Boundary Integral Analysis for the Non-homogeneous 3D Stokes Equation
For computational fluid dynamics researchers, this provides a more efficient volume integration technique for Stokes flow, though it is an incremental improvement over existing boundary integral methods.
The paper presents a method to convert volume integrals in the non-homogeneous 3D Stokes equation into boundary integrals plus a remainder, enabling efficient regular-grid evaluation. Test results with linear element Galerkin approximation validate the implementation.
A regular-grid volume-integration algorithm is developed for the non-homogeneous 3D Stokes equation. Based upon the observation that the Stokeslet ${\mathcal U}$ is the Laplacian of a function ${\mathcal H}$, the volume integral is reformulated as a simple boundary integral, plus a remainder domain integral. The modified source term in this remainder integral is everywhere zero on the boundary and can therefore be continuously extended as zero to a regular grid covering the domain. The volume integral can then be evaluated on the grid. Applying this method to the Navier-Stokes equations will require obtaining velocity gradients, and thus an efficient algorithm for post-processing these derivatives is also discussed. To validate the numerical implementation, test results employing a linear element Galerkin approximation are presented.