A separation between the boundary shape and the boundary functions in the parametric integral equation system for the 3D Stokes equation

Abstract

The paper introduces the analytical modification of the classic boundary integral equation (BIE) for Stokes equation in 3D. The performed modification allows us to obtain separation of the approximation boundary shape from the approximation of the boundary functions. As a result, the equations, called the parametric integral equation system (PIES) with formal separation between the boundary geometry and the boundary functions, are obtained. It enables us to independently choose the most convenient methods of boundary geometry modeling depending on its complexity without any intrusion into the approximation of boundary functions and vice versa. Furthermore, we investigated the possibility of the modeling of the domains bounded by rectangular and triangular parametric Bézier patches. The boundary functions are approximated by generalized Chebyshev series. In addition, numerical techniques for solving the obtained PIES have been developed. The effectiveness of the presented strategy for boundary representation by surface patches in connection with PIES has been studied in numerical examples.

T-splines and Bézier extraction in modeling shape of boundary in Parametric Integral Equation System for 3D problems of linear elasticity

“…Moreover, PIES separates the boundary shape represented by the aforementioned curves or surfaces from the approximation of field values on the boundary, so the accuracy improvement is independent from the boundary description. Previous research about the practical application of boundary representation by parametric surfaces in PIES in the case of 3D problems are concerned mainly with the boundary value problems described for example by the Laplace (Zieniuk et al 2014), Navier-Lamé (Zieniuk et al 2018), and Stokes (Zieniuk et al 2019) equations with the boundary described by Coons and Bézier surfaces.…”