35

CHAPTER 4

The Locally-Corrected Nyström

Method

In this chapter, the Nyström method is extended in order to better handle singularities in the

kernel of the integral operator. The extension, known as the “locally corrected” Nyström (LCN)

method, was originally proposed in [1–3]. In contrast to the conventional Nyström method, the

LCN enables a high accuracy discretization of integral operators whose kernels have either weak

or strong singularities. The LCN will be applied to two-dimensional electromagnetic scattering

problems using the electric-ﬁeld and magnetic-ﬁeld integral equations.

4.1 THE LOCALLY-CORRECTED NYSTRÖM METHOD

A fundamental difﬁculty with the classical Nyström method is that it fails if the kernel of the

integral operator is inﬁnite at one of the quadrature sample points. Although the MFIE considered

in Chapter 3 has a bounded kernel, most of the operators of interest in electromagnetics have kernels

with stronger singularities (in fact,most are inﬁnite when the source and observer locations coincide).

As demonstrated in Chapter 3, even if the kernel is bounded, the fact that it is not analytic degrades

the accuracy of the overall Nyström solution.

One remedy to this situation was proposed in [1–3], based on an approach for enhancing the

accuracy of quadrature rules [4]. The original idea was to synthesize a new quadrature rule when

the source and observer locations are in close proximity. An essentially equivalent approach is to

synthesize a new (bounded and smooth) kernel to be sampled by the original quadrature rule for

near-ﬁeld interactions. In equation form, the original approximation of the integral operator over a

cell (using the notation from Chapter 3)

cell n

J

z

(t

)K(t

mj

,t

)dt

∼

=

q

i=1

w

ni

J

z

(t

ni

)K(t

mj

,t

ni

) (4.1)

is replaced by

cell n

J

z

(t

)K(t

mj

,t

)dt

∼

=

q

i=1

w

ni

J

z

(t

ni

) L(t

mj

,t

ni

) (4.2)

where L is the new “corrected” kernel, which must be synthesized at the necessary samples. One

way to accomplish that is to derive L so that the near ﬁelds of some set of hypothetical current

distributions are correct at the necessary sample points. The hypothetical currents are, in essence,

36 CHAPTER 4. THE LOCALLY-CORRECTED NYSTRÖM METHOD

just a set of basis functions

{

B

k

(t)

}

. By imposing the condition that

q

i=1

w

ni

B

k

(t

ni

) L(t

mj

,t

ni

)

=

q

i=1

B

k

(t

ni

)

w

ni

L(t

mj

,t

ni

)

=

cell n

B

k

(t

)K(t

mj

,t

)dt

(4.3)

for each basis function in the set, and each location t

mj

where L is needed, the near ﬁeld of the

source B

k

obtained via the summation is forced to be the same as the true near ﬁeld of B

k

, obtained

from the original integral operator. If there are q sample points in the quadrature rule, and q basis

functions in the set, this leads to the square system of equations

⎡

⎢

⎢

⎢

⎢

⎣

B

1

(t

n1

)B

1

(t

n2

) ··· B

1

(t

nq

)

B

2

(t

n1

)B

2

(t

n2

)B

2

(t

nq

)

.

.

.

.

.

.

B

q

(t

n1

)B

q

(t

n2

) ··· B

q

(t

nq

)

⎤

⎥

⎥

⎥

⎥

⎦

⎡

⎢

⎢

⎢

⎢

⎣

w

n1

L(t

mj

,t

n1

)

w

n2

L(t

mj

,t

n2

)

.

.

.

w

nq

L(t

mj

,t

nq

)

⎤

⎥

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎢

⎣

cell n

B

1

(t

)K(t

mj

,t

)dt

cell n

B

2

(t

)K(t

mj

,t

)dt

.

.

.

cell n

B

q

(t

)K(t

mj

,t

)dt

⎤

⎥

⎥

⎥

⎥

⎦

. (4.4)

This system can be solved for the numerical values of L(t

mj

,t

n1

) through L(t

mj

,t

nq

)

1

.A

similar system must be solved for each observer location mj in the near ﬁeld of cell n. By creating L

in this manner, the summation in (4.2) should produce accurate near ﬁelds for any current density

J

z

(t) that can be adequately represented by the basis functions. Since the synthetic kernel L is only

needed at a ﬁnite number of source/observer locations, and the system of (4.4) is only of order q, the

required set of computations is relatively inexpensive. The most expensive part of the calculation is

the accurate evaluation of the integrals appearing on the right-hand side of Equation (4.4) because

of the kernel singularity.

In the electromagnetic equations of interest, the kernel of the integral operator is complex

valued. In that situation, both the real and imaginary parts of the kernel are modiﬁed by the above

procedure.Since it is only necessary to“correct” the non-analytic part of the original kernel,computa-

tional efﬁciency suggests that we separate the real and imaginary parts and apply the LCN procedure

only to the non-analytic part. Although that is common practice, in the interest of simplicity, we

present the development in the following sections as though we correct both parts.

1

In practice, we solve for

˜

L

mj ni

= w

ni

L(t

mj

,t

ni

), which are the actual matrix entries required in (4.12).

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