Applied Ocean Research 28 (2006) 193–207www.elsevier.com/locate/apor
A semianalytical solution method for twodimensional Helmholtz equation
Boning Li, Liang Cheng
∗
, Andrew J. Deeks, Ming Zhao
School of Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
Received 18 November 2005; received in revised form 31 March 2006; accepted 15 June 2006Available online 25 January 2007
Abstract
A semianalytical solution method, the socalled Scaled Boundary Finite Element Method (SBFEM), is developed for the twodimensionalHelmholtz equation. The new method is applicable to twodimensional computational domains of any shape including unbounded domains. Theaccuracy and efﬁciency of this method are illustrated by numerical examples of wave diffraction around vertical cylinders and harbour oscillationproblems. The computational results are compared with those obtained using analytical methods, numerical methods and physical experiments. Itis found that the present method is completely free from the irregular frequency difﬁculty that the conventional Green’s Function Method (GFM)often encounters. It is also found that the present method does not suffer from computational stability problems at sharp corners, is able to resolvevelocity singularities analytically at such corners by choosing the structure surfaces as sidefaces, and produces more accurate solutions thanconventional numerical methods with far less number of degrees of freedom. With these attractive attributes, the scaled boundary ﬁnite elementmethod is an excellent alternative to conventional numerical methods for solving the twodimensional Helmholtz equation.c
2007 Published by Elsevier Ltd
Keywords:
Scaled boundary ﬁnite element method; Helmholtz equation; Wave diffraction; Harbour oscillation; Irregular frequency; Singularity
1. Introduction
Surface water wave diffraction around surface piercingstructures and harbour oscillations excited by waves havebeen the topics of many research projects in the past.These two classes of problems, in general, are governedby the twodimensional Helmholtz equation associated withboundary conditions on the structure surface and a radiationcondition at inﬁnity. Analytical solutions exist only fora few simple objects such as a circular cylinder [16].Most problems of engineering signiﬁcance, however, needto be solved numerically. Conventional numerical methods,including the Finite Element Method (FEM)/Inﬁnite ElementMethod (IEM) [2,3] and the Boundary Element Method
(BEM)/Green Function Method (GFM) [1,4,11,21] have been
employed for solving these problems.The FEM was applied to solve wave diffraction problemsin the early 1970s [3,30]. The FEM is well known for its
ﬂexibility in handling irregular boundary problems. This isparticularly attractive for wave diffraction around offshore
∗
Corresponding author. Tel.: +61 8 9380 3073; fax: +61 8 9380 1018.
Email address:
cheng@civil.uwa.edu.au (L. Cheng).
structures, because most offshore structures are of irregularshape. However, one of the difﬁculties encountered in usingthe ﬁnite element method to solve wave diffraction problemsin an unbounded domain is the implementation of the radiationboundary condition at the inﬁnity. In the FEM, a calculationdomain of a ﬁnite size is normally used to approximate theinﬁnite domain on which the wave diffraction problem isdeﬁned. To satisfy the radiation boundary condition at the outerboundary of the truncated domain, this outer boundary has tobe far away from the object investigated. The further awaythe outer boundary is, the more the nodes needed inside thedomain to maintain the level of accuracy of the solution. Agreaternumberofnodesnormallyimplieshighercomputationalcost. This problem becomes more severe for threedimensionalproblems. To avoid this difﬁculty, the socalled hybrid elementmethod has been used by many researchers with moderatesuccess [5]. The hybrid element method combines the ﬁniteelement solution of the problem in a ﬁnite domain next to theobject with an analytical (or inﬁnite element) solution at theouter boundary of the ﬁnite domain. It employs the advantagesof both the ﬁnite element method and the analytical method forthis particular kind of problems. Zienkiewicz et al. (1978) [30]provides a detailed review of such methods.
01411187/$  see front matter c
2007 Published by Elsevier Ltddoi:10.1016/j.apor.2006.06.003
194
B. Li et al. / Applied Ocean Research 28 (2006) 193–207
TheBEM/GFMhavealsobeenwidelyusedforwavediffraction problems. These methods make use of a singular solutionthat satisﬁes the free surface, seabed and radiation boundaryconditions.Thescatteredwavepotentialisrepresentedbyanintegral of the singular solution multiplied by a distribution function of singularities on the surface of the object. The strength of the singularities is then determined by enforcing the boundarycondition on the surface of the object. The GFM is normallyvery efﬁcient, because only the surface of the object needs tobe discretized, and is widely used for calculating wave forceson offshore structures. However, the GFM does suffer fromsome fundamental problems such as irregular frequencies dueto the use of the source distribution representation of the solution [9,19], and some numerical difﬁculties such as modelling
reentrant structure geometries or structures with sharp cornersor small openings [20].The Scaled Boundary Finite Element Method (SBFEM) isa semianalytical computational procedure for solving linearpartial differential equations, combining the advantages of the ﬁnite element and the boundary element methods withappealing features of its own. The method, based on the socalled multicell cloning technique, was srcinally establishedto model dynamic problems in an unbounded domain byWolf and Song [26], and has gained considerable success in
modelling structural mechanics problems. The key advantageof the method is the derivation of fundamental equations basedon a scaled boundary coordinate transformation that leads toa system of linear secondorder ordinary differential equationsin displacements with the radial coordinate as the independentvariable [22]. Recent developments in the solution of problems
of elastostatics and elastodynamics have demonstrated itsabilities and advantages for soilstructure interaction problemsin unbounded domains [27,29].
Two recent extensions of the srcinal scaled boundary ﬁniteelement method [27] to ﬂuid ﬂow problems have demonstrated
the promising potential of the method in solving ﬂuid ﬂowproblems. Deeks and Cheng [6] established a scaled boundary
ﬁnite element solution to twodimensional Laplace equationfor potential ﬂow around obstacles. Their work demonstratedthe excellent ability and advantages of the technique to themodel velocity singularities at the corners of a square cylinder.Flow around multiple obstacles is solved using a substructuretechnique where the scaled boundary ﬁnite element solutionsfor bounded and unbounded domain problems are matched atthe domain interface. However, this solution is not applicableto computational domains with parallel sidefaces because theconventional SBFEM requires existence of a scaling centre.Li et al. [14] proposed a modiﬁed SBFEM for problems withparallel sidefaces. Li et al. [15] also applied this modiﬁedmethod to calculate wave diffraction by ﬁxed structures andwaveradiationexcitedbyoscillatingstructuresinwaterofﬁnitedepth.However, the previous studies [6,14,15] of the SBFEM
are only applicable to problems governed by the Laplaceequation. The objective of the present paper is to extend theSBFEM to solving the twodimensional Helmholtz equationthat governs many practical problems such as wave diffraction
Fig. 1. Deﬁnition sketch of wave diffraction around obstacles.
problems around vertical structures. In the following sections,commencing with the summary of the linearized wavediffraction formulae and the fundamental theory of thescaled boundary ﬁnite element method, the solution of twodimensional Helmholtz equation for a bounded domain andan unbounded domain are derived. Then the substructuretechniqueisintroduced. Finally, theproposedmethod isappliedto several examples and the numerical results are comparedwith those obtained using other methods.
2. Formulation of the boundaryvalue problem
The typical wave diffraction problem shown in Fig. 1 isconsidered here. Applying the assumption of linearized wavetheory, the total velocity potential
Φ
T
may be expressed as thesummation of the incident wave potential
Φ
I
and the scatteredwave potential
Φ
S
. Employing the method of the separation of variables, the velocity potential may be written as
Φ
(
x
,
y
,
z
,
t
)
=
φ(
x
,
y
)
Z
(
z
)
e
−
i
ω
t
(1)where
Φ
denotes any one of
Φ
T
,
Φ
I
and
Φ
S
,
Z
(
z
)
isthe corresponding water depth function,
ω
is the angularfrequency,
t
is the time and i is the imaginary unit. Thevelocity potential
φ(
x
,
y
)
is governed by the twodimensionalHelmholtz equation
∇
2
φ
+
k
2
φ
=
0 in domain
Ω
(2)where
k
is the wave number. On the boundaries (wettedbody boundary
Γ
c
and inﬁnity boundary
Γ
∞
) of the domain
Ω
, Neumann boundary condition and Sommerfeld radiationcondition are applied respectively:
∂φ
T
∂
n
=
0 at the body surface
Γ
c
(3)lim
r
→∞
√
r
∂φ
S
∂
r
−
i
k
φ
S
=
0 at inﬁnity
Γ
∞
(4)in which
n
designates the unit normal to the boundary and
r
is the radial coordinate. The boundary condition speciﬁed byEq. (3) will result in a homogeneous scaled boundary ﬁniteelementequationthatisconvenienttosolve,whiletheboundary
B. Li et al. / Applied Ocean Research 28 (2006) 193–207
195Fig. 2. Substructuring conﬁguration and scaled boundary coordinate deﬁnition.
condition speciﬁed by Eq. (4) will result in satisfaction of theradiation condition at inﬁnity.The scaled boundary ﬁnite element solution of wave diffraction problems around multiple obstacles in an unbounded domain can be constructed in a way similar to that illustrated inFig. 2(a). The entire computational domain is divided into anunbounded domain and a bounded domain with a common interface of
Γ
b
. Thescaled boundary ﬁnite elementsolution of thetwodimensional Helmholtz equation is modiﬁed to seek solutions of Eq. (2) in the unbounded domain with relevant boundary conditions on
Γ
b
and
Γ
∞
and in the bounded domain withappropriate boundary conditions on
Γ
b
and
Γ
c
. The boundarycondition on
Γ
b
can be eliminated by matching the unboundeddomain solution and bounded domain solution on
Γ
b
.
3. Scaled boundary ﬁnite element solutions
3.1. Scaled boundary ﬁnite element equation
As illustrated in Fig. 2, the computational domain can bedivided into a number of subdomains, denoted as
Ω
i
(
i
=
1
,
2
,
3
,...,
N
+
1
)
. A typical subdomain (
Ω
i
in Fig. 2(a)) hasa scaling centre (solid point in Fig. 2(b)) and is bounded by twosidefaces (
s
0
and
s
1
in Fig. 2(b)) or
Γ
ci
,
Γ
bi
,
Γ
i
and
Γ
i
+
1
. Thescaled boundary ﬁnite element solution in the bounded domaincan be obtained by matching the solutions of Eq. (2) on internalboundaries
Γ
i
(
i
=
1 to
N
). The scaled boundary ﬁnite elementsolution in a typical subdomain
Ω
i
is presented here and thesolutions in the other subdomains can be obtained using thesame approach.Suppose
Γ
Ω
i
is comprised of
Γ
bi
,
Γ
i
and
Γ
i
+
1
. The totalvelocity potential on
Γ
Ω
i
satisﬁes
∂φ
T
∂
n
=−¯
v
n
on
Γ
Ω
i
(5)where
¯
v
n
is the velocity component outward normal to theboundary
Γ
Ω
i
.ApplyingtheweightedresidualapproachtoEqs.(2) and (3) and employing Green’s identity result in
Ω
i
∇
T
w
∇
φ
T
d
Ω
−
Ω
i
w
k
2
φ
T
d
Ω
−
Γ
Ω
i
w
¯
v
n
d
Γ
=
0 (6)where
w
is a weighting function.To apply the SBFEM to Eq. (6), the socalled scaledboundary coordinate system is introduced. A typical scaledboundary coordinate system [28] is shown in Fig. 2(b). The
scaled radial coordinate
ξ
is 1 on
ξ
=
ξ
e
(
Γ
bi
∪
Γ
i
∪
Γ
i
+
1
) and zero at the scaling centre for a bounded domain.Consequently, each value of
ξ
deﬁnes a scaled version of thecurve
S
. Thecircumferentialcoordinate
s
measuresthedistanceanticlockwise around a deﬁning curve
S
. The bounded domain
Ω
i
is deﬁned as the region enclosed by 0
≤
ξ
≤
1 and
s
0
≤
s
≤
s
1
. For the case of unbounded domain, the scaledradial coordinate
ξ
equals 1 at the boundary
Γ
bi
∪
Γ
i
∪
Γ
i
+
1
and
ξ
e
approaches to inﬁnity (see Ref. [6] for more details). The
deﬁning curve
S
can be discretized by shape functions
[
N
(
s
)
]
in the classical ﬁnite element manner. Thus, an approximatesolution
φ
h
(ξ,
s
)
to Eq. (6) is sought in the form
φ
h
(ξ,
s
)
=[
N
(
s
)
]{
a
(ξ)
}
(7)wherevector
{
a
(ξ)
}
representsradialnodalfunctionsanalogousto nodal values in the standard ﬁnite element method. Ateach node
i
the function
a
i
(ξ)
designates the variation of thetotal potential in the radial direction. Since no discretization iscarried out in the radial direction, the scaled boundary ﬁniteelement method keeps the radial solution
{
a
(ξ)
}
analytical.Using the scaled boundary transformation detailed by Deeksand Cheng (2003) [6], the Laplace operator
∇
can be expressedas
∇ ={
b
1
(
s
)
}
∂∂ξ
+
1
ξ
{
b
2
(
s
)
}
∂∂
s
(8)where
{
b
1
(
s
)
}
and
{
b
2
(
s
)
}
are vectors that are only dependenton the deﬁnition of the curve
S
. The approximate velocityvector
{
v
h
(ξ,
s
)
}
can be formulated as
{
v
h
(ξ,
s
)
}=[
B
1
(
s
)
]{
a
(ξ)
}
,
ξ
+
1
ξ
[
B
2
(
s
)
]{
a
(ξ)
}
(9)where, for convenience,
[
B
1
(
s
)
]={
b
1
(
s
)
}[
N
(
s
)
]
(10)
[
B
2
(
s
)
]={
b
2
(
s
)
}[
N
(
s
)
]
,
s
.
(11)Applying the Galerkin approach, the weighting function
w
canbe formulated by employing the same shape functions as theapproximation for the potential (Eq. (7)).
w(ξ,
s
)
=[
N
(
s
)
]{
w(ξ)
}={
w(ξ)
}
T
[
N
(
s
)
]
T
(12)The following SBFEM equation can be obtained by substitutingEqs. (7), (8) and (12) into (6) and integrating all terms
containing
{
w(ξ)
}
,
ξ
by parts with respect to
ξ
:
{
w(ξ
e
)
}
T
[
E
0
]
ξ
e
{
a
(ξ
e
)
}
,
ξ
+[
E
1
]
T
{
a
(ξ
e
)
}−
s
[
N
(
s
)
]
T
¯
v
n
τ
ξ
d
s
−{
w(ξ
i
)
}
T
[
E
0
]
ξ
i
{
a
(ξ
i
)
}
,
ξ
+[
E
1
]
T
{
a
(ξ
i
)
}+
s
[
N
(
s
)
]
T
¯
v
n
τ
ξ
d
s
−
ξ
e
ξ
i
{
w(ξ)
}
T
[
E
0
]
ξ
{
a
(ξ)
}
,
ξξ
196
B. Li et al. / Applied Ocean Research 28 (2006) 193–207
+
(
[
E
0
]+[
E
1
]
T
−[
E
1
]
)
{
a
(ξ)
}
,
ξ
−[
E
2
]
1
ξ
{
a
(ξ)
}+
k
2
ξ
[
M
0
]{
a
(ξ)
}
d
ξ
=
0 (13)where
[
E
0
]=
s
[
B
1
(
s
)
]
T
[
B
1
(
s
)
]
J

d
s
(14)
[
E
1
]=
s
[
B
2
(
s
)
]
T
[
B
1
(
s
)
]
J

d
s
(15)
[
E
2
]=
s
[
B
2
(
s
)
]
T
[
B
2
(
s
)
]
J

d
s
(16)
[
M
0
]=
s
[
N
(
s
)
]
T
[
N
(
s
)
]
J

d
s
.
(17)The following conditions hold if Eq. (13) is satisﬁed for all setsof weighting functions
{
w(ξ)
}
.
[
E
0
]
ξ
i
{
a
(ξ
i
)
}
,
ξ
+[
E
1
]
T
{
a
(ξ
i
)
}=
s
N
T
(
s
)(
−¯
v
n
)τ
ξ
d
s
(18)
[
E
0
]
ξ
e
{
a
(ξ
e
)
}
,
ξ
+[
E
1
]
T
{
a
(ξ
e
)
}=
s
[
N
(
s
)
]
T
¯
v
n
τ
ξ
d
s
(19)
[
E
0
]
ξ
2
{
a
(ξ)
}
,
ξξ
+
(
[
E
0
]+[
E
1
]
T
−[
E
1
]
)ξ
{
a
(ξ)
}
,
ξ
−[
E
2
]{
a
(ξ)
}+
k
2
ξ
2
[
M
0
]
a
(ξ)
=
0
.
(20)Eqs. (18) and (19) indicate the relationships between the nodal
potential and the integrated nodal ﬂow on the boundary withconstant radial coordinate
ξ
. For the case of bounded domainconsidered in Fig. 2(b),
ξ
e
represents the sum of
Γ
bi
Γ
i
and
Γ
i
+
1
and
ξ
i
is actually the scaling centre with
ξ
i
=
0. For the caseof unbounded domain,
ξ
i
represents the sum of
Γ
bi
Γ
i
and
Γ
i
+
1
and
ξ
e
tends to inﬁnity. Eq. (20) is the scaled boundary ﬁniteelement equation. It can be seen that Eq. (20) is a homogeneoussecondorder ordinary differential equation, due to the use of the total velocity potential as an unknown in the boundeddomain.
3.2. Bounded domain solution
Song and Wolf [23] presented the solution to Eq. (20)
for elastodynamic problems in a bounded domain. However,this solution is based on a speciﬁc property of rigid bodymovements in solid mechanics that is not available for thewave diffraction problem considered in this study. Thereforethe application of this solution to the problem investigated hereis not straightforward. An improved method has been proposedin this study to obtain an analytical solution to Eq. (20).To solve Eq. (20), a transformation of Eq. (20) to a set of
ﬁrstorder ordinary differential equations of order 2
n
can beperformed [23]. Considering the satisfaction of Eqs. (18) and
(19), no generality is lost by assuming
{
q
(ξ)
}=[
E
0
]
ξ
{
a
(ξ)
}
,
ξ
+[
E
1
]
T
{
a
(ξ)
}
.
(21)Combining Eq. (20) with Eq. (21) and introducing the
independent variables
¯
ξ
=
k
ξ
(22)and
{
X
(
¯
ξ)
}=
{
a
(ξ)
}{
q
(ξ)
}
=[
X
(
¯
ξ)
]{
c
}
(23)result in
¯
ξ
[
X
(
¯
ξ)
]
,
¯
ξ
=−[
Z
][
X
(
¯
ξ)
]− ¯
ξ
2
[
M
][
X
(
¯
ξ)
]
(24)with the coefﬁcient matrices
[
Z
]=
[
E
0
]
−
1
[
E
1
]
T
−[
E
0
]
−
1
−[
E
2
]+[
E
1
][
E
0
]
−
1
[
E
1
]
T
−[
E
1
][
E
0
]
−
1
(25)and
[
M
]=
0 0
[
M
0
]
0
.
(26)For the Hamiltonian matrix
[
Z
]
, the eigenvalues consist of twogroups of values with opposite signs
[¯
Λ
]=
[
λ
j
]
00
−[
λ
j
]
,
j
=
1
,
2
,...,
n
(27)where Re
(λ
j
)
≥
0. The corresponding eigenvalue problem canbe formulated as
[
Z
][
Φ
]=[
Φ
][¯
Λ
]
(28)where
[
Φ
]
denotes the eigenvector matrix.Based on the physical meaning of Eqs. (27) and (28), zero
eigenvalues appear in pairs in Eq. (27). A zero eigenvalue leadsto a constant potential in the entire ﬂuid domain. This meansthat there is no ﬂow in the ﬂuid domain. Thus the eigenvectorsof zero eigenvalues are not linearly independent any more,leading to a singular matrix
[
Φ
]
. The method proposed by Songand Wolf [23] to solve Eq. (24) is not applicable here, because
the eigenvectors of zero eigenvalue are linearly independent forthe solid mechanics problems [23] investigated. Therefore, to
obtain the solution for wave problems, Jordan decompositionof the matrix
[
Z
]
is performed, namely,
[
Z
][
T
]=[
T
][
Λ
]
(29)with
[
Λ
]=
[
λ
j
]
0 10 0
−[
λ
j
]
,
j
=
1
,
2
,...,
n
−
1(30)in which the transform matrix
[
T
]
is invertible. Althoughprogramming of Jordan decomposition for general matrix couldbe tedious and unstable, Jordan decomposition of the matrix
[
Z
]
does not reduce the efﬁciency of the computer program toomuch when advantage is taken of the simple structural form of the matrix
[
Λ
]
and known eigenvalues. In fact, based on theeigenvector matrix
[
Φ
]
, the matrix
[
Z
]
can be easily formedusing the characteristic of the Jordan chain.Following the procedures provided by Ref. [7], through a
series of matrix transformations and the solution of a system of recursion equations, the analytical solution of Eq. (24) may be
B. Li et al. / Applied Ocean Research 28 (2006) 193–207
197
expressed as
[
X
(
¯
ξ)
]=[
T
][
R
(
¯
ξ)
]¯
ξ
[
Λ
]
¯
ξ
[
U
]
(31)where the matrix
[
R
(
¯
ξ)
]
is formulated as a power series in
¯
ξ
with a leading identity matrix
[
I
][
R
(
¯
ξ)
]=[
R
0
]+ ¯
ξ
2
[
R
1
]+ ¯
ξ
4
[
R
2
]+···+ ¯
ξ
2
m
[
R
m
]+···
(32)with
[
R
0
]=[
I
]
(33)and the matrix
[
Λ
]
is an upper triangular matrix witheigenvalues being on the diagonal entries and
[
U
]
is alsoan upper triangular matrix with zero diagonal entries. Forconvenience, the matrix function
¯
ξ
[
U
]
may be written as
¯
ξ
[
U
]
=[
Y
(
¯
ξ)
]
.
(34)Partitioning all the matrices on the right hand side of Eq. (31)and constant vector
{
c
}
into block matrix with
n
×
n
dimensionsand block vector with
n
×
1 dimensions respectively, thensubstituting Eqs. (31) and (34) into Eq. (23) yields
{
X
(
¯
ξ)
}=
[
T
11
] [
T
12
][
T
21
] [
T
22
]
[
R
11
(
¯
ξ)
] [
R
12
(
¯
ξ)
][
R
21
(
¯
ξ)
] [
R
22
(
¯
ξ)
]
×
¯
ξ
[
λ
]
¯
ξ
[
P
]
0
¯
ξ
−[
λ
]
[
Y
11
(
¯
ξ)
] [
Y
12
(
¯
ξ)
][
Y
21
(
¯
ξ)
] [
Y
22
(
¯
ξ)
]
{
c
1
}{
c
2
}
.
(35)To obtain a ﬁnite solution for
¯
ξ
=
0,
{
c
2
}
must be zero. Forbrevity, introducing
[
K
11
(
¯
ξ)
] [
K
12
(
¯
ξ)
][
K
21
(
¯
ξ)
] [
K
22
(
¯
ξ)
]
=
[
T
11
] [
T
12
][
T
21
] [
T
22
]
×
[
R
11
(
¯
ξ)
] [
R
12
(
¯
ξ)
][
R
21
(
¯
ξ)
] [
R
22
(
¯
ξ)
]
(36)into Eq. (35) yields
{
a
(ξ)
}=[
A
(
¯
ξ)
]{
c
1
}
(37)
{
q
(ξ)
}=[
Q
(
¯
ξ)
]{
c
1
}
(38)with
[
A
(
¯
ξ)
]=[
K
11
(
¯
ξ)
]¯
ξ
[
λ
]
[
Y
11
(
¯
ξ)
]{
c
1
}
(39)
[
Q
(
¯
ξ)
]=[
K
21
(
¯
ξ)
]¯
ξ
[
λ
]
[
Y
11
(
¯
ξ)
]{
c
1
}
.
(40)Eliminating the constant vector
{
c
1
}
in Eqs. (37) and (38) leads
to
{
q
(ξ)
}=[
H
b
(
¯
ξ)
]{
a
(ξ)
}
(41)with
[
H
b
(
¯
ξ)
]=[
Q
(
¯
ξ)
][
A
(
¯
ξ)
]
−
1
.
(42)Once the nodal potential vector
{
a
(
¯
ξ)
}
is obtained, theintegration constant vector
{
c
1
}
can be determined as
{
c
1
}=[
A
(
¯
ξ)
]
−
1
{
a
(ξ)
}
.
(43)With the known integration constant vector
{
c
1
}
, the entirepotential ﬁeld can be found by substituting Eq. (37) into Eq. (7).
The velocity ﬁeld can be determined by substituting theintegration constant vector
{
c
1
}
into Eq. (38) to determinethe nodal ﬂow vector
{
q
(ξ)
}
, then employing Eq. (21) toderive the derivative of the nodal potential vector
{
a
(
¯
ξ)
}
withrespect to the radial coordinate
ξ
and substituting the derivativeinto Eq. (9).
3.3. Unbounded domain solution
For an unbounded domain
(
1
≤
ξ
≤ ∞
)
, Eq. (24) is moredifﬁcult to solve because it has an irregular singular point at
ξ
=∞
, and all eigenvalues of the matrix
[
M
]
are zero. In solidmechanics problems, a highfrequency asymptotic expansion of the dynamic stiffness can be constructed using the radiationcondition [23]. The displacements obtained from the high
frequency asymptotic expansion method outlined do not satisfythe Sommerfeld radiation condition automatically in the sameway as the Hankel function does for the scalar case. Therefore,the solution procedure for an unbounded domain provided bySong and Wolf (1998) [23] cannot be applied directly to the
problem at hand.In the work discussed here, an asymptotic expansionfor the scattered wave potential as
ξ
→ ∞
is obtainedusing procedures suggested by Wasow [25]. The resulting
solution is able to satisfy the Sommerfeld radiation conditionautomatically. This is one of the major contributions of thecurrent work. Due to its generality, the asymptotic expansionpresented here is also valid in the solid mechanics problems.Since the derivation of the procedure is rather long, only asummary is presented here. For full details the reader shouldrefer to Li et al. (2004) [13], or to the text by Wasow [25].
Eq. (24) can be rewritten as
¯
ξ
−
1
[
X
(
¯
ξ)
]
,
¯
ξ
=[−[
M
]− ¯
ξ
−
2
[
Z
]][
X
(
¯
ξ)
]
.
(44)Since the eigenvalues of the square matrix
[
M
]
are all zero, itis not possible to obtain an asymptotic expansion for
[
X
(
¯
ξ)
]
directly. Instead a transformation is introduced so that
[
X
(
¯
ξ)
]
is replaced by
[
G
(
¯
ξ)
]
, where
[
X
(
¯
ξ)
]=[¯
T
][
P
(
¯
ξ)
][
G
(
¯
ξ)
]
(45)with
[ ¯
P
(
¯
ξ)
]=[
I
]+
∞
m
=
1
[ ¯
P
m
]¯
ξ
−
m
.
(46)The coefﬁcient matrices
[ ¯
P
m
]
and
[
T
]
are selected so that thetransformed differential equation (44) becomes
¯
ξ
−
1
[ ¯
G
(
¯
ξ)
]
,
¯
ξ
=[¯
B
(
¯
ξ)
][ ¯
G
(
¯
ξ)
]
(47)where the matrix function
[¯
B
(
¯
ξ)
]
may be expressed as
[¯
B
(
¯
ξ)
]=
∞
m
=
0
[¯
B
m
]¯
ξ
−
m
(48)and each of the coefﬁcient matrices
[¯
B
m
]
in this expansionare block diagonal. Insertion of Eq. (46) into Eq. (48) with