then multiplying (11.51) by a constant η 0,1, and adding to (11.52) yields the combined equation

The parameter η is tunable; a common choice is 0.5. Discrete versions of the BIE are now in place and the task at hand is the definition of the surface tiles, basis elements, and the evaluation of the coupling integrals. A technical point worth mentioning here is that the evaluation of the matrix elements will require a quadruple integral, or a double double integral, over two patches. In contrast, the FEM only requires integration over a single basis element region. This is due to the FEM being a discretization of a PDE, whereas here in the BEM it is a BIE being discretized.

11.4Basis Elements and Test Functions

The development of the discretized BIE introduced basis elements for expanding the fields on a set of flat tiles and a set of test functions for sampling the equations. Nothing was said about the nature of these functions except that they may cover more than one triangular tile, as in the FEM case. Some comments regarding the test functions are made, followed by a discussion on basis elements. There are two popular choices of test function, or test method. The first is called a point matching. This amounts to sampling the equations at a single point, or weighted average over points, in the interior of the tile. The point match test function can be expressed in the following notation as a Dirac delta:

The other popular choice is to let the test function be equal to the basis function on each tile:

This choice is referred to as Galerkin’s method.

The basis element is meant to model the expected behavior of the surface field. When choosing a basis element for evaluating a BEM system, one needs to consider several competing factors. On the one hand, simpler functions lead to easier calculation of the matrix elements. On the other hand, over-trivializing the problem can lead to poor results and slow convergence of the method, requiring a greater number of tiles to improve fidelity. The simplest is the pulse basis, defined in a single patch and constant on that patch, zero everywhere else:

A better choice is to use the rooftop function introduced in the chapter on FEM. This identifies one basis element for each sample point on the surface. The rooftop function previously

introduced uses linear functions, but in general higher-order polynomials can be used. One defines a shape function in simplex coordinates on each patch in the basis, φnj ξ, where n is the basis index and j labels the specific triangle in that basis. The most common shape functions are linear and quadratic. In terms of the shape function operator, matrices can be expressed as

In (11.57) Tnm represents any operator matrix and g ξ , ξ is Green’s function or one of its derivatives. This generic form represents (11.36)–(11.38).

11.5Coupling Integrals

This section deals with the evaluation of the coupling integrals encountered in the various operator matrices used in BEM. Numerical integration over triangles was presented in the chapter on FEM and the reader is directed there for appropriate formulae. The operator matrix elements involve integration over two triangular patches. The primed integration will be referred to in this section as the inner integration, while the unprimed the outer integration. Some texts refer to the inner integral as a source integral and the outer integration points as test, or receiver points. This is appropriate considering that these matrix elements actually couple pairs of points by a propagator. Time will not be spent on integrations that can be performed readily using triangle quadrature. Of particular interest are the integrations involving Green’s function and its derivatives. For pairs of patches at large distances, numerical integration is reliable. However, when two patches are close to each other, the singular behavior of Green’s function makes numerical integration unstable. Even worse is the case of selfcoupling. Some of the integrals will vanish when the inner and outer integration domains are identical, but others will not. Even when patches do not overlap but share an edge or vertex, the same problem arises. To deal with the singular integrals, a technique called singularity extraction is introduced. Specific closed-form expressions for some of these integrals are presented.

11.5.1 Derivation of Coupling Terms

This section focuses on evaluating the quadruple integrals that are encountered in (11.36)– (11.41). First note that the integrals in (11.40) and (11.41) are almost trivial for any choice of basis. For an incident plane wave, these integrals can be completed in closed form. For more complicated source distributions, these are well behaved but require numerical integration to evaluate. The methods introduced for evaluating FEM matrix elements carry over here without modification. The integral (11.39) can be done by hand for almost any reasonable choice of basis and test functions. The hard work comes in evaluating integrals (11.36)–(11.38). These involve Green’s function and its first and second derivative. All of these integrands contain

some order of singular behavior. When evaluating the coupling between separate nonoverlapping basis elements, Green’s function will never be evaluated at the singular point x = x, so these cases are not as much of an issue. The real problem occurs when self-coupling is evaluated. To evaluate these integrals, explicit expressions for the normal derivatives of Green’s function are developed. From the chain rule

and GR = − GR. The normal derivatives are

To evaluate the second derivative of Green’s function, the operator n∙ is applied to (11.60). Steps in the derivation are shown as follows:

11.5.2 Singularity Extraction

An interesting feature of these integrals is the presence of n∙ R. For self-coupling on flat patches, these terms will vanish. While the singular integrals will not directly contribute to the evaluation of all self-coupling terms, singular behavior will arise when evaluating the coupling between elements that share an edge or vertex. As the integrations are performed, points exactly on a shared edge or vertex will be problematic. However, it turns out that for the integral of Green’s function and its first derivative, these results are known to be well behaved. One approach to avoiding the issue is to realize that quadrature rules for integration do not include the boundary points. Therefore, even a pure numerical integration of these types of cases will not evaluate the singularity. This is not really a viable path to numerically integrating these terms as the behavior of the integrals will prevent accurate quadrature evaluation at reasonable orders. Some of these integrals can be performed analytically to yield closed-form results that are functions only of the geometry of the two integration regions and their relative orientation. For the case of integrating Green’s function, there is a well-known result for the quadruple integral. For other singular integrals, a popular technique is to perform one of the two integrations analytically and then use numerical quadrature for the other region. The analytic integration removes the singular behavior for all but one of the integrals, leaving a finite well-behaved function defined on the other region. An algorithm for evaluating these integrals will be presented in this section. Lastly, the only real culprit in the evaluation of self-coupling is in the evaluation of the elements of M, (11.36). When both integrations are on the same patch, there is one surviving term:

− G,R

(11.62)

R

This is known as a hypersingular integral with an R− 3 order singularity. Evaluating the integral of (11.62) has been the subject of research in BEM.

Continuing with the presentation of singularity extraction, the first and second derivatives of G(R) with respect to R are evaluated:

The behavior of the integrand becomes singular when dealing with self-coupling. If a numerical integration is attempted for self-coupling terms, contributions where x = x will be encountered. The singular behavior of Green’s function in this limit is governed by the real part. Separating Green’s function into real and imaginary parts,

The limit of the imaginary part is

while the real part diverges. The integral of G(R) over a two-dimensional region is well behaved, as are many of the more singular integrals coming from derivatives of G(R). Singularity extraction involves splitting Green’s function into a part that is well behaved and a part that can be integrated over the patch to yield a closed form for one or both of the double integrations. The well-behaved part will still need to be integrated numerically but will be relatively easy to perform. It is clear that singular behavior occurs in the following functions: G, G,R, and G,RR. The specific singularities can be expressed as

where n = 0, 1, 2, …. The behavior of these functions near R = 0 is determined by looking at the Taylor series of the exponential:

For simplicity, a shorthand notation for the truncated Taylor series out to finite order n = N is introduced:

k = 0 k

By definition

The singularities can be successfully extracted from all of the terms appearing in the BIE integrals by subtracting out terms up to and including order n, leaving a leadingorder term of order Rn + 1 to cancel out the denominator in (11.67). In general,

This is written as the sum of a singular term, denoted, for example, by FnS, and a remainder,

Fn, which is well behaved:

The definition of each is obvious from the context. This strategy pays off for two reasons. The first is that the quantity in brackets is finite at R = 0. In fact, the value of this term at R = 0 for any order n can easily be determined:

The second payoff is that, for some values of n, integrals of the singular term over triangular domains can be done analytically over one triangle, in some cases both triangles. When the integration over one triangle is possible, the result is typically a non-singular function over the second triangular integration domain. Hence the self-coupling and near coupling terms may be performed with good precision and quickly by a combination of numerical and analytical methods.

The singularity extraction is performed on Green’s function and its first and second derivatives to illustrate exactly how the procedure works. Based on (11.63) and (11.64), the following are required for this example:

The remainders of each term need to be evaluated at R = 0 explicitly to make use of them in a software routine:

What remains is combining the following functions with the rest of the integrand and integration over pairs of triangular patches, or the same patch twice:

It is left as an exercise for the reader to work out explicit formulae for the other terms needed to complete the BIE calculations.

It is clear that well-behaved function for numerical quadrature, FnR, has been produced. However, it turns out that this is not always the best choice. In simplex coordinates, there is a cusp in the function FnR, that is, the derivative is discontinuous along a subset of the integration domain. An alternate approach is to generate singularity extractions that give functions with continuous first derivatives. This can be achieved by removing more terms than required in the extraction process:

with m > n. Figure 11.2 shows an example of this applied to Green’s function. Shown are the real part of Green’s function (solid line), the singular part (dotted line), and the regular part (dashed).

Figure 11.2 Real part of Green’s function

11.5.3 Evaluation of the Singular Part

All the integrals for the operator matrix elements are of the form

In (11.83) fi is a non-singular scalar function defined on the patch and gx − x is a singular function, for example, Green’s function or any of its normal derivatives. Separating the singular function into its regular and singular parts gives the following:

The first integral in (11.84) can be evaluated using the triangle quadrature methods introduced for FEM analysis. Depending on the order of the singularity and the nature of the function fj, these integrals may have a closed form. This is the case for integrals of Green’s function and linear shape functions integrated over two triangular patches. In more complex cases it may still be possible to extract a closed form for the quadruple integral using Maple or other symbolic math software. The expressions involve multiple occurrences of the logarithm function with complex arguments. Expressing the integrand as the product of a singular and nonsingular part allows one to evaluate the singular integrals by evaluating the non-singular part at one point and removing it from the integrand. Thus, attention is focused on integrals of the following form:

11.5.3.1Closed-Form Expression for the Singular Part of K

A fairly simple closed-form solution for the quadruple integral of G,SR times two linear basis functions over the same triangular patch twice is known. These formulae are presented here for the reader to use as they see fit. They were derived by Eibert and Hansen [2, 3] and used by the authors in electromagnetic calculations. Closed-form expressions are provided for the integration of linear basis functions over Green’s function in Galerkin’s method. The evaluation of these integrals makes use of the simplex coordinates defined in triangle patch integration for FEM. The three simplex coordinates are constrained, leaving two independent coordinates, defined here as (s, t). For self-coupling the relative distance can be expressed as a quadratic form:

The index n may take values, n = − 1 1,3 5,…. The integration region T is a triangle and the vector V is one of the vertices. The vertices are labeled Vi, i = 1,2 3. Points in space relative to this triangle are referenced to a local coordinate frame. This frame is constructed using the normal vector of the tile, n, and two orthogonal vectors, û and υ, in the plane of the triangle. Following the notation of the reference, the n coordinate is labeled w, and a point in space relative to this frame is labeled (u, υ, w). For simplicity V1 is placed at the origin and V2 along the horizontal axis, u, in two-dimensional space. From the vertex vectors three vectors along each edge of the triangle are defined:

Note that the edges of the triangle are labeled with the same index as the vertex opposite each edge, that is, the vertex excluded from the edge. The unit vectors may be written in terms of the triangle vertices:

The coordinate vector, x , of any point in space, p, may be written in this local patch coordinate as follows:

and similarly for υp and wp. The projection of x into the plane of the triangle is defined, ρ = x − wpn. The coordinates of the third vertex, V3, are determined by setting x = V3. An outward normal vector for each edge of the triangle is defined:

Figure 11.3 shows the variables and all geometric quantities defined on the triangle. Capital letters for the normal in the plane are used to avoid confusion with the normal to the surface.

Figure 11.3 Variables used for evaluating singular integrals

The solution to the integrals involves a recursive formula, allowing higher-order integrals to be evaluated from the results of lower-order integrals. With these geometric quantities defined, the following parameters are introduced:

The indexing in these equations obeys the constraint j = 0 3 j = 4 1. Using these quantities several well-known results for developing a procedure to evaluate the singular integrals are presented, starting with the following two integrals:

terms will occasionally generate small negative numbers. Focusing on the logarithm function is helpful. Rather, the complete integral expression needs to be investigated. From (11.94),

T

with wp = 0. Now from (11.108) and (11.113),

Before getting too involved in trying to regulate the behavior of Ij− 1, take a deeper look at the combination, tj0Ij−1, for one value of the index:

Referring to Figure 11.3, the singular behavior of the logarithm will occur inside the triangle for points along the line defined by L1. For any point, x , on this line, and between the points V2 and V3, the vector x − V2 will be parallel to L1. This is really just the point x relative to V2 as the origin of a new coordinate system. The vector x − V3 will be antiparallel to L1. Hence the

numerator of the logarithm argument will be 2 x − V3, while the denominator will be zero. This is the culprit preventing an easy evaluation of the integral. Now consider the factor t10. Based on previous comments it is clear that this term vanishes identically for any point on the L1 axis. In reality the term evaluates to 0∞ along the edge. For proper evaluation, a small amount, ε, is added in the − N1 to a point x on the edge, but not a vertex, and the behavior as ε 0 investigated. Applying L’Hôpital’s rule twice gives a term that vanishes in the limit ε 0. This term is identically zero for any point on the L1 edge. The same results are obtained for the second and third terms in (11.119) for points on the L2 and L3 edges, respectively.

It is worth mentioning that the analysis of the behavior was made easier by the vector notation of the terms and their geometric interpretation. It is rare, if not impossible, to have a true edge point present in a numerical integration procedure, but it is wise to plan for them. Figure 11.4 illustrates the removal of the singularity by showing a plot of K1− 1 for a test triangle with vertices

V1 = 0,0,0T , V2 = 01,0,0T , and V3 = 07,05,0T . The horizontal coordinates are (u, υ), which are potential locations of outer triangle integration points. Figure 11.5 is a contour plot version of Figure 11.4.

The equation for K 3− 1 can be used to evaluate the terms containing the first-order normal derivatives of Green’s function, that is, N. The method presented in this subsection also applies to near coupling that can regulate integration when the two patches share an edge or vertex, whereas the closed form cannot. As for equation (11.96) only being defined for points outside the plane in the inner triangle, this is not a deficit since N will vanish for these configurations.

Figure 11.4 Evaluation of K1−1

0.15

0.1

0.05

0

Figure 11.5 Contour plot of Figure 11.4

11.5.3.3The Hypersingular Integral

So far details on the regularization of singular integrals have been discussed along with specific formulae form the literature that are useful for most of the integrals encountered. The remaining integral that occurs in self-coupling is the hypersingular integral. This is the remaining term in the second normal derivative of Green’s function in the operator M. Evaluating (11.61) for selfcoupling yields