## 3 Finite Element Method
### The \({}_{1}\)I Local Problem: Weak Formulation  
The differential equation used to calculate the thermal conductivity for the problem \({}_{1}\)I can be obtained from Equation (13) and it is given by  
\[(\kappa_{11}+\kappa_{11}\;{}_{1}M_{,1})_{,1}+(\kappa_{22}\;{}_{1}M_{,2})_{,2}=0. \tag{25}\]  
Perfect contact conditions (14) for problem \(qI\) are given by  
\[||(\kappa_{11}+\kappa_{11}\;{}_{1}M_{,1})_{n}+\kappa_{22}\;{}_{1}M _{,2}n_{2}||_{\Gamma} = 0,\] \[||\kappa_{11}n_{1}+\kappa_{11}n_{1}+\tau_{22}n_{2}||_{\Gamma} = 0,\] \[||\tau_{11}n_{1}+\tau_{22}n_{2}||_{\Gamma} = -||\kappa_{11}n_{1}||_{\Gamma}, \tag{26}\]  
where \(\tau_{11}=\kappa_{11}\;{}_{1}M_{,1}\), \(\tau_{22}=\kappa_{22}\;{}_{1}M_{,2}\) and \(||g||\) denotes the discontinuity or jump of the function \(g\) across the interface. The jump at the interface \(\Gamma\) for the conductivity problem is defined by  
\[||\kappa_{11}n_{1}||_{\Gamma}=(\kappa_{11}^{m}-\kappa_{11}^{f})n_{1},\]  
where \(\kappa_{11}^{m}\) and \(\kappa_{11}^{f}\) are the matrix and fiber thermal conductivity \(\kappa_{11}\) coefficient, respectively. By using symmetry and periodicity conditions, the boundary conditions for problem \({}_{1}\)I are obtained  
\[{}_{1}M=0,\quad\text{in}\;\partial\Omega_{L},\quad{}_{1}M=0, \quad\text{in}\;\partial\Omega_{R},\] \[{}_{1}\tau_{2}=0,\quad\text{in}\;\partial\Omega_{U},\quad{}_{1} \tau_{2}=0,\quad\text{in}\;\partial\Omega_{D},\]  
where \(\partial\Omega_{L}\), \(\partial\Omega_{R}\), \(\partial\Omega_{U}\) and \(\partial\Omega_{D}\) represent the left, right, upper and lower boundaries of the cell quarter. Assuming that we look for the approximate solution \(\tilde{M}\approx M\) of the problem (25) within a certain space of functions \(\mathcal{H}\) with dimension \(n\),  
\[\tilde{M}=\mathbf{N}\mathbf{q}, \tag{27}\]  
where \(\mathbf{N}=[N_{1}\quad N_{2}\;\cdots\;N_{n}]\) is a basis of \(\mathcal{H}\) and \(\mathbf{q}=[q_{1}\quad q_{2}\;\cdots\;q_{n}]^{t}\) are the nodal values of the unknown field. The weak formulation of problem \({}_{1}\)I is obtained by applying the weighted residual method and [PERSON]'s method [30; 31]. The differential problem (25) becomes  
\[\sum_{\xi=1}^{n}\int\int_{\Omega^{\varepsilon}}\mathbf{D}^{t}\; \mathbf{C}\;\mathbf{D}\;d\Omega^{\varepsilon}\;\mathbf{q} = \sum_{\xi=1}^{n}\int\int_{\Omega^{\varepsilon}}\frac{\partial \kappa_{11}}{\partial y_{1}}\mathbf{N}^{t}\;d\Omega^{\varepsilon}+\sum_{\xi=1 }^{n}\oint_{\Gamma^{\varepsilon}}(\tau_{11}\mathbf{n}_{1}+\tau_{22}\mathbf{n}_{ 2})\mathbf{N}^{t}\;d\Gamma^{\varepsilon}, \tag{28}\]where  
\[\mathbf{D}=\left( \begin{array}{c}\frac{\partial\mathbf{N}}{ \partial y_{1}}\\ \frac{\partial\mathbf{N}}{\partial y_{2}}\end{array} \right), \mathbf{C}=\left[ \begin{array}{cc}\kappa_{11}&0\\ 0&\kappa_{22}\end{array} \right],\] \[\bar{\tau}_{11}=\kappa_{11}\frac{\partial\tilde{M}}{\partial y_ {1}}, \bar{\tau}_{22}=\kappa_{22}\frac{\partial\tilde{M}}{\partial y_{2}},\]  
\(\bar{\tau}_{11}\) and \(\bar{\tau}_{22}\) represent the jump between different materials at the interface \(\Gamma\) and \(\mathrm{n}_{1},\mathrm{n}_{2}\) constitute the components of the unitary normal vector at the interface. The stiffness matrix, the force vector and the traction at the boundary of each element can be obtained from Equation (28)  
\[\mathbf{K}^{\epsilon} = \int\int_{\Gamma^{\epsilon}}\mathbf{D}^{t}\ \mathbf{C}\ \mathbf{D}\ d\Omega^{ \epsilon}, \tag{29}\] \[\mathbf{F}^{\epsilon} = \int\int_{\Gamma^{\epsilon}}\frac{\partial\kappa_{11}}{\partial y _{1}}\mathbf{N}^{t}d\Omega^{\epsilon},\] (30) \[\mathbf{B}^{\epsilon} = \oint_{\Gamma^{\epsilon}}(\bar{\tau}_{11}\mathrm{n}_{1}+\bar{ \tau}_{22}\mathrm{n}_{2})\mathbf{N}^{t}\ d\Gamma^{\epsilon}. \tag{31}\]  
Expressions (29)-(31) are the element stiffness matrix, the element force vector and the integral vector that corresponds to the boundary of each element. The vector \(\mathbf{B}^{\epsilon}\) must be evaluated only at the edges that belong to the interface \(\Gamma\). Equation (26) is used to determine the vector \(\mathbf{B}^{\epsilon}\).