206 research outputs found
On the Identification of Symmetric Quadrature Rules for Finite Element Methods
In this paper we describe a methodology for the identification of symmetric
quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms,
pyramids, and hexahedra. The methodology is free from manual intervention and
is capable of identifying an ensemble of rules with a given strength and a
given number of points. We also present polyquad which is an implementation of
our methodology. Using polyquad we proceed to derive a complete set of
symmetric rules on the aforementioned domains. All rules possess purely
positive weights and have all points inside the domain. Many of the rules
appear to be new, and an improvement over those tabulated in the literature.Comment: 17 pages, 6 figures, 1 tabl
Analytical computation of moderate-degree fully-symmetric cubature rules on the triangle
A method is developed to compute analytically fully symmetric cubature rules
on the triangle by using symmetric polynomials to express the two kinds of
invariance inherent in these rules. Rules of degree up to 15, some of them new
and of good quality, are computed and presented.Comment: 13 pages, submitted to Journal of Computational and Applied
Mathematic
Programming of Finite Element Methods in MATLAB
We discuss how to implement the linear finite element method for solving the
Poisson equation. We begin with the data structure to represent the
triangulation and boundary conditions, introduce the sparse matrix, and then
discuss the assembling process. We pay special attention to an efficient
programming style using sparse matrices in MATLAB
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
An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Tetrahedral Elements
The flux reconstruction (FR) approach offers an efficient route to high-order accuracy on unstructured grids. In this work we study the effect of solution point placement on the stability and accuracy of FR schemes on tetrahedral grids. To accomplish this we generate a large number of solution point candidates that satisfy various criteria at polynomial orders â=3,4,5â=3,4,5 . We then proceed to assess their properties by using them to solve the non-linear Euler equations on both structured and unstructured meshes. The results demonstrate that the location of the solution points is important in terms of both the stability and accuracy. Across a range of cases it is possible to outperform the solution points of Shunn and Ham for specific problems. However, there appears to be a degree of problem-dependence with regards to the optimal point set, and hence overall it is concluded that the Shunn and Ham points offer a good compromise in terms of practical utility
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