Complex analytic integration in Galerkin boundary element method for 2-D potential problems

In this paper, a novel technique is proposed for evaluating integrals in the boundary element method for the 2D Laplace equation. Linear elements with linear shape functions are assumed. All integrals in collocation or Galerkin methods, whether singular or regular, are evaluated in closed form as real or imaginary parts of complex analytic functions. The complex reformulation redefines real operations in complex terms, e.g., the Euclidean norm as the absolute value and the inner product as the real part of a complex product. Thus, coordinates are written directly in Cartesian form. No coordinate transformation is employed; results are expressed directly in terms of distance vectors. Complex results are concise due to the generality of complex analysis, where division is defined for complex numbers but undefined for real vectors. Galerkin integrals are classified into eight kinds, all obtained as a linear combination of a basic integral. The closed form of collocation integrals facilitates the evaluation of the Galerkin projection. Singular integrals are handled using the approach-to-boundary directional limit without regularization. For 460 boundary elements, computational times were 2066, 865, and 33 s for numerical, semi-analytical, and analytical methods, respectively. Finally, a complicated geometry is simulated for validation.