SCTBEM: A scaled coordinate transformation boundary element method with 99-line MATLAB code

TL;DR

This paper presents SCTBEM, a boundary element method with a 99-line MATLAB implementation, that uses a novel coordinate transformation to solve 3D potential problems efficiently and accurately without domain integrals.

Contribution

It introduces the Scaled Coordinate Transformation to eliminate domain integrals in BEM, maintaining dimensionality reduction and improving accuracy for complex problems.

Findings

High numerical accuracy demonstrated on complex models

Elimination of lower-order singularities via coordinate translation

Simple 99-line MATLAB code facilitates wider adoption

Abstract

This paper introduces the Scaled Coordinate Transformation Boundary Element Method (SCTBEM), a novel boundary-type method for solving 3D potential problems. To address the challenges of applying the Boundary Element Method (BEM) to complex problems, it is common practice to use the fundamental solution corresponding to the partial governing equation operator to establish the integral equation. However, this approach introduces domain integral, which may jeopardize the dimensionality reduction advantages of BEM. To preserve the benefits of dimensionality reduction, this paper proposes a novel domain integral transformation method known as the Scaled Coordinate Transformation (SCT). The SCT is purely a mathematical operation that does not rely on particular solution of operators, which requires only discretization on the structure's surface while remaining analytical in the radial direction. An even better novelty is that the lower-order singularity can be eliminated by coordinate translation technique. To facilitate the wider adoption of BEM, the authors present 99-line MATLAB code. Numerical results confirm that the SCTBEM exhibits high numerical accuracy even when dealing with complex model.