The Fictitious Boundary Method for the Implicit Treatment of Dirichlet Boundary Conditions with Applications to Incompressible Flow Simulations

Abstract

A ‘fictitious boundary method’ for computing incompressible flows with complicated small-scale and/or time-dependent geometric details is presented. The underlying technique is based on a special treatment of Dirichlet boundary conditions, particularly for FEM discretizations, together with so-called `iterative filtering techniques’ in the context of hierarchical multigrid approaches such that the flow can be efficienctly computed on a fixed computational mesh while the solid boundaries are allowed to move freely through the given mesh. The presented method provides an easy way of incorporating geometrically complicated objects and time-dependent boundary components into standard CFD codes to simulate (at least) the qualitative flow behaviour of complex configurations. Furthermore, higher accuracy can be reached via local mesh adaptation techniques which might be based on local (coarse) mesh adaptation or mesh deformation techniques to avoid expensive (global) grid reconstruction.

We explain the mathematical and algorithmic details and provide numerical examples based on the FeatFlow [13] software for incompressible flow to illustrate qualitatively and to examine quantitatively the presented fictitious boundary method, for various stationary and time-dependent configurations. In particular, we compare with standard approaches which use geometrically adapted meshes, and with a `viscosity-density blockage) method which describes internal objects via appropriate settings of the density and viscosity parameters in the Navier-Stokes equations. Moreover, we also discuss implementation details and software techniques which can provide very high MFLOP/s rates for such techniques in combination with special hierarchical data, matrix and solver structures.