Adaptive integration techniques for almost singular functions in the boundary element method

Integration of Singular Enrichment Functions in the Generalized/Extended Finite Element Method for Three-Dimensional Problems

A mapping method is developed to integrate weak singularities which result from enrichment functions in the generalized/extended FEM. The integration scheme is applicable to 2D and 3D problems including arbitrarily shaped triangles and tetrahedra. Implementation of the proposed scheme in existing codes is straightforward. Numerical examples for 2D and 3D problems demonstrate the accuracy and co...

A finite element method using singular functions for Poisson equations: Mixed boundary conditions

In [Z. Cai, S. Kim, A finite element method using singular functions for the Poisson equation: corner singularities, SIAM J. Numer. Anal. 39 (2001) 286–299], we proposed a new finite element method to compute singular solutions of Poisson equations on a polygonal domain with re-entrant angles. Singularities are eliminated and only the regular part of the solution that is in H is computed. The s...

Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method

For the simple layer potential V that is associated with the 3D Laplacian, we consider the weakly singular integral equation V φ = f . This equation is discretized by the lowest order Galerkin boundary element method. We prove convergence of an h-adaptive algorithm that is driven by a weighted residual error estimator. Moreover, we identify the approximation class for which the adaptive algorit...

On the evaluation of hyper-singular integrals arising in the boundary element method for linear elasticity

The boundary element method (BEM) for linear elasticity in its curent usage is based on the boundary integral equation for displacements. The stress field in the interior of the body is computed by differentiating the displacement field at the source point in the BEM formulation, via the strain field. However, at the boundary, this method gives rise to a hypersingular integral relation which be...

A Regular Indirect Boundary Element Method for Thermal Analysis