Reduced Basis

Heat Diffusion on a 3D thermal fin $\Omega$: Heat flux enter from $\Gamma_{root}$ and outs $\Gamma_{top}$. (Download all here : RBCODES.zip )

Thermal fin is represented as a union of blocks $\Omega =\cup B_b$ for $b=0,1,2,...,11$. Each block with conductivity $\kappa_b>0$   Thermal Fin Conductivities Triangulation of Thermal Fin

Heat flux Input on root is 1 ...... Output to measure is $s(u)=\int_{\Gamma_{top}} u(x)dx$

\begin{array}{rcl} -\kappa_b\Delta u & =& 0 \quad\mbox{on each block } B_b,\quad b=1,2,...\\ -\kappa_b\frac{\partial u}{\partial n}&=& Bi \cdot\,u \quad \mbox{on } \Gamma_b=\partial B_b\cap \partial\Omega,\quad b=1,2,... (Bi>0 \mbox{ Biot number})\\ -\kappa_b\frac{\partial u}{\partial n}&=& -q \quad \mbox{on } \Gamma_{root} \end{array}

Variational formulation:

Find $u\in H^1(\Omega)$ such that for all test $v\in H^1(\Omega)$ \begin{array}{rcl} \sum_{b=0}^{11}\kappa_b\int_{B_b}\nabla u \cdot \nabla v+Bi\sum_{b=0}^{11}\int_{\Gamma_b} u \,v& =& \int_{\Gamma_{root}} q \,v. \end{array}  Example of solution (using medit) Example of solution (using plot)