Boundary Element Methods in flow problems governed by Navier-Stokes equations

In this presentation will be discussed recent advances of Boundary Element Method (BEM) in Computational Fluid Dynamics (CFD). Unlike other methods, BEM is a a multi-angle numerical technique, that permits the approach to a partial differential equation (PDE) in completely different ways. In Navier-Stokes equations in particular, many different test functions can be used in the weak form, as the Laplace, the Stokeslet, the convective parabolic-diffusion or other convective fundamental solutions, among others. Apart from that, it is found recently that hypersingular BEM in Navier-Stokes equations have a broad area of applicability, as they provide the gradients of the field. These gradients can further be applied to numerous cases as impovement of system's condition number, enforcing continuity, computation of wall quantities such as wall vorticities, strain and stress tensors, and pressure calculation, among others. However, derivation of such equations is not always simple since they are accompanied with extra terms, mainly in convection. Another important finding is that hypersingular equations can permit the use of constant elements simplifying immensely the preparation of the computational model. Another part of the talk will be dedicated to the transformation of the BEM system to Finite Element (FEM) or Finite Volume (FVM) equivalent in terms of sparsity. A system produced by BEM with domain unknowns cannot be solved efficiently, but with proper transformations it can be changed to a sparse system, which can be solved remarkably faster. Other accelerating techniques like hypersingular BEM/ Fast multipole (FMM) and meshless Local Boundary integral equation (LBIE) will be discussed.