Lecture 37

Computational

Electromagnetics, Numerical

Methods

37.1 Computational Electromagnetics and Numerical Meth-

ods

Numerical methods exploit the blinding speed of modern digital computers to perform calcu-

lations, and hence to solve large system of equations. These equations are partial differential

equations or integral equations. When these methods are applied to solving Maxwell’s equa-

tions and related equations, the field is known as computational electromagnetics.

Maxwell’s equations are a form of partial differential equations (PDE). Boundary con-

ditions have to be stipulated for these PDE’s, and the solution can be sought by solving a

boundary value problem (BVP). In solving PDE’s the field in every point in space is solved

for.

On the other hand, the Green’s function method can be used to convert a PDE into an

integral equation (IE). Then the fields are expressed in terms of the sources, and the sources

are the unknowns to be solved. Sources in IEs are supported by a finite part of space, for

instance on the surface of the scatterer, whereas fields from PDEs permeate all of space.

Therefore, sources in IEs can be represented by a smaller set of unknowns, and therefore, are

easier to solve for compared to fields in PDEs.

37.1.1 Examples of Differential Equations

An example of differential equations written in terms of sources are the scalar wave equation:

(∇2 + k2) φ(r) = Q(r), (37.1.1)

373

374 Electromagnetic Field Theory

An example of vector differential equation for vector electromagnetic field is

∇ × μ −1 · ∇ × E(r) − ω2 · E(r) = iωJ(r) (37.1.2)

These equations are linear equations. They have one commonality, i.e., they can be

abstractly written as

L f = g (37.1.3)

where L is the differential operator which is linear, and f is the unknown, and g is the

driving source. Differential equations, or partial differential equations, as mentioned before,

have to be solved with boundary conditions. Otherwise, there is no unique solution to these

equations.

In the case of the scalar wave equation (37.1.1), L = (∇2 + k2) is a differential operator.

In the case of the electromagnetic vector wave equation (37.1.2), L = (∇ × μ −1 · ∇×) − ω2·.

Furthermore, f will be φ(r) for the scalar wave equation (37.1.1), while it will be E(r) in the

case of vector wave equation for an electromagnetic system (37.1.2). The g on the right-hand

side can represent Q in (37.1.1) or iωJ(r) in (37.1.2).

37.1.2 Examples of Integral Equations

This course is replete with PDE’s, but we have not come across too many integral equations.

Therefore, we shall illustrate the derivation of some integral equations. Since the acoustic

wave problem is homomorphic to the electromagnetic wave problem, we will illustrate the

derivation of integral equation of scattering using acoustic wave equation.1

The surface integral equation method is rather popular in a number of applications, be-

cause it employs a homogeneous-medium Green’s function which is simple in form, and the

unknowns reside on a surface rather than in a volume. In this section, the surface integral

equations2 for scalar and will be studied first. Then, the volume integral equation will be

discussed next.

Surface Integral Equations

In an integral equation, the unknown to be sought is embedded in an integral. An integral

equation can be viewed as an operator equation as well, just as are differential equations. We

shall see how such integral equations with only surface integrals are derived, using the scalar

wave equation.

1The cases of electromagnetic wave equations can be found in Chew, Waves and Fields in Inhomogeneous

Media [34].

2These are sometimes called boundary integral equations [208, 209].

Computational Electromagnetics, Numerical Methods 375

Figure 37.1: A two-region problem can be solved with a surface integral equation.

Consider a scalar wave equation for a two-region problem as shown in Figure 37.1. In

region 1, the governing equation for the total field is

(∇2 + k2

1 ) φ1(r) = Q(r), (37.1.4)

For simplicity, we will assume that the scatterer is impenetrable, meaning that the field in

region 2 is zero. Therefore, we need only define Green’s functions for regions 1 to satisfy the

following equations:

(∇2 + k2

1 ) g1(r, r′) = −δ(r − r′), (37.1.5)

The derivation here is similar to the that of Huygens’ principle. On multiplying Equation

(37.1.1) by g1(r, r′) and Equation (37.1.5) by φ1(r), subtracting the two resultant equations,

and integrating over region 1, we have, for r′ ∈ V1,



V1

dV [g1(r, r′)∇2φ1(r) − φ1(r)∇2g1(r, r′)]

=



V1

dV g1(r, r′)Q(r) + φ1(r′), r′ ∈ V1. (37.1.6)

Since ∇ · (g∇φ − φ∇g) = g∇2φ − φ∇2g, by applying Gauss’ theorem, the volume integral

on the left-hand side of (37.1.6) becomes a surface integral over the surface bounding V1.

Consequently,3

−



S+Sinf

dS ˆn · [g1(r, r′)∇φ1(r) − φ1(r)∇g1(r, r′)]

= −φinc(r′) + φ1(r′), r′ ∈ V1. (37.1.7)

3The equality of the volume integral on the left-hand side of (37.1.6) and the surface integral on the

left-hand side of (37.1.7) is also known as Green’s theorem.

376 Electromagnetic Field Theory

In the above, we have let

φinc(r′) = −



V1

dV g1(r, r′)Q(r), (37.1.8)

since it is the incident field generated by the source Q(r).

Note that up to this point, g1(r, r′) is not explicitly specified, as long as it is a solution of

(37.1.5). A simple choice for g1(r, r′) that satisfies the radiation condition is

g1(r, r′) = eik1|r−r′|

4π|r − r′| , (37.1.9)

It is the unbounded, homogeneous medium scalar Green’s function. In this case, φinc(r) is

the incident field generated by the source Q(r) in the absence of the scatterer. Moreover, the

integral over Sinf vanishes when Sinf → ∞ by virtue of the radiation condition. Then, after

swapping r and r′, we have

φ1(r) = φinc(r) −



S

dS′ ˆn′ · [g1(r, r′)∇′φ1(r′) − φ1(r′)∇′g1(r, r′)], r ∈ V1. (37.1.10)

But if r′ /∈ V1 in (37.1.6), the second term, φ1(r), on the right-hand side of (37.1.6) would be

zero, for r′ would be in V2 where the integration is not performed. Therefore, we can write

(37.1.10) as

if r ∈ V1, φ1(r)

if r ∈ V2, 0

}

= φinc(r) −



S

dS′ ˆn′ · [g1(r, r′)∇′φ1(r′) − φ1(r′)∇′g1(r, r′)]. (37.1.11)

The above equation is evocative of Huygens’ principle. It says that when the observation

point r is in V1, then the total field φ1(r) consists of the incident field, φinc(r), and the

contribution of field due to surface sources on S, which is the second term on the right-hand

side of (37.1.11). But if the observation point is in V2, then the surface sources on S generate

a field that exactly cancels the incident field φinc(r), making the total field in region 2 zero.

This fact is the core of the extinction theorem as shown in Figure 37.2 (see Born and Wolf

1980).

In (37.1.11), ˆn · ∇φ1(r) and φ1(r) act as surface sources. Moreover, they are impressed on

S, creating a field in region 2 that cancels exactly the incident field in region 2 (see Figure

37.2).

Computational Electromagnetics, Numerical Methods 377

Figure 37.2: The illustration of the extinction theorem.

Applying the extinction theorem, integral equations can now be derived. So, using the

lower parts of Equations (37.1.11), we have

φinc(r) =



S

dS′ ˆn′ · [g1(r, r′)∇′φ1(r′) − φ1(r′)∇′g1(r, r′)], r ∈ V2, (37.1.12)

The integral equations above still has two independent unknowns, φ1 and ˆn · ∇φ1. Next,

boundary conditions can be used to eliminate one of these two unknowns.

An acoustic scatterer which is impenetrable either has a hard surface boundary condition

where normal velocity is zero, or it has soft surface where the pressure is zero (also called a

pressure release surface). Since the force or the velocity of the particle is proportional to the

∇φ, a hard surface will have ˆn · ∇φ1 = 0, or a homogeneous Neumann boundary condition,

while a soft surface will have φ1 = 0, a homogeneous Dirichlet boundary condition.

φinc(r) =



S

dS′ ˆn′ · [g1(r, r′)∇′φ1(r′)], r ∈ V2, soft boundary condition

(37.1.13)

φinc(r) = −



S

dS′ φ1(r′)∇′g1(r, r′], r ∈ V2, hard boundary condition

(37.1.14)

More complicated surface integral equations (SIEs) for penetrable scatterers, as well as

vector surface integral equations for the electromagnetics cases are derived in Chew, Waves

and Fields in Inhomogeneous Media [34,210]. Also, there is another class of integral equations

called volume integral equations (VIEs) [211]. They are also derived in [34].

Nevertheless, all the linear integral equations can be unified under one notation:

L f = g (37.1.15)

where L is a linear operator. This is similar to the differential equation case. The difference

is that the unknown f represents the source of the problem, while g is the incident field

impinging on the scatterer or object. Furthermore, f does not need to satisfy any boundary

condition, since the field radiated via the Green’s function satisfies the radiation condition.

378 Electromagnetic Field Theory

37.2 Subspace Projection Methods

Several operator equations have been derived in the previous sections. They are all of the

form

L f = g (37.2.1)

37.2.1 Function as a Vector

In the above, f is a functional vector which is the analogy of the vector f in matrix theory

or linear algebra. In linear algebra, the vector f is of length N in an N dimensional space. It

can be indexed by a set of countable index, say i, and we can described such a vector with

N numbers such as fi, i = 1, . . . , N explicitly. This is shown in Figure 37.3(a).

A function f (x), however, can be thought of as being indexed by x in the 1D case.

However, the index in this case is a continuum, and countably infinite. Hence, it corresponds

to a vector of infinite dimension and it lives in an infinite dimensional space.4

To make such functions economical in storage, for instance, we replace the function f (x)

by its sampled values at N locations, such that f (xi), i = 1, . . . , N . Then the values of the

function in between the stored points f (xi) can be obtained by interpolation. Therefore, a

function vector f (x), even though it is infinite dimensional, can be approximated by a finite

length vector, f . This concept is illustrated in Figure 37.3(b) and (c). This concept can be

generalized to a function of 3D space f (r). If r is sampled over a 3D volume, it can provide

an index to a vector fi = f (ri), and hence, f (r) can be thought of as a vector as well.

4When these functions are square integrable implying finite “energy”, these infinite dimensional spaces are

called Hilbert spaces.

Computational Electromagnetics, Numerical Methods 379

Figure 37.3: A function can be thought of as a vector.

37.2.2 Operator as a Map

An operator like L above can be thought of as a map or a transformation. It maps a function

f defined in a Hilbert space V to g defined in another Hilbert space W . Mathematically, this

is written as

L : V → W (37.2.2)

Indicating that L is a map of vectors in the space V to the space W . Here, V is also called

the domain space (or domain) of L while W is the range space (or range) of L .

37.2.3 Approximating Operator Equations with Matrix Equations

One main task of numerical method is first to approximate an operator equation L f = g by

a matrix equation L · f = g. To convert the above, we first let

f ∼ =

N ∑

n=1

anfn = g (37.2.3)

380 Electromagnetic Field Theory

In the above, fn, n, . . . , N are known functions called basis functions. Now, an’s are the

new unknowns to be sought. Also the above is an approximation, and the accuracy of the

approximation depends very much on the original function f . A set of very popular basis

functions are functions that form a piece-wise linear interpolation of the function from its

nodes. These basis functions are shown in Figure 37.4.

Figure 37.4: Examples of basis function in (a) one dimension, (b) two dimension. Each of

these functions are define over a finite domain. Hence, they are also called sub-domain basis

functions.

Upon substituting (37.2.3) into (37.2.1), we obtain

N ∑

n=1

anL fn = g (37.2.4)

Then, multiplying (37.2.4) by wm and integrate over the space that wm(r) is defined, then

we have N ∑

n=1

an 〈wm, L fn〉 = 〈wm, g〉 , m = 1, . . . , N (37.2.5)

In the above, the inner product is defined as

〈f1, f2〉 =



drf1(r)f2(r) (37.2.6)

where the integration is over the support of the functions, or the space over which the functions

are defined.5 For PDEs these functions are defined over a 3D space, while in SIEs, these

5This is known as the reaction inner product [34, 47, 120]. As oppose to most math and physics literature,

the energy inner product is used [120].

Computational Electromagnetics, Numerical Methods 381

functions are defined over a surface. In a 1D problems, these functions are defined over a 1D

space.

The functions wm, m = 1, . . . , N is known as the weighting functions or testing functions.

The testing functions should be chosen so that they can approximate well a function that

lives in the range space W of the operator L . Such set of testing functions lives in the dual

space of the range space. For example, if fr lives in the range space of the operator L , the

set of function fd, such that the inner product 〈fd, fr 〉 exists, forms the dual space of W .

The above is a matrix equation of the form

L · a = g (37.2.7)

where [L]

mn = 〈wm, L fn〉

[a]n = an, [g]m = 〈wm, g〉 (37.2.8)

What has effectively happened here is that given an operator L that maps a function that lives

in an infinite dimensional Hilbert space V , to another function that lives in another infinite

dimensional Hilbert space W , via the operator equation L f = g, we have approximated

the Hilbert spaces with finite dimensional spaces (subspaces), and finally, obtain a finite

dimensional matrix equation that is the representation of the original infinite dimensional

operator equation.

In the above, L is the matrix representation of the operator L in the subspaces, and a

and g are the vector representations of f and g, respectively, in their respective subspaces.

When such a method is applied to integral equations, it is usually called the method

of moments (MOM). (Surface integral equations are also called boundary integral equations

(BIEs) in other fields [209]. When finite discrete basis are used to represent the surface

unknowns, it is also called the boundary element method (BEM) [212]. But when this method

is applied to solve PDEs, it is called the finite element method (FEM) [213–216], which is a

rather popular method due to its simplicity.

37.2.4 Mesh Generation

In order to approximate a function defined on an arbitrary shaped surface or volume by

sum of basis functions, it is best to mesh (tessellate or discretize) the surface and volume by

meshes. In 2D, all shapes can be tessellated by unions of triangles, while a 3D volume can be

meshed (tessellated) by unions of tetrahedrons. Such meshes are used not only in CEM, but

in other fields such as solid mechanics. Hence, there are many commercial software available

to generate sophisticated meshes.

When a surface is curved, or of arbitrary shape, it can be meshed by union of triangles as

shown in Figure 37.5. When a volume is of arbitrary shape of a volume is around an arbitrary

shape object, it can be meshed by tetrahedrons as shown in Figure 37.6. Then basis functions

are defined to interpolate the field between nodal values or values defined on the edges of a

triangle or a tetrahedron.

382 Electromagnetic Field Theory

Figure 37.5: An arbitrary surface can be meshed by a union of triangles.

Figure 37.6: A volume region can be meshed by a union of tetrahedra. But the surface of the

aircraft is meshed with a union of trianlges (courtesy of gmsh.info).

37.3 Solving Matrix Equation by Optimization

When a matrix system get exceedingly large, it is preferable that a direct inversion of the

matrix equation not performed. Direct inversions (e.g., using Gaussian elimination [217]

or Kramer’s rule [218]) have computational complexity6 of O(N 3), and requiring storage of

O(N 2). Hence, when N is large, other methods have to be sought.

To this end, it is better to convert the solving of a matrix equation into an optimiza-

tion problem. These methods can be designed so that a much larger system can be solved

with a digital computer. Optimization problem results in finding the stationary point of a

functional.7 First, we will figure out how to find such a functional.

Consider a matrix equation given by

L · f = g (37.3.1)

6The scaling of computer time with respect to the number of unknowns (degrees of freedom) is known in

the computer parlance as computational complexity.

7Functional is usually defined as a function of a function [34, 43]. Here, we include a function of a vector

to be a functional as well.

Computational Electromagnetics, Numerical Methods 383

For simplicity, we consider L as a symmetric matrix.8 Then one can define a functional

I = f t · L · f − 2f t · g (37.3.2)

Such a functional is called a quadratic functional because it is analogous to I = Lx2 − 2xg,

which is quadratic, in its simplest 1D rendition.

Taking the first variation with respect to f , namely, we let f = fo +δf , and find the leading

order approximation of the functional. Therefore, one gets

δI = δf t · L · fo + f t

o · L · δf − 2δf t · g (37.3.3)

If L is symmetric, the first two terms are the same, and the above becomes

δI = 2δf t · L · fo − 2δf t · g (37.3.4)

For fo to be the optimal point or the stationary point, then its first variation has to be zero,

or that δI = 0. Thus we conclude that

L · fo = g (37.3.5)

Hence, the optimal point to the functional I in (37.3.2) is the solution to (37.3.1).

Such method, when applied to an infinite dimensional Hilbert space problem, is called

variational method, but the main ideas are similar. The wonderful idea about such a method

is that instead of doing direct inversion, one can search for the optimal point or stationary

point of the quadratic functional using gradient search or gradient descent methods or some

optimization method.

37.3.1 Gradient of a Functional

It turns out that the gradient of a quadratic functional can be found quite easily. Also it

is cheaper to computer the gradient of a functional than to find the inverse of a matrix

operator. To do this, it is better to write out functional using index (or indicial, or Einstein)

notation [219]. In this notation, the functional first variation δI in (37.3.4) becomes

δI = 2δfj Lij fi − 2δfj gj (37.3.6)

In the above, we neglect to distinguish between fo and f and f represents the optimal point

is implied. Also, in this notation, the summation symbol is dropped, and summations over

repeated indices are implied. In this notation, it is easier to see what a functional derivative

is. We can differentiate the above with respect to fj easily to arrive at

∂I

∂fj

= 2Lij fi − 2gj (37.3.7)

8Functional for the asymmetric case can be found in Chew, Waves and Fields in Inhomogeneous Media,

Chapter 5 [34].

384 Electromagnetic Field Theory

Notice that the remaining equation has one index j remaining in index notation, meaning

that it is a vector equation. We can reconstitute the above using our more familiar matrix

notation that

∇f I = 2L · f − 2g (37.3.8)

The left-hand side is a notation for the gradient of a functional in a multi-dimensional space

defined by f , and the right-hand side is the expression for calculating this gradient. One needs

only to perform a matrix-vector product to find this gradient. Hence, the computational

complexity of finding this gradient is O(N 2) at worst, and O(N ) for many sparse matrices.

In a gradient search method, such a gradient is calculated repeated until the optimal point is

found. Such methods are called iterative methods.

If the optimal point can be found in Niter iterations, then the CPU time scales as Niter N α

where 1 < α < 2. There is a clever gradient search algorithm, called the conjugate gradient

method that can find the optimal point in Niter in exact arithmetics. In many gradient search

methods, Niter  N resulting in great savings in computer time.

What is more important is that this method does not require the storage of the matrix L,

but a computer code that produces the vector go = L · f as an output, with f as an input.

Both f and go require only O(N ) memory storage. Such methods are called matrix-free

methods. Even when L is a dense matrix, but is the matrix representation of some Green’s

function, fast methods now exist to perform the dense matrix-vector product in O(N log N )

operations.9

The value I is also called the cost function, and its minimum is sought in the seeking of

the solution by gradient search methods. Detail discussion of these methods is given in [220].

Figure 37.7 shows the contour plot of a cost function in 2D. When the condition number10

of the matrix L is large (implying that the matrix is ill-conditioned), the contour plot will

resemble a deep valley. And hence, the gradient search method will tend to zig-zag along the

way as it finds the solution. Therefore, convergence is slow for matrices with large condition

numbers

9Chew et al, Fast and Efficient Algorithms in CEM [9].

10This is the ratio of the largest eigenvalue of the matrix to its smallest eigenvalue.

Computational Electromagnetics, Numerical Methods 385

Figure 37.7: Plot of a 2D cost function for an ill-conditioned system (courtesy of Numerical

Recipe [220]).

Figure 37.8 shows a cartoon picture in 2D of the histories of different search paths from a

machine-learning example where a cost functional similar to I has to be minimized. Finding

the optimal point or the minimum point of a general functional is still a hot topic of research:

it is important in artificial intelligence.

Figure 37.8: Gradient search or gradient descent method is finding an optimal point (courtesy

of Y. Ioannou: https://blog.yani.io/sgd/).

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