Efficient FEM-BEM Coupling Based on Argyris Element for Axi-Symmetric Open Boundary Magnetostatic Problems

Abstract

This article describes a novel and accurate implementation of finite element method-boundary element method (FEM-BEM) coupling based on Argyris element (AE), for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C^{1}$ </tex-math></inline-formula> solution of axi-symmetric magnetostatic problems in open boundary domains. The aim of the work is twofold: 1) to provide novel, easy, and accurate way to compute the basis functions that is strongly not affected by accuracy errors and 2) to give a clear mathematical and numerical description of the FEM-BEM coupling based on AE. The first goal is achieved by shifting to the origin the physical triangular elements and computing the basis function in the shifted coordinate system. Second, AE-based FEM-BEM coupling relies on the boundary integral equations (BIEs) for poloidal flux and its partial derivatives of both first and second order, accurately computed by combining the analytical formula based on elliptic integrals together with high degree Gaussian cubature. The application of the proposed method to an advanced plasma equilibrium magnetic configuration, typical of magnetic confinement fusion (MCF) (i.e., <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">snowflake configuration</i> ), highlights that, even in case of coarse triangulation, satisfactory results can be achieved, and the error on the boundary conditions (BCs) is well below machine working precision.