1.3. Multi-materials
Given a domain \Omega \subset \mathbb{R}^d, d=1,2,3, \Omega is partitioned into N_r regions \Omega_i,i=1,\ldots,N_r corresponding to different materials (solid or fluid). We consider \rho_i, C_{p,i} and k_i the material properties defined in each regions \Omega_i. We define also \boldsymbol{n}_i the outward unit normal vector associated to the boundary \partial \Omega_i.
\begin{eqnarray}
\rho_i C_{p,i} \frac{\partial T}{\partial t} - \nabla \cdot \left( k_i \nabla T \right) &=& Q, \quad &\text{ in }& \Omega_i,i=1,\ldots,N_r \\
T_{|_{\Omega_i}} &=& T_{|_{\Omega_j}}, \quad &\text{ on }& \partial \Omega_i \cap \Omega_j = \Gamma_{ ij}, \forall i \neq j \\
-k_i \nabla T \cdot \boldsymbol{n}_i &=& k_j \nabla T \cdot \boldsymbol{n}_j, \quad &\text{ on }& \partial \Omega_i \cap \Omega_j = \Gamma_{ ij}, \forall i \neq j \\
\end{eqnarray}
We assume the operator \mathcal{L} tel que \mathcal{L} T = \rho_i C_{p,i} \frac{\partial T}{\partial t} - \nabla \cdot \left( k_i \nabla T \right) is elliptical.
We multiply \mathcal{L} u = Q by a function test v \in \mathbf{V} and integrates by part on \Omega_i. Which give:
\rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{\partial T}{\partial t} v - \int_{\Omega_i} \nabla \cdot \left[ k_i \nabla T \right] v = \int_{\Omega_i} Qv, \quad \forall v \in H^1_0(\Omega) \quad for i = 1, \cdots , N_r
By the formula of Green, we get
\rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{\partial T}{\partial t} v + \int_{\Omega_i} k_i(y) \nabla T \cdot \nabla v- \int_{\partial \Omega} k_i \nabla T \cdot \boldsymbol{n}_i v = \int_{\Omega_i} Qv \quad \forall v \in \mathbf{V}
Additivity of the integral, we have
\sum_{ i=1}^{N_r} \left( \rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{\partial T}{\partial t} v + \int_{\Omega_i} k_i \nabla T \cdot \nabla v- \int_{\partial \Omega_i} k_i \nabla T \cdot \boldsymbol{n}_i v \right) = \sum_{ i=1}^{N_r} \left( \int_{\Omega_i} Qv \right) \forall v \in \mathbf{V}
\bigcup_{ i=1}^{ N } \partial \Omega_i = \bigcup_{ i,j} \Gamma_{ ij} \cup \partial \Omega
Use the conditions in the interfaces, we get
\sum_{ i=1}^{N_r} \left( \rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{\partial T}{\partial t} v + \int_{\Omega_i} k_i \nabla T \cdot \nabla v- \int_{\partial \Omega} k_i \nabla T \cdot \boldsymbol{n} v \right) = \sum_{ i=1}^{N_r} \left( \int_{\Omega_i} Qv \right) \forall v \in \mathbf{V}
Using the implicit Euler method for the time term:
\frac{\partial T}{\partial t} (t^{ k+1}) \approx \frac{ T (t^{ k+1}) - T(t^k)}{ dt} \quad \forall t^k \in \mathbb{ R^+} \text{ et } k \in \mathbb{N}
Denoting T^k = T(t^k), we write the formula in t^{ k+1}, we obtain:
\sum_{ i=1}^{N_r} \left( \rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{ T^{k+1}}{dt} v + \int_{\Omega_i} k_i \nabla T^{k+1} \cdot \nabla v - \int_{\partial \Omega} k_i \nabla T^{k+1} \cdot \boldsymbol{n} v \right) = \sum_{ i=1}^{N_r} \left( \int_{\Omega_i} \frac{T^{k}}{dt} v + \int_{\Omega_i} Qv \right) \quad \forall v \in \mathbf{V}
So, the weak wording becomes:
The weak formulation
\text{ On cherche } T \in \mathbf{H} \text{ telle que:}
\\
a(T^{k+1}, v) = l(v) \quad \forall v \in \mathbf{V}
\\
\text{ and} \quad
a(T^{k+1}, v) = \sum_{ i=1}^{N_r} \left( \rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{ T^{k+1}}{dt} v + \int_{\Omega_i} k_i \nabla T^{k+1} \cdot \nabla v - \int_{\partial \Omega} k_i \nabla T^{k+1} \cdot \boldsymbol{n} v \right)
\\
l(v) = \sum_{ i=1}^{N_r} \left( \int_{\Omega_i} \frac{T^{k}}{dt} v + \int_{\Omega_i} Qv \right)
So we have a(u_{k+1},v) a continuous bilinear form coercive in v \in \mathbf{V} and l(\phi) a continuous linear form . We are in a Hilbert space, so we have all the conditions for the application of the Lax-Milgram theorem. So this problem is well posed.
We use the Galerkin approximation method:
Let \{ \mathcal{T}_h \} a family of meshes of :\Omega.
Let \{ \mathcal{K}, P, \sum \} a finite element of Lagrange of reference of the degree k \geq 1.
Let P^k_{c,h} the conforming approximation space defined by
P^k_{ c,h} = \{ v \in C^0(\Omega), \forall \mathcal{K} \in \mathcal{T}_h, v|_{\mathcal{K}} \in \mathbb{P}_k(\mathcal{K}) \}
To obtain a conformal approximation in V, we add the boundary conditions
Discrete problem is written:
Problème discrète
\text{ Find } T_h \in V_h \text{ such that}
\\
a(T_h, v_h) = l(v_h) \quad \forall v_h \in V_h
Let \{ \varphi_1, \varphi_2, ..., \varphi_N \} the base of V_h. An element T_h \in V_h is written as
T_h = \sum^{N}_{l=1} T_l \varphi_l
Using v as a basic function of V_h, our problem becomes
\sum_{ i=1}^{N_r} \left( \rho C_{p,i} \displaystyle \int_{\Omega_{i}} \sum_{ l=1}^N T^{k+1}_l \frac{ \varphi_l }{dt} \varphi_j + \int_{\Omega_i} k_i \sum_{ l=1}^N T^{k+1}_l \nabla \varphi_l \cdot \nabla \varphi_j - \int_{\partial \Omega} k_i \sum_{ l=1}^N T^{k+1}_l \nabla \varphi_l \cdot \boldsymbol{n} \varphi_j \right) = \sum_{ i=1}^{N_r} \left( \int_{\Omega_i} \sum_{ l=1}^N T^{k}_l \frac{ \varphi_l }{dt} \varphi_j + \int_{\Omega_i} Q \varphi_j \right)
The variational problem of approximation is then equivalent to a linear system
Algebraic problem
\text{Determine } T_l \text{ satisfying}
\\
\sum_{ l=1}^N a(\varphi_l, \varphi_j) T^{k+1}_l = l(\varphi_j) \forall j = 1, \cdots , N
A = (a(\varphi_i , \varphi_j)), \quad 1 \leq i,j \leq N ,
\\
U^{k+1} = (T_1^{k+1}, T_2^{k+1}, ..., T_N^{k+1}) \in \mathbb{R}^{N},
\\
F = (l(\varphi_1), l(\varphi_2), ..., l(\varphi_N)) \in \mathbb{R}^{N}
We write the system in matrix form