Application of the Direct Interpolation Boundary Element Technique with Self-Adaptative Integration to Bidimensional Diffusive-Advective Problems

Resumo

The application of the Boundary Element Method (BEM) in the formulation of advective-diffusive models
experiences a solid scientific revival and finds its main challenges in the numerical treatment of the transport term.
Despite the classic formulation, which is based on the fundamental solution related to the problem, being able
to adequately describe physical situations dominated by advection, it does not present flexibility when dealing
with variable velocity fields. The recent Direct Interpolation technique (DIBEM), also based on the robustness
of approximations by radial basis, handles in a more balanced way the challenges imposed to BEM, while able
to represent hydrodynamic fields with spatial variation and maintaining numerical stability up to moderate Peclet
numbers. Basically, the technique procedure transforms domain integrals into boundary integrals, approaching
the entire kernel using radial basis functions. The DIBEM’s mathematical structure is very similar to a classic
interpolation, that in turn requires a significant number of interpolating points in the domain. In this context,
the performance of the Direct Integration formulation as well as it is numerical stability depend, among other
factors, on the quality of the approximation of ordinary, quasi-singular and singular integrals that are inherent to
the discrete linear system. The current article proposes a comparison and analysis of the influence of the numerical
integration scheme selection, testing the classic Gaussian quadrature against a self-adaptive scheme, aiming to
evaluate possible accuracy gains of the Direct Interpolation Technique. Quantitative assessment of the impacts of
integration schemes are evaluated using well-known analytical solutions.