Acoustic and electromagnetic scattering analysis using discrete sources v null field method in acoustic theory

Acoustic and electromagnetic scattering analysis using discrete sources v null field method in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources v null field method in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources v null field method in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources v null field method in acoustic theory

V

NULL-FIELD METHOD IN

ACOUSTIC THEORY

In this chapter we will present the fundaments of the null-field method (NFM) for solving the Dirichlet and Neumann boundary-value problems

We begin by showing that the scattering problem reduces to the approxi-mation problem of the surface densities by convergent sequences We then present convergent projection methods for the general null-field equations Next we will investigate the conventional null-field method with discrete sources The foundations of the method include convergence analysis fol-lowing Ramm's treatment [128] and derivation of sufficient conditions which guarantee the convergence of the approximate solution The conclusion of this analysis is that the null-field method converges if the systems of

ex-pansion and testing functions form a Riesz basis in L'^{S) Finally, we will

present the equivalence between the null-field method and the auxiliary current method

1 BASIC CONCEPTS

Let hg solve the null-field equation for the exterior Dirichlet problem and let hsN be an approximation of h Define the approximate scattered field

93

d5(y),xeA (5.2)

by

«,w(x) = - 1 kN(y)5(x,y,fc) + t x o ( y ) ^ ^ ^ ] d5(y), X e £>„

(5.1) and the residual field by

SuNix) =J L;v(y)5(x,y,fc) + «o(y)^^^j^]

s

Then, the estimates

11^5 - ^SNWOO.GS ^ ^ W^^ - f^sNh^S (5-3)

and

¥uN\UG,<C'\\hs-hsN\\2,S (5.4)

hold in any closed sets Gs C Dg and d C Di Similar estimates are valid

for the exterior Neumann problem and therefore we may conclude that the

exterior boundary-value problems reduce to the approximation problem of

the surface fields on 5

Let {?A^}^^i be a system of radiating waves such that {tp^}^^^ and

{9'ipl/dn}^_^ are complete and linear independent in L^{S) Specifically,

the functions tpl stand for the localized and distributed radiating spherical

wave functions and for the distributed point sources

In the following we will construct convergent approximations of the

surface fields by using the least-squares method To this end let us rewrite

the null-field equations for the Dirichlet problem as

iK,^l)2,s = -{<'^)^ ,« = 1,2, , (5.5)

and let the sequence

KN = Y.a^,^l (5.6)

solve the null-field equations

{KN^i>l)2,s = - { < ^ ) ^ ,«/ = !, 2, AT (5.7)

Set ftg = /i* Retaining the first N equations in (5.5) and subtracting (5.7)

from (5.5) we obtain

{K - KN i^l)2,s = 0, 1/ = 1,2, , N, (5.8)

The system (5.8) is the system of normal equations for the ampHtudes

a^,/x = 1,2, ,iV Consequently, ft^^y is the projection of h'g onto the

N-dimensional space HN = Sp{V^i, , ^ ^ } ; whence \\hs — h^sN\\2 s ~^ 0 as

iV —• 00 follow The same arguments hold for the Neumann problem: the

sequence

K^ = t < ^ (5.9)

subject to the null-field equation

,.3

' 2,5 \ *^" / 2,5

^-^L=-(^'*"L'^-''^ ^' '^•'°'

converges to h^ = h^^ where hg is the unique solution to the null-field

equations

KM) =-(^,^t) ,-1,2, (5.11)

^ ^ /2,5 \ ^^ I 2,5

A comparison between the above projection method and the conventional

null-field method will presented in the electromagnetic case

2 CONVENTIONAL NULL-FIELD METHOD

2.1 Formulations

The expansion or the basis functions employed in (5.6) and (5.9) are the

complex conjugates of the testing functions This special choice leads to

a system of normal equations for amplitude determination On the other

hand, with h standing for du/dn^ we may rewrite (5.5) as

and, with h standing for u, we may rewrite (5.11) as

Assuming that k is not an irregular frequency, we can approximate h by

linear combinations of regular fields in Di, that means

for the Dirichlet problem, and

N

'^^ = E<^i (5.15)

/ x = l

for the Neumann problem When the localized spherical wave functions are

used as expansion and testing functions, the above projection methods are

equivalent to the single spherical coordinate-based null-field method For

this reason, the projection methods (5.12), (5.14) and (5.13), (5.15) with

localized and distributed sources will be referred to as the conventional

null-field method with discrete sources Before going any further, we mention

that in the single spherical coordinate-based null-field method, the matrices

corresponding to systems (5.12) and (5.13) can be expressed in a symmetric

form Indeed, let

^^ul^dS (5.16)

and

n' = I ^I'n'^dS (5.17)

be the corresponding matrices for the Dirichlet and Neumann problems,

respectively Then, application of second Green's theorem in a domain

bounded outside by S and inside by the inscribed sphere S^ gives

J [Umn Q^ U^>^ g^ Jdb-^ ^^^^dm^-m'^n.W- ( 5 1 8 )

S

Consequently

^mnm'n' = ~Z2V ^ ^ - ' ^ ' ^ " ^ ' "^ 2 / fl*n V ^ ^ ' " ' ^ ^ ^ / ^^ ( 5 1 9 )

mn ^

and

^mnm'n' = " T o p ^m,-m'On,n' "^ n / 5>ri \^^''f^'^'^'^) ( 5 2 0 )

As noted by Waterman [157] the last expressions have an interesting

fea-ture from a numerical point of view For indices n, n' > ka where a equals

the maximum radius of the obstacle, the product of radial functions in the

integrand can be approximated by the leading term arising from the

appro-priate power series expansion Similarly, the product of angular functions

can be expanded in a finite set of Fourier harmonics The dominant

numer-ical contribution to the off-diagonal elements generally would be expected

to arise from the first and more slowly varying of these terms However,

(5.19) and (5.20) reveal that the contribution in question consists of the

surface integral of the normal gradient of a potential function, and hence

by the divergence theorem vanishes identically for all off^-diagonal elements

Because of the term containing 6n,n'^ this cancellation does not occur with

diagonal elements, which might therefore be expected to dominate their

off-diagonal neighbors, resulting in a matrix better suited for inversion by

numerical techniques

2.2 Convergence analysis

The projection schemes employed in the conventional null-field method

contain different kind of basis and testing functions Consequently, the

convergence analysis has some peculiarities which we now describe For

simplicity we restrict our analysis to the projection method for the Dirichlet

problem

<^'^'*)2.5 = / - ^ = 1.2, , (5.21)

with the approximation hj\f of the form

N

^iv = X^<c/>^ (5.22)

/.3*\ / n./.3* ,

Here /^ = {duo/dn,il)l*)^^g - {uo^d^jif /dn)^^ , and {<^^}^i is a linearly

independent and complete system of functions in L'^{S) Obviously, this

analysis is relevant for all cases under consideration We are interested

to answer the following questions: is the truncated system (5.21) unique

solvable, does the sequence HN converges to the exact solution h ?

As stated in the introduction, these questions were discussed in detail

by Ramm [128] Ramm's theoretical foundations of the null-field approach

include convergence analysis, stability of the numerical scheme towards

small perturbations of the data, and estimates of the rate of convergence

We restrict our presentation to the convergence analysis and the interested

reader is referred to this reference for further details

Let us rewrite the null-field equations (5.21) in an operator form

Ah = f, (5.23)

where f = [/i, /2, ] ^ , f G /^, and the operator A : L^{S) —• /^ transforms

functions u 6 L'^{S) into vectors (^j^i*)2 5) (^'^2*)2 5 ' •••* • Hence,

theorem 3.3 of Chapter 1 can be appUed to the operator equation (5.23)

with H = L 2 ( 5 ) and G = l^

It is easy to check that A and A~^ are both bounded if

ci M2.S < \\M\12 < C2 ||ti|l2,5 ci > 0, (5.24) for all u € I'^(S'); and the inequalities hold if the system {'^t}i^=i ^^ ^ Riesz

basis of L'^iS) The convergence condition (1.44) may be written in matrix

form as

A^N^N > cGyv, 0 0 , (5.25)

for all N > NQ, Here A^ is the truncated matrix A^v = [^i^^j]^ ^i^fj, =

{<l>^i '^l*)2 5» ^» /^ = 1» 2, , N, and GN denotes the Gramm matrix GAT =

[Gi/;x], Gun — {^^^^1^)28' ^'^ ~ 1,2, jAT Note that the matrix

in-equality A < B means that {Ax,x)^2 < (Bx,x)^2 for any x eP Let A(A)

and A(A) denote the minimal and the maximal eigenvalues of a Hermitian

matrix A Then

A(A) = inf < ^ ^ , A(A) = sup i ^ ^ (5.26)

^''^ 11x11^2 x ^ o iixii;^

and therefore (5.25) holds if, for example

inf A (Air AN) > A > 0 and sup A (G/^) < A < 00 (5.27)

N \ / N

Thus, assuming that { 0 ^ } _^ is a Riesz basis of 1/^(5), {0^}°*li is

a complete and linearly independent system of elements of L'^{S) and the

inequalities (5.27) holds we deduce that the truncated system of equations

corresponding to (5.23) is unique solvable for all N > NQ and \\hN ~ /i|l2,5 ~^

0 as AT —• 00

Now, we assume that {0^} ^, is also a Riesz basis of L'^{S) Then, the

second inequality (5.27) holds, and h solving (5.23) can be represented by

h = Y^^=i ^fi^n' Furthermore, if ^ </)^ > is biorthogonal to the system

{ ^ / i } ^ i ^^^^ ^^PM \^^'\\2 g^^<^' ^^^ ^N = E^=i ^^<l>fi we obtain

a^ = (/iAr,0^) and therefore |a^ —a^| < c\\h]s[ — h\\2S' The last

inequality implies uniform convergence in fx of the sequence a^ provided

that ||/ijv ~ /i||2,5 -^ 0 as AT —> 00

Si

FIGURE 5.1 The spherical surface S^^

Conditions (5.27) are spectral conditions which can be verified directly

if the systems {^^} ^^ and {</>^}^i are given The practical conclusion is

that the projection scheme converges if the systems {^^} _j and {(t>^} _.j

form a Riesz basis of L^{S) and the first inequality (5.27) holds These

conditions will be numerically verified in the electromagnetic case for the

systems of localized and distributed spherical vector wave functions

As it was shown by Ramm, the spherical waves, although complete in

L^(5), do not form a Riesz basis on S unless S is a sphere This conclusion

follows from the fact that the condition number of the truncated Gramm

matrix grows to infinity as the truncation index grows to infinity In general

these systems do not form a Schauder basis either A counterexample

showing that the spherical wave functions do not form a basis in L'^(S) can

be given as follows Consider a spherical surface 5^ dividing S into exactly

two parts: the first one, 5 i , is in the exterior of S'^^ and the second one,

52, is contained in the interior of S^ This situation is depicted in Figure

5.1 Define the scalar field u{x) = ^(x,y,fc), x 7*^ y, where y is fixed point

on S^ Using the multipole expansion of the Green function we obtain for

x € 5 i , | y | < | x | ,

00 n

"W = I ] 53 ^mnuLW (5.28)

n=Om=—n

and for x € S2,|y| > | x | ,

00 n

" W = E E «mn"mn(x), (5.29) n=Om=—n

where

amn = ^T>mnUlrnn{y) and a'^^ = ^ P m n u l m n ( y )

-Thus, assuming that the representations (5.28) and (5.29) are vahd on the

entire surface S we will contradict the spherical wave expansion of the

Green function

In the single spherical coordinate-based null-field method, the infinite

set of null-field equations guarantees that the total field will be zero inside

the maximal inscribed sphere Because of its analyticity, the total field

vanishes throughout the entire interior volume If we consider a finite

se-quence of null-field equations, we guarantee that the residual field tends to

zero inside the maximal inscribed sphere But in general this result does

not imply that the residual field converges to zero within the entire interior

volume If instead of localized multipoles we use distributed sources

(spher-ical multipoles and point sources) it is possible to overcome the numer(spher-ical

instabilities associated with the single spherical coordinate-based null-field

method The explanation is that the null-field conditions will be satisfied

in the interior of the support of discrete sources, whose form and position

can be correlated with the boundary geometry A similar technique was

used by Bates and Wall [11] Using the bilinear expansion of the Green

function in the spheroidal coordinate system Bates and Wall imposed the

null-field condition inside the inscribed spheroid In this way it was

possi-ble to reduce numerical instabilities by decreasing the part of the null-field

region not included in D^ Although this method enables many bodies to

be analyzed satisfactorily, one can devise shapes for which this method is

not particularly suitable In this context, the method based on discrete

sources appears to be more flexible

We conclude this section by presenting the equivalence between the

null-field method and the auxiliary current method Let us consider the

Dirichlet boundary-value problem and let S~ be a surface of class C^^

enclosed in Di Define the operator Ti : L^{S) —> L^{S~) by

)=Jh{y)g{x,y,k)dS(y) (5.30)

(W/i)(x

T H E O R E M 2.1: Consider Di a bounded domain of class C^; let the

surface S~ be enclosed in Di and assume k ^ p{D~), where D~ is the

interior of S~ Let h solve the integral equation of the first kind

Hh = UQ (5.31) Then h solves the general null-field equation (2.82) and conversely

Proof: Let

ii(x) = u o ( x ) - /"/i(y)^(x,y,fc)d5(y), x € R^ - 5 (5.32)

Since h solves (5.31) we see that u = 0 on S~ From k ^ p[D~) we find

that w = 0 in D^\ whence, by the analyticity of iz, i/ = 0 in Di follows

Hence, h satisfies the general null-field equation The converse theorem is

immediate

Because of this equivalence the integral equation (5.31) has precisely

one solution and this solution belongs to C^'^{S) The following theorem

is the analog of theorem 2.3 given in Chapter 4 for the operator H

T H E O R E M 2.2: Consider Di a hounded domain of class C^ and let the

surface S~ he enclosed in Di The operator H defined hy (5.30) is injective

and has a dense range provided that k is not an eigenvalue for the interior

ofS-

Proof: The injectivity of H follows from the assumption k ^ p{D~)

and theorem 2.2 of Chapter 2 For proving the second part of the theorem

we have to show that N{V)) = {0}, where H^ is the adjoint operator of

H and N{H^) is the null space of H ^ Since the adjoint operator H^ :

L^{S-) -^ L'^(S) is given hy

{n^a) (y) = J a(x)p*(x,y,fc)d5(x), (5.33)

s-we may proceed as in theorem 2.3 of the precedent chapter to conclude

The operator 7i has an analytic kernel and therefore the integral

equa-tion is severely ill-posed Actually, the integral operator H acting from

L^{S) into 1/^(5"") is a compact operator with an open range of values

The integral equation (5.31) may be solved by using the Tikhonov

regular-ization, that is by solving

Xhx-hH^nhx = n^uo (5.34)

with the regularization parameter A > 0 From the classical theory of the

Tikhonov regularization scheme, we know that the operator AJ 4- H^H :

L^{S) —• L^{S) is bijective and has a bounded inverse Furthermore, since

H is injective TZx = {XI4- H^H)~^H^ defines a regularization scheme with

||72,;^|| < 1 / 2 \ / A (cf Colton and Kress [35] for a detailed discussion) Note

that the Tikhonov regularization can be interpreted as a penalized residual

minimization since hx solving (5.34) minimizes the Tikhonov functional

/iA = argmin{||W/i-uo|l2,5- + A||ft||2,5} (5.35) Projection methods for the integral equation (5.34) are given by

the-orem 3.4 of Chapter 1 with A = XI and B = V)H The approximate

solution hxN is sought in the form of a linear combination of regular fields

hxN = X^^=i ^^i^^, a-nd assumed to satisfy the projection relations

({XI + V)U) hxN - W^uo, i^D^s = 0, 1/ = 1,2, , N (5.36)

In the electromagnetic case, we will analyze the Tikhonov regularization

from a computational point of view

A projection scheme for the integral equation (5.31) is

<H/iN-t*o,W^^>2 5- = 0 , ^ = l,2, ,iV, (5.37)

where h^ = Y^^z^i ^^V'^- Note that the system {'Hipl}^_^ is complete in

1/^(5"") provided the system {i^l}^^^ is complete in L^{S) and k ^ p{D^),

The projection scheme (5.37) is equivalent to the minimization problem

a = argmin \\HhN — tio||2 s- » (5.38) Il2,5

i T

with a = [a^y ,ti = 1,2, ,N Now, let /ijv(y) = E n = i « n % - y n ) j

y € 5 , where { y n } ^ i is a dense set of points on 5 Choose a mesh on S

with Xj, j = 1,2, , J, located at the center of each cell Then, we may

compute a by solving the discrete version of the minimization problem

(5.38), i.e

a = arg min || A x - f Ufa, (5.39)

where A = [ajn]y ajn = ^(xj,y„,A:), j = 1,2, , J, n = 1,2, , AT, is a

design matrix and f = [fj] , fj = uo(xj), j = 1,2, , J The above

least-squares problem is similar to that obtained in the auxiliary sources method

but with the collocation points replaced by the source points and conversely

In other words, the auxiliary sources are kept on the surface and the

collo-cation surface is shifted inside the body Numerical simulations performed

by Zaridze et al [171] demonstrate that the computation of the surface

fields by the above scheme does not avoid the problem of scattered field

singularities

In Figures 5.2 and 5.3, the dependence of the far-field pattern on the

geometry of the auxiliary surface is shown The plotted data show that the

convergence occurs when the auxiliary and extended boundary surfaces

contain the scattered field singularities Conversely, when the singularities

are outside of these surfaces the solution diverges

2.3 Transition matrix

The transition matrix relates the expansion coeflScients of the incident and

the scattered fields and plays an important role in multiple scattering and

orientation averaging problems Let us assume that the systems {^J,} _i

and {^V^i/5n}^_j form a Schauder basis in L^{S) Then, h can be

repre-sented as