The method of fundamental solutions for Brinkman flows. Part II. Interior domains

Abstract

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 optimization toolbox routine lsqnonlin.

Appendix

In this appendix we provide the expressions for the partial derivatives $\big ({\partial G_{ij}}/{\partial x_k} \big )_{i,j,k=1,2}$ needed for calculating the matrix $\left( D_{ij}\right) _{i,j=1,2}$ appearing in the stress force MFS approximation (3.8). First, using the identity

$$\begin{aligned} \dfrac{2 K_1(z)}{z}=K_2(z)-K_0(z), \quad z\ne 0, \end{aligned}$$

we can rewrite (3.3) in the equivalent form, see [34],

$$\begin{aligned} G_{ik}({\varvec{x}}, {\varvec{x}}^\prime )\!=\!\dfrac{1}{2 \pi \mu } \left[ \left( -\dfrac{1}{\kappa ^2 r^2}\!+\! \dfrac{K_0(\kappa r) + K_2(\kappa r)}{2}\right) \delta _{ik} \!+\! \dfrac{(x_i-x_i^\prime ) (x_k-x_k^\prime ) }{r^2} \left( \dfrac{2}{\kappa ^2 r^2} \!-\! K_2(\kappa r) \right) \right] , \quad i,k\!=\!1,2.\nonumber \\ \end{aligned}$$

(A.1)

Using that

$$\begin{aligned} K_0^\prime (s)=-K_1(s), \quad K_2^\prime (s)=-\dfrac{1}{2} \left( K_1(s)+K_3(s)\right) , \end{aligned}$$

(A.2)

we obtain (checked using the symbolic computation package MAPLE^™)

$$\begin{aligned} \dfrac{\partial G_{11}}{\partial x_1}= & {} \dfrac{1}{2 \pi \mu } \left\{ \dfrac{2(x_1-x_1^\prime )}{\kappa ^2 r^4} - \dfrac{\kappa (x_1-x_1^\prime ) \left( 3 K_1(\kappa r) + K_3(\kappa r) \right) }{4 r} \right. \\&+\dfrac{(x_1-x_1^\prime )^2}{r^2} \left[ -\dfrac{4 (x_1-x_1^\prime ) }{\kappa ^2 r^4 } + \dfrac{\kappa (x_1-x_1^\prime ) \left( K_1(\kappa r) + K_3(\kappa r) \right) }{2 r} \right] \\&\left. +\dfrac{2 (x_1-x_1^\prime )(x_2-x_2^\prime )^2}{ r^4} \left( \dfrac{2}{\kappa ^2 r^2} - K_2(\kappa r) \right) \right\} , \\ \dfrac{\partial G_{12}}{\partial x_1}= & {} \dfrac{\partial G_{21}}{\partial x_1} = \dfrac{1}{2 \pi \mu } \left\{ \dfrac{(x_1-x_1^\prime )(x_2-x_2^\prime )}{r^2}\left[ -\dfrac{4 (x_1-x_1^\prime ) }{\kappa ^2 r^4} + \dfrac{\kappa (x_1-x_1^\prime )\left( K_1(\kappa r) + K_3(\kappa r) \right) }{2 r} \right] \right. \\&\left. + \dfrac{(x_2-x_2^\prime )\left[ (x_2-x_2^\prime )^2- (x_1-x_1^\prime )^2\right] }{ r^4} \left( \dfrac{2}{\kappa ^2 r^2} - K_2(\kappa r) \right) \right\} , \\ \dfrac{\partial G_{22}}{\partial x_1}= & {} \dfrac{1}{2 \pi \mu } \left\{ \dfrac{2(x_1-x_1^\prime )}{\kappa ^2 r^4} - \dfrac{\kappa (x_1-x_1^\prime ) \left( 3 K_1(\kappa r) + K_3(\kappa r) \right) }{4 r}\right. \\&+\dfrac{(x_2-x_2^\prime )^2}{r^2} \left[ -\dfrac{4 (x_1-x_1^\prime )}{\kappa ^2 r^4 } + \dfrac{\kappa (x_1-x_1^\prime ) \left( K_1(\kappa r) + K_3(\kappa r) \right) }{2 r} \right] \\&\left. - \dfrac{2(x_1-x_1^\prime )(x_2-x_2^\prime )^2}{r^4} \left( \dfrac{2}{\kappa ^2 r^2} - K_2(\kappa r) \right) \right\} , \\ \dfrac{\partial G_{11}}{\partial x_2}= & {} \dfrac{1}{2 \pi \mu } \left\{ \dfrac{2(x_2-x_2^\prime )}{\kappa ^2 r^4} - \dfrac{\kappa (x_2-x_2^\prime ) \left( 3 K_1(\kappa r) + K_3(\kappa r) \right) }{4 r}\right. \\&+\dfrac{(x_1-x_1^\prime )^2}{r^2} \left[ -\dfrac{4 (x_2-x_2^\prime ) }{\kappa ^2 r^4 } + \dfrac{\kappa (x_2-x_2^\prime ) \left( K_1(\kappa r) + K_3(\kappa r) \right) }{2 r} \right] \\&\left. -\dfrac{2(x_1-x_1^\prime )^2(x_2-x_2^\prime )}{r^4} \left( \dfrac{2}{\kappa ^2 r^2} - K_2(\kappa r) \right) \right\} , \\ \dfrac{\partial G_{12}}{\partial x_2}= & {} \dfrac{\partial G_{21}}{\partial x_2} = \dfrac{1}{2 \pi \mu } \left\{ \dfrac{(x_1-x_1^\prime )(x_2-x_2^\prime )}{r^2}\left[ -\dfrac{4 (x_2-x_2^\prime ) }{\kappa ^2 r^4} + \dfrac{\kappa (x_2-x_2^\prime ) \left( K_1(\kappa r) + K_3(\kappa r) \right) }{2 r } \right] \right. \\&\left. + \dfrac{(x_1-x_1^\prime )\left[ (x_1-x_1^\prime )^2- (x_2-x_2^\prime )^2\right] }{r^4} \left( \dfrac{2}{\kappa ^2 r^2} - K_2(\kappa r) \right) \right\} , \\ \dfrac{\partial G_{22}}{\partial x_2}= & {} \dfrac{1}{2 \pi \mu } \left\{ \dfrac{2(x_2-x_2^\prime )}{\kappa ^2 r^4} - \dfrac{\kappa (x_2-x_2^\prime ) \left( 3 K_1(\kappa r) + K_3(\kappa r) \right) }{4r }\right. \\&+\dfrac{(x_2-x_2^\prime )^2}{r^2} \left[ -\dfrac{4 (x_2-x_2^\prime ) }{\kappa ^2 r^4 } + \dfrac{\kappa (x_2-x_2^\prime ) \left( K_1(\kappa r) + K_3(\kappa r) \right) }{2 r } \right] \\&\left. + \dfrac{2(x_1-x_1^\prime )^2(x_2-x_2^\prime )}{r^4} \left( \dfrac{2}{\kappa ^2 r^2} - K_2(\kappa r) \right) \right\} . \end{aligned}$$