Steady State MultiGroup Diffusion for ANL11-A2 benchmark
The ANL11-A2 benchmark is taken from the Argonne Code Center - Supplement 2. It’s 2D version of a Pressurised Water Reactor with 4 regions as depicted in the following figure.
The governing equations are the eigenvalue problem for the 2-groups Neutron Diffusion:
\[\begin{split}\left\{
\begin{array}{ll}
-\nabla\cdot \left(D_1\nabla \phi_1 \right)+(\Sigma_{a,1}+\Sigma_{s,1\rightarrow 2} + D_1\,B_{z,1}^2)\phi_1 =\frac{1}{k_{eff}}\nu\Sigma_{f,2}\phi_2 & \mathbf{x}\in\Omega\\
-\nabla\cdot \left(D_2\nabla \phi_2 \right)+(\Sigma_{a,2}+D_2\,B_{z,2}^2) \phi_2 - \Sigma_{s,1\rightarrow 2}\phi_1=0 & \mathbf{x}\in\Omega\\
\nabla \phi_g\cdot \mathbf{n} = -\frac{0.4692}{D_g}\,\phi_g & \mathbf{x}\in\Gamma_{void} \\
\nabla \phi_g\cdot \mathbf{n} = 0 & \mathbf{x}\in\Gamma_{sym}
\end{array}
\right.\end{split}\]
given \(\Omega\) as the domain and \(\partial\Omega\) as its boundary, composed by \(\partial\Omega = \Gamma_{sym}\cup\Gamma_{void}\) where \(\Gamma_{sym}\) is the symmetry boundary and \(\Gamma_{void}\) is the end of the reactor.