The PIES for solving 3D potential problems with domains bounded by rectangular Bézier patches

Abstract

Purpose

The purpose of this paper is to apply rectangular Bézier surface patches directly into the mathematical formula used to solve boundary value problems modeled by Laplace's equation. The mathematical formula, called the parametric integral equation systems (PIES), will be obtained through the analytical modification of the conventional boundary integral equations (BIE), with the boundary mathematically described by rectangular Bézier patches.

Design/methodology/approach

The paper presents the methodology of the analytic connection of the rectangular patches with BIE. This methodology is a generalization of the one previously used for 2D problems.

Findings

In PIES the paper separates the necessity of performing simultaneous approximation of both boundary shape and the boundary functions, as the boundary geometry has been included in its mathematical formalism. The separation of the boundary geometry from the boundary functions enables to achieve an independent and more effective improvement of the accuracy of both approximations. Boundary functions are approximated by the Chebyshev series, whereas the boundary is approximated by Bézier patches.

Originality\value

The originality of the proposed approach lies in its ability to automatic adapt the PIES formula for modified shape of the boundary modeled by the Bézier patches. This modification does not require any dividing the patch into elements and creates the possibility for effective declaration of boundary geometry in continuous way directly in PIES.

Citation

Zieniuk, E. and Szerszen, K. (2014), "The PIES for solving 3D potential problems with domains bounded by rectangular Bézier patches", Engineering Computations, Vol. 31 No. 4, pp. 791-809. https://doi.org/10.1108/EC-08-2012-0197