
\nonumber \\
\mathcal{Z}_1 &= \int_{z_T}^{z_B}\int_{R_B}^{R_F}\big[\hat{\ub}(z)\otimes \bar{\ub}(r)\big]\vec{b}_1^T\big(\hat{b}(z)^T\otimes B(r)\big) \ud r \ud z \nonumber \\[6pt]
&= \bigg[\int_{z_T}^{z_B}\hat{\ub}(z)\hat{b}(z)^T \ud z\bigg]\otimes \bigg[\int_{R_B}^{R_F}\bar{\ub}(r)\vec{b}_1^TB(r)\ud r\bigg] \label{eq:ffem3d113}\\
\nonumber \\
\mathcal{Z}_2 &= \int_{z_T}^{z_B}\int_{R_B}^{R_F}\big[\hat{\ub}_z(z)\otimes r\bar{\ub}(r)\big]\vec{b}_1^T\big(\hat{b}(z)^T\otimes B(r)\big) \ud r \ud z \nonumber \\[6pt]
&= \bigg[\int_{z_T}^{z_B}\hat{\ub}_z(z)\hat{b}(z)^T \ud z\bigg]\otimes \bigg[\int_{R_B}^{R_F}r\bar{\ub}(r)\vec{b}_1^TB(r)\ud r\bigg] \label{eq:ffem3d114}
\end{align} 
%
%
\newpage
\noindent
\centerline{$M_rM_z\times 3M_{Dz}$ matrices:}
\begin{align}
Y_{B0}& = \int_{z_T}^{z_B}\int_{R_B}^{R_F}\big[\hat{\ub}(z)\otimes \big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\big]\frac{R_F-r}{R_F-R_B}\vec{b}_1^T \nonumber \\[6pt]
&\qquad \cdot\Big[\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big] \ud r \ud z \nonumber \\[6pt]
& = \bigg[\int_{z_T}^{z_B}\hat{\ub}(z)\vec{b}_1^T\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big) \ud z\bigg] \nonumber \\[6pt]
& \qquad \otimes \bigg[\int_{R_B}^{R_F}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\frac{R_F-r}{R_F-R_B} \ud r\bigg] \label{eq:ffem3d115}\\
\nonumber \\
\text{Similarly,} \nonumber\\
Y_{F0}& = \bigg[\int_{z_T}^{z_B}\hat{\ub}(z)\vec{b}_1^T\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big) \ud z\bigg] \nonumber \\[6pt]
& \qquad \otimes \bigg[\int_{R_B}^{R_F}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\frac{r-R_B}{R_F-R_B} \ud r\bigg] \label{eq:ffem3d116}\\
\nonumber \\
Y_{B1}& = \bigg[\int_{z_T}^{z_B}\hat{\ub}(z)\vec{b}_1^T\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big) \ud z\bigg] \nonumber \\[6pt]
& \qquad \otimes \bigg[\int_{R_B}^{R_F}\bar{\ub}(r)\frac{R_F-r}{R_F-R_B} \ud r\bigg] \label{eq:ffem3d117}\\
\nonumber \\
Y_{F1}& = \bigg[\int_{z_T}^{z_B}\hat{\ub}(z)\vec{b}_1^T\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big) \ud z\bigg] \nonumber \\[6pt]
& \qquad \otimes \bigg[\int_{R_B}^{R_F}\bar{\ub}(r)\frac{r-R_B}{R_F-R_B} \ud r\bigg] \label{eq:ffem3d118} \\
\nonumber \\
Y_{B2}& = \bigg[\int_{z_T}^{z_B}\hat{\ub}_z(z)\vec{b}_1^T\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big) \ud z\bigg] \nonumber \\[6pt]
& \qquad \otimes \bigg[\int_{R_B}^{R_F}r\bar{\ub}(r)\frac{R_F-r}{R_F-R_B} \ud r\bigg] \label{eq:ffem3d119}\\
\nonumber \\
Y_{F2}& = \bigg[\int_{z_T}^{z_B}\hat{\ub}_z(z)\vec{b}_1^T\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big) \ud z\bigg] \nonumber \\[6pt]
& \qquad \otimes \bigg[\int_{R_B}^{R_F}r\bar{\ub}(r)\frac{r-R_B}{R_F-R_B} \ud r\bigg] \label{eq:ffem3d120} \\
\nonumber \\
S_0& = R_B\int_{z_T}^{z_B}\big(\hat{\ub}(z) \otimes \bar{\ub}(R_B)P_3^T\hat{B}(z) \ud z \nonumber \\[6pt]
& = R_B\bigg[\int_{z_T}^{z_B}\hat{\ub}(z)P_3^T\hat{B}(z) \ud z\bigg] \otimes \bar{\ub}(R_B) \label{eq:ffem3d121}
\end{align} \\
%
\centerline{$M_rM_z\times 3M_{Dr}$ matrices:}
\begin{align}
\tilde{Y}_{B0}& = \int_{z_T}^{z_B}\int_{R_B}^{R_F}\big[\hat{\ub}(z)\otimes \big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\big]\frac{z-z_T}{z_B-z_T}\vec{b}_1^TB(r) \ud r \ud z \nonumber \\[6pt]
& = \bigg[\int_{z_T}^{z_B}\hat{\ub}(z)\frac{z-z_T}{z_B-z_T} \ud z\bigg] \otimes \bigg[\int_{R_B}^{R_F}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\vec{b}_1^TB(r) \ud r\bigg] \label{eq:ffem3d122}\\
%
\text{Similarly,} \nonumber\\
\tilde{Y}_{T0}& = \bigg[\int_{z_T}^{z_B}\hat{\ub}(z)\frac{z_B-z}{z_B-z_T} \ud z\bigg] \otimes \bigg[\int_{R_B}^{R_F}\big(r\bar{\ub}_r(r)+\bar{\ub}(r)\big)\vec{b}_1^TB(r) \ud r\bigg] \label{eq:ffem3d123}\\[9pt]
\tilde{Y}_{B1}& = \bigg[\int_{z_T}^{z_B}\hat{\ub}(z)\frac{z-z_T}{z_B-z_T} \ud z\bigg] \otimes \bigg[\int_{R_B}^{R_F}\bar{\ub}(r)\vec{b}_1^TB(r) \ud r\bigg] \label{eq:ffem3d124}\\[9pt]
\tilde{Y}_{T1}& = \bigg[\int_{z_T}^{z_B}\hat{\ub}(z)\frac{z_B-z}{z_B-z_T} \ud z\bigg] \otimes \bigg[\int_{R_B}^{R_F}\bar{\ub}(r)\vec{b}_1^TB(r) \ud r\bigg] \label{eq:ffem3d125} \\[9pt]
\tilde{Y}_{B2}& = \bigg[\int_{z_T}^{z_B}\hat{\ub}_z(z)\frac{z-z_T}{z_B-z_T} \ud z\bigg] \otimes \bigg[\int_{R_B}^{R_F}r\bar{\ub}_r(r)\vec{b}_1^TB(r) \ud r\bigg] \label{eq:ffem3d126}\\[9pt]
\tilde{Y}_{T2}& = \bigg[\int_{z_T}^{z_B}\hat{\ub}_z(z)\frac{z_B-z}{z_B-z_T} \ud z\bigg] \otimes \bigg[\int_{R_B}^{R_F}r\bar{\ub}_r(r)\vec{b}_1^TB(r) \ud r\bigg] \label{eq:ffem3d127}
\end{align}  \\
%
%
\centerline{$M_rM_z\times M_{Dz}$ matrices:}
\begin{align}
S_{RF}& = R_Fg_0^{(z)} \otimes \bar{\ub}(R_F) \label{eq:ffem3d128}\\[9pt]
S_1& = \Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \bigg[\int_{R_B}^{R_F}\big(1-\frac{R_B^2}{r^2}\big)r\bar{\ub}(r)\ud r\bigg] \label{eq:ffem3d129} \\[9pt]
S_2& = -\Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \bigg[\int_{R_B}^{R_F}\big(1+\frac{R_B^2}{r^2}\big)r\bar{\ub}(r)\ud r\bigg] \label{eq:ffem3d130} \\[9pt]
S_3& = \Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \bigg[\int_{R_B}^{R_F}r\bar{\ub}(r)\ud r\bigg] \label{eq:ffem3d131} \\[9pt]
S_4& = -4\nu R_B^2 \Big(\hat{\ub}(z_B)\hat{\ub}(z_B)^T-\hat{\ub}(z_T)\hat{\ub}(z_T)^T\Big)\otimes \bigg[\int_{R_B}^{R_F}r\bar{\ub}(r)\ud r\bigg] \label{eq:ffem3d132}
\end{align} \\
%
\indent
Applying (\ref{eq:ffem3d111})-(\ref{eq:ffem3d132}) to (\ref{eq:ffem3d97})-(\ref{eq:ffem3d110}), we obtain the following matrix equations:
%%% mu_0, lambda_0
\begin{gather}
\begin{bmatrix} \\[-2.5em]
\mathcal{M}_0 & \mathcal{M}_3 \\
\mathcal{M}_3^T & \mathcal{L}_0  
\end{bmatrix} \begin{bmatrix} \\[-2.5em]
\bar{\mu}_0(t) \\
\bar{\lambda}_0(t)  
\end{bmatrix} = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
\mathcal{Z}_2   
\end{bmatrix}\bar{c}_0(t)+\begin{bmatrix} \\[-2.5em]
S_{RF} \\
0   
\end{bmatrix}\bar{\sigma}_R+\begin{bmatrix} \\[-2.5em]
0 \\
S_3   
\end{bmatrix}\bar{\sigma}_{zz} \nonumber \\[12pt]
+\begin{bmatrix} \\[-2.5em]
Y_{B0}-S_0 \\
Y_{B2}   
\end{bmatrix}\bar{\nu}_{B0}(t)+\begin{bmatrix} \\[-2.5em]
Y_{F0} \\
Y_{F2}   
\end{bmatrix}\bar{\nu}_{F0}+\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{B0} \\
\tilde{Y}_{B2}   
\end{bmatrix}\bar{\nu}_{\tilde{B}0}(t)+\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{T0} \\
\tilde{Y}_{T2}   
\end{bmatrix}\bar{\nu}_{\tilde{T}0}(t) \label{eq:ffem3d133}
\end{gather}
%
%%%% mu_1, eta_2, lambda_1
\begin{gather}
\begin{bmatrix} \\[-2.5em]
\mathcal{M}_1 & -\mathcal{N}_1 & \mathcal{M}_3 \\
-\mathcal{N}_1^T & \mathcal{N}_2 & -\mathcal{L}_1 \\
\mathcal{M}_3^T & -\mathcal{L}_1^T & \mathcal{L}_2  
\end{bmatrix} \begin{bmatrix} \\[-2.5em]
\bar{\mu}_1(t) \\
\bar{\eta}_2(t) \\
\bar{\lambda}_1(t)  
\end{bmatrix} = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
-\mathcal{Z}_1 \\
\mathcal{Z}_2   
\end{bmatrix}\bar{c}_1(t)+\begin{bmatrix} \\[-2.5em]
S_1 \\
-S_2 \\
S_{RF}   
\end{bmatrix}\bar{\sigma}_{yz} \nonumber \\[12pt]
+\begin{bmatrix} \\[-2.5em]
Y_{F0} \\
-Y_{F1} \\
Y_{F2}   
\end{bmatrix}\bar{\nu}_{F1}+\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{B0} \\
-\tilde{Y}_{B1} \\
\tilde{Y}_{B2}   
\end{bmatrix}\bar{\nu}_{\tilde{B}1}(t)+\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{T0} \\
-\tilde{Y}_{T1} \\
\tilde{Y}_{T2}   
\end{bmatrix}\bar{\nu}_{\tilde{T}1}(t) \label{eq:ffem3d134}
\end{gather}
\\
%%%% mu_2, eta_1, lambda2
\begin{gather}
\begin{bmatrix} \\[-2.5em]
\mathcal{M}_1 & \mathcal{N}_1 & \mathcal{M}_3 \\
\mathcal{N}_1^T & \mathcal{N}_2 & \mathcal{L}_1 \\
\mathcal{M}_3^T & \mathcal{L}_1^T & \mathcal{L}_2  
\end{bmatrix} \begin{bmatrix} \\[-2.5em]
\bar{\mu}_2(t) \\
\bar{\eta}_1(t) \\
\bar{\lambda}_2(t)  
\end{bmatrix} = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
\mathcal{Z}_1 \\
\mathcal{Z}_2   
\end{bmatrix}\bar{c}_2(t)+\begin{bmatrix} \\[-2.5em]
S_1 \\
S_2 \\
S_{RF}   
\end{bmatrix}\bar{\sigma}_{xz} \nonumber \\[12pt]
+\begin{bmatrix} \\[-2.5em]
Y_{F0} \\
Y_{F1} \\
Y_{F2}   
\end{bmatrix}\bar{\nu}_{F2}+\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{B0} \\
\tilde{Y}_{B1} \\
\tilde{Y}_{B2}   
\end{bmatrix}\bar{\nu}_{\tilde{B}2}(t)+\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{T0} \\
\tilde{Y}_{T1} \\
\tilde{Y}_{T2}   
\end{bmatrix}\bar{\nu}_{\tilde{T}2}(t) \label{eq:ffem3d135}
\end{gather}
\\
%%%% mu_3, eta_4, lambda3
\begin{gather}
\begin{bmatrix} \\[-2.5em]
\mathcal{M}_2 & -2\mathcal{N}_1 & \mathcal{M}_3 \\
-2\mathcal{N}_1^T & \mathcal{N}_3 & -2\mathcal{L}_1 \\
\mathcal{M}_3^T & -2\mathcal{L}_1^T & \mathcal{L}_3  
\end{bmatrix} \begin{bmatrix} \\[-2.5em]
\bar{\mu}_3(t) \\
\bar{\eta}_4(t) \\
\bar{\lambda}_3(t)  
\end{bmatrix} = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
-2\mathcal{Z}_1 \\
\mathcal{Z}_2   
\end{bmatrix}\bar{c}_3(t)+\begin{bmatrix} \\[-2.5em]
S_{RF} \\
S_{RF} \\
S_4   
\end{bmatrix}\bar{\sigma}_{xy} \nonumber \\[12pt]
+\begin{bmatrix} \\[-2.5em]
Y_{F0} \\
-2Y_{F1} \\
Y_{F2}   
\end{bmatrix}\bar{\nu}_{F3}+\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{B0} \\
-2\tilde{Y}_{B1} \\
\tilde{Y}_{B2}   
\end{bmatrix}\bar{\nu}_{\tilde{B}3}(t)+\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{T0} \\
-2\tilde{Y}_{T1} \\
\tilde{Y}_{T2}   
\end{bmatrix}\bar{\nu}_{\tilde{T}3}(t) \label{eq:ffem3d136}
\end{gather}
%
%%%% mu_4, eta_3, lambda_4
\begin{gather}
\begin{bmatrix} \\[-2.5em]
\mathcal{M}_2 & 2\mathcal{N}_1 & \mathcal{M}_3 \\
2\mathcal{N}_1^T & \mathcal{N}_3 & 2\mathcal{L}_1 \\
\mathcal{M}_3^T & 2\mathcal{L}_1^T & \mathcal{L}_3  
\end{bmatrix} \begin{bmatrix} \\[-2.5em]
\bar{\mu}_4(t) \\
\bar{\eta}_3(t) \\
\bar{\lambda}_4(t)  
\end{bmatrix} = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
2\mathcal{Z}_1 \\
\mathcal{Z}_2   
\end{bmatrix}\bar{c}_4(t)+\begin{bmatrix} \\[-2.5em]
S_{RF} \\
S_{RF} \\
S_4   
\end{bmatrix}\bar{\sigma}_D \nonumber \\[12pt]
+\begin{bmatrix} \\[-2.5em]
Y_{F0} \\
2Y_{F1} \\
Y_{F2}   
\end{bmatrix}\bar{\nu}_{F4}+\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{B0} \\
2\tilde{Y}_{B1} \\
\tilde{Y}_{B2}   
\end{bmatrix}\bar{\nu}_{\tilde{B}4}(t)+\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{T0} \\
2\tilde{Y}_{T1} \\
\tilde{Y}_{T2}   
\end{bmatrix}\bar{\nu}_{\tilde{T}4}(t) \label{eq:ffem3d137}
\end{gather}
%
\newpage
Define the following matrices:
%
\begin{gather*}
\mathcal{Q}_0 = \begin{bmatrix} \\[-2.5em]
\mathcal{M}_0 & \mathcal{M}_3 \\
\mathcal{M}_3^T & \mathcal{L}_0  
\end{bmatrix}
\\[9pt]
%
J = \begin{bmatrix} \\[-2.5em]
I_{M_rM_z} & 0 & 0 \\
0 & -I_{M_rM_z} & 0 \\
0 & 0 & I_{M_rM_z}  
\end{bmatrix}
\\[9pt]
%
\mathcal{Q}_1 = \begin{bmatrix} \\[-2.5em]
\mathcal{M}_1 & \mathcal{N}_1 & \mathcal{M}_3 \\
\mathcal{N}_1^T & \mathcal{N}_2 & \mathcal{L}_1 \\
\mathcal{M}_3^T & \mathcal{L}_1^T & \mathcal{L}_2  
\end{bmatrix} \quad \Rightarrow \quad 
\begin{bmatrix} \\[-2.5em]
\mathcal{M}_1 & -\mathcal{N}_1 & \mathcal{M}_3 \\
-\mathcal{N}_1^T & \mathcal{N}_2 & -\mathcal{L}_1 \\
\mathcal{M}_3^T & -\mathcal{L}_1^T & \mathcal{L}_2  
\end{bmatrix} = J\mathcal{Q}_1 J
\\[9pt]
%
\mathcal{Q}_2 = \begin{bmatrix} \\[-2.5em]
\mathcal{M}_2 & 2\mathcal{N}_1 & \mathcal{M}_3 \\
2\mathcal{N}_1^T & \mathcal{N}_3 & 2\mathcal{L}_1 \\
\mathcal{M}_3^T & 2\mathcal{L}_1^T & \mathcal{L}_3  
\end{bmatrix} \quad \Rightarrow \quad 
\begin{bmatrix} \\[-2.5em]
\mathcal{M}_2 & -2\mathcal{N}_1 & \mathcal{M}_3 \\
-2\mathcal{N}_1^T & \mathcal{N}_3 & -2\mathcal{L}_1 \\
\mathcal{M}_3^T & -2\mathcal{L}_1^T & \mathcal{L}_3  
\end{bmatrix} = J\mathcal{Q}_2 J
\\[9pt]
%
\bar{\mathcal{Z}}_0 = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
\mathcal{Z}_2   
\end{bmatrix}, \qquad \bar{\mathcal{Z}}_1 = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
\mathcal{Z}_1 \\
\mathcal{Z}_2   
\end{bmatrix}, \qquad \bar{\mathcal{Z}}_2 = \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
2\mathcal{Z}_1 \\
\mathcal{Z}_2   
\end{bmatrix} \qquad \Rightarrow
\\[9pt]
%
\begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
-\mathcal{Z}_1 \\
\mathcal{Z}_2   
\end{bmatrix} = J\bar{\mathcal{Z}}_1, \qquad \begin{bmatrix} \\[-2.5em]
\mathcal{Z}_0 \\
-2\mathcal{Z}_1 \\
\mathcal{Z}_2   
\end{bmatrix} = J\bar{\mathcal{Z}}_2
\\[9pt]
%
X_{B0} = \begin{bmatrix} \\[-2.5em]
Y_{B0}-S_0 \\
Y_{B2}   
\end{bmatrix}, \quad 
X_{F0} = \begin{bmatrix} \\[-2.5em]
Y_{F0} \\
Y_{F2}   
\end{bmatrix}, \quad
X_{F1} = \begin{bmatrix} \\[-2.5em]
Y_{F0} \\
Y_{F1} \\
Y_{F2}   
\end{bmatrix}, \quad
X_{F2} = \begin{bmatrix} \\[-2.5em]
Y_{F0} \\
2Y_{F1} \\
Y_{F2}   
\end{bmatrix} \quad \Rightarrow
\\[9pt]
%
\begin{bmatrix} \\[-2.5em]
Y_{F0} \\
-Y_{F1} \\
Y_{F2}   
\end{bmatrix} = JX_{F1}, \qquad
\begin{bmatrix} \\[-2.5em]
Y_{F0} \\
-2Y_{F1} \\
Y_{F2}   
\end{bmatrix} = JX_{F2}
\\[9pt]
%
\tilde{X}_{B0} = \begin{bmatrix} \\[-2.5em]
\tilde{Y}_{B0} \\
\tilde{Y}_{B2}   
\end{bmatrix}, \qquad
\tilde{X}_{B1} = \begin{bmatrix} \\[-2.5em]
\tilde{Y}_{B0} \\
\tilde{Y}_{B1} \\
\tilde{Y}_{B2}   
\end{bmatrix}, \qquad
\tilde{X}_{B2} = \begin{bmatrix} \\[-2.5em]
\tilde{Y}_{B0} \\
2\tilde{Y}_{B1} \\
\tilde{Y}_{B2} \end{bmatrix} \qquad \Rightarrow
\\[9pt]
%
\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{B0} \\
-\tilde{Y}_{B1} \\
\tilde{Y}_{B2}   
\end{bmatrix} = J\tilde{X}_{B1}, \qquad
\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{B0} \\
-2\tilde{Y}_{B1} \\
\tilde{Y}_{B2} \end{bmatrix} = J\tilde{X}_{B2} 
\\[9pt]
%
\tilde{X}_{T0} = \begin{bmatrix} \\[-2.5em]
\tilde{Y}_{T0} \\
\tilde{Y}_{T2}   
\end{bmatrix}, \qquad
\tilde{X}_{T1} = \begin{bmatrix} \\[-2.5em]
\tilde{Y}_{T0} \\
\tilde{Y}_{T1} \\
\tilde{Y}_{T2}   
\end{bmatrix}, \qquad
\tilde{X}_{T2} = \begin{bmatrix} \\[-2.5em]
\tilde{Y}_{T0} \\
2\tilde{Y}_{T1} \\
\tilde{Y}_{T2} \end{bmatrix} \qquad \Rightarrow
\\[9pt]
%
\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{T0} \\
-\tilde{Y}_{T1} \\
\tilde{Y}_{T2}   
\end{bmatrix} = J\tilde{X}_{T1}, \qquad
\begin{bmatrix} \\[-2.5em]
\tilde{Y}_{T0} \\
-2\tilde{Y}_{T1} \\
\tilde{Y}_{T2} \end{bmatrix} = J\tilde{X}_{T2} 
\\[9pt]
%
\bar{S}_0 = \begin{bmatrix} \\[-2.5em]
S_{RF} \\
0   
\end{bmatrix}, \quad
\bar{S}_1 = \begin{bmatrix} \\[-2.5em]
S_1 \\
S_2 \\
S_{RF}   
\end{bmatrix} \Rightarrow
\begin{bmatrix} \\[-2.5em]
S_1 \\
-S_2 \\
S_{RF}   
\end{bmatrix} = J\bar{S}_1, \quad
\bar{S}_2 = \begin{bmatrix} \\[-2.5em]
S_{RF} \\
S_{RF} \\
S_4   
\end{bmatrix}, \quad
\bar{S}_3 = \begin{bmatrix} \\[-2.5em]
0 \\
S_3  
\end{bmatrix}
\end{gather*}
%
\begin{align*}
P_{100} & = \begin{bmatrix} \\[-2.6em]
1 & 0 & 0 \\[-0.3em]
\end{bmatrix} \otimes I_{M_rM_z\times M_rM_z} \\
P_{010} & = \begin{bmatrix} \\[-2.6em]
0 & 1 & 0 \\[-0.3em]
\end{bmatrix} \otimes I_{M_rM_z\times M_rM_z} \\
P_{001} & = \begin{bmatrix} \\[-2.6em]
0 & 0 & 1 \\[-0.3em]
\end{bmatrix} \otimes I_{M_rM_z\times M_rM_z} \\
P_{10} & = \begin{bmatrix} \\[-2.6em]
1 & 0 \\[-0.3em]
\end{bmatrix} \otimes I_{M_rM_z\times M_rM_z} \\
P_{01} & = \begin{bmatrix} \\[-2.6em]
0 & 1 \\[-0.3em]
\end{bmatrix} \otimes I_{M_rM_z\times M_rM_z}
\end{align*}
%
From (\ref{eq:ffem3d133})-(\ref{eq:ffem3d137}) we have
%
\begin{align}
\mathcal{Q}_0 \begin{bmatrix} \\[-2.5em]
\bar{\mu}_0(t) \\
\bar{\lambda}_0(t)  
\end{bmatrix} &= \bar{\mathcal{Z}}_0\bar{c}_0(t)+\bar{S}_0\bar{\sigma}_R+\bar{S}_3\bar{\sigma}_{zz} \nonumber \\[-12pt]
&\quad+X_{B0}\bar{\nu}_{B0}(t)+X_{F0}\bar{\nu}_{F0}+\tilde{X}_{B0}\bar{\nu}_{\tilde{B}0}(t)+\tilde{X}_{T0}\bar{\nu}_{\tilde{T}0}(t) \label{eq:ffem3d138}\\[6pt]
%
J\mathcal{Q}_1 J \begin{bmatrix} \\[-2.5em]
\bar{\mu}_1(t) \\
\bar{\eta}_2(t) \\
\bar{\lambda}_1(t)  
\end{bmatrix} &= J\bar{\mathcal{Z}}_1\bar{c}_1(t)+J\bar{S}_1\bar{\sigma}_{yz} \nonumber \\[-24pt]
&\quad+JX_{F1}\bar{\nu}_{F1}+J\tilde{X}_{B1}\bar{\nu}_{\tilde{B}1}(t)+J\tilde{X}_{T1}\bar{\nu}_{\tilde{T}1}(t) \label{eq:ffem3d139}\\[6pt]
%
\mathcal{Q}_1 \begin{bmatrix} \\[-2.5em]
\bar{\mu}_2(t) \\
\bar{\eta}_1(t) \\
\bar{\lambda}_2(t)  
\end{bmatrix} &= \bar{\mathcal{Z}}_1\bar{c}_2(t)+\bar{S}_1\bar{\sigma}_{xz} \nonumber \\[-24pt]
&\quad+X_{F1}\bar{\nu}_{F2}+\tilde{X}_{B1}\bar{\nu}_{\tilde{B}2}(t)+\tilde{X}_{T1}\bar{\nu}_{\tilde{T}2}(t) \label{eq:ffem3d140}\\[6pt]
%
J\mathcal{Q}_2 J \begin{bmatrix} \\[-2.5em]
\bar{\mu}_3(t) \\
\bar{\eta}_4(t) \\
\bar{\lambda}_3(t)  
\end{bmatrix} &= J\bar{\mathcal{Z}}_2\bar{c}_3(t)+JJ\bar{S}_2\bar{\sigma}_{xy} \nonumber \\[-24pt]
&\quad+JX_{F2}\bar{\nu}_{F3}+J\tilde{X}_{B2}\bar{\nu}_{\tilde{B}3}(t)+J\tilde{X}_{T2}\bar{\nu}_{\tilde{T}3}(t) \label{eq:ffem3d141}\\[6pt]
%
\mathcal{Q}_2 \begin{bmatrix} \\[-2.5em]
\bar{\mu}_4(t) \\
\bar{\eta}_3(t) \\
\bar{\lambda}_4(t)  
\end{bmatrix} &= \bar{\mathcal{Z}}_2\bar{c}_4(t)+\bar{S}_2\bar{\sigma}_D \nonumber \\[-24pt]
&\quad+X_{F2}\bar{\nu}_{F4}+\tilde{X}_{B2}\bar{\nu}_{\tilde{B}4}(t)+\tilde{X}_{T2}\bar{\nu}_{\tilde{T}4}(t) \label{eq:ffem3d142}
\end{align}\\
%
From (\ref{eq:ffem3d138})-(\ref{eq:ffem3d142}) we obtain
\begin{align}
\bar{\mu}_0(t)& = P_{10}\mathcal{Q}_0^{-1}\Big[\bar{\mathcal{Z}}_0\bar{c}_0(t)+\bar{S}_0\bar{\sigma}_R+\bar{S}_3\bar{\sigma}_{zz} \nonumber \\
&\quad \qquad \qquad +X_{B0}\bar{\nu}_{B0}(t)+X_{F0}\bar{\nu}_{F0}+\tilde{X}_{B0}\bar{\nu}_{\tilde{B}0}(t)+\tilde{X}_{T0}\bar{\nu}_{\tilde{T}0}(t)\Big] \label{eq:ffem3d143} \\[6pt]
%
\bar{\lambda}_0(t)& = P_{01}\mathcal{Q}_0^{-1}\Big[\bar{\mathcal{Z}}_0\bar{c}_0(t)+\bar{S}_0\bar{\sigma}_R+\bar{S}_3\bar{\sigma}_{zz} \nonumber \\
&\quad \qquad \qquad +X_{B0}\bar{\nu}_{B0}(t)+X_{F0}\bar{\nu}_{F0}+\tilde{X}_{B0}\bar{\nu}_{\tilde{B}0}(t)+\tilde{X}_{T0}\bar{\nu}_{\tilde{T}0}(t)\Big] \label{eq:ffem3d144} \\[6pt]
%
\bar{\mu}_1(t)& = P_{100}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\bar{c}_1(t)+\bar{S}_1\bar{\sigma}_{yz} \nonumber \\
&\quad \qquad \qquad +X_{F1}\bar{\nu}_{F1}+\tilde{X}_{B1}\bar{\nu}_{\tilde{B}1}(t)+\tilde{X}_{T1}\bar{\nu}_{\tilde{T}1}(t) \Big] \label{eq:ffem3d145}\\[6pt]
%
\bar{\eta}_2(t)& = -P_{010}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\bar{c}_1(t)+\bar{S}_1\bar{\sigma}_{yz} \nonumber \\
&\quad \qquad \qquad \quad +X_{F1}\bar{\nu}_{F1}+\tilde{X}_{B1}\bar{\nu}_{\tilde{B}1}(t)+\tilde{X}_{T1}\bar{\nu}_{\tilde{T}1}(t) \Big] \label{eq:ffem3d146}\\[6pt]
%
\bar{\lambda}_1(t)& = P_{001}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\bar{c}_1(t)+\bar{S}_1\bar{\sigma}_{yz} \nonumber \\
&\quad \qquad \qquad {}+X_{F1}\bar{\nu}_{F1}+\tilde{X}_{B1}\bar{\nu}_{\tilde{B}1}(t)+\tilde{X}_{T1}\bar{\nu}_{\tilde{T}1}(t) \Big] \label{eq:ffem3d147}\\[6pt]
%
\bar{\mu}_2(t)& = P_{100}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\bar{c}_2(t)+\bar{S}_1\bar{\sigma}_{xz} \nonumber \\
&\quad \qquad \qquad +X_{F1}\bar{\nu}_{F2}+\tilde{X}_{B1}\bar{\nu}_{\tilde{B}2}(t)+\tilde{X}_{T1}\bar{\nu}_{\tilde{T}2}(t)\Big] \label{eq:ffem3d148}\\[6pt]
%
\bar{\eta}_1(t)& = P_{010}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\bar{c}_2(t)+\bar{S}_1\bar{\sigma}_{xz} \nonumber \\
&\quad \qquad \qquad +X_{F1}\bar{\nu}_{F2}+\tilde{X}_{B1}\bar{\nu}_{\tilde{B}2}(t)+\tilde{X}_{T1}\bar{\nu}_{\tilde{T}2}(t)\Big] \label{eq:ffem3d149}\\[6pt]
%
\bar{\lambda}_2(t)& = P_{001}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\bar{c}_2(t)+\bar{S}_1\bar{\sigma}_{xz} \nonumber \\
&\quad \qquad \qquad +X_{F1}\bar{\nu}_{F2}+\tilde{X}_{B1}\bar{\nu}_{\tilde{B}2}(t)+\tilde{X}_{T1}\bar{\nu}_{\tilde{T}2}(t)\Big] \label{eq:ffem3d150}\\[6pt]
%
\bar{\mu}_3(t)& = P_{100}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\bar{c}_3(t)+J\bar{S}_2\bar{\sigma}_{xy} \nonumber \\
&\quad \qquad \qquad +X_{F2}\bar{\nu}_{F3}+\tilde{X}_{B2}\bar{\nu}_{\tilde{B}3}(t)+\tilde{X}_{T2}\bar{\nu}_{\tilde{T}3}(t)\Big] \label{eq:ffem3d151}\\[6pt]
%
\bar{\eta}_4(t)& = -P_{010}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\bar{c}_3(t)+J\bar{S}_2\bar{\sigma}_{xy} \nonumber \\
&\quad \qquad \qquad +X_{F2}\bar{\nu}_{F3}+\tilde{X}_{B2}\bar{\nu}_{\tilde{B}3}(t)+\tilde{X}_{T2}\bar{\nu}_{\tilde{T}3}(t)\Big] \label{eq:ffem3d152}\\[6pt]
%
\bar{\lambda}_3(t)& = P_{001}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\bar{c}_3(t)+J\bar{S}_2\bar{\sigma}_{xy} \nonumber \\
&\quad \qquad \qquad +X_{F2}\bar{\nu}_{F3}+\tilde{X}_{B2}\bar{\nu}_{\tilde{B}3}(t)+\tilde{X}_{T2}\bar{\nu}_{\tilde{T}3}(t)\Big] \label{eq:ffem3d153}\\[6pt]
%
\bar{\mu}_4(t)& = P_{100}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\bar{c}_4(t)+\bar{S}_2\bar{\sigma}_D \nonumber \\
&\quad \qquad \qquad +X_{F2}\bar{\nu}_{F4}+\tilde{X}_{B2}\bar{\nu}_{\tilde{B}4}(t)+\tilde{X}_{T2}\bar{\nu}_{\tilde{T}4}(t)\Big] \label{eq:ffem3d154}\\[6pt]
%
\bar{\eta}_3(t)& = P_{010}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\bar{c}_4(t)+\bar{S}_2\bar{\sigma}_D \nonumber \\
&\quad \qquad \qquad +X_{F2}\bar{\nu}_{F4}+\tilde{X}_{B2}\bar{\nu}_{\tilde{B}4}(t)+\tilde{X}_{T2}\bar{\nu}_{\tilde{T}4}(t)\Big] \label{eq:ffem3d155}\\[6pt]
%
\bar{\lambda}_4(t)& = P_{001}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\bar{c}_4(t)+\bar{S}_2\bar{\sigma}_D \nonumber \\
&\quad \qquad \qquad +X_{F2}\bar{\nu}_{F4}+\tilde{X}_{B2}\bar{\nu}_{\tilde{B}4}(t)+\tilde{X}_{T2}\bar{\nu}_{\tilde{T}4}(t)\Big] \label{eq:ffem3d156}
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM Approximation of TCP System %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM Approximation of TCP System I (radially symmetric case) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newpage
\noindent
\centerline{Finite element approximation of the TCP system}
%
\centerline{System I (radially symmetric case)}
%
System I is given by (\ref{eq:ffem3d28})-(\ref{eq:ffem3d30}):
\begin{gather*} 
M\dot{V}_0-A\Big[\frac{1}{r}(rV_{0_r})_r+(V_{0_z})_z\Big] = -M\dot{W}_0+A\Big(\frac{1}{r}(rW_{0_r})_r+(W_{0_z})_z\Big) \\[6pt]
-\vec{b}_0\Big( \dot{u}_{0_r}+\frac{1}{r}\dot{u}_0+\dot{w}_{0_z}\Big)\tag{\ref{eq:ffem3d28}}
\end{gather*}
with boundary conditions
\begin{equation*}
V_0(r,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \tag{\ref{eq:ffem3d29}} 
\end{equation*}
and initial conditions
\begin{equation*}
V_0(r,z,0)=\tilde{V}_{I0}(r,z)-W_0(r,z,0) \tag{\ref{eq:ffem3d30}}
\end{equation*}
%
\indent
Let
\begin{equation*}
\mathcal{A}=M^{-1}A \\[-6pt]
\end{equation*}
Then (\ref{eq:ffem3d28}) has the form
\begin{gather} 
\dot{V}_0-\mathcal{A}\Big[\frac{1}{r}(rV_{0_r})_r+(V_{0_z})_z\Big] = -\dot{W}_0+\mathcal{A}\Big[\frac{1}{r}(rW_{0_r})_r+(W_{0_z})_z\Big] \nonumber \\[6pt]
-M^{-1}\vec{b}_0\Big( \dot{u}_{0_r}+\frac{1}{r}\dot{u}_0+\dot{w}_{0_z}\Big)  \label{eq:ffem3d157}
\end{gather}
%
\indent
Using (\ref{eq:ffem3d89}), the left-hand side of (\ref{eq:ffem3d157}) is
\begin{equation}
\Big(\hat{b}(z)^T\otimes B(r)\Big)\dot{\bar{c}}_0(t)-\mathcal{A}\frac{1}{r}\Big(\hat{b}(z)^T\otimes rB_r(r)\Big)_r\bar{c}_0(t)-\mathcal{A}\Big(\hat{b}_z(z)^T\otimes B(r)\Big)_z\bar{c}_0(t) \label{eq:ffem3d158}
\vspace{0.2in}
\end{equation}
%
%
Multiplying (\ref{eq:ffem3d158}) by $\Big(\hat{b}(z)^T\otimes B(r)\Big)^T = \Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have\\
\vspace{-0.2in}
\begin{align}
&\Big[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\Big(\hat{b}(z)^T\otimes B(r)\Big)r\ud r\ud z\Big]\dot{\bar{c}}_0(t) \nonumber \\[6pt]
-& \Big[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\mathcal{A}\frac{1}{r}\Big(\hat{b}(z)^T\otimes rB_r(r)\Big)_rr\ud r\ud z\Big]\bar{c}_0(t) \nonumber \\[6pt]
-& \Big[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\mathcal{A}\Big(\hat{b}_z(z)^T\otimes B(r)\Big)_zr\ud r\ud z\Big]\bar{c}_0(t) \nonumber \\ 
\nonumber\\
=&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\} \dot{\bar{c}}_0(t) \nonumber \\[6pt]
+& \Big[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B_r(r)^T\Big)\mathcal{A}\Big(\hat{b}(z)^T\otimes rB_r(r)^T\Big) \ud r \ud z\Big]\bar{c}_0(t) \nonumber \\[6pt]
+& \Big[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}_z(z)\otimes B(r)^T\Big)\mathcal{A}\Big(\hat{b}_z(z)^T\otimes B_r(r)^T\Big)r \ud r \ud z\Big]\bar{c}_0(t) \nonumber \\
\nonumber\\
=& \bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\dot{\bar{c}}_0(t) \nonumber \\[6pt]
+& \bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B_r(r)^T\mathcal{A}B_r(r)r\ud r\Big]\bigg\}\bar{c}_0(t) \nonumber \\[6pt]
+& \bigg\{\Big[\int_{z_T}^{z_B}\hat{b}_z(z)\hat{b}_z(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^T\mathcal{A}B(r)r\ud r\Big]\bigg\}\bar{c}_0(t) \nonumber \\
\nonumber \\
=& \bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\dot{\bar{c}}_0(t) \nonumber \\[6pt]
+& \bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big]\bigg\}\bar{c}_0(t) \nonumber \\[6pt]
+& \bigg\{\Big[\int_{z_T}^{z_B}\hat{b}_z(z)\hat{b}_z(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^Tr \ud r\Big]\bigg\}\bar{c}_0(t) \label{eq:ffem3d159}
\end{align}\\[-18pt]
%
Using (\ref{eq:ffem3d91})-(\ref{eq:ffem3d93}), the last term of the right-hand side of (\ref{eq:ffem3d157}) is
\begin{align*}
&-M^{-1}\vec{b}_0\Big( \dot{u}_{0_r}+\frac{1}{r}\dot{u}_0+\dot{w}_{0_z}\Big) \\[6pt]
=&-M^{-1}\vec{b}_0\Big[\Big(\hat{\ub}(z)^T\otimes \bar{\ub}_r(r)^T\Big)\dot{\bar{\mu}}_0(t)+\Big(\hat{\ub}(z)^T\otimes \frac{1}{r}\bar{\ub}(r)^T\Big)\dot{\bar{\mu}}_0(t) \\[6pt]
&\qquad \qquad +\Big(\hat{\ub}_z(z)^T\otimes \bar{\ub}(r)^T\Big)\dot{\bar{\lambda}}_0(t)\Big]
\end{align*}
%
Multiplying by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have
%
\begin{align}
-&\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)M^{-1}\vec{b}_0\Big(\hat{\ub}(z)^T\otimes \bar{\ub}_r(r)^T\Big)r \ud r \ud z \nonumber \\[6pt]
&{}+\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)M^{-1}\vec{b}_0\Big(\hat{\ub}(z)^T\otimes \frac{1}{r}\bar{\ub}(r)^T\Big) r \ud r \ud z\bigg]\dot{\bar{\mu}}_0(t) \nonumber \\[6pt]
-&\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)M^{-1}\vec{b}_0\Big(\hat{\ub}_z(z)^T\otimes \bar{\ub}(r)^T\Big)r \ud r \ud z\bigg]\dot{\bar{\lambda}}_0(t) \nonumber \\
\nonumber \\
=-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0(r\bar{\ub}_r(r)+\bar{\ub}(r))^T \ud r\Big]\bigg\}\dot{\bar{\mu}}_0(t) \nonumber \\[6pt]
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}_z(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^Tr \ud r\Big]\bigg\}\dot{\bar{\lambda}}_0(t) \label{eq:ffem3d160}
\end{align}\\ 
%
\indent
From (\ref{eq:ffem3d20}) and (\ref{eq:ffem3d90})
\begin{align}
W_i(r,z,t) &= \frac{R_F-r}{R_F-R_B}\hat{B}(z)\bar{\nu}_{Bi}(t)+\frac{r-R_B}{R_F-R_B}\hat{B}(z)\bar{\nu}_{Fi} \nonumber \\[6pt]
& +\frac{z-z_T}{z_B-z_T}\Big(B(r)\bar{\nu}_{\tilde{B}i}(t)-\frac{R_F-r}{R_F-R_B}\hat{B}(z_B)\bar{\nu}_{Bi}(t)-\frac{r-R_B}{R_F-R_B}\hat{B}(z_B)\bar{\nu}_{Fi}\Big) \nonumber \\[6pt]
& +\frac{z_B-z}{z_B-z_T}\Big(B(r)\bar{\nu}_{\tilde{T}i}(t)-\frac{R_F-r}{R_F-R_B}\hat{B}(z_T)\bar{\nu}_{Bi}(t)-\frac{r-R_B}{R_F-R_B}\hat{B}(z_T)\bar{\nu}_{Fi}\Big)  \nonumber \\
\nonumber \\
& = \frac{R_F-r}{R_F-R_B}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\bar{\nu}_{Bi}(t) \nonumber \\[6pt]
& + \frac{r-R_B}{R_F-R_B}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\bar{\nu}_{Fi} \nonumber \\[6pt]
& + \frac{z-z_T}{z_B-z_T}B(r)\bar{\nu}_{\tilde{B}i}(t)+\frac{z_B-z}{z_B-z_T}B(r)\bar{\nu}_{\tilde{T}i}(t), \quad i=0,\ldots,4 \label{eq:ffem3d161}
\end{align}
and $\bar{\nu}_{Bi}(t)=0,\quad i=1,\ldots,4.$\\[6pt] 
%
Multiplying (\ref{eq:ffem3d161}) by $\Big(\hat{b}(z)\otimes B(r)^T\Big) = \Big(\hat{b}(z)\otimes I_{3\times 3}\Big)\otimes \bar{b}(r)$ and integrating with respect to $r$ and $z$, we have\\
%
\vspace{-0.3in}
\begin{align}
&\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\frac{R_F-r}{R_F-R_B}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B) \nonumber \\[6pt]
&\qquad \qquad \quad -\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)r \ud r\ud z\bigg]\bar{\nu}_{Bi}(t) \nonumber \\[6pt]
+ &\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\frac{r-R_B}{R_F-R_B}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B) \nonumber \\[6pt]
&\qquad \qquad \quad -\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)r \ud r\ud z\bigg] \bar{\nu}_{Fi} \nonumber \\[6pt]
+ &\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\frac{z-z_T}{z_B-z_T}B(r)r\ud r\ud z\bigg] \bar{\nu}_{\tilde{B}i}(t) \nonumber \\[6pt]
+ &\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\frac{z_B-z}{z_B-z_T}B(r)r\ud r\ud z\bigg] \bar{\nu}_{\tilde{T}i}(t) \nonumber \\
\nonumber \\
=&\bigg[\int_{z_T}^{z_B}\hat{b}(z)\otimes \Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\ud z\bigg] \nonumber \\[6pt]
&\otimes \bigg\{\Big[\int_{R_B}^{R_F}\bar{b}(r)\frac{R_F-r}{R_F-R_B}r\ud r\Big]\bar{\nu}_{Bi}(t) + \Big[\int_{R_B}^{R_F}\bar{b}(r)\frac{r-R_B}{R_F-R_B}r\ud r\Big]\bar{\nu}_{Fi}\bigg\} \nonumber \\[6pt]
+&{}\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z-z_T}{z_B-z_T}\ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\bar{\nu}_{\tilde{B}i}(t) \nonumber \\[6pt]
+&{}\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z_B-z}{z_B-z_T}\ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\bar{\nu}_{\tilde{T}i}(t) \label{eq:ffem3d162}
\end{align} \\
%
From (\ref{eq:ffem3d23}) and (\ref{eq:ffem3d90})
\begin{align}
W_{i_r}(r,z,t) &= -\frac{1}{R_F-R_B}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\bar{\nu}_{Bi}(t) \nonumber \\[6pt]
& + \frac{1}{R_F-R_B}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\bar{\nu}_{Fi} \nonumber \\[6pt]
& + \frac{z-z_T}{z_B-z_T}B_r(r)\bar{\nu}_{\tilde{B}i}(t)+\frac{z_B-z}{z_B-z_T}B_r(r)\bar{\nu}_{\tilde{T}i}(t), \quad i=0,\ldots,4 \label{eq:ffem3d163}
\end{align}
and $\bar{\nu}_{Bi}(t)=0,\quad i=1,\ldots,4.$
%
\newpage
Multiplying $\mathcal{A}\dfrac{1}{r}(rW_{i_r})_r$ by $\Big(\hat{b}(z)\otimes B(r)^T\Big) = \Big(\hat{b}(z)\otimes I_{3\times 3}\Big)\otimes \bar{b}(r)$ and integrating with respect to $r$ and $z$, we have
%
\begin{align}
& \bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\mathcal{A}\frac{1}{R_F-R_B}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B) \nonumber \\[6pt]
&\qquad \qquad \quad-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\ud r\ud z\bigg]\big(-\bar{\nu}_{Bi}(t)+\bar{\nu}_{Fi}\big) \nonumber \\[6pt]
& -\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B_r(r)^T\Big)\mathcal{A}\frac{z-z_T}{z_B-z_T}B_r(r)r\ud r\ud z\bigg]\bar{\nu}_{\tilde{B}i}(t) \nonumber \\[6pt]
& -\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B_r(r)^T\Big)\mathcal{A}\frac{z_B-z}{z_B-z_T}B_r(r)r\ud r\ud z\bigg]\bar{\nu}_{\tilde{T}i}(t) \nonumber \\
\nonumber \\
=& \bigg\{\bigg[\int_{z_T}^{z_B}\hat{b}(z)\otimes \Big[\mathcal{A}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\Big] \ud z\bigg] \nonumber \\[6pt]
&\qquad \otimes \bigg[\int_{R_B}^{R_F}\bar{b}(r)\frac{1}{R_F-R_B}\ud r\bigg]\bigg\}\big(-\bar{\nu}_{Bi}(t)+\bar{\nu}_{Fi}\big) \nonumber \\[6pt]
& -\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z-z_T}{z_B-z_T}\ud z\Big]\otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big]\bigg\}\bar{\nu}_{\tilde{B}i}(t) \nonumber \\[6pt]
& -\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z_B-z}{z_B-z_T}\ud z\Big]\otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big]\bigg\}\bar{\nu}_{\tilde{T}i}(t) ,\quad i=0,\ldots,4 \label{eq:ffem3d164}
\end{align}
and $\bar{\nu}_{Bi}(t)=0,\quad i=1,\ldots,4.$\\[6pt]
%
From (\ref{eq:ffem3d26}) and (\ref{eq:ffem3d90})
\begin{align}
W_{i_z}(r,z,t)& = \frac{R_F-r}{R_F-R_B}\Big(\hat{B}_z(z)-\frac{1}{z_B-z_T}\hat{B}(z_B)+\frac{1}{z_B-z_T}\hat{B}(z_T)\Big)\bar{\nu}_{Bi}(t) \nonumber \\[6pt]
& + \frac{r-R_B}{R_F-R_B}\Big(\hat{B}_z(z)-\frac{1}{z_B-z_T}\hat{B}(z_B)+\frac{1}{z_B-z_T}\hat{B}(z_T)\Big)\bar{\nu}_{Fi} \nonumber \\[6pt]
& + \frac{1}{z_B-z_T}B(r)\big(\bar{\nu}_{\tilde{B}i}(t)-\bar{\nu}_{\tilde{T}i}(t)\big), \quad i=0,\ldots,4 \label{eq:ffem3d165}
\end{align}
and $\bar{\nu}_{Bi}(t)=0,\quad i=1,\ldots,4.$
%
\newpage
Multiplying $\mathcal{A}(W_{i_z})_z$ by $\Big(\hat{b}(z)\otimes B(r)^T\Big) = \Big(\hat{b}(z)\otimes I_{3\times 3}\Big)\otimes \bar{b}(r)$ and integrating with respect to $r$ and $z$, we have
%
\begin{align}
& -\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}_z(z)\otimes B(r)^T\Big)\mathcal{A}\frac{R_F-r}{R_F-R_B}\Big(\hat{B}_z(z)-\frac{1}{z_B-z_T}\hat{B}(z_B) \nonumber \\[6pt]
&\qquad \qquad \quad+\frac{1}{z_B-z_T}\hat{B}(z_T)\Big) r\ud r\ud z\bigg] \bar{\nu}_{Bi}(t) \nonumber \\[6pt]
& -\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}_z(z)\otimes B(r)^T\Big)\mathcal{A}\frac{r-R_B}{R_F-R_B}\Big(\hat{B}_z(z)-\frac{1}{z_B-z_T}\hat{B}(z_B) \nonumber \\[6pt]
&\qquad \qquad \quad+\frac{1}{z_B-z_T}\hat{B}(z_T)\Big) r\ud r\ud z\bigg] \bar{\nu}_{Fi} \nonumber \\[6pt]
& -\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}_z(z)\otimes B(r)^T\Big)\mathcal{A}\frac{1}{z_B-z_T}B(r)r\ud r\ud z\bigg]\bar{\nu}_{\tilde{B}i}(t) \nonumber \\[6pt]
& +\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}_z(z)\otimes B(r)^T\Big)\mathcal{A}\frac{1}{z_B-z_T}B_r(r)r\ud r\ud z\bigg]\bar{\nu}_{\tilde{T}i}(t) \nonumber \\
\nonumber \\
=& -\bigg[\int_{z_T}^{z_B}\hat{b}_z(z)\otimes \Big[\mathcal{A}\Big(\hat{B}_z(z)-\frac{1}{z_B-z_T}\hat{B}(z_B)+\frac{1}{z_B-z_T}\hat{B}(z_T)\Big)\Big] \ud z\bigg] \nonumber \\[6pt]
&\qquad \otimes \bigg\{\bigg[\int_{R_B}^{R_F}\bar{b}(r)\frac{R_F-r}{R_F-R_B}r\ud r\bigg]\bar{\nu}_{Bi}(t)+\bigg[\int_{R_B}^{R_F}\bar{b}(r)\frac{r-R_B}{R_F-R_B}r\ud r\bigg]\bar{\nu}_{Fi}\bigg\} \nonumber \\[6pt]
& -\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}_z(z)\frac{1}{z_B-z_T}\ud z\Big]\otimes \mathcal{A} \otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^Tr\ud r\Big]\bigg\}\big(\bar{\nu}_{\tilde{B}i}(t)-\bar{\nu}_{\tilde{T}i}(t)\big) \label{eq:ffem3d166}
\end{align}
%
\indent
Define the following matrices:\\
\centerline{$3M_{Dr}M_{Dz}\times 3M_{Dr}M_{Dz}$ matrices:}
\begin{align*}
G_0&=\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big] \\[6pt]
G_2(r)&=\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big] \\[6pt]
G_2(z)&=\Big[\int_{z_T}^{z_B}\hat{b}_z(z)\hat{b}_z(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^Tr \ud r\Big] 
\end{align*}
%
\centerline{$3M_{Dr}M_{Dz}\times M_rM_z$ matrices:}
\begin{align*}
H_0&=\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0(r\bar{\ub}_r(r)+\bar{\ub}(r))^T \ud r\Big] \\[6pt]
H_1&=\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}_z(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^Tr \ud r\Big]
\end{align*}
\\
\centerline{$3M_{Dr}M_{Dz}\times 3M_{Dz}$ matrices:}
\begin{align*}
\mathcal{R}_B&=\int_{z_T}^{z_B}\hat{b}(z)\otimes \Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\ud z\bigg] \\[6pt]
&\qquad \otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\frac{R_F-r}{R_F-R_B}r\ud r\Big] \\[18pt]
\mathcal{R}_F&=\int_{z_T}^{z_B}\hat{b}(z)\otimes \Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\ud z\bigg] \\[6pt]
&\qquad \otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\frac{r-R_B}{R_F-R_B}r\ud r\Big] \\[18pt]
\mathcal{R}_F(a)&=\bigg[\int_{z_T}^{z_B}\hat{b}(z)\otimes \Big[\mathcal{A}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\Big] \ud z\bigg] \\[6pt]
&\qquad \otimes \bigg[\int_{R_B}^{R_F}\bar{b}(r)\frac{1}{R_F-R_B}\ud r\bigg] \\[18pt]
\mathcal{R}_B(z)&=\bigg[\int_{z_T}^{z_B}\hat{b}_z(z)\otimes \Big[\mathcal{A}\Big(\hat{B}_z(z)-\frac{1}{z_B-z_T}\hat{B}(z_B)+\frac{1}{z_B-z_T}\hat{B}(z_T)\Big)\Big] \ud z\bigg] \\[6pt]
&\qquad \otimes \bigg[\int_{R_B}^{R_F}\bar{b}(r)\frac{R_F-r}{R_F-R_B}r\ud r\bigg] \\[18pt]
\mathcal{R}_F(z)&=\bigg[\int_{z_T}^{z_B}\hat{b}_z(z)\otimes \Big[\mathcal{A}\Big(\hat{B}_z(z)-\frac{1}{z_B-z_T}\hat{B}(z_B)+\frac{1}{z_B-z_T}\hat{B}(z_T)\Big)\Big] \ud z\bigg] \\[6pt]
&\qquad \otimes \bigg[\int_{R_B}^{R_F}\bar{b}(r)\frac{r-R_B}{R_F-R_B}r\ud r\bigg] 
\end{align*}
\\
\centerline{$3M_{Dr}M_{Dz}\times 3M_{Dr}$ matrices:}
\begin{align*}
\tilde{\mathcal{R}}_B&=\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z-z_T}{z_B-z_T}\ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big] \\[6pt]
\tilde{\mathcal{R}}_T&=\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z_B-z}{z_B-z_T}\ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big] \\[6pt]
\tilde{\mathcal{R}}_B(a)&=\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z-z_T}{z_B-z_T}\ud z\Big]\otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big] \\[6pt]
\tilde{\mathcal{R}}_T(a)&=\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z_B-z}{z_B-z_T}\ud z\Big]\otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big] \\[6pt]
\tilde{\mathcal{R}}_0(z)&=\Big[\int_{z_T}^{z_B}\hat{b}_z(z)\frac{1}{z_B-z_T}\ud z\Big]\otimes \mathcal{A} \otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^Tr\ud r\Big]
\end{align*}
%
\indent
Using these matrices, (\ref{eq:ffem3d157}) multiplied by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrated with respect to $r$ and $z$, has the form 
\begin{align}
G_0\dot{\bar{c}}_0(t)&+\big(G_2(r)+G_2(z)\big)\bar{c}_0(t)= -H_0\dot{\bar{\mu}}_0(t)-H_1\dot{\bar{\lambda}}_0(t) \nonumber \\
&-\mathcal{R}_B\dot{\bar{\nu}}_{B0}(t)-\tilde{\mathcal{R}}_B\dot{\bar{\nu}}_{\tilde{B}0}(t)-\tilde{\mathcal{R}}_T\dot{\bar{\nu}}_{\tilde{T}0}(t) \nonumber \\
&-\mathcal{R}_F(a)\bar{\nu}_{B0}(t)+\mathcal{R}_F(a)\bar{\nu}_{F0}-\tilde{\mathcal{R}}_B(a)\bar{\nu}_{\tilde{B}0}(t)-\tilde{\mathcal{R}}_T(a)\bar{\nu}_{\tilde{T}0}(t) \nonumber \\
&-\mathcal{R}_B(z)\bar{\nu}_{B0}(t)-\mathcal{R}_F(z)\bar{\nu}_{F0}-\tilde{\mathcal{R}}_0(z)\bar{\nu}_{\tilde{B}0}(t)+\tilde{\mathcal{R}}_0(z)\bar{\nu}_{\tilde{T}0}(t) \label{eq:ffem3d167}
\end{align}
%
From (\ref{eq:ffem3d143}), (\ref{eq:ffem3d144})
\begin{align}
\dot{\bar{\mu}}_0(t)&=P_{10}\mathcal{Q}_0^{-1}\Big[\bar{\mathcal{Z}}_0\dot{\bar{c}}_0(t)+X_{B0}\dot{\bar{\nu}}_{B0}(t)+\tilde{X}_{B0}\dot{\bar{\nu}}_{\tilde{B}0}(t)+\tilde{X}_{T0}\dot{\bar{\nu}}_{\tilde{T}0}(t)\Big] \label{eq:ffem3d168} \\
%
\dot{\bar{\lambda}}_0(t)&=P_{01}\mathcal{Q}_0^{-1}\Big[\bar{\mathcal{Z}}_0\dot{\bar{c}}_0(t)+X_{B0}\dot{\bar{\nu}}_{B0}(t)+\tilde{X}_{B0}\dot{\bar{\nu}}_{\tilde{B}0}(t)+\tilde{X}_{T0}\dot{\bar{\nu}}_{\tilde{T}0}(t)\Big] \label{eq:ffem3d169}
\end{align}
%
Applying (\ref{eq:ffem3d168}) and (\ref{eq:ffem3d169}) to (\ref{eq:ffem3d167}), we have
\begin{align}
\mathcal{H}_0\dot{\bar{c}}_0(t)+\mathcal{G}_0\bar{c}_0(t)&=\mathcal{D}_{B0}\dot{\bar{\nu}}_{B0}(t)+\mathcal{D}_{\tilde{B}0}\dot{\bar{\nu}}_{\tilde{B}0}(t)+\mathcal{D}_{\tilde{T}0}\dot{\bar{\nu}}_{\tilde{T}0}(t) \nonumber \\
&+\mathcal{F}_{B0}\bar{\nu}_{B0}(t)+\mathcal{F}_{F0}\bar{\nu}_{F0}+\mathcal{F}_{\tilde{B}0}\bar{\nu}_{\tilde{B}0}(t)+\mathcal{F}_{\tilde{T}0}\bar{\nu}_{\tilde{T}0}(t) \label{eq:ffem3d170}
\end{align}
where $\quad \mathcal{H}_0=G_0+\big(H_0P_{10}+H_1P_{01}\big)\mathcal{Q}_0^{-1}\bar{\mathcal{Z}}_0$\\
${}\qquad \qquad \mathcal{G}_0=G_2(r)+G_2(z)$\\
${}\qquad \qquad \mathcal{D}_{B0}=-\big(H_0P_{10}+H_1P_{01}\big)\mathcal{Q}_0^{-1}X_{B0}-\mathcal{R}_B$\\
${}\qquad \qquad \mathcal{D}_{\tilde{B}0}=-\big(H_0P_{10}+H_1P_{01}\big)\mathcal{Q}_0^{-1}\tilde{X}_{B0}-\tilde{\mathcal{R}}_B$\\
${}\qquad \qquad \mathcal{D}_{\tilde{T}0}=-\big(H_0P_{10}+H_1P_{01}\big)\mathcal{Q}_0^{-1}\tilde{X}_{T0}-\tilde{\mathcal{R}}_T$\\
${}\qquad \qquad \mathcal{F}_{B0}=-\mathcal{R}_F(a)-\mathcal{R}_B(z)$\\
${}\qquad \qquad \mathcal{F}_{F0}=\mathcal{R}_F(a)-\mathcal{R}_F(z)$\\
${}\qquad \qquad \mathcal{F}_{\tilde{B}0}=-\tilde{\mathcal{R}}_B(a)-\tilde{\mathcal{R}}_0(z)$\\
${}\qquad \qquad \mathcal{F}_{\tilde{T}0}=-\tilde{\mathcal{R}}_T(a)+\tilde{\mathcal{R}}_0(z)$\\
%
\indent
Now we approximate initial conditions. ~From (\ref{eq:ffem3d20}), (\ref{eq:ffem3d30}) and (\ref{eq:ffem3d90}), the coefficients $V_i(r,z,0)$ of $\theta$-approximation of initial conditions have the form
\begin{equation*}
V_i(r,z,0)=\tilde{V}_{Ii}(r,z)-W_i(r,z,0)
\end{equation*} 
where $\qquad \qquad \tilde{V}_{Ii}(r,z)=\Big(\hat{b}(z)^T\otimes B(r)\Big)\bar{\nu}_{Ii}$
\begin{align}
W_i(r,z,0)&=\frac{R_F-r}{R_F-R_B}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\bar{\nu}_{Bi}(0) \nonumber \\[6pt]
& + \frac{r-R_B}{R_F-R_B}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\bar{\nu}_{Fi} \nonumber \\[6pt]
& + \frac{z-z_T}{z_B-z_T}B(r)\bar{\nu}_{\tilde{B}i}(0)+\frac{z_B-z}{z_B-z_T}B(r)\bar{\nu}_{\tilde{T}i}(0), \quad i=0,\ldots,4 \label{eq:ffem3d171}
\end{align}
and $\bar{\nu}_{Bi}(0)=0,\quad i=1,\ldots,4.$\\[6pt]
%
Multiplying $V_i(r,z,0)$ by $\hat{b}(z)\otimes B(r)^T$ and integrating with respect to $r$ and $z$, we have
\begin{align}
&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\bar{c}_i(0) \nonumber \\[6pt]
=&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\bar{\nu}_{Ii} \nonumber \\[6pt]
-&\bigg[\int_{z_T}^{z_B}\hat{b}(z)\otimes \Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\ud z\bigg] \nonumber \\[6pt]
&{}\otimes\bigg\{\Big[\int_{R_B}^{R_F}\bar{b}(r)\frac{R_F-r}{R_F-R_B}r\ud r\Big]\bar{\nu}_{Bi}(0)+\Big[\int_{R_B}^{R_F}\bar{b}(r)\frac{r-R_B}{R_F-R_B}r\ud r\Big]\bar{\nu}_{Fi}\bigg\} \nonumber \\[6pt]
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z-z_T}{z_B-z_T}\ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\bar{\nu}_{\tilde{B}i}(0) \nonumber \\[6pt]
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z_B-z}{z_B-z_T}\ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\bar{\nu}_{\tilde{T}i}(0) \label{eq:ffem3d172}
\end{align}
%
Using matrices defined above, (\ref{eq:ffem3d172}) multiplied by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrated with respect to $r$ and $z$, has the form
\begin{equation}
G_0\bar{c}_i(0)=G_0\bar{\nu}_{Ii}-\mathcal{R}_B\bar{\nu}_{Bi}(0)-\mathcal{R}_F\bar{\nu}_{Fi}-\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}i}(0)-\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}i}(0) \label{eq:ffem3d173}
\end{equation}
%
From (\ref{eq:ffem3d170}) and (\ref{eq:ffem3d173}) with $i=0$ we obtain the following system.\\
%
\noindent
\centerline{System of ODEs for TCP I (radially symmetric case):}
\begin{align*}
\mathcal{H}_0\dot{\bar{c}}_0(t)+\mathcal{G}_0\bar{c}_0(t)&=\mathcal{D}_{B0}\dot{\bar{\nu}}_{B0}(t)+\mathcal{D}_{\tilde{B}0}\dot{\bar{\nu}}_{\tilde{B}0}(t)+\mathcal{D}_{\tilde{T}0}\dot{\bar{\nu}}_{\tilde{T}0}(t) \nonumber \\
&+\mathcal{F}_{B0}\bar{\nu}_{B0}(t)+\mathcal{F}_{F0}\bar{\nu}_{F0}+\mathcal{F}_{\tilde{B}0}\bar{\nu}_{\tilde{B}0}(t)+\mathcal{F}_{\tilde{T}0}\bar{\nu}_{\tilde{T}0}(t) \tag{\ref{eq:ffem3d170}}
\end{align*}
with initial conditions
\begin{equation}
\bar{c}_0(0)=\bar{\nu}_{I0}-G_0^{-1}\big[\mathcal{R}_B\bar{\nu}_{B0}(0)+\mathcal{R}_F\bar{\nu}_{F0}+\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}0}(0)+\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}0}(0)\big] \label{eq:ffem3d174}
\end{equation}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM Approximation of TCP System II (sin \theta) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\noindent
\centerline{System II (corresponds to $\sin\theta$ approximation)}
%
System II is given by (\ref{eq:ffem3d31})-(\ref{eq:ffem3d33}):
\begin{gather*} 
M\dot{V}_1-A\Big[\frac{1}{r}(rV_{1_r})_r-\frac{1}{r^2}V_1+(V_{1_z})_z\Big] = -M\dot{W}_1+A\Big(\frac{1}{r}(rW_{1_r})_r-\frac{1}{r^2}W_1+(W_{1_z})_z\Big)  \\[6pt]
-\vec{b}_0\Big(\dot{u}_{1_r}+\frac{1}{r}(\dot{u}_1-\dot{v}_2)+\dot{w}_{1_z}\Big) \tag{\ref{eq:ffem3d31}}
\end{gather*}
with boundary conditions
\begin{equation*}
V_1(r,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \tag{\ref{eq:ffem3d32}}
\end{equation*}
and initial conditions
\begin{equation*}
V_1(r,z,0)=\tilde{V}_{I1}(r,z)-W_1(r,z,0) \tag{\ref{eq:ffem3d33}}
\end{equation*}
%
Using $\mathcal{A}=M^{-1}A$, (\ref{eq:ffem3d31}) has the form
\begin{gather} 
\dot{V}_1-\mathcal{A}\Big[\frac{1}{r}(rV_{1_r})_r-\frac{1}{r^2}V_1+(V_{1_z})_z\Big] =-M^{-1}\vec{b}_0\Big( \dot{u}_{1_r}+\frac{1}{r}\dot{u}_1-\frac{1}{r}\dot{v}_2+\dot{w}_{1_z}\Big) \nonumber \\[6pt]
-\dot{W}_1+\mathcal{A}\Big[\frac{1}{r}(rW_{1_r})_r-\frac{1}{r^2}W_1+(W_{1_z})_z\Big] \label{eq:ffem3d175}
\end{gather}
%
Using (\ref{eq:ffem3d88}), the left-hand side of (\ref{eq:ffem3d175}) is
\begin{align}
\Big(\hat{b}(z)^T\otimes B(r)\Big)\dot{\bar{c}}_1(t)&-\mathcal{A}\frac{1}{r}\Big(\hat{b}(z)^T\otimes rB_r(r)\Big)_r\bar{c}_1(t)+\mathcal{A}\frac{1}{r^2}\Big(\hat{b}(z)^T\otimes B(r)\Big)\bar{c}_1(t) \nonumber \\[6pt]
& -\mathcal{A}\Big(\hat{b}_z(z)^T\otimes B(r)\Big)_z\bar{c}_0(t) \label{eq:ffem3d176}
\end{align}
%
Multiplying (\ref{eq:ffem3d176}) by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$ we have
\begin{align}
&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\dot{\bar{c}}_1(t) \nonumber \\[6pt]
+& \bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \bigg(\Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big]+\Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r} \ud r\Big]\bigg) \nonumber \\[6pt]
&{}+ \Big[\int_{z_T}^{z_B}\hat{b}_z(z)\hat{b}_z(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^Tr \ud r\Big]\bigg\}\bar{c}_1(t) \label{eq:ffem3d177}
\end{align}
%
Using (\ref{eq:ffem3d91})-(\ref{eq:ffem3d93}), the first term of the right-hand side of (\ref{eq:ffem3d175}) is
\begin{align*}
-&M^{-1}\vec{b}_0\Big[\Big(\hat{\ub}(z)^T\otimes \bar{\ub}_r(r)^T\Big)\dot{\bar{\mu}}_1(t)+\Big(\hat{\ub}(z)^T\otimes \frac{1}{r}\bar{\ub}(r)^T\Big)\dot{\bar{\mu}}_1(t) \\[6pt]
-&\Big(\hat{\ub}(z)^T\otimes \frac{1}{r}\bar{\ub}(r)^T\Big)\dot{\bar{\eta}}_2(t)+\Big(\hat{\ub}_z(z)^T\otimes \bar{\ub}(r)^T\Big)\dot{\bar{\lambda}}_1(t)\Big]
\end{align*}
%
Multiplying by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have
%
\begin{align}
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0(r\bar{\ub}_r(r)+\bar{\ub}(r))^T \ud r\Big]\bigg\}\dot{\bar{\mu}}_1(t) \nonumber \\[6pt]
+&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^T \ud r\Big]\bigg\}\dot{\bar{\eta}}_2(t) \nonumber \\[6pt]
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}_z(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^Tr \ud r\Big]\bigg\}\dot{\bar{\lambda}}_1(t) \label{eq:ffem3d178}
\end{align} 
%
Multiplying the corresponding term in (\ref{eq:ffem3d175}) by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have the following results\\
for $\dot{W}_1$: (\ref{eq:ffem3d162}) with $\dot{\bar{\nu}}_{B1}(t)=0$\\
for $\mathcal{A}\dfrac{1}{r}(rW_{1_r})_r$: (\ref{eq:ffem3d164}) with $\bar{\nu}_{B1}(t)=0$\\
for $\mathcal{A}(rW_{1_z})_z$: (\ref{eq:ffem3d166}) with $\bar{\nu}_{B1}(t)=0$.\\
%
$W_i(r,z,t)$ is given by (\ref{eq:ffem3d161}) with $\bar{\nu}_{Bi}(t)=0, \quad i=1,\ldots,4.$\\[9pt]
%
Multiplying $\mathcal{A}\dfrac{1}{r^2}W_i$ by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have
%
\begin{align}
&\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\mathcal{A}\frac{1}{r^2}\frac{r-R_B}{R_F-R_B}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B) \nonumber \\[6pt]
&\qquad \qquad \quad -\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)r \ud r\ud z\bigg] \bar{\nu}_{Fi} \nonumber \\[6pt]
+ &\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\mathcal{A}\frac{1}{r^2}\frac{z-z_T}{z_B-z_T}B(r)r\ud r\ud z\bigg] \bar{\nu}_{\tilde{B}i}(t) \nonumber \\[6pt]
+ &\bigg[\int_{z_T}^{z_B}\int_{R_B}^{R_F}\Big(\hat{b}(z)\otimes B(r)^T\Big)\mathcal{A}\frac{1}{r^2}\frac{z_B-z}{z_B-z_T}B(r)r\ud r\ud z\bigg] \bar{\nu}_{\tilde{T}i}(t) \nonumber \\
\nonumber \\
=&\bigg[\int_{z_T}^{z_B}\hat{b}(z)\otimes \bigg(\mathcal{A}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\bigg)\ud z\bigg] \nonumber \\[6pt]
&\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\frac{r-R_B}{R_F-R_B}\frac{1}{r}\ud r\Big]\bar{\nu}_{Fi}\nonumber \\[6pt]
+&{}\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z-z_T}{z_B-z_T}\ud z\Big]\otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r}\ud r\Big]\bigg\}\bar{\nu}_{\tilde{B}i}(t) \nonumber \\[6pt]
+&{}\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z_B-z}{z_B-z_T}\ud z\Big]\otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r}\ud r\Big]\bigg\}\bar{\nu}_{\tilde{B}i}(t), \qquad i=1,\ldots,4. \label{eq:ffem3d179}
\end{align}
%
\newpage
Define the following matrices:\\
\centerline{$3M_{Dr}M_{Dz}\times 3M_{Dr}M_{Dz}$ matrix:}
\begin{equation*}
G_0\big(\frac{1}{r}\big)=\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r} \ud r\Big] 
\vspace{0.2in}
\end{equation*}
%
\centerline{$3M_{Dr}M_{Dz}\times M_rM_z$ matrix:}
\begin{equation*}
H_0(r)=\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^T \ud r\Big] 
\vspace{0.2in}
\end{equation*}
%
\centerline{$3M_{Dr}M_{Dz}\times 3M_{Dz}$ matrix:}
\begin{align*}
\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)&=\bigg\{\int_{z_T}^{z_B}\hat{b}(z)\otimes \Big[\mathcal{A}\Big(\hat{B}(z)-\frac{z-z_T}{z_B-z_T}\hat{B}(z_B)-\frac{z_B-z}{z_B-z_T}\hat{B}(z_T)\Big)\Big] \ud z\bigg\} \\[6pt]
&\qquad \otimes \bigg[\int_{R_B}^{R_F}\bar{b}(r)\frac{r-R_B}{R_F-R_B}\frac{1}{r}\ud r\bigg] 
\end{align*}
\\[-6pt]
\centerline{$3M_{Dr}M_{Dz}\times 3M_{Dr}$ matrices:}
\begin{align*}
\tilde{\mathcal{R}}_B^{(a)}\big(\frac{1}{r}\big)&=\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z-z_T}{z_B-z_T}\ud z\Big]\otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r} \ud r\Big] \\[6pt]
\tilde{\mathcal{R}}_T^{(a)}\big(\frac{1}{r}\big)&=\Big[\int_{z_T}^{z_B}\hat{b}(z)\frac{z_B-z}{z_B-z_T}\ud z\Big]\otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r} \ud r\Big]
\end{align*}
\\[-12pt]
\indent
Using these matrices together with matrices defined for System I, (\ref{eq:ffem3d175}) multiplied by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrated with respect to $r$ and $z$, has the form 
\begin{align}
G_0\dot{\bar{c}}_1(t)+&\big(G_2(r)+G_0\big(\frac{1}{r}\big)+G_2(z)\big)\bar{c}_1(t)= \nonumber \\
=&-H_0\dot{\bar{\mu}}_1(t)+H_0(r)\dot{\bar{\eta}}_2(t)-H_1\dot{\bar{\lambda}}_1(t)-\tilde{\mathcal{R}}_B\dot{\bar{\nu}}_{\tilde{B}1}(t)-\tilde{\mathcal{R}}_T\dot{\bar{\nu}}_{\tilde{T}1}(t) \nonumber \\
&+\mathcal{R}_F(a)\bar{\nu}_{F1}-\tilde{\mathcal{R}}_B(a)\bar{\nu}_{\tilde{B}1}(t)-\tilde{\mathcal{R}}_T(a)\bar{\nu}_{\tilde{T}1}(t) \nonumber \\
&-\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{F1}-\tilde{\mathcal{R}}_B^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{\tilde{B}1}(t)-\tilde{\mathcal{R}}_T^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{\tilde{T}1}(t) \nonumber \\
&-\mathcal{R}_F(z)\bar{\nu}_{F1}-\tilde{\mathcal{R}}_0(z)\bar{\nu}_{\tilde{B}1}(t)+\tilde{\mathcal{R}}_0(z)\bar{\nu}_{\tilde{T}1}(t) \label{eq:ffem3d180}
\end{align}
%
From (\ref{eq:ffem3d145})-(\ref{eq:ffem3d147})
\begin{align}
\dot{\bar{\mu}}_1(t)&=P_{100}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\dot{\bar{c}}_1(t)+\tilde{X}_{B1}\dot{\bar{\nu}}_{\tilde{B}1}(t)+\tilde{X}_{T1}\dot{\bar{\nu}}_{\tilde{T}1}(t)\Big] \label{eq:ffem3d181} \\
%
\dot{\bar{\eta}}_2(t)&=-P_{010}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\dot{\bar{c}}_1(t)+\tilde{X}_{B1}\dot{\bar{\nu}}_{\tilde{B}1}(t)+\tilde{X}_{T1}\dot{\bar{\nu}}_{\tilde{T}1}(t)\Big] \label{eq:ffem3d182} \\
%
\dot{\bar{\lambda}}_1(t)&=P_{001}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\dot{\bar{c}}_1(t)+\tilde{X}_{B1}\dot{\bar{\nu}}_{\tilde{B}1}(t)+\tilde{X}_{T1}\dot{\bar{\nu}}_{\tilde{T}1}(t)\Big] \label{eq:ffem3d183} 
\end{align}
%
Applying (\ref{eq:ffem3d181})-(\ref{eq:ffem3d183}) to (\ref{eq:ffem3d180}), we have
%
\begin{align}
\mathcal{H}_1\dot{\bar{c}}_1(t)+\mathcal{G}_1\bar{c}_1(t)&=\mathcal{D}_{\tilde{B}1}\dot{\bar{\nu}}_{\tilde{B}1}(t)+\mathcal{D}_{\tilde{T}1}\dot{\bar{\nu}}_{\tilde{T}1}(t) \nonumber \\
&+\mathcal{F}_{F1}\bar{\nu}_{F1}+\mathcal{F}_{\tilde{B}1}\bar{\nu}_{\tilde{B}1}(t)+\mathcal{F}_{\tilde{T}1}\bar{\nu}_{\tilde{T}1}(t) \label{eq:ffem3d184}
\end{align}
where 
\begin{align}
\mathcal{H}_1&=G_0+\big(H_0P_{100}+H_0(r)P_{010}+H_1P_{001}\big)\mathcal{A}_1^{-1}\bar{\mathcal{Z}}_1 \nonumber \\
\mathcal{G}_1&=G_2(r)+G_0\big(\frac{1}{r}\big)+G_2(z) \nonumber \\
\mathcal{D}_{\tilde{B}1}&=-\big(H_0P_{100}+H_0(r)P_{010}+H_1P_{001}\big)\mathcal{A}_1^{-1}\tilde{X}_{B1}-\tilde{\mathcal{R}}_B \nonumber \\
\mathcal{D}_{\tilde{T}1}&=-\big(H_0P_{100}+H_0(r)P_{010}+H_1P_{001}\big)\mathcal{A}_1^{-1}\tilde{X}_{T1}-\tilde{\mathcal{R}}_T \nonumber \\
\mathcal{F}_{F1}&=-\big(-\mathcal{R}_F(a)+\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)+\mathcal{R}_F(z)\big) \nonumber \\
\mathcal{F}_{\tilde{B}1}&=-\big(\mathcal{R}_B(a)+\mathcal{R}_B^{(a)}\big(\frac{1}{r}\big)+\tilde{\mathcal{R}}_0(z)\big) \nonumber \\
\mathcal{F}_{\tilde{T}1}&=-\big(\mathcal{R}_T(a)+\mathcal{R}_T^{(a)}\big(\frac{1}{r}\big)+\tilde{\mathcal{R}}_0(z)\big) \label{eq:ffem3d185}
\end{align}
%
\indent
Initial conditions are given by (\ref{eq:ffem3d173}) with $i=1$ and $\bar{\nu}_{B1}(0)=0$.\\
From (\ref{eq:ffem3d173}) and (\ref{eq:ffem3d184}) with $i=1$ and $\bar{\nu}_{B1}(0)=0$ we obtain the following system.\\
%
\centerline{System of ODEs for TCP II (corresponds to $\sin\theta$ approximation):}
\begin{align*}
\mathcal{H}_1\dot{\bar{c}}_1(t)+\mathcal{G}_1\bar{c}_1(t)&=\mathcal{D}_{\tilde{B}1}\dot{\bar{\nu}}_{\tilde{B}1}(t)+\mathcal{D}_{\tilde{T}1}\dot{\bar{\nu}}_{\tilde{T}1}(t) \nonumber \\
&+\mathcal{F}_{F1}\bar{\nu}_{F1}+\mathcal{F}_{\tilde{B}1}\bar{\nu}_{\tilde{B}1}(t)+\mathcal{F}_{\tilde{T}1}\bar{\nu}_{\tilde{T}1}(t) \tag{\ref{eq:ffem3d184}}
\end{align*}
with initial conditions
\begin{equation}
\bar{c}_1(0)=\bar{\nu}_{I1}+G_0^{-1}\big[\mathcal{R}_F\bar{\nu}_{F1}+\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}1}(0)+\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}1}(0)\big] \label{eq:ffem3d186}
\end{equation}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM Approximation of TCP System III (cos \theta) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\noindent
\centerline{System III (corresponds to $\cos\theta$ approximation)}
%
System III is given by (\ref{eq:ffem3d34})-(\ref{eq:ffem3d36}):
\begin{gather*} 
M\dot{V}_2-A\Big[\frac{1}{r}(rV_{2_r})_r-\frac{1}{r^2}V_2+(V_{2_z})_z\Big] = -M\dot{W}_2+A\Big(\frac{1}{r}(rW_{2_r})_r-\frac{1}{r^2}W_2+(W_{2_z})_z\Big) \nonumber \\[6pt]
-\vec{b}_0\Big(\dot{u}_{2_r}+\frac{1}{r}(\dot{u}_2+\dot{v}_1)+\dot{w}_{2_z}\Big)\tag{\ref{eq:ffem3d34}}
\end{gather*}
with boundary conditions
\begin{equation*}
V_2(r,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \tag{\ref{eq:ffem3d35}}
\end{equation*}
and initial conditions
\begin{equation*}
V_2(r,z,0)=\tilde{V}_{I2}(r,z)-W_2(r,z,0) \tag{\ref{eq:ffem3d36}}
\end{equation*}
%
Using $\mathcal{A}=M^{-1}A$, (\ref{eq:ffem3d34}) has the form
\begin{gather} 
\dot{V}_2-\mathcal{A}\Big[\frac{1}{r}(rV_{2_r})_r-\frac{1}{r^2}V_2+(V_{2_z})_z\Big] =-M^{-1}\vec{b}_0\Big( \dot{u}_{2_r}+\frac{1}{r}\dot{u}_2+\frac{1}{r}\dot{v}_1+\dot{w}_{2_z}\Big) \nonumber \\[6pt]
-\dot{W}_2+\mathcal{A}\Big[\frac{1}{r}(rW_{2_r})_r-\frac{1}{r^2}W_2+(W_{2_z})_z\Big] \label{eq:ffem3d187}
\end{gather}
%
Applying (\ref{eq:ffem3d88}) to the left-hand side of (\ref{eq:ffem3d187}), multiplying the result by \\
$\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we obtain
\vspace{0.1in}
\begin{align}
&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\dot{\bar{c}}_2(t) \nonumber \\[6pt]
+& \bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \bigg(\Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big]+\Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r} \ud r\Big]\bigg) \nonumber \\[6pt]
&{}+ \Big[\int_{z_T}^{z_B}\hat{b}_z(z)\hat{b}_z(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^Tr \ud r\Big]\bigg\}\bar{c}_2(t) \label{eq:ffem3d188}
\end{align}
%
Using (\ref{eq:ffem3d91})-(\ref{eq:ffem3d93}), the last term of the right-hand side of (\ref{eq:ffem3d187}) is
\begin{align*}
&-M^{-1}\vec{b}_0\Big( \dot{u}_{2_r}+\frac{1}{r}\dot{u}_2+\frac{1}{r}\dot{v}_1+\dot{w}_{2_z}\Big) \\[6pt]
=&-M^{-1}\vec{b}_0\Big[\Big(\hat{\ub}(z)^T\otimes \bar{\ub}_r(r)^T\Big)\dot{\bar{\mu}}_2(t)+\Big(\hat{\ub}(z)^T\otimes \frac{1}{r}\bar{\ub}(r)^T\Big)\dot{\bar{\mu}}_2(t) \\[6pt]
&\qquad \qquad +\Big(\hat{\ub}(z)^T\otimes \frac{1}{r}\bar{\ub}(r)^T\Big)\dot{\bar{\eta}}_1(t)+\Big(\hat{\ub}_z(z)^T\otimes \bar{\ub}(r)^T\Big)\dot{\bar{\lambda}}_2(t)\Big]
\end{align*}
%
Multiplying by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have
%
\begin{align}
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0(r\bar{\ub}_r(r)+\bar{\ub}(r))^T \ud r\Big]\bigg\}\dot{\bar{\mu}}_2(t) \nonumber \\[6pt]
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^T \ud r\Big]\bigg\}\dot{\bar{\eta}}_1(t) \nonumber \\[6pt]
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}_z(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^Tr \ud r\Big]\bigg\}\dot{\bar{\lambda}}_2(t) \label{eq:ffem3d189}
\end{align} 
%
Multiplying the corresponding term in (\ref{eq:ffem3d187}) by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have the following results\\
for $\dot{W}_2$: (\ref{eq:ffem3d162}) with $\dot{\bar{\nu}}_{B2}(t)=0$\\
for $\mathcal{A}\dfrac{1}{r}(rW_{2_r})_r$: (\ref{eq:ffem3d164}) with $\bar{\nu}_{B2}(t)=0$\\
for $\mathcal{A}(rW_{2_z})_z$: (\ref{eq:ffem3d166}) with $\bar{\nu}_{B2}(t)=0$\\
for $\mathcal{A}\dfrac{1}{r^2}W_2$: (\ref{eq:ffem3d179}).\\[6pt] 
%
Using matrices defined for Systems I and II, (\ref{eq:ffem3d187}) multiplied by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrated with respect to $r$ and $z$, has the form 
\begin{align}
G_0\dot{\bar{c}}_2(t)+&\big(G_2(r)+G_0\big(\frac{1}{r}\big)+G_2(z)\big)\bar{c}_2(t)= \nonumber \\
=&-H_0\dot{\bar{\mu}}_2(t)-H_0(r)\dot{\bar{\eta}}_1(t)-H_1\dot{\bar{\lambda}}_2(t) -\tilde{\mathcal{R}}_B\dot{\bar{\nu}}_{\tilde{B}2}(t)-\tilde{\mathcal{R}}_T\dot{\bar{\nu}}_{\tilde{T}2}(t) \nonumber \\
&+\mathcal{R}_F(a)\bar{\nu}_{F2}-\tilde{\mathcal{R}}_B(a)\bar{\nu}_{\tilde{B}2}(t)-\tilde{\mathcal{R}}_T(a)\bar{\nu}_{\tilde{T}2}(t) \nonumber \\
&-\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{F2}-\tilde{\mathcal{R}}_B^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{\tilde{B}2}(t)-\tilde{\mathcal{R}}_T^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{\tilde{T}2}(t) \nonumber \\
&-\mathcal{R}_F(z)\bar{\nu}_{F2}-\tilde{\mathcal{R}}_0(z)\bar{\nu}_{\tilde{B}2}(t)+\tilde{\mathcal{R}}_0(z)\bar{\nu}_{\tilde{T}2}(t) \label{eq:ffem3d190}
\end{align}
%
From (\ref{eq:ffem3d148})-(\ref{eq:ffem3d150})
\begin{align}
\dot{\bar{\mu}}_2(t)&=P_{100}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\dot{\bar{c}}_2(t)+\tilde{X}_{B1}\dot{\bar{\nu}}_{\tilde{B}2}(t)+\tilde{X}_{T1}\dot{\bar{\nu}}_{\tilde{T}2}(t)\Big] \label{eq:ffem3d191} \\
%
\dot{\bar{\eta}}_1(t)&=P_{010}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\dot{\bar{c}}_2(t)+\tilde{X}_{B1}\dot{\bar{\nu}}_{\tilde{B}2}(t)+\tilde{X}_{T1}\dot{\bar{\nu}}_{\tilde{T}2}(t)\Big] \label{eq:ffem3d192} \\
%
\dot{\bar{\lambda}}_2(t)&=P_{001}\mathcal{Q}_1^{-1}\Big[\bar{\mathcal{Z}}_1\dot{\bar{c}}_2(t)+\tilde{X}_{B1}\dot{\bar{\nu}}_{\tilde{B}2}(t)+\tilde{X}_{T1}\dot{\bar{\nu}}_{\tilde{T}2}(t)\Big] \label{eq:ffem3d193} 
\end{align}
%
Applying (\ref{eq:ffem3d191})-(\ref{eq:ffem3d193}) to (\ref{eq:ffem3d190}), we have
%
\begin{align}
\mathcal{H}_1\dot{\bar{c}}_2(t)+\mathcal{G}_1\bar{c}_2(t)&=\mathcal{D}_{\tilde{B}1}\dot{\bar{\nu}}_{\tilde{B}2}(t)+\mathcal{D}_{\tilde{T}1}\dot{\bar{\nu}}_{\tilde{T}2}(t) \nonumber \\
&+\mathcal{F}_{F1}\bar{\nu}_{F2}+\mathcal{F}_{\tilde{B}1}\bar{\nu}_{\tilde{B}2}(t)+\mathcal{F}_{\tilde{T}1}\bar{\nu}_{\tilde{T}2}(t) \label{eq:ffem3d194}
\end{align}
where matrices $\quad \mathcal{H}_1, \mathcal{G}_1, \mathcal{D}_{\tilde{B}1}, \mathcal{D}_{\tilde{T}1}, \mathcal{F}_{F1}, \mathcal{F}_{\tilde{B}1}, \mathcal{F}_{\tilde{T}1}$ are given by (\ref{eq:ffem3d184}).\\
%
\indent
Initial conditions are given by (\ref{eq:ffem3d173}) with $i=2$ and $\bar{\nu}_{B2}(0)=0$.\\
From (\ref{eq:ffem3d194}) and (\ref{eq:ffem3d173}) with $i=2$ and $\bar{\nu}_{B2}(0)=0$ we obtain the following system.\\ 
%
\centerline{System of ODEs for TCP III (corresponds to $\cos\theta$ approximation):}
\begin{align*}
\mathcal{H}_1\dot{\bar{c}}_2(t)+\mathcal{G}_1\bar{c}_2(t)&=\mathcal{D}_{\tilde{B}1}\dot{\bar{\nu}}_{\tilde{B}2}(t)+\mathcal{D}_{\tilde{T}1}\dot{\bar{\nu}}_{\tilde{T}2}(t) \\
&+\mathcal{F}_{F1}\bar{\nu}_{F2}+\mathcal{F}_{\tilde{B}1}\bar{\nu}_{\tilde{B}2}(t)+\mathcal{F}_{\tilde{T}1}\bar{\nu}_{\tilde{T}2}(t) \tag{\ref{eq:ffem3d194}}
\end{align*}
with initial conditions
\begin{equation}
\bar{c}_2(0)=\bar{\nu}_{I2}+G_0^{-1}\big[\mathcal{R}_F\bar{\nu}_{F2}+\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}2}(0)+\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}2}(0)\big] \label{eq:ffem3d195}
\end{equation}\\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM Approximation of TCP System IV (sin2\theta) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
\centerline{System IV (corresponds to $\sin2\theta$ approximation)}
%
System IV is given by (\ref{eq:ffem3d37})-(\ref{eq:ffem3d39}):
\begin{gather*} 
M\dot{V}_3-A\Big[\frac{1}{r}(rV_{3_r})_r-\frac{4}{r^2}V_3+(V_{3_z})_z\Big] = -M\dot{W}_3+A\Big(\frac{1}{r}(rW_{3_r})_r-\frac{4}{r^2}W_3+(W_{3_z})_z\Big) \\[6pt]
-\vec{b}_0\Big(\dot{u}_{3_r}+\frac{1}{r}(\dot{u}_3-2\dot{v}_4)+\dot{w}_{3_z}\Big) \tag{\ref{eq:ffem3d37}}
\end{gather*}
with boundary conditions
\begin{equation*}
V_3(r,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \tag{\ref{eq:ffem3d38}}
\end{equation*}
and initial conditions
\begin{equation*}
V_3(r,z,0)=\tilde{V}_{I3}(r,z)-W_3(r,z,0) \tag{\ref{eq:ffem3d39}}
\end{equation*}
%
Using $\mathcal{A}=M^{-1}A$, (\ref{eq:ffem3d37}) has the form
\begin{gather} 
\dot{V}_3-\mathcal{A}\Big[\frac{1}{r}(rV_{3_r})_r-\frac{4}{r^2}V_3+(V_{3_z})_z\Big] =-M^{-1}\vec{b}_0\Big( \dot{u}_{3_r}+\frac{1}{r}\dot{u}_3-\frac{2}{r}\dot{v}_4+\dot{w}_{3_z}\Big) \nonumber \\[6pt]
-\dot{W}_3+\mathcal{A}\Big[\frac{1}{r}(rW_{3_r})_r-\frac{4}{r^2}W_3+(W_{3_z})_z\Big] \label{eq:ffem3d196}
\end{gather}
%
Applying (\ref{eq:ffem3d88}) to the left-hand side of (\ref{eq:ffem3d196}), multiplying the result by\\
$\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we obtain
\begin{align}
&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\dot{\bar{c}}_3(t) \nonumber \\[6pt]
+& \bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \bigg(\Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big]+4\Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r} \ud r\Big]\bigg) \nonumber \\[6pt]
&{}+ \Big[\int_{z_T}^{z_B}\hat{b}_z(z)\hat{b}_z(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^Tr \ud r\Big]\bigg\}\bar{c}_2(t) \label{eq:ffem3d197}
\end{align}
%
Using (\ref{eq:ffem3d91})-(\ref{eq:ffem3d93}), the first term of the right-hand side of (\ref{eq:ffem3d196}) is
\begin{align*}
&-M^{-1}\vec{b}_0\Big( \dot{u}_{3_r}+\frac{1}{r}\dot{u}_3-\frac{2}{r}\dot{v}_4+\dot{w}_{3_z}\Big) \\[6pt]
=&-M^{-1}\vec{b}_0\Big[\Big(\hat{\ub}(z)^T\otimes \bar{\ub}_r(r)^T\Big)\dot{\bar{\mu}}_3(t)+\Big(\hat{\ub}(z)^T\otimes \frac{1}{r}\bar{\ub}(r)^T\Big)\dot{\bar{\mu}}_3(t) \\[6pt]
&\qquad \qquad -\Big(\hat{\ub}(z)^T\otimes \frac{2}{r}\bar{\ub}(r)^T\Big)\dot{\bar{\eta}}_4(t)+\Big(\hat{\ub}_z(z)^T\otimes \bar{\ub}(r)^T\Big)\dot{\bar{\lambda}}_3(t)\Big]
\end{align*}
%
Multiplying by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have
%
\begin{align}
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0(r\bar{\ub}_r(r)+\bar{\ub}(r))^T \ud r\Big]\bigg\}\dot{\bar{\mu}}_3(t) \nonumber \\[6pt]
+2&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^T \ud r\Big]\bigg\}\dot{\bar{\eta}}_4(t) \nonumber \\[6pt]
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}_z(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^Tr \ud r\Big]\bigg\}\dot{\bar{\lambda}}_3(t) \label{eq:ffem3d198}
\end{align} 
%
Multiplying the corresponding term in (\ref{eq:ffem3d196}) by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have the following results\\
for $\dot{W}_3$: (\ref{eq:ffem3d162}) with $\dot{\bar{\nu}}_{B3}(t)=0$\\
for $\mathcal{A}\dfrac{1}{r}(rW_{3_r})_r$: (\ref{eq:ffem3d164}) with $\bar{\nu}_{B3}(t)=0$\\
for $\mathcal{A}(rW_{3_z})_z$: (\ref{eq:ffem3d166}) with $\bar{\nu}_{B3}(t)=0$\\
for $\mathcal{A}\dfrac{4}{r^2}W_3=4\mathcal{A}\dfrac{1}{r^2}W_3$: (\ref{eq:ffem3d179}).\\[6pt]
%
Using matrices defined for Systems I and II, (\ref{eq:ffem3d196}) multiplied by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrated with respect to $r$ and $z$, has the form 
\begin{align}
G_0\dot{\bar{c}}_3(t)+&\big(G_2(r)+4G_0\big(\frac{1}{r}\big)+G_2(z)\big)\bar{c}_3(t)= \nonumber \\
=&-H_0\dot{\bar{\mu}}_3(t)+2H_0(r)\dot{\bar{\eta}}_4(t)-H_1\dot{\bar{\lambda}}_3(t) -\tilde{\mathcal{R}}_B\dot{\bar{\nu}}_{\tilde{B}3}(t)-\tilde{\mathcal{R}}_T\dot{\bar{\nu}}_{\tilde{T}3}(t) \nonumber \\
&+\mathcal{R}_F(a)\bar{\nu}_{F3}-\tilde{\mathcal{R}}_B(a)\bar{\nu}_{\tilde{B}3}(t)-\tilde{\mathcal{R}}_T(a)\bar{\nu}_{\tilde{T}3}(t) \nonumber \\
&-4\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{F3}-4\tilde{\mathcal{R}}_B^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{\tilde{B}3}(t)-4\tilde{\mathcal{R}}_T^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{\tilde{T}3}(t) \nonumber \\
&-\mathcal{R}_F(z)\bar{\nu}_{F3}-\tilde{\mathcal{R}}_0(z)\bar{\nu}_{\tilde{B}3}(t)+\tilde{\mathcal{R}}_0(z)\bar{\nu}_{\tilde{T}3}(t) \label{eq:ffem3d199}
\end{align}
%
From (\ref{eq:ffem3d151})-(\ref{eq:ffem3d153})
\begin{align}
\dot{\bar{\mu}}_3(t)&=P_{100}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\dot{\bar{c}}_3(t)+\tilde{X}_{B2}\dot{\bar{\nu}}_{\tilde{B}3}(t)+\tilde{X}_{T2}\dot{\bar{\nu}}_{\tilde{T}3}(t)\Big] \label{eq:ffem3d200} \\
%
\dot{\bar{\eta}}_4(t)&=-P_{010}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\dot{\bar{c}}_3(t)+\tilde{X}_{B2}\dot{\bar{\nu}}_{\tilde{B}3}(t)+\tilde{X}_{T2}\dot{\bar{\nu}}_{\tilde{T}3}(t)\Big] \label{eq:ffem3d201} \\
%
\dot{\bar{\lambda}}_3(t)&=P_{001}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\dot{\bar{c}}_3(t)+\tilde{X}_{B2}\dot{\bar{\nu}}_{\tilde{B}3}(t)+\tilde{X}_{T2}\dot{\bar{\nu}}_{\tilde{T}3}(t)\Big] \label{eq:ffem3d202} 
\end{align}
%
Applying (\ref{eq:ffem3d200})-(\ref{eq:ffem3d202}) to (\ref{eq:ffem3d199}), we have
%
\begin{align}
\mathcal{H}_2\dot{\bar{c}}_3(t)+\mathcal{G}_2\bar{c}_3(t)&=\mathcal{D}_{\tilde{B}2}\dot{\bar{\nu}}_{\tilde{B}3}(t)+\mathcal{D}_{\tilde{T}2}\dot{\bar{\nu}}_{\tilde{T}3}(t) \nonumber \\
&+\mathcal{F}_{F2}\bar{\nu}_{F3}+\mathcal{F}_{\tilde{B}2}\bar{\nu}_{\tilde{B}3}(t)+\mathcal{F}_{\tilde{T}2}\bar{\nu}_{\tilde{T}3}(t) \label{eq:ffem3d203}
\end{align}
where 
\begin{align}
\mathcal{H}_2&=G_0+\big(H_0P_{100}+2H_0(r)P_{010}+H_1P_{001}\big)\mathcal{A}_2^{-1}\bar{\mathcal{Z}}_2 \nonumber \\
\mathcal{G}_2&=G_2(r)+4G_0\big(\frac{1}{r}\big)+G_2(z) \nonumber \\
\mathcal{D}_{\tilde{B}2}&=-\big(H_0P_{100}+2H_0(r)P_{010}+H_1P_{001}\big)\mathcal{A}_2^{-1}\tilde{X}_{B2}-\tilde{\mathcal{R}}_B \nonumber \\
\mathcal{D}_{\tilde{T}2}&=-\big(H_0P_{100}+2H_0(r)P_{010}+H_1P_{001}\big)\mathcal{A}_2^{-1}\tilde{X}_{T2}-\tilde{\mathcal{R}}_T \nonumber \\
\mathcal{F}_{F2}&=-\big(-\mathcal{R}_F(a)+4\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)+\mathcal{R}_F(z)\big) \nonumber \\
\mathcal{F}_{\tilde{B}2}&=-\big(\mathcal{R}_B(a)+4\mathcal{R}_B^{(a)}\big(\frac{1}{r}\big)+\tilde{\mathcal{R}}_0(z)\big) \nonumber \\
\mathcal{F}_{\tilde{T}2}&=-\big(\mathcal{R}_T(a)+4\mathcal{R}_T^{(a)}\big(\frac{1}{r}\big)+\tilde{\mathcal{R}}_0(z)\big) \label{eq:ffem3d204}
\end{align}
%
\indent
Initial conditions are given by (\ref{eq:ffem3d173}) with $i=3$ and $\bar{\nu}_{B3}(0)=0$.\\
From (\ref{eq:ffem3d203}) and (\ref{eq:ffem3d173}) with $i=3$ and $\bar{\nu}_{B3}(0)=0$ we obtain the following system.\\ 
%
\centerline{System of ODEs for TCP IV (corresponds to $\sin2\theta$ approximation):}
\begin{align*}
\mathcal{H}_2\dot{\bar{c}}_3(t)+\mathcal{G}_2\bar{c}_3(t)&=\mathcal{D}_{\tilde{B}2}\dot{\bar{\nu}}_{\tilde{B}3}(t)+\mathcal{D}_{\tilde{T}2}\dot{\bar{\nu}}_{\tilde{T}3}(t) \nonumber \\
&+\mathcal{F}_{F2}\bar{\nu}_{F3}+\mathcal{F}_{\tilde{B}2}\bar{\nu}_{\tilde{B}3}(t)+\mathcal{F}_{\tilde{T}2}\bar{\nu}_{\tilde{T}3}(t) \tag{\ref{eq:ffem3d203}}
\end{align*}
with initial conditions
\begin{equation}
\bar{c}_3(0)=\bar{\nu}_{I3}+G_0^{-1}\big[\mathcal{R}_F\bar{\nu}_{F3}+\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}3}(0)+\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}3}(0)\big] \label{eq:ffem3d205}
\end{equation}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FEM Approximation of TCP System V (cos2\theta) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
\centerline{System V (corresponds to $\cos2\theta$ approximation)}
%
System V is given by (\ref{eq:ffem3d40})-(\ref{eq:ffem3d42}):
\begin{gather*} 
M\dot{V}_4-A\Big[\frac{1}{r}(rV_{4_r})_r-\frac{4}{r^2}V_4+(V_{4_z})_z\Big] = -M\dot{W}_4+A\Big(\frac{1}{r}(rW_{4_r})_r-\frac{4}{r^2}W_4+(W_{4_z})_z\Big) \nonumber \\[6pt]
-\vec{b}_0\Big(\dot{u}_{4_r}+\frac{1}{r}(\dot{u}_4+2\dot{v}_3)+\dot{w}_{4_z}\Big) \tag{\ref{eq:ffem3d40}}
\end{gather*}
with boundary conditions
\begin{equation*}
V_4(r,z,t)=0 \qquad \text{on } \Gamma \times[0,t_f) \tag{\ref{eq:ffem3d41}}
\end{equation*}
and initial conditions
\begin{equation*}
V_4(r,z,0)=\tilde{V}_{I4}(r,z)-W_4(r,z,0) \tag{\ref{eq:ffem3d42}}
\end{equation*}
%
Using $\mathcal{A}=M^{-1}A$, (\ref{eq:ffem3d40}) has the form
\begin{gather} 
\dot{V}_4-\mathcal{A}\Big[\frac{1}{r}(rV_{4_r})_r-\frac{4}{r^2}V_4+(V_{4_z})_z\Big] =-M^{-1}\vec{b}_0\Big( \dot{u}_{4_r}+\frac{1}{r}\dot{u}_4+\frac{2}{r}\dot{v}_3+\dot{w}_{4_z}\Big) \nonumber \\[6pt]
-\dot{W}_4+\mathcal{A}\Big[\frac{1}{r}(rW_{4_r})_r-\frac{4}{r^2}W_4+(W_{4_z})_z\Big] \label{eq:ffem3d206}
\end{gather}
%
Applying (\ref{eq:ffem3d88}) to the left-hand side of (\ref{eq:ffem3d206}), multiplying the result by\\
$\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we obtain
\begin{align}
&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \Big[\int_{R_B}^{R_F}B(r)^TB(r)r\ud r\Big]\bigg\}\dot{\bar{c}}_4(t) \nonumber \\[6pt]
+& \bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{b}(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \bigg(\Big[\int_{R_B}^{R_F}\bar{b}_r(r)\bar{b}_r(r)^Tr \ud r\Big]+4\Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^T\frac{1}{r} \ud r\Big]\bigg) \nonumber \\[6pt]
&{}+ \Big[\int_{z_T}^{z_B}\hat{b}_z(z)\hat{b}_z(z)^T\ud z\Big] \otimes \mathcal{A}\otimes \Big[\int_{R_B}^{R_F}\bar{b}(r)\bar{b}(r)^Tr \ud r\Big]\bigg\}\bar{c}_4(t) \label{eq:ffem3d207}
\end{align}
%
Using (\ref{eq:ffem3d91})-(\ref{eq:ffem3d93}), the last term of the right-hand side of (\ref{eq:ffem3d206}) is
\begin{align*}
&-M^{-1}\vec{b}_0\Big( \dot{u}_{4_r}+\frac{1}{r}\dot{u}_4+\frac{2}{r}\dot{v}_3+\dot{w}_{4_z}\Big) \\[6pt]
=&-M^{-1}\vec{b}_0\Big[\Big(\hat{\ub}(z)^T\otimes \bar{\ub}_r(r)^T\Big)\dot{\bar{\mu}}_4(t)+\Big(\hat{\ub}(z)^T\otimes \frac{1}{r}\bar{\ub}(r)^T\Big)\dot{\bar{\mu}}_4(t) \\[6pt]
&\qquad \qquad -\Big(\hat{\ub}(z)^T\otimes \frac{2}{r}\bar{\ub}(r)^T\Big)\dot{\bar{\eta}}_3(t)+\Big(\hat{\ub}_z(z)^T\otimes \bar{\ub}(r)^T\Big)\dot{\bar{\lambda}}_4(t)\Big]
\end{align*}
%
Multiplying by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have
%
\begin{align}
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0(r\bar{\ub}_r(r)+\bar{\ub}(r))^T \ud r\Big]\bigg\}\dot{\bar{\mu}}_4(t) \nonumber \\[6pt]
-2&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^T \ud r\Big]\bigg\}\dot{\bar{\eta}}_3(t) \nonumber \\[6pt]
-&\bigg\{\Big[\int_{z_T}^{z_B}\hat{b}(z)\hat{\ub}_z(z)^T \ud z\Big]\otimes \Big[\int_{R_B}^{R_F}B(r)^TM^{-1}\vec{b}_0\bar{\ub}(r)^Tr \ud r\Big]\bigg\}\dot{\bar{\lambda}}_4(t) \label{eq:ffem3d208}
\end{align} 
%
Multiplying the corresponding term in (\ref{eq:ffem3d206}) by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrating with respect to $r$ and $z$, we have the following results\\
for $\dot{W}_4$: (\ref{eq:ffem3d162}) with $\dot{\bar{\nu}}_{B4}(t)=0$\\
for $\mathcal{A}\dfrac{1}{r}(rW_{4_r})_r$: (\ref{eq:ffem3d164}) with $\bar{\nu}_{B4}(t)=0$\\
for $\mathcal{A}(rW_{4_z})_z$: (\ref{eq:ffem3d166}) with $\bar{\nu}_{B4}(t)=0$\\
for $\mathcal{A}\dfrac{4}{r^2}W_4=4\mathcal{A}\dfrac{1}{r^2}W_4$: (\ref{eq:ffem3d179}).\\[6pt]
%
Using matrices defined for Systems I and II, (\ref{eq:ffem3d196}) multiplied by $\Big(\hat{b}(z)\otimes B(r)^T\Big)$ and integrated with respect to $r$ and $z$, has the form 
\begin{align}
G_0\dot{\bar{c}}_4(t)+&\big(G_2(r)+4G_0\big(\frac{1}{r}\big)+G_2(z)\big)\bar{c}_4(t)= \nonumber \\
=&-H_0\dot{\bar{\mu}}_4(t)-2H_0(r)\dot{\bar{\eta}}_3(t)-H_1\dot{\bar{\lambda}}_4(t) -\tilde{\mathcal{R}}_B\dot{\bar{\nu}}_{\tilde{B}4}(t)-\tilde{\mathcal{R}}_T\dot{\bar{\nu}}_{\tilde{T}4}(t) \nonumber \\
&+\mathcal{R}_F(a)\bar{\nu}_{F4}-\tilde{\mathcal{R}}_B(a)\bar{\nu}_{\tilde{B}4}(t)-\tilde{\mathcal{R}}_T(a)\bar{\nu}_{\tilde{T}4}(t) \nonumber \\
&-4\mathcal{R}_F^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{F4}-4\tilde{\mathcal{R}}_B^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{\tilde{B}4}(t)-4\tilde{\mathcal{R}}_T^{(a)}\big(\frac{1}{r}\big)\bar{\nu}_{\tilde{T}4}(t) \nonumber \\
&-\mathcal{R}_F(z)\bar{\nu}_{F4}-\tilde{\mathcal{R}}_0(z)\bar{\nu}_{\tilde{B}4}(t)+\tilde{\mathcal{R}}_0(z)\bar{\nu}_{\tilde{T}4}(t) \label{eq:ffem3d209}
\end{align}
%
From (\ref{eq:ffem3d154})-(\ref{eq:ffem3d156})
\begin{align}
\dot{\bar{\mu}}_4(t)&=P_{100}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\dot{\bar{c}}_4(t)+\tilde{X}_{B2}\dot{\bar{\nu}}_{\tilde{B}4}(t)+\tilde{X}_{T2}\dot{\bar{\nu}}_{\tilde{T}4}(t)\Big] \label{eq:ffem3d210} \\
%
\dot{\bar{\eta}}_3(t)&=-P_{010}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\dot{\bar{c}}_4(t)+\tilde{X}_{B2}\dot{\bar{\nu}}_{\tilde{B}4}(t)+\tilde{X}_{T2}\dot{\bar{\nu}}_{\tilde{T}4}(t)\Big] \label{eq:ffem3d211} \\
%
\dot{\bar{\lambda}}_4(t)&=P_{001}\mathcal{Q}_2^{-1}\Big[\bar{\mathcal{Z}}_2\dot{\bar{c}}_4(t)+\tilde{X}_{B2}\dot{\bar{\nu}}_{\tilde{B}4}(t)+\tilde{X}_{T2}\dot{\bar{\nu}}_{\tilde{T}4}(t)\Big] \label{eq:ffem3d212} 
\end{align}
%
Applying (\ref{eq:ffem3d210})-(\ref{eq:ffem3d212}) to (\ref{eq:ffem3d209}), we have
%
\begin{align}
\mathcal{H}_2\dot{\bar{c}}_4(t)+\mathcal{G}_2\bar{c}_4(t)&=\mathcal{D}_{\tilde{B}2}\dot{\bar{\nu}}_{\tilde{B}4}(t)+\mathcal{D}_{\tilde{T}2}\dot{\bar{\nu}}_{\tilde{T}4}(t) \nonumber \\
&+\mathcal{F}_{F2}\bar{\nu}_{F4}+\mathcal{F}_{\tilde{B}2}\bar{\nu}_{\tilde{B}4}(t)+\mathcal{F}_{\tilde{T}2}\bar{\nu}_{\tilde{T}4}(t) \label{eq:ffem3d213}
\end{align}
where matrices $\quad \mathcal{H}_2, \mathcal{G}_2, \mathcal{D}_{\tilde{B}2}, \mathcal{D}_{\tilde{T}2}, \mathcal{F}_{F2}, \mathcal{F}_{\tilde{B}2}, \mathcal{F}_{\tilde{T}2}$ are given by (\ref{eq:ffem3d204}).\\
%
Initial conditions are given by (\ref{eq:ffem3d173}) with $i=4$ and $\bar{\nu}_{B4}(0)=0$.\\
From (\ref{eq:ffem3d213}) and (\ref{eq:ffem3d173}) with $i=4$ and $\bar{\nu}_{B4}(0)=0$ we obtain the following system.\\ 
%
\centerline{System of ODEs for TCP V (corresponds to $\cos2\theta$ approximation):}
\begin{align*}
\mathcal{H}_2\dot{\bar{c}}_4(t)+\mathcal{G}_2\bar{c}_4(t)&=\mathcal{D}_{\tilde{B}2}\dot{\bar{\nu}}_{\tilde{B}4}(t)+\mathcal{D}_{\tilde{T}2}\dot{\bar{\nu}}_{\tilde{T}4}(t) \nonumber \\
&+\mathcal{F}_{F2}\bar{\nu}_{F4}+\mathcal{F}_{\tilde{B}2}\bar{\nu}_{\tilde{B}4}(t)+\mathcal{F}_{\tilde{T}2}\bar{\nu}_{\tilde{T}4}(t) \tag{\ref{eq:ffem3d213}}
\end{align*}
with initial conditions
\begin{equation}
\bar{c}_4(0)=\bar{\nu}_{I4}+G_0^{-1}\big[\mathcal{R}_F\bar{\nu}_{F4}+\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}4}(0)+\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}4}(0)\big]  \label{eq:ffem3d214}
\vspace{0.4in}
\end{equation}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Five Systems of ODEs %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We have the following five systems of ODEs:\\[6pt]
%
\centerline{System of ODEs for TCP I (radially symmetric case):}
\begin{align*}
\mathcal{H}_0\dot{\bar{c}}_0(t)+\mathcal{G}_0\bar{c}_0(t)&=\mathcal{D}_{B0}\dot{\bar{\nu}}_{B0}(t)+\mathcal{D}_{\tilde{B}0}\dot{\bar{\nu}}_{\tilde{B}0}(t)+\mathcal{D}_{\tilde{T}0}\dot{\bar{\nu}}_{\tilde{T}0}(t) \nonumber \\
&+\mathcal{F}_{B0}\bar{\nu}_{B0}(t)+\mathcal{F}_{F0}\bar{\nu}_{F0}+\mathcal{F}_{\tilde{B}0}\bar{\nu}_{\tilde{B}0}(t)+\mathcal{F}_{\tilde{T}0}\bar{\nu}_{\tilde{T}0}(t) \tag{\ref{eq:ffem3d170}}
\end{align*}
with initial conditions
\begin{equation*}
\bar{c}_0(0)=\bar{\nu}_{I0}+G_0^{-1}\big[\mathcal{R}_B\bar{\nu}_{B0}(0)+\mathcal{R}_F\bar{\nu}_{F0}+\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}0}(0)+\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}0}(0)\big] \tag{\ref{eq:ffem3d174}}
\end{equation*}
System I gives coefficients $~V_0~,~u_0,~\omega_0~$ of the $~\theta$-approximation (\ref{eq:ffem3d18}). \\ 
\\
%
\centerline{System of ODEs for TCP II (corresponds to $\sin\theta$ approximation):}
\begin{align*}
\mathcal{H}_1\dot{\bar{c}}_1(t)+\mathcal{G}_1\bar{c}_1(t)&=\mathcal{D}_{\tilde{B}1}\dot{\bar{\nu}}_{\tilde{B}1}(t)+\mathcal{D}_{\tilde{T}1}\dot{\bar{\nu}}_{\tilde{T}1}(t) \nonumber \\
&+\mathcal{F}_{F1}\bar{\nu}_{F1}+\mathcal{F}_{\tilde{B}1}\bar{\nu}_{\tilde{B}1}(t)+\mathcal{F}_{\tilde{T}1}\bar{\nu}_{\tilde{T}1}(t) \tag{\ref{eq:ffem3d184}}
\end{align*}
with initial conditions
\begin{equation*}
\bar{c}_1(0)=\bar{\nu}_{I1}+G_0^{-1}\big[\mathcal{R}_F\bar{\nu}_{F1}+\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}1}(0)+\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}1}(0)\big] \tag{\ref{eq:ffem3d186}}
\end{equation*}
%
System II gives $~\sin\theta$-coefficients $~V_1,~u_1,~w_1~$ and $~\cos\theta$-coefficient $~v_2~$ of the $~\theta$-approximation (\ref{eq:ffem3d18}).\\
\\
\centerline{System of ODEs for TCP III (corresponds to $\cos\theta$ approximation):}
\begin{align*}
\mathcal{H}_1\dot{\bar{c}}_2(t)+\mathcal{G}_1\bar{c}_2(t)&=\mathcal{D}_{\tilde{B}1}\dot{\bar{\nu}}_{\tilde{B}2}(t)+\mathcal{D}_{\tilde{T}1}\dot{\bar{\nu}}_{\tilde{T}2}(t) \\
&+\mathcal{F}_{F1}\bar{\nu}_{F2}+\mathcal{F}_{\tilde{B}1}\bar{\nu}_{\tilde{B}2}(t)+\mathcal{F}_{\tilde{T}1}\bar{\nu}_{\tilde{T}2}(t) \tag{\ref{eq:ffem3d194}}
\end{align*}
with initial conditions
\begin{equation*}
\bar{c}_2(0)=\bar{\nu}_{I2}+G_0^{-1}\big[\mathcal{R}_F\bar{\nu}_{F2}+\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}2}(0)+\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}2}(0)\big] \tag{\ref{eq:ffem3d195}}
\end{equation*}
%
System III gives $~\sin\theta$-coefficient $~v_1~$ and $~\cos\theta$-coefficients $~V_2,~u_2,~w_2~$ of the $~\theta$-approximation (\ref{eq:ffem3d18}).\\\\
%
\centerline{System of ODEs for TCP IV (corresponds to $\sin2\theta$ approximation):}
\begin{align*}
\mathcal{H}_2\dot{\bar{c}}_3(t)+\mathcal{G}_2\bar{c}_3(t)&=\mathcal{D}_{\tilde{B}2}\dot{\bar{\nu}}_{\tilde{B}3}(t)+\mathcal{D}_{\tilde{T}2}\dot{\bar{\nu}}_{\tilde{T}3}(t) \nonumber \\
&+\mathcal{F}_{F2}\bar{\nu}_{F3}+\mathcal{F}_{\tilde{B}2}\bar{\nu}_{\tilde{B}3}(t)+\mathcal{F}_{\tilde{T}2}\bar{\nu}_{\tilde{T}3}(t) \tag{\ref{eq:ffem3d203}}
\end{align*}
with initial conditions
\begin{equation*}
\bar{c}_3(0)=\bar{\nu}_{I3}+G_0^{-1}\big[\mathcal{R}_F\bar{\nu}_{F3}+\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}3}(0)+\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}3}(0)\big] \tag{\ref{eq:ffem3d205}}
\end{equation*}
System IV gives $~\sin2\theta$-coefficients $~V_3,~u_3,~w_3~$ and $~\cos2\theta$-coefficient $~v_4~$ of the $~\theta$-approximation (\ref{eq:ffem3d18}).\\
\\
\centerline{System of ODEs for TCP V (corresponds to $\cos2\theta$ approximation):}
\begin{align*}
\mathcal{H}_2\dot{\bar{c}}_4(t)+\mathcal{G}_2\bar{c}_4(t)&=\mathcal{D}_{\tilde{B}2}\dot{\bar{\nu}}_{\tilde{B}4}(t)+\mathcal{D}_{\tilde{T}2}\dot{\bar{\nu}}_{\tilde{T}4}(t) \nonumber \\
&+\mathcal{F}_{F2}\bar{\nu}_{F4}+\mathcal{F}_{\tilde{B}2}\bar{\nu}_{\tilde{B}4}(t)+\mathcal{F}_{\tilde{T}2}\bar{\nu}_{\tilde{T}4}(t) \tag{\ref{eq:ffem3d213}}
\end{align*}
with initial conditions
\begin{equation*}
\bar{c}_4(0)=\bar{\nu}_{I4}+G_0^{-1}\big[\mathcal{R}_F\bar{\nu}_{F4}+\tilde{\mathcal{R}}_B\bar{\nu}_{\tilde{B}4}(0)+\tilde{\mathcal{R}}_T\bar{\nu}_{\tilde{T}4}(0)\big] \tag{\ref{eq:ffem3d214}}
\end{equation*}
%
System V gives $~\sin2\theta$-coefficient $~v_3~$ and $~\cos2\theta$-coefficients $~V_4,~u_4,~w_4~$ of the $~\theta$-approximation (\ref{eq:ffem3d18}).\\[12pt]
%
\indent
Thus, in this chapter, numerical methods for solving fully coupled parabolic-elliptic initial-boundary value problems for two- and three-dimensional TCPu processes, including radially non-symmetric processes in an inclined borehole were developed. These methods are based on a hybrid Fourier-finite-element approximation technique proposed in this research. The technique also involves a boundary penalization strategy and captures borehole geometry and mechanics. The proposed numerical methods reduce the solution of fully coupled parabolic-elliptic initial-boundary value problems to the solution of systems of time-dependent ODEs supplemented by initial conditions. The systems of ODEs explicitly contain data of the underlying boundary conditions, thereby allowing numerical analysis of a control problem in which the temperature and pressure on the borehole boundary are considered as control parameters used to achieve desirable stresses in the neighborhood of the borehole.    
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Experimental Validation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\chapter{Experimental Validation of the Fourier-Finite Element\\ Method} 
%
In this chapter, we experimentally validate the Fourier-finite element method for the TCPu models developed in Chapter 4. Numerical experiments are performed using a MATLAB code that we have written to implement the above numerical method. In this code, the backward Euler approximation is used to discretize the temporal domain. The validation is done by comparing experimental results to other available analytical and previously validated numerical solutions for thermo-chemo-poroelastic wellbore problems. \\
\indent
For our numerical experiments, we focus on the two-dimensional radially symmetric fully coupled TCPu model. In this case, the temperature $T$, the solute mass fraction $C$, and the pore pressure $p$ on the far-field boundary are considered as independent of the polar angle $~\theta$. However, applied far-field boundary stresses remain $~\theta$-dependent in accordance with realistic geomechanical behavior around the borehole. We assume that an annular region exterior to the borehole has the inner (wellbore) radius $~R_B=0.1~\text{m}$ and the outer (far-field) radius $~R_F=1~\text{m}.$ The values of physical constants used in the computations are presented in Appendix A.\\
\indent
Below, we present the experimental results for the temperature $T$, the solute mass fraction $C$, and the pore pressure $p$, as well as for total radial stress $\tau_r$ and total tangential stress $\tau_\theta$ at different polar angles $\theta$. Each figure contains four solution curves that correspond to time intervals of 1 hour, 12 hours, 24 hours, and 120 hours. 
%
\clearpage
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{Temperature1.png}
\caption{Temperature distribution around the borehole}
\label{Temp}
%
\vspace{0.7in}
\includegraphics[scale=1]{Solute1.png}
\caption{Solute mass fraction distribution around the borehole}
\label{Solute}
\end{figure}
%
\clearpage
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{pressure1.png}
\caption{Pore pressure distribution around the borehole}
\label{Pressure}
%
\vspace{0.7in}
\includegraphics[scale=1]{Tot_rad_stress_01.png}
\caption{Total radial stress distribution around the borehole at the polar angle of $\theta=0$}
\label{Trs0}
\end{figure}
%
\clearpage
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{Tot_tan_stress_01.png}
\caption{Total tangential stress distribution around the borehole at the polar angle of $\theta=0$}
\label{Tts0}
%
\vspace{0.7in}
\includegraphics[scale=1]{Tot_rad_stress_301.png}
\caption{Total radial stress distribution around the borehole at the polar angle of $\theta=30$}
\label{Trs30}
\end{figure}
%
\clearpage
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{Tot_tan_stress_301.png}
\caption{Total tangential stress distribution around the borehole at the polar angle of $\theta=30$}
\label{Tts30}
%
\vspace{0.7in}
\includegraphics[scale=1]{Tot_rad_stress_451.png}
\caption{Total radial stress distribution around the borehole at the polar angle of $\theta=45$}
\label{Trs45}
\end{figure}
%
\clearpage
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{Tot_tan_stress_451.png}
\caption{Total tangential stress distribution around the borehole at the polar angle of $\theta=45$}
\label{Tts45}
%
\vspace{0.7in}
\includegraphics[scale=1]{Tot_rad_stress_601.png}
\caption{Total radial stress distribution around the borehole at the polar angle of $\theta=60$}
\label{Trs60}
\end{figure}
%
\clearpage
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{Tot_tan_stress_601.png}
\caption{Total tangential stress distribution around the borehole at the polar angle of $\theta=60$}
\label{Tts60}
%
\vspace{0.7in}
\includegraphics[scale=1]{Tot_rad_stress_901.png}
\caption{Total radial stress distribution around the borehole at the polar angle of $\theta=90$}
\label{Trs90}
\end{figure}
%
\clearpage
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{Tot_tan_stress_901.png}
\caption{Total tangential stress distribution around the borehole at the polar angle of $\theta=90$}
\label{Tts90}
\end{figure}

The obtained experimental results are consistent with available analytical and previously validated numerical solutions for thermo-chemo-poroelastic wellbore problems presented, for instance, in Zhou and Ghassemi \cite{Zhou} or Rafieepour at el. \cite{Rafieepour}. The existing discrepancy between our results and those presented in \cite{Rafieepour, Zhou} is due to the essential difference between the  TCPu model and the models considered in \cite{Rafieepour, Zhou}. The TCPu model includes all coupling mechanisms, whereas both models \cite{Rafieepour, Zhou} deal with fully decoupled thermal diffusion, and \cite{Rafieepour} neglects couplings in the solute diffusion equation, as well. This allows us to conclude that the developed Fourier-finite element method provides correct and satisfactory numerical approximation to the coupled TCPu models.\\

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Conclusions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter {Conclusions and Future Work}
%
This dissertation has addressed the questions of well-posedness and approximation for a specific class of coupled parabolic-elliptic initial-boundary value problems with applications to geomechanics. The underlying thermo-chemo-poroelastic (TCPu) model constitutes the general theory of chemical thermo-poro-elasticity that fully couples thermal, chemical, hydrolic, and mechanical processes in homogeneous isotropic fluid-saturated porous media. The main results obtained within the dissertation research are summarized below. These results are fundamental to posing optimal control problems using borehole parameters as controls to achieve desired stress distributions in the neighborhood of the borehole. \\[12pt]
%
\indent
A sufficient condition for well-posedness in the weak sense of the coupled parabolic-elliptic initial-boundary value problems for the two-and three-dimensional TCPu models has been established. This condition depends on physical parameters of the system, coupling vectors, and the Korn's constant. For a two-dimensional annular region that does not experience pure rotation and has the ratio of inner to outer radii not exceeding $e^{-1}$, the sufficient condition for the well-posedness of the problem has been formulated in terms of physical parameters, coupling vectors, and the ratio of the inner and outer radii. The obtained results have provided a mathematical basis for the general theory of fully coupled chemo-thermo-poroelasticity in homogeneous isotropic porous media saturated by a fluid. \\[12pt]
%
\indent
Numerical methods for solving fully coupled parabolic-elliptic initial-boundary value problems for two- and three-dimensional TCPu processes, including radially non-symmetric processes in an inclined borehole have been developed. These methods are based on a hybrid Fourier-finite-element approximation technique proposed in this research. The technique also involves a boundary penalization strategy and captures borehole geometry and mechanics. The developed numerical methods allow numerical analysis of a control problem in which the temperature and pressure on the borehole boundary are considered as control parameters used to achieve desirable stresses in the neighborhood of the borehole. \\[12pt]
%
\indent
The experimental validation of the developed numerical method was performed in MATLAB using the two-dimensional radially symmetric fully coupled TCPu model with real data. The experimental results have confirmed that the hybrid Fourier-finite-element method provides correct and satisfactory numerical approximation to the coupled TCPu models for the purpose of boundary control and comprehensive numerical investigation of the corresponding fully coupled reservoir-geomechanical processes that impact borehole stability. \\[12pt]
%
\indent
There are several promising directions for further research on coupled parabolic-elliptic systems with applications to geomechanics:
\begin{enumerate}[leftmargin = 1.2em]
\item An open question that immediately arises from this dissertation is establishing necessary conditions for well-posedness of the coupled parabolic-elliptic initial-boundary value problems for the two- and three-dimensional TCPu models. 
\item The second direction is the development of a well-posedness theory and numerical methods for coupled parabolic-elliptic systems with variable coefficients. The variability of coefficients results in nonlinearity of the coupled system but allows modeling fully coupled reservoir-geomechanical processes in inhomogeneous fluid-saturated porous media.
\item The other project is the development of the optimal boundary control theory for systems governed by coupled linear and nonlinear parabolic-elliptic partial differential equations. This research direction is motivated by geomechanics and wellbore stability problems in which the temperature and pressure at the borehole are considered as control parameters used to obtain desirable stresses in the neighborhood of the borehole. 
\item Another project is also dictated by the geomechanical nature of the problem and includes the development of the mathematical theory of fully coupled thermo-chemo-poroelastic fracture initiation and propagation processes in fluid-saturated porous media.
\end{enumerate}

%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Bibliography %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\addcontentsline{toc}{chapter}{Bibliography}
\singlespacing
\begin{thebibliography}{99}
%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendices %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\addcontentsline{toc}{chapter}{Appendices}
\doublespacing
\appendix 
\chapter {Physical Parameters of the Coupled TCPu System} \label{App:AppendixA}
%
The coefficients in the matrices and vectors (\ref{eq:uneq2d17}) and (\ref{eq:uneq2d30}),
%
\begin{equation*}
M=\begin{bmatrix} \\[-2.4em]
\Lambda & \Sigma & \Phi \\[-6pt]
0 & \phi & 0 \\[-6pt]
\Gamma & \chi & \Psi
\end{bmatrix}, \qquad 
A=\begin{bmatrix} \\[-1.6em]
\dfrac{k^T}{T_F} & \dfrac{\rho_fRT_FD^T}{M^s(1-C_F)} & -K^T \\[9pt]
C_FD^T & D & -\dfrac{k\mathfrak{R}}{\eta} \\[9pt]
-K^T & -\dfrac{\rho_f\mathfrak{R}\Omega k}{\eta} & \dfrac{k}{\eta}
\end{bmatrix}
\vspace{0.1in}
\end{equation*}
%
\begin{equation*}
\vec{b}_0=\begin{bmatrix} \\[-2.4em]
\zeta \\[-6pt]
0 \\[-6pt]
\alpha
\end{bmatrix}, \qquad   
\vec{b}_1=\begin{bmatrix} \\[-2.4em]
\tilde{\zeta} \\[-6pt]
-\xi \\[-6pt]
\tilde{\alpha}
\end{bmatrix}
\end{equation*}\\[-6pt]
are the following:
%
\begin{gather*}
\Lambda = \frac{c}{T_F}-s_F\beta_s\tilde{\omega}, \qquad c=(1-\phi)\rho_sc_s+\phi\rho_fc_f, \quad s_F=c_f-\frac{\beta_fp_F}{\rho_f}, \quad \tilde{\omega}=\frac{\omega}{1-C_F} \nonumber \\[6pt]
\Sigma=\xi\beta_s, \qquad \xi=\frac{RT_F\tilde{\omega}(1-2C_F)}{M^sC_F} \nonumber \\[6pt]
\Phi=\Xi-\phi \beta_f+\frac{\beta_s\tilde{\omega}}{\rho_f}, \qquad \Xi=(\alpha-\phi)\beta_s, \qquad \alpha=1-\frac{K}{K_s} \nonumber \\[6pt]
\Gamma = \Xi-\phi\beta_f-\frac{s_F\tilde{\omega}}{K_s}, \qquad \chi=-\frac{\xi}{K_s}\nonumber \\[6pt]
\Psi= \psi +\frac{\phi}{K_f}-\frac{\tilde{\omega}}{\rho_fK_s}, \qquad \psi=\frac{\alpha-\phi}{K_s} \nonumber \\[6pt]
\Omega = \frac{RT_F}{M^s(1-C_F)C_F} \nonumber \\[6pt]
\zeta=K\beta_s, \qquad \tilde{\zeta}=\zeta+s_F\tilde{\omega}, \qquad \tilde{\alpha}=\alpha-\frac{\tilde{\omega}}{\rho_f}
\end{gather*}
where $~s_F~$ is specific fluid entropy, respectively, $~R~$ is the universal gas constant, $~\alpha~$ is Biot effective stress coefficient, and $~K~$ is the rock's bulk modulus. Other physical properties of the rock/fluid system and input parameters for examples are given in Table \ref{table:phys_prop} and Table \ref{table:input_param}, respectively. 

\begin{table}[ht] 
\caption{Physical properties of the rock/fluid system}
\begin{center}
\begin{tabular}{|l|c|}
\hline
Drained elastic modulus, $~E$ & $45\times 10^9$ Pa \\
\hline
Solid bulk modulus, $~K_s$ & $65\times 10^9$ Pa \\
\hline  
Drained Poisson's ratio, $~\nu$ & 0.27 \\
\hline
Permeability/Fluid viscosity ratios, $k/\eta$ & $10^{-17}$ m$^2$/(Pa$\cdot$s) \\
\hline
Reference porosity, $\phi$ & 0.15 \\
\hline
Fluid density, $\rho_f$ & 1111 kg/m$^3$ \\
\hline
Fluid specific heat capacity, $~c_f$ & 4186 J/(Kg$^{\circ}$K) \\
\hline
Fluid bulk modulus, $~K_f$ & $3.3 \times 10^9$ Pa \\
\hline
Fluid volumetric thermal expansion coefficient, $~\beta_f$ & $3\times 10^{-4 ~\circ}$K$^{-1}$ \\
\hline
Solid density, $~\rho_s$ & $2.83 \times 10^3$ kg/m$^3$ \\
\hline
Solid specific heat capacity, $~c_s$ & 920 J/(Kg$^{\circ}$K) \\
\hline
Solid volumetric thermal expansion coefficient, $~\beta_s$ & $2.4 \times 10^{-5 ~\circ}$K$^{-1}$ \\
\hline
Rock thermal conductivity coefficient, $~k^T$ & 4 W/(m$^{\circ}$K) \\
\hline
Thermal osmosis coefficient, $~K^T$ & $10^{-11}$ m$^2$/(s$^{\circ}$K) \\
\hline
Solute molar mass, $~M^s$ & 0.1111 kg/m$^3$ \\
\hline
Solute reflected fraction, $~\mathfrak{R}$ & 0.01 \\
\hline
Solute chemical diffusion coefficient, $~D$ & $10^{-9}$ m$^2$/s \\
\hline
Solute thermal diffusion coefficient, $~D^T$ & $10^{-10}$ m$^2$/(s$^{\circ}$K) \\
\hline
Chemical stress coupling parameter, $~\omega$ & 100 kg/m$^3$ \\
\hline
Universal gas constant, $~R$ & 8.3 JK$^{-1}$/mol$^{-1}$ \\
\hline
\end{tabular}
\end{center} 
\label{table:phys_prop} 
\end{table} 
%
\begin{table}[ht!] 
\caption{Input parameters for experimental validation}
\begin{center}
\begin{tabular}{|l|c|}
\hline
In situ hydrostatic stress, & 91MPa (Therm. ex.) \\
$\sigma_x=\sigma_y$ & 88.5 MPa (Chem. ex.)\\
\hline
In situ non-hydrostatic stress  & $\sigma_x= 91$ MPa \\
& $~\sigma_y=86$ MPa \\
\hline
In situ formation pore pressure, $~p_F$ & 52 MPa \\
\hline
Mud pressure, $~p_B$ & 60 MPa \\
\hline
Mud solute mass fraction, $~C_B$ & 0.2 \\
\hline
Reference formation solute mass fraction, $~C_F$ & 0.1 \\
\hline
In situ formation temperature, $~T_F$ & 135$^\circ$C \\
\hline
Mud temperature, $~T_B$ & 85 $^\circ C$ \\
\hline
Wellbore radius, $R_B$ & 0.1 m \\
\hline
\end{tabular} 
\end{center}
\label{table:input_param} 
\end{table} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

