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We present a generic technique, automated by computer-algebra systems and available as open-source software cite{scuff-em}, for efficient numerical evaluation of a large family of singular and nonsingular 4-dimensional integrals over triangle-product domains, such as those arising in the boundary-element method (BEM) of computational electromagnetism. To date, practical implementation of BEM solvers has often required the aggregation of multiple disparate integral-evaluation schemes to treat all of the distinct types of integrals needed for a given BEM formulation; in contrast, our technique allows many different types of integrals to be handled by the emph{same} algorithm and the same code implementation. Our method is a significant generalization of the Taylor--Duffy approach cite{Taylor2003,Duffy1982}, which was originally presented for just a single type of integrand; in addition to generalizing this technique to a broad class of integrands, we also achieve a significant improvement in its efficiency by showing how the emph{dimension} of the final numerical integral may often be reduced by one. In particular, if $n$ is the number of common vertices between the two triangles, in many cases we can reduce the dimension of the integral from $4-n$ to $3-n$, obtaining a closed-form analytical result for $n=3$ (the common-triangle case).
I present an accurate and efficient technique for numerical evaluation of singular 6-dimensional integrals over tetrahedon-product domains, with applications to calculation of Galerkin matrix elements for discretized volume-integral-equation (VIE) so
We developed a fast numerical algorithm for solving the three dimensional vectorial Helmholtz equation that arises in electromagnetic scattering problems. The algorithm is based on electric field integral equations and is essentially a boundary eleme
The displacement field for three dimensional dynamic elasticity problems in the frequency domain can be decomposed into a sum of a longitudinal and a transversal part known as a Helmholtz decomposition. The Cartesian components of both the longitudin
A monolithic coupling between the material point method (MPM) and the finite element method (FEM) is presented. The MPM formulation described is implicit, and the exchange of information between particles and background grid is minimized. The reduced
We propose a weak Galerkin(WG) finite element method for solving the one-dimensional Burgers equation. Based on a new weak variational form, both semi-discrete and fully-discrete WG finite element schemes are established and analyzed. We prove the ex