A polar coordinate integration scheme with a hierachical correction procedure to improve numerical accuracy on the boundary element method

Abstract

In order to calculate accurate physical values of interior region by the boundary element method, a new approach is proposed to singular kernel integration which is applicable to general isoparametric elements and never fails, no matter where the internal point under consideration may be located. The integration scheme consists of two parts.

First, the singular kernel functions are assumed to be “reciprocal of the distance” types. Then the present scheme describes on the quadrate boundary element. The element is subdivided into four triangular regions for which Gauss-Legendre numerical quadrature is applied.

Secondly, a method is proposed to reduce residual errors in the application of the above mentioned numerical scheme. The boundary integrals to calculate interior physical values are expressed formally with exact and numerical error terms, and boundary values in error terms are expanded by a Taylor series around the interior point. To evaluate the coefficient of each derivative in the series, a boundary integral form of an identity with respect to a vector from the interior point to the boundary surface is derived. Error resulting from numerical integration of the identity is found to coincide with the coefficient of the derivative in the Taylor series. Thus, the correction factor for numerical errors is obtained.

The present scheme was verified to be quite effective such that both the numerical error and CPU time became 1/100 less than those by the double exponential quadrature. Moreover, the present numerical scheme is applicable to general curved elements.