6. Spatial discretization

Let \({P}_{h}\) be a division of space \(\Omega\), let’s designate by \(N\) the number of nodes in the mesh, \({p}_{i}\) the shape function associated with the node \(i\). \(J\) refers to all the nodes belonging to the \({\Gamma }_{1}\) border.

Let’s be:

\(\begin{array}{}{V}_{{t}^{+}}^{h}=\{v=\sum _{i=\mathrm{1,}N}{v}_{i}{p}_{i}(x)\text{};\text{}{v}_{j}={T}_{1}({x}_{j},{t}^{+})\text{}j\in J\}\\ {V}_{{t}^{-}}^{h}=\{v=\sum _{i=\mathrm{1,}N}{v}_{i}{p}_{i}(x)\text{};\text{}{v}_{j}={T}_{1}({x}_{j},{t}^{-})\text{}j\in J\}\\ {V}_{0}^{h}=\{v=\sum _{i=\mathrm{1,}N}{v}_{i}{p}_{i}(x)\text{};\text{}{v}_{j}=0\text{}j\in J\}\end{array}\)

Let’s say:

\(\begin{array}{}{K}_{\text{ij}}{T}_{i}=\underset{{\Omega }_{h}}{\int }\frac{\rho {C}_{p}}{\Delta t}{T}_{i}{p}_{i}{p}_{j}d{\Omega }_{h}+\underset{{\Omega }_{h}}{\int }\theta \lambda {T}_{i}\nabla {p}_{i}\text{.}\nabla {p}_{j}d{\Omega }_{h}+\underset{{\Gamma }_{\mathrm{h3}}}{\int }\theta {h}^{+}{T}_{i}{p}_{i}d{\Gamma }_{\mathrm{h3}}\\ \\ \begin{array}{cc}{L}_{j}=& \underset{{\Omega }_{h}}{\int }\frac{\rho {C}_{p}}{\Delta t}{T}^{-}{p}_{j}d{\Omega }_{h}-\underset{{\Omega }_{h}}{\int }(1-\theta )\lambda \nabla {T}^{-}\text{.}\nabla {p}_{j}d{\Omega }_{h}+\underset{{\Gamma }_{\mathrm{h2}}}{\int }{f}^{\theta }{p}_{j}d{\Gamma }_{\mathrm{h2}}\\ & +\underset{{\Gamma }_{\mathrm{h3}}}{\int }(({\text{hT}}_{\text{ext}}{)}^{\theta }-(1-\theta ){h}^{-}{T}^{-}){p}_{j}d{\Gamma }_{\mathrm{h3}}+\underset{{\Omega }_{h}}{\int }(\theta {s}^{+}+(1-\theta ){s}^{-}){p}_{j}d{\Omega }_{h}\end{array}\\ \end{array}\)

By dualizing the limit conditions in terms of imposed temperature ([R3.03.01]), we make appear the operator \(B\) defined by:

\((\text{Bv}{)}_{j}=\{\begin{array}{ccc}0& \text{si}& j\notin J\\ {v}_{j}& \text{si}& j\notin J\end{array}\)

The following system is finally obtained:

\(\{\begin{array}{ccccccc}\sum _{i=1}^{N}{K}_{\text{ij}}{T}_{i}& +& {({}^{t}\text{}B\lambda )}_{j}& =& {L}_{j}& & \forall j\\ & & {(\mathrm{BT})}_{j}& =& {T}_{1}({x}_{j},t)& & j\in J\end{array}\)