The method of fundamental solutions for Brinkman flows. Part I. Exterior domains

Abstract

This is a repository copy of The method of fundamental solutions for Brinkman flows. Part I. Exterior domains.

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

“…For the Brinkman flow we took the parameters µ = 1 and κ = 1. Numerical experiments carried out in Part I [17] in the case of exterior Brinkman flows revealed that there is no significant change in accuracy of the numerical results when the parameters R, µ and κ are varied. For the arbitrary star-shaped obstacle (3.5) the additional traction (stress force) data (2.4) is numerically simulated by first solving the direct problem (1.1)-(2.3) with known D using the MFS (with different numbers of degrees of freedom M and N than those employed in the inverse problem to avoid committing an inverse crime).…”

“…Another possibility for the inhomogeneous condition (2.3) is to represent an internal fluid velocity measurement in the exterior domain R 2 \D, as previously considered in Part I [17].…”

“…In the companion paper (Part I), see [17], the method of fundamentals solutions (MFS) [18] with Tikhonov's regularization was developed for solving both direct and inverse problems for the Brinkman flow in the porous medium exterior to an obstacle of known or unknown shape. In the exterior inverse problem which was concerned with the identification of the obstacle, the extra data was the fluid velocity specification/measurement on a curve surrounding the obstacle.…”

The identification of obstacles immersed in a steady incompressible viscous fluid

“…For solving the inverse problem (1)-( 5) and (11) to detect the obstacle (12), we minimize the least-squares functional…”

“…In such a situation, inverse modeling offers a valuable approach to overcome these challenges by utilizing observed effects or indirect measurements to infer the properties, shapes, sizes and locations of submerged obstacles. There are several papers about the identification of obstacles immersed in some different types of fluids such as, potential [5], Stokes [18,3,2], Oseen [15,11], Brinkman [12,13] or Navier-Stokes [1,7]. The studies in [1,7] were mainly theoretical and they dealt with the simpler Dirichlet boundary conditions for which the existence and uniqueness of solution theory is available [23].…”