Electric potential and field calculation of charged BEM triangles and rectangles by Gaussian cubature

It is a widely held view that analytical integration is more accurate than the numerical one. In some special cases, however, numerical integration can be more advantageous than analytical integration. In our paper we show this benefit for the case of electric potential and field computation of charged triangles and rectangles applied in the boundary element method (BEM). Analytical potential and field formulas are rather complicated (even in the simplest case of constant charge densities), they have usually large computation times, and at field points far from the elements they suffer from large rounding errors. On the other hand, Gaussian cubature, which is an efficient numerical integration method, yields simple and fast potential and field formulas that are very accurate far from the elements. The simplicity of the method is demonstrated by the physical picture: the triangles and rectangles with their continuous charge distributions are replaced by discrete point charges, whose simple potential and field formulas explain the higher accuracy and speed of this method. We implemented the Gaussian cubature method for the purpose of BEM computations both with CPU and GPU, and we compare its performance with two different analytical integration methods. The ten different Gaussian cubature formulas presented in our paper can be used for arbitrary high-precision and fast integrations over triangles and rectangles.Comment: 28 pages, 13 figure

On "marcov" inequalities

As colleagues and friends we wish to dedicate these pages to Marco Vianello on the occasion of his 60th birthday, which is on October 26, 2021. Marco has made many important contributions to approximation theory and beyond. Here we briefly summarize some of them in the spirit of the occasion

A one point integration rule over star convex polytopes

In this paper, the recently proposed linearly consistent one point integration rule for the meshfree methods is extended to arbitrary polytopes. The salient feature of the proposed technique is that it requires only one integration point within each n-sided polytope as opposed to 3n in Francis et al. (2017) and 13n integration points in the conventional approach for numerically integrating the weak form in two dimensions. The essence of the proposed technique is to approximate the compatible strain by a linear smoothing function and evaluate the smoothed nodal derivatives by the discrete form of the divergence theorem at the geometric center. This is done by Taylor's expansion of the weak form which facilitates the use of the smoothed nodal derivatives acting as the stabilization term. This translates to 50% and 30% reduction in the overall computational time in the two and three dimensions, respectively, whilst preserving the accuracy and the convergence rates. Th

A generalized precorrected-FFT method for electromagnetic analysis

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008.Includes bibliographical references (p. 117-119).Boundary Element Methods (BEM) can be ideal approaches for simulating the behavior of physical systems in which the volumes have homogeneous properties. These, especially the so-called "fast" or "accelerated" BEM approaches often have significant computational advantages over other well-known methods which solve partial differential equations on a volume domain. However, the implementation of techniques used to accelerate BEM approaches often comes at a loss of some generality, reducing their applicability to many problems and preventing engineers and researchers from easily building on a common, popular base of code. In this thesis we create a BEM solver which uses the Pre-Corrected FFT technique for accelerating computation, and uses a novel approach which allows users to provide arbitrary basis functions. We demonstrate its utility for both electrostatic and full-wave electromagnetic problems in volumes with homogeneous isotropic permittivity, bounded by arbitrarily complex surface geometries. The code is shown to have performance characteristics similar to the best known approaches for these problems. It also provides an increased level of generality, and is designed in such a way that should allow it to easily be extended by other researchers.by Stephen Gerald Leibman.S.M