Integration of singular Galerkin-type boundary element integrals for 3D elasticity problems

Abstract

A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals. Subject Classification (1991): 65N38,65D30,65R20,65N30 Mathematics

Evaluation of Singular Electric Field Integral Equation (EFIE) Matrix Elements

“…Recently, the computational mechanics community has begun formulating boundary integral equation (BEE) solutions of Fredholm integral equations using a Galerkin descretization [10,11]. The focus of their research was to obtain high accuracy in the numerical treatment of the resulting singular 4-dimensional integrals.…”

“…This method has also been applied to formulations with curved geometry where the singularity extraction is the first term in a Taylor's series, however this approximate method can be improved upon with the formulation outlined here. Erichsen and Andra [10,11] encountered integrals of the type shown in (3), and others with higher order singularities, and were able to completely remove the singularity using a combination of relative coordinates, changing the order of integration and Duffy's transformations [13]. In addition to completely removing the singularity their method also allowed some parts of the 4-dimensional Galerkin integrals to be evaluated analytically, thereby reducing the effort required to numerically evaluate the entire inner product.…”

Galerkin BEM with Direct Evaluation of Hypersingular Integrals

“…It takes advantage of certain symmetry property exhibited by all Cauchy singular kernel functions, even when mapped onto the parameter space. To apply it in the SGBEM, that is to hypersingular kernels, an analytic regularization is therefore necessary, as shown in [26] and [27], where simple solutions and Stokes theorem are employed, respectively.…”

“…in Ref. [26]) through the introduction of suitable coordinate transformations in the two-dimensional space of intrinsic coordinates. The proposed algorithm is in particular applicable to the symmetric Galerkin BEM (SGBEM) and is devised so as to define in that case a perfectly symmetric integration procedure, even when the numerical quadrature is not exact.…”

Numerical Integration of Functions with Endpoint Singularities and/or Complex Poles in 3D Galerkin Boundary Element Methods

“…Many papers have been devoted to this topic (see, for example, [1], [5], [12], [13]), although none of them seems to treat the case (ii), which is considered by some users the most "difficult" case. However in our opinion none of them are fully satisfactory because generally they either require some analytical calculation which is by non means trivial, or do not take into account the presence of complex poles which may effect adversely the numerical computation.…”