In part I, we considered the application of the method of fundamental solutions (MFS) for solving numerically the Brinkman fluid flow in the unbounded porous medium outside obstacles of known or unknown shapes. In this companion paper we consider the corresponding interior problem for the Brinkman flow in a bounded porous medium which contains an unknown rigid inclusion \(D \subset \Omega \). The inclusion D is to be identified by a pair of Cauchy data represented by the fluid velocity and traction on the boundary \(\partial \Omega \). The fluid velocity and pressure of the incompressible viscous flow in the porous medium \(\Omega \backslash \overline{D}\) are approximated by linear combinations of fundamentals solutions for the Brinkman system with sources (singularities) placed outside the closure of the solution domain, i.e. in \(D \cup \big ({\mathbb {R}}^2\backslash \overline{\Omega } \big )\), assuming, for simplicity, that we analyse planar domains. By further assuming that the unknown obstacle D is star-shaped (with respect to the origin), the inverse problem recasts as the minimization of the nonlinear Tikhonov’s regularization functional with respect to the MFS expansion coefficients and the discretized polar radii defining D. This minimization subject to simple bounds on the variables is solved numerically using the MATLAB

在第一部分中，我们考虑了基本解法（MFS）在数值上求解已知或未知形状的障碍物之外的无边界多孔介质中的Brinkman流体流的应用。在这篇伴随论文中，我们考虑了有界多孔介质中Brinkman流的相应内部问题，该介质包含未知的刚性夹杂物\（D \ subset \ Omega \）。夹杂物D将通过一对柯西数据来识别，该柯西数据由边界\（\ partial \ Omega \）上的流体速度和牵引力表示。多孔介质中不可压缩粘性流的流体速度和压力\（\ Omega \反斜杠\ overline {D} \）通过Brinkman系统基本解决方案的线性组合近似得到，其中源（奇点）位于解决方案域的封闭之外，即\（D \ cup \ big（{\ mathbb {R}} ^ 2 \反斜杠\ overline { \ Omega} \ big）\），为简单起见，假设我们分析平面域。通过进一步假设未知障碍物D为星形（相对于原点），逆问题将重现为非线性Tikhonov正则化函数相对于MFS膨胀系数和定义D的离散极化半径的最小值。使用MATLAB以数值方式解决了对变量进行简单限制的最小化问题