10.5Expressing the Helmholtz Equation in the FEM Basis

points of this basis define the other basis functions that couple to this one. This will always be a finite number much smaller than the total number of points but will not be the same for every mesh. This can be verified by inspection of Figure 10.4. If a regular lattice is chosen, then the number of couplings can be the same for all interior points. In 1-dim there can only be two overlaps for every interior basis, for example, basis n will couple to n + 1 and n−1. Including self-coupling gives a total of three entries for each row of B corresponding to an interior point. In 2-dim and 3-dim different meshes will produce different size rows. For example, in Figure 10.2 each interior point, not near the boundary, couples to six other basis elements. Interior points near the boundary will have fewer couplings. Inspection of Figure 10.4 shows that in this case some interior elements are coupled to as few as four neighbors and some to six. Despite irregularity, it is fairly straightforward to identify these couplings and keep track of them. It is useful to consider the number of basis elements that share a particular triangle, or tetrahedron. Since each point is a basis and a triangle (tetrahedron) has only three (four) points, this must be the number of elements that share that triangle (tetrahedron). One can evaluate the contributions to the operator matrix elements on a per triangle basis (different use of the word) and store these values. Evaluating the complete coupling amounts to picking and summing the correct precalculated values. The boundary conditions are already taken care of with the boundary function. A take-away is that the FEM matrix will be sparse but not necessary tri-diagonal.

10.6Numerical Integration over Triangular and Tetrahedral Domains

The remaining ingredient for completing the operator matrix elements is numerical integration over the basis elements. In the application of the Galerkin method, the original PDE was “projected” onto the same set of functions used to expand the unknown field. The integral over the basis element is a sum of integrals over the triangles in the basis element. Due to the nature of the basis element, this projection will yield a nontrivial result only when the two basis elements in the integrand share common triangles, that is, when the two basis elements overlap:

In (10.74) T is a triangle, dT is a differential area element on the triangle, and the notation T I means sum over all triangles in the intersection of the two basis elements. For the source term the sum will be over all triangles in the basis bn. The good news is that these integrals are not likely to be poorly behaved. Despite this they are also not likely to be known in closed form in terms of ordinary, or special, functions. A numerical integration technique is needed for evaluating these integrals. Following the theme of this chapter, numerical integration will be introduced for 1-dim intervals. The generalization to integration over rectangular domains in 2-dim and 3-dim is immediate, but for integration over triangular and tetrahedral domains, some additional steps will be needed. The 1-dim treatment will not be comprehensive, defaulting to Gaussian quadrature. This will serve as the foundation to developing higher-dimensional integration, with the exception of a special technique for integration over triangles introduced by Dunavant.

10.6.1 Gaussian Quadrature

Gaussian quadrature is introduced here as a foundation for building up integration over triangular and tetrahedral domains. For a complete introduction to numerical integration, the reader is

referred to Ref. [2]. Classic numerical integration methods use evenly spaced samples of points on the interval [a, b], {xj}. Various methods of approximating the integral to a desired order lead to a choice of weighting factors {wj}, associated with each point. The integral of the function f(x), over the interval [a, b], is then approximated by

Depending on the number of sample points taken, the approximation (10.75) will be exact for polynomials up to some order N. It often happens that one encounters integrands with an integrable singularity, expressed as W(x):

b

a

Gaussian quadrature provides a procedure for choosing not only the weights but also the sample points {xj} using a set of orthogonal polynomial functions, Pn(x) with n = 0,1 2,…, on (a, b). Rather than equally sampling the interval, the quadrature method takes these points to be the roots of an Nth-order polynomial from the set to construct an order integration scheme. Using an Nth-order polynomial yields an integration scheme that is exact for polynomials of degree less than or equal to 2N −1. Once the roots of the polynomial {xj} are found to a desired accuracy, the weights are given by

The numerator in (10.77) is defined by the following:

And in the denominator PN = dPNdx. The weights can be chosen to incorporate the function W(x).

10.6.2 Integration over Triangular Domains

In this section two standard approaches for numerical integration over triangles are presented. Consider numerical integration on a two-dimensional rectangular region, R =

x,yx ≤ a, y ≤ b:

b a

−b −a

A 2-dim quadrature routine for (10.79) can be built from two sets of sample points and

weights for the x and y coordinates be xn,wnx , n = 1, …, N, and ym,wmy , m = 1, …, M, respectively. The value of the integral can be approximated by the following combination of two 1-dim integrations:

The function values at the sample points are denoted as fnm. Defining a single index for the points on the 2-dim grid, J(n, m), j = 1, …, NM, and new weights, Wj = wnwm, (10.80) is written as a single sum:

j = 1

Integrating over triangles is a little trickier. One technique is to map the triangle into a square and then perform the integration over the square (see Hussain et al. [3]). This is accomplished by mapping the triangle to an isosceles right triangle with unit length legs along the x and y axes, followed by a mapping of this triangle into a square region. Let the vertices of the triangle be given by the vectors V k, k = 1,2 3. A set of parameters ξi, called simplex coordinates, is defined. An arbitrary point in the triangle is expressed as a weighted sum of the vertices in these coordinates:

Each parameter is constrained: 0 ≤ ξi ≤ 1, and ξ1 + ξ2 + ξ3 = 1. In this space all triangles are equilateral triangles with vertices, V i, at ξi = 1. The constraint can be used to eliminate a degree of freedom. For example, solving for ξ1, ξ1 = 1−ξ2 −ξ3, gives the following:

From Figure 10.8 it is clear that this is a projection of the equilateral triangle onto the ξ2, ξ3 plane with the vertex V 1 mapped to the origin. In this representation, the standard triangle is

Figure 10.8 Mapping of a generic triangle (left) into the standard triangle (center) and to a square (right)

should lead to better precision and less processing time when used in numerical integration routines.

This quadrature technique works well but it tends to cluster points in one corner of the triangle, creating an asymmetric sampling that oversamples one part of the triangle and undersamples other parts. An alternate quadrature is provided by Dunavant [4]. This approach samples the standard triangle in simplex space in a manner that is symmetric with respect to vertex exchange. The resulting quadrature points and weights provide high fidelity numerical integration with much fewer points than the previous technique requires. The theoretical details regarding the technique are beyond the scope of this text. The interested reader is referred to Ref. [5] for details. The original reference provides tables with quadrature points and weights up to order 20, a complete description of the algorithm, and sample code, for evaluating higher orders. Points occur in groups of either 1, 3, or 6, with each member of the group having the same weight and the coordinates of each related by a permutation. The unique parameter values are presented as rows of a matrix of this text, along with the weights as a column vector. An example is provided as follows for p = 6 in [5]:

The results of Dunavant are rounded to seven decimal places. A single row of P is a unique (ξ1, ξ2, ξ3). There are 12 points in all. They are generated by taking all unique permutations of these values. The first two rows each have a repeated value. There are only three unique permutations of these. The last point has three independent values for (ξ1, ξ2, ξ3) and produces a total of six integration points. The weights are ordered to match the rows of P. All points generated by permutation have the same weight as the original point. Figure 10.9 provides an

Figure 10.9 Example of Dunavant quadrature points in simplex space, order = 12