A fast multipole boundary element method based on higher order elements for analyzing 2-D potential problems

A novel fast multipole boundary element method (FM-BEM) is proposed to analyze 2-D potential problems by using linear and three-node quadratic elements. In FM-BEM, fast multipole expansions are used for the integrals on elements which are far away from the source point, whereas direct evaluations are used for the integrals on elements which are close to the source point. The use of higher-order elements results in more complex forms of the integrands, which increases the burden in direct evaluations, especially for singular and nearly singular integrals. Herein, the complex notation is introduced to simplify the computational formulations in boundary integral equations for 2-D potential problems. The singular and nearly singular integrals on linear elements are calculated by the analytic formulas, and those on three-node quadratic elements are evaluated by a robust semi-analytical algorithm. Numerical examples show that the proposed FM-BEM possesses higher accuracy than the conventional FM-BEM. Besides, the present method can analyze thin structures and evaluate accurately the physical quantities at interior points much closer to the boundary.