Abstract
The collocation version of the Method of Fundamental Solutions (MFS) with subdomains is introduced in the present work for the solution of the 2D Stokes flow in backward-facing-step geometry, including Dirichlet and Neumann boundary conditions. The motivation for the present work is the inability of the MFS to solve such problems and the problems with slits and cracks due to the discretization of a single domain. The inability stems from the artificial boundary that is difficult or impossible to properly geometrically set in such cases. The solution for such problems is found by splitting such domains into subdomains. The MFS equations for the equilibrium conditions at the collocation points on the interface between the adjacent subdomains are derived for the Stokes equation. A matrix that simultaneously solves the collocation problem on all the subdomains is formed and solved. A sensitivity study of the MFS results is performed by comparing the relative root mean square error with the reference solution obtained by the classical mesh-based finite volume method on a very fine mesh. The subdomain technique is verified by dividing the domain into 2, 3 and 5 subdomains. The velocity, vorticity and pressure compare very well with the reference solution in all three cases while the solution for the single domain approach is outstandingly poor and inappropriate. The paper shows that the proposed subdomain technique maintains the simplicity, true meshless character and accuracy of the MFS for the Stokes flow in cases where the domain topology requires the use of the subdomain technique.
Highlights
The Stokes flow is a type of steady flow where body forces can be disregarded and the inertial forces are negligible compared to the viscous forces [1]
The resulting Stokes flow equations have been solved in the past using many numerical techniques such as the following: domain discretization methods, boundary discretization methods and meshless methods, among others
We focus on the implementation of the method of fundamental solutions (MFS) for the Stokes flow in special geometrically nontrivial situations in the present paper
Summary
The Stokes flow is a type of steady flow where body forces can be disregarded and the inertial forces are negligible compared to the viscous forces [1]. A simple and very accurate direct evaluation of the derivatives of the field variables is another main advantage of the MFS This is especially important when implementing equilibrium conditions on fixed, free or moving interfaces inside a domain [14,19]. On the interface between adjacent subdomains, we consider the values of the variables from the adjacent (overlapping) subdomain as a boundary condition This approach makes sense when the leading equations are nonlinear, e.g., Navier-Stokes equations. It is better to use the nonoverlapping subdomains and apply interface condition equations to collocation points that are common to adjacent subdomains In this case, the system of linear equations becomes sparse, and one of the dedicated direct solvers can be used to solve it. The Stokeslets derivatives required for the interface condition equations at the subdomain boundaries of the domain are:
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