The method of fundamental solutions (MFS) is developed for solving numerically the Brinkman flow in the porous medium outside obstacles of known or unknown shapes. The MFS uses the fundamental solution of the Brinkman equation as in the boundary element method (BEM), but the single-layer representation is desingularized by moving the boundary sources to fictitious points outside the solution domain. In the case of unbounded flow past obstacles, these source points are placed in the domain inside the obstacle on a contracted fictitious pseudo-boundary. When the obstacle is known, then the fluid flow in porous media problem is direct, linear and well-posed. In the case of Brinkman flow in the porous medium outside an infinitely long circular cylinder, the MFS numerical solution is found to be in very good agreement with the available analytical solution. However, when the obstacle is unknown and has to be determined from fluid velocity measurements at some points inside the fluid, the resulting problem becomes inverse, nonlinear and ill-posed. The \(\hbox {MATLAB}^{\copyright }\) optimization toolbox routine lsqnonlin is employed for minimizing the least-squares gap between the computed and measured fluid velocity which is further penalized with extra smoothness regularization terms in order to overcome the instability of the solution. For proper choices of the regularization parameters involved, accurate and stable numerical reconstructions are achieved for various star-shaped obstacles.

开发了基本解法（MFS），用于数值求解已知或未知形状的障碍物外部的多孔介质中的Brinkman流。MFS使用边界元素方法（BEM）中的Brinkman方程的基本解，但是通过将边界源移动到解域之外的虚拟点，单层表示被分解为单数形式。在无障碍流经过障碍物的情况下，这些源点将在收缩的虚拟伪边界上放置在障碍物内部的区域中。当已知障碍物时，则多孔介质中的流体流动问题是直接的，线性的且位置良好。如果布林克曼流在无限长圆柱体外部的多孔介质中，发现MFS数值解与可用的解析解非常吻合。但是，当障碍物是未知的并且必须根据流体内部某些点的流速测量来确定时，所产生的问题将变成逆向，非线性和不适定。的\（\ hbox {MATLAB} ^ {\ copyright} \）优化工具箱例程lsqnonlin用于最小化计算和测得的流体速度之间的最小二乘差距，并用额外的平滑度正则项进一步惩罚，以克服解决方案。为了正确选择所涉及的正则化参数，可以对各种星形障碍物进行准确而稳定的数值重建。