This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Applied Numerical Analysis. By C.F. Gerald. Pp. xii, 340. £3·60. 1970. (Addison-Wesley.)

In this paper, we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons. These quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required. We have used this algorithm for the construction of symmetric and non-symmetric quadrature rules over convex and concave polygons. While in the case of symmetric quadratures our results are comparable to available rules, the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions. To demonstrate the efficiency of the new quadrature rules, we have tested them for the integration of rational polygonal shape functions over a regular hexagon. For a relative error of 10−8 in the computation of stiffness matrix entries, one needs at least 198 evaluation points when the region is partitioned, whereas 85 points suffice with our quadrature rule. Copyright © 2009 John Wiley & Sons, Ltd.

/pdf/generalized-gaussian-quadrature-rules-on-arbitrary-polygons-2qlgm61qk5.pdf

Generalized Gaussian quadrature rules on arbitrary polygons

This work assumes a basic familiarity with the finite element method. For those readers desiring a better understanding of the method, there are a number of excellent books varying from an introductory treatment to advanced topics listed in the bibliography [1] [2] [3] [4] [5]. In this chapter, we will review the fundamentals of the finite element method and introduce the notation which will be used throughout the book. For the examples shown in this book, either first or second order elements were used. For the worked examples first order triangles are used. The use of first order triangular elements simplifies the arguments and in no way reduces generality. In all cases the procedures given in this book can be used with high order and/or curved elements.

Introduction to Finite Elements

We seek accurate, fast and reliable computations of the confluent and Gauss hypergeometric functions 1F1(a; b; z) and 2F1(a, b; c; z) for different parameter regimes within the complex plane for the parameters a and b for 1F1 and a, b and c for 2F1, as well as different regimes for the complex variable z in both cases. In order to achieve this, we implement a number of methods and algorithms using ideas such as Taylor and asymptotic series computations, quadrature, numerical solution of differential equations, recurrence relations, and others. These methods are used to evaluate 1F1 for all z ∈ C and 2F1 for |z| < 1. For 2F1, we also apply transformation formulae to generate approximations for all z ∈ C. We discuss the results of numerical experiments carried out on the most effective methods and use these results to determine the best methods for each parameter and variable regime investigated. We find that, for both the confluent and Gauss hypergeometric functions, there is no simple answer to the problem of their computation, and different methods are optimal for different parameter regimes. Our conclusions regarding the best methods for computation of these functions involve techniques from a wide range of areas in scientific computing, which are discussed in this dissertation. We have also prepared MATLAB code that takes account of these conclusions.

Computation of Hypergeometric Functions

Symmetric Gauss Legendre quadrature formulas for composite numerical integration over a triangular surface

The use of parabolic arcs in matching curved boundaries by point transformations for some higher order triangular elements

http://www.m-hikari.com/ijma/ijma-2011/ijma-1-4-2011/venkateshIJMA1-4-2011.pdf

B. Venkatesudu

Papers

Symmetric Gauss Legendre quadrature formulas for composite numerical integration over a triangular surface

The use of parabolic arcs in matching curved boundaries by point transformations for some higher order triangular elements

Gauss Legendre-Gauss Jacobi quadrature rules over a tetrahedral region

On the application of two Gauss–Legendre quadrature rules for composite numerical integration over a tetrahedral region

Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space