## 3 Finite Element Calculations
### Boundary Conditions
#### 3.2.2 Thermal Conductivity  
For the thermal problem, the temperature, its gradient, and the heat flux vector are denoted by \(T\), \(\
abla T\), and \(q\) respectively. The heat flux vector and the temperature gradient are related by [PERSON]'s law, which reads:  
\[\underline{q}=-\lambda\
abla T \tag{19}\]in the isotropic case. The scalar \(\lambda\) is the thermal conductivity coefficient of the considered phase. A volume \(V\) of heterogeneous material is considered again. Three types of boundary conditions are used in the study of the effective thermal conductivity:  
* Uniform gradient of temperature at the boundary (UGT): \[T=\underline{G}\cdot\underline{x}\quad\forall\underline{x}\in\partial V\] (20) \(\underline{G}\) is a constant vector independent of \(\underline{x}\). This implies that: \[\langle\underline{\
abla T}\rangle=\frac{1}{V}\int_{V}\underline{\
abla T}\, \mathrm{d}V=\underline{G}\] (21) The macroscopic flux vector is defined by the spatial average: \[\underline{Q}=\langle\underline{q}\rangle=\frac{1}{V}\int_{V}\underline{q}\, \mathrm{d}V\] (22)
* Uniform heat flux at the boundary (UHF): \[\underline{q}\cdot\underline{n}=\underline{Q}\cdot\underline{x}\quad\forall \underline{x}\in\partial V\] (23) \(\underline{Q}\) is a constant vector independent of \(\underline{x}\). This implies that: \[\langle\underline{q}\rangle=\frac{1}{V}\int_{V}\underline{q}\,\mathrm{d}V= \underline{Q}\] (24) The macroscopic temperature gradient is given by the spatial average: \[\underline{G}=\langle\
abla T\rangle=\frac{1}{V}\int_{V}\
abla T\,\mathrm{d}V\] (25)
* Periodic boundary conditions (PERIODIC): the temperature field takes the form \[T=\underline{G}\cdot\underline{x}+t\quad\forall\underline{x}\in V\] (26)  
Apparent conductivities coincide with the wanted effective properties for sufficiently large volumes \(V\). To evaluate the effective thermal conductivity under both UGT and periodic boundary conditions, the procedure entails defining specific test temperature gradients and fluxes imposed on the elementary volume, outlined as follows:  
\[\underline{G}=\begin{pmatrix}1\\ 1\\ 1\end{pmatrix}\quad\text{and}\quad\underline{Q}=\begin{pmatrix}1\\ 1\\ 1\end{pmatrix} \tag{27}\]  
They are used respectively to define the following _apparent conductivities_:  
\[\lambda^{\mathrm{app}}=\frac{1}{3}\,\mathrm{trace}\langle\underline{q}\rangle \quad\text{and}\quad\lambda^{\mathrm{app}}=\frac{1}{3}\,\mathrm{trace}\langle \
abla\underline{T}\rangle \tag{28}\]  
After an extensive comparison of results derived from various methods, we decided to employ the periodic method for both linear elasticity and thermal conductivity analyses. This method demonstrated a significant advantage in computation speed while delivering results that were on par with those obtained from other methods.  
Beyond its computational efficiency, the periodic method also excels at accurately calculating the elastic properties of complex materials. Its relative simplicity in implementation and its versatility make it suitable for a diverse array of materials and geometries. This combination of speed, accuracy, and ease of application underscores the periodic method's utility in advanced material analysis.