Fast multipole accelerated boundary
integral equation methods
N Nishimura
Department of Civil Engineering, Kyoto University, Kyoto 606-8501, Japan;
nchml@gee.kyoto-u.ac.jp
Fundamentals of Fast Multipole Method 共FMM兲 and FMM accelerated Boundary Integral Equation Method 共BIEM兲 are presented. Developments of FMM accelerated BIEM in the Laplace
and Helmholtz equations, wave equation, and heat equation are reviewed. Applications of
these methods in computational mechanics are surveyed. This review article contains 173
references. 关DOI: 10.1115/1.1482087兴
1
INTRODUCTION
The first step in the Boundary Integral Equation Method
共BIEM兲 关1兴, or the Boundary Element Method 共BEM—in
this article, we shall call this method BIEM兲, is to reduce the
共initial兲 boundary value problem in question into the solution
of boundary integral equations. In this way one converts the
problem posed in a domain to another defined on the boundary of the domain, thus reducing the dimensionality of the
problem by one. Nobody would imagine, at first sight, that
this reduction leads to increased computational complexities
in comparison with domain methods such as finite element
methods 共FEM兲 or finite difference methods 共FDM兲. However, this is the case. Indeed, suppose that one introduces N
unknowns to discretize a boundary integral equation. The
conventional BIEM will produce an N times N full matrix,
whose construction will undoubtedly require operations of
complexity proportional to N 2 . Such an approach, unfortunately, is considered expensive in large problems since other
major numerical tools such as FDM or FEM do the equivalent jobs with O(N) operations thanks to their banded coefficient matrices. One may argue that the Ns for BIEM and
domain methods are different by an order of magnitude.
Even with this difference, one sees that BIEM is really inferior to domain methods, at least in 3D problems. Indeed,
suppose that one solves a boundary value problem for a cube
using O(n) nodes on an edge. In that case the computational
complexity of BIEM is O(n 4 ) since N⫽O(n 2 ), while those
for FDM or FEM is O(n 3 ) since N⫽O(n 3 ). This is the
reason why BIEM has been considered a loser in large problems. However, recent developments of the fast BIEM have
revealed that the discretized equation for BIEM may possibly be solved with O(N)⫽O(n 2 ) operations, at least in integral equations for Laplace’s equation, with the help of the
Fast Multipole Method 共FMM兲. Although the constant multiplying N in the operation count is quite large, FMM accelerated BIEM usually becomes faster than the conventional
BIEM when N is larger than a few hundreds to thousands.
BIEM is thus back once again as a practical solver for large
scale problems.
With fast BIEMs, one can now really appreciate the high
accuracy of BIEM in large 3D problems. Also the numerical
stability of some BIE formulations 共eg, those which use
Fredholm’s second kind of integral equations兲 can now be
really enjoyed in problems requiring very high accuracy
共hence very fine discretization兲 or in highly ill-conditioned
problems such as those with closely spaced boundaries.
FMM was introduced by Rokhlin 关2兴 as an O(N) numerical method for solving an integral equation for 2D Laplace’s
equation. This method was further developed and made famous by Greengard 关3,4兴 as he applied FMM to many body
problems. FMM and related methods found applications in
various fields in science and engineering, such as astrophysics 共eg, 关5兴兲, molecular dynamics 共eg, 关6兴兲, fluid mechanics
共vortex methods, eg, 关7兴兲, etc. The influence of FMM to science and engineering was so profound that this method is
ranked among the top ten algorithms of the 20th century
along with Dantzig’s simplex method for linear programming, Krylov subspace iteration, QR algorithm, FFT, etc, in
a recent article by Board and Schulten 关8兴 共See also the criticism by Makino 关9兴 directed to this reference and the reply
from the authors兲. It was after some time that FMM regained
popularity as a fast solver of integral equations. Particularly
impressive are the developments of FMM accelerated integral equation solvers in electrical engineering where people
needed numerical tools for huge boundary value problems
related to radar technology, chip industry, etc. Indeed, problems such as the scattering of high frequency electromagnetic waves, or the design of complicated ICs yield boundary
value problems with possibly millions of unknowns, and
these problems have been solved successfully by these fast
BIEMs.
The BIEM community in applied mechanics, however,
seems to remain rather indifferent to such developments.
Transmitted by Associate Editor DE Beskos
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300
Nishimura: Fast multipole accelerated BIEMs
Some people may argue that the size of the boundary value
problems in applied mechanics has not been that large. This
might have been true. However, this is simply because it
was impossible to solve such huge problems so far. Indeed,
with conventional BIEM, for example, one could not go
much beyond several thousands of unknowns with a desktop
computer. Engineers therefore had to either give up the
analysis of large problems or reduce the size of problems
by introducing simplifying assumptions such as the twodimensionality of the phenomena, or by using approximate
theories such as the theories of plates and shells, approximate theories of composite materials, etc. This does not
mean that we do not need solvers for large problems in applied mechanics. Actually, methods of solution of problems
of the size of more than 108 unknowns 共which roughly correspond to O(106 ) unknowns with BIEM兲 are investigated in
FEM with massively parallel computers. With fast multipole
accelerated BIEM, problems of the size of 106 unknowns are
well within reach even for users of desktop computers. With
increased ability of solvers thus obtained, one is able to investigate problems which have so far been impossible. For
example, one may think of computing earthquake motions
of a whole regional area, or one may use the ordinary theory
of 3D elasticity in place of the theories of plates and shells
in problems where effects of supports, edges, connections,
etc are important, or one may determine the behavior of
composite materials from the analysis of all the constituents,
etc. The present author therefore believes that it is worth
the efforts to put together important developments in FMM
accelerated BIEM made so far in various fields of science
and engineering for the benefit of those wishing to use fast
BIEMs in their applications in applied mechanics.
In doing this, the present author found it necessary to cite
many articles from applied mathematics, computational
physics, and above all, electrical engineering, because these
are where important developments are taking place. Citation
of electrical engineering papers in 2D, however, has been
limited to those addressing fundamental issues, since applications directed to electrical engineering per se will not be of
interest to an audience in applied mechanics. In 3D problems, however, we have cited papers on the Maxwell equations since otherwise we will miss important techniques
which should be useful in acoustics and in elastodynamics,
as well. Also, we have cited some Japanese articles when
there was no other choice. Since some cited papers are from
outside applied mechanics or in Japanese, the present author
tried to explain the contents of cited papers as thoroughly as
possible using words and concepts familiar to those from
applied mechanics. For example, the author used the word
element even when the original paper uses Nyström’s
method 共which does not seem to be a favored choice by
engineers兲, replaced the electrical engineering technical term
MoM 共Method of Moment兲 by the essentially synonymous
BIEM, and showed formulas for direct BIEs, if admissible,
instead of the original for indirect BIEs. With such efforts
this article may have a flavor of tutorial review.
As a matter of fact, FMM accelerated BIEM is not the
Appl Mech Rev vol 55, no 4, July 2002
only fast BIEM available. Indeed, the treecode by Barnes
and Hut 关10兴, and Hackbusch and Nowak’s panel clustering
methods 关11兴 are considered effective, and are closely related
to FMM. Because of their similarity to FMM, we shall include discussions on these methods in this article. Also
promising are the use of wavelet basis and multigrid methods. Indeed, the use of wavelet basis with BIEM is known to
make the coefficient matrix approximately sparse, thus leading to an O(N log␣ N) ( ␣ ⭓0) algorithm. Multigrid methods
are also known to accelerate the solution of a certain class
of integral equations to an O(N log N) work. Due to the limited space and the inability of the present author, however,
detailed descriptions of these methods will not be made
in this article. The reader is referred to Beylkin et al 关12兴,
Alpert et al 关13兴, Damen et al 关14兴, Wang 关15兴, Petersdorff
et al 关16兴, Rathsfeld 关17兴, Lage and Schwab 关18兴, etc for
the use of wavelets, and Brandt and Lubrecht 关19兴, etc, for
multigrids.
Also, we have restricted our attention to papers related
directly to BIEM, except for a few fundamental papers, because it is far beyond the ability of the present author to
cover all the developments in FMM. Hence, we do not cite
papers on astrophysics, molecular dynamics, and vortex
methods, although some papers in these fields have relevance
to applied mechanics.
This article begins with a brief description of the algorithm of the original FMM, which forms the basis of all the
subsequent developments. Some explicit formulas used in
the FMM for the Laplace and Helmholtz equations are given.
Sections 3 and 4 present fundamentals of FMM accelerated
BIEM. Section 3 reviews developments of FMM in the
Laplace and Helmholtz equations in three parts. The first part
presents the fundamentals and applications of the original
FMM. The original FMM, however, is found to be inefficient
in Helmholtz’ equation with high frequency, and diagonal
forms are introduced to remedy this inefficiency. The second
part is devoted to the review of these diagonal forms. The
diagonal forms, however, are known to have instability problems in the Laplace and the low frequency Helmholtz equations. The third part discusses new FMMs which are diagonal, but are free of the low frequency instability problems.
Section 4 is concerned with FMM in the time domain. The
first part reviews progress made in the wave equation, and
the second part is for the heat equation. We then proceed
to applications of fast FMM in computational mechanics in
section 5. We shall survey applications in elastostatics, elastodynamics, Stokes flow, etc. This article concludes with
some remarks concerning additional information and future
directions.
Notation
In this article, we shall use standard diadic notations for vectors. Also, the position vector of a point x will be denoted by
either x or ជ
Ox , the latter being the preferred choice when
one has to indicate the origin explicitly.
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Appl Mech Rev vol 55, no 4, July 2002
Nishimura: Fast multipole accelerated BIEMs
2 WHAT IS FMM?
This section describes the original FMM.
2.1 Basic ideas
In BIEM, one converts the 共initial-兲 boundary value problem
of interest into an equivalent boundary integral equation, and
solves this equation to obtain the solution of the original
problem. Suppose that one has thus obtained an integral
equation of the following form:
f 共 x 兲⫽
冕
S
K 共 x,y 兲 ␾ 共 y 兲 dy
x苸S
(1)
where f is a given function defined on a set S, K is a given
kernel function defined on S⫻S, and ␾ is an unknown function on S. The kernel K may include Dirac’s delta and its
derivatives, so 共1兲 may not necessarily be of the Fredholm
1st kind. As a matter of fact, the local behavior of K near
x⫽y is not quite relevant to our discussion, since we are
interested in fast methods of evaluating contributions to the
integral in 共1兲 from parts of S located away from x. In BIEM
one discretizes the unknown function ␾, and uses collocation, Galerkin’s method, etc, to reduce 共1兲 into a certain algebraic equation, which one solves for the N unknown nodal
values for ␾ i (i⫽1,...,N). In this statement, N is the number
of discretized unknowns. Suppose that one uses a certain
iterative method, such as CG, GMRES, Bi-CGSTAB, etc, to
solve this discretized equation numerically. In that case, one
can solve 共1兲 if one has a method of computing the discretized RHS of 共1兲 for a given ␾. Indeed, these iterative
solvers work as one provides a method to compute the
matrix–trial vector product. The conventional approach provides an O(N 2 ) algorithm to this end, which is considered
expensive in large problems. On the other hand, the fast multipole method proposed by Rokhlin gives a fast method to
compute this matrix–trial solution product with O(N) operations. One therefore sees that FMM gives a fast method of
solving boundary integral equations provided the employed
iterative solver leads to convergence with a small number of
iterations. In this section we shall show how one can achieve
an order N matrix vector multiplication with the original
FMM.
2.2 Mathematical tools for FMM
The original FMM is built using the following mathematical
tools:
Expansion of the kernel function
The kernel function K is expanded into the following form
K 共 x,y 兲 ⫽
(2)
兺n k (1)
n 共 x⫺y0 兲 k n 共 y⫺y0 兲 ,
(2)
k (1)
n
where y 0 is a certain point. The functions
are usually
are
usually
entire
functions.
singular at the origin, and k (2)
n
The most obvious expansion of this type is the Taylor series
of K with respect to x or y, which is the commonest choice
in the panel clustering method proposed by Hackbusch and
Nowak 关11兴. Some authors even say that FMM is based on
the Taylor expansion of the kernel functions. That this is not
301
true is easily seen in examples for Helmholtz’ equation, etc.
This confusion seems to have arisen from the fact that the
first FMM paper 关2兴 was for Laplace’s equation in 2D, and
Rokhlin used complex variables and the Taylor expansion to
derive necessary formulas. Notice, however, that polynomials in complex variables are harmonic.
In BIEM, K is related to the fundamental solution which
satisfies the given PDE with respect to x and/or y. It is
therefore possible to choose k (1)
and/or k (2)
in a way that
n
n
these functions satisfy the governing equation. The formulas
given in Section 2.4 provide such examples.
Other possible examples of 共2兲 are:
• A kernel function K(x,y) can be viewed as an infinite
dimensional matrix. As in matrices, some kernel functions
possess a singular value decomposition given by
K共x,y 兲⫽
兺i ui共x⫺y0 兲 s i共 y 0 兲v i共 y⫺y0 兲 ,
where u i , v i , and s i are certain functions 关20兴. One may arrange
this into the form in 共2兲.
• Let D be a certain bounded domain, and u i (x) form a
complete set of orthonormal eigensolutions of a certain
boundary value problem in D with the eigenvalues ␭ i . The
Green function for this problem is written as
ui共x兲ui共y 兲
G共x,y 兲⫽
␭i
i
兺
if ␭ i ⫽0 holds for all the relevant i. This gives a decomposition of
the form of 共2兲 if K⫽G.
Usually an expansion of the form in 共2兲 is valid under a
certain condition. A typical one is
兩 x⫺y0 兩 ⬎ 兩 y⫺y0 兩 .
(3)
usually satisfy equalities of the
Also, the functions k (2)
n
following forms
k (2)
n 共 y⫺y1 兲 ⫽
R
共 y 1 ,y 0 兲 ,
兺m k m(2)共 y⫺y0 兲 c n,m
(4)
R
where y 1 is a point and c n,m
(y 1 ,y 0 ) are numbers. Indeed, if
(2)
k n are chosen as a complete set of independent interior
solutions 共entire solutions兲 of a certain PDE, for example,
(2)
one would be able to expand k (2)
n (y⫺y1 ) in terms of k m (y
R
⫺y0 ), the coefficients of such expansion being c n,m (y 1 ,y 0 ).
When one uses Taylor’s expansion with respect to y to obtain 共2兲, 共4兲 is simply the binomial expansion of a polynomial.
Multipole expansion
For a set S 0 which is a part of S and a point x which is not
contained in S 0 共see Fig. 1兲, we use 共2兲 to obtain
冕
S0
K 共 x,y 兲 ␾ 共 y 兲 dy⫽
兺n k (1)
n 共 x⫺y0 兲 M n 共 y 0 兲 ,
(5)
where M n (y 0 ) stands for the multipole moment centered at
y 0 defined by
M n共 y 0 兲 ⫽
冕
S0
k (2)
n 共 y⫺y0 兲 ␾ 共 y 兲 dy.
(6)
From 共4兲 and 共6兲 one has the following formula called M2M
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302
Nishimura: Fast multipole accelerated BIEMs
M n共 y 1 兲 ⫽
Appl Mech Rev vol 55, no 4, July 2002
R
共 y 1 ,y 0 兲 .
兺m M m共 y 0 兲 c n,m
(7)
The function k (1)
n also allows the following expansion
S
共 x 0 ,y 0 兲
兺m k m(3)共 x⫺x0 兲 c m,n
k (1)
n 共 x⫺y0 兲 ⫽
(8)
(3)
near x 0 when x 0 and y 0 are far apart, where x 0 is a point, k m
S
are functions which are usually entire, and c m,n (x 0 ,y 0 ) are
(3)
is
constants. The commonest choice for k m
(3)
(2)
km
⫽k m
.
(3)
If k (1)
n satisfies the governing PDE and k n are chosen to be
a complete set of interior solutions of the governing PDE,
(3)
then 共8兲 is simply the expansion of k (1)
n (x⫺y0 ) by k m (x
S
⫺x0 ) with c m,n (x 0 ,y 0 ) being the coefficients of the expansion. Or, one may simply expand k (1)
n (x⫺y0 ) into a Taylor
series at x 0 to obtain an expansion of the form in 共8兲. As in
(3)
the case of k (2)
n , k n satisfies
k (3)
n 共 y⫺y1 兲 ⫽
⬘ R 共 y 1 ,y 0 兲 .
兺m k m(3)共 y⫺y0 兲 c n,m
(9)
With 共5兲 and 共8兲 one obtains the local expansion of the
potential function given by
冕
S0
K 共 x,y 兲 ␾ 共 y 兲 dy⫽
兺n L n共 x 0 兲 k (3)
n 共 x⫺x0 兲 ,
(10)
where the coefficient of local expansion L m (x 0 ) is related to
the multipole moment by M2L
L m共 x 0 兲 ⫽
S
共 x 0 ,y 0 兲 M n 共 y 0 兲 .
兺n c m,n
(11)
• Discretize S into N elements, and discretize ␾ in an ordinary manner with boundary elements.
• Obtain a hierarchical tree structure of elements 共See Fig.
2兲. Namely, take a square which contains S and call it a
cell of level 0. Now, we take a cell 共a parent cell兲 of level
l (l⭓0) and divide it into four equal sub squares whose
size is half of that of the parent cell. A sub square which
contains more than one boundary element is called a child
cell of the parent. These children are cells of level l⫹1. A
cell of level l⫹1 is further subdivided into four sub
squares if this cell contains more than a given number
共denoted by M 兲 of boundary elements. Otherwise, one terminates the subdivision. In this manner one determines a
quad-tree structure of cells containing boundary elements
共See the lower part of Fig. 2兲. A childless cell is called a
leaf. Obviously, the numbers of leaves and other cells are
estimated to be O(N/M ) and O(N/M )⫻((1/4)⫹(1/4) 2
⫹...)⫽O(N/M ), respectively. Also the number of levels
are O(log(N/M)).
• 共Upward pass兲: Compute M n (y c ) in each cell starting from
the leaves and tracing the tree structure of cells upward
共decreasing level number兲, where y c is the centroid of the
cell. For leaves, we use the definition in 共6兲 to compute the
multipole moments associated with them. For other cells,
we add all the moments from their children after shifting
the origin to that of the parent by using 共7兲. In this manner,
we compute all the multipole moments associated with
cells of level l satisfying l⭓2. In this computation, we
truncate the infinite sums with respect to n in 共5兲 at n
⫽ p, where p is a certain number. Accordingly, we will
compute a finite number of multipole moments for n
⫽0,...,p. The computation of moments at leaves takes
As in M2M, we have L2L
L n共 y 1 兲 ⫽
2.3
⬘ R 共 y 0 ,y 1 兲 .
兺m L m共 y 0 兲 c m,n
(12)
Algorithm
We now describe the FMM algorithm assuming 2D geometry
for simplicity. The 3D algorithm is obtained simply by replacing quad-tree by oct-tree, etc, in the following description. Also, the distribution of boundary elements are assumed
to be relatively uniform. See Nabors et al 关21兴 for more precise discussions.
The FMM algorithm for BIEM goes as follows.
Fig. 1
Domain and points
Fig. 2
Tree structure
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Appl Mech Rev vol 55, no 4, July 2002
O(pN) operations. A non-leaf cell uses M2M four times
with its four children, and the results are added together to
obtain its moment. This process obviously takes O(p 2
⫻(N/M )) operations. Hence, the total complexity of the
upward pass is O(pN)⫹O(p 2 (N/M )).
• 共Downward pass兲: Compute L n (x c ) in each cell recursively starting from the level 2 cells and tracing the tree
structure of cells downward 共increasing level number兲,
where x c is the centroid of the cell. To describe how we
prepare a few definitions: A cell C ⬘ of level l is said to be
far from another level l cell C if they share no vertex. A
cell C ⬘ of level l is said to be in the interaction list of
another level l cell C if they are far from each other but
their parents are not. Finally the local expansion of a cell
stands for the sum of the terms of the form given in 共10兲
with x 0 ⫽x c and S 0 taken consecutively to be one of the
cells far from C. Consider, for example the black cell in
Fig. 3, whose level is assumed to be l. The gray cells form
the interaction list of the black cell. Now suppose that the
coefficients of the local expansion for the cells of the level
less than l are already computed. The local expansion of
the black cell C is computed in two steps. Namely, the
contributions from the interaction list of C is evaluated
using M2L with 共11兲. The contributions from other far
cells 共children of the white cells with thick boundaries in
Fig. 3兲 are evaluated by using the coefficients of the local
expansion of the parent of C 共hatched cell兲 by shifting the
center of expansion from the centroid of the parent to that
of C. This is considered to be the most ingenious part of
the whole FMM algorithm. In the interaction list of each
cell, we have at most 27 cells. Hence the M2L operation
used for each cell is of the order of O(p 2 ). Also, the inheritance from the parent cell is obtained in an L2L operation per cell, which is another O( p 2 ) operation. Hence the
total complexity of the downward pass is O(p 2 (N/M )).
• Finally, in leaves, contributions from nearby cells are
evaluated directly. Contributions from far cells are evaluated with the help of the local expansion. Since a leaf has
at most nine nearby cells, the direct computation takes
Nishimura: Fast multipole accelerated BIEMs
303
O((N/M )M 2 )⫽O(NM ) operations. The evaluation of the
local expansion at N points takes O(Np) operations.
The complexity of the whole process is thus seen to be
O(N) if the number of terms p in the multipole expansion
and the number of elements in a leaf M are taken constant.
The algorithm given above is a simplified version of the
more efficient original FMM proposed by Greengard 关3兴.
This algorithm is characterized by the use of hierarchical
structure of cells consisting of several levels. Therefore, an
algorithm of this type is said to be a multilevel one. However, it is possible to design a fast method without using
structures of this type. Indeed, suppose that one divides the
N boundary elements into m groups each containing N/m
elements. Each group has less than a finite number n n of
nearby groups with which the interactions have to be evaluated directly. With other groups, however, the interactions
are computed with the multipole expansion. In this case the
number of operations needed for computing multipole moments in all the groups will be O(pN). The interactions
共M2L兲 between far groups are computed with O(p 2 m 2 ) operations. The contributions from nearby groups take
O(n n m(N/m) 2 ) operations, and the local expansion is evaluated with O(pN) operations. For constant p and n n the total
complexity becomes O(N 4/3) as one takes m to be O(N 2/3).
An approach of this type is called a single stage algorithm or
two level algorithm, somewhat confusingly.
2.4
Examples
2.4.1 Laplace’s equation in 3D [3,22,23]
In the direct BIEM, one writes the solution for Laplace’s
equation in the following form
u共 x 兲⫽
冕
⳵D
G 共 x⫺y兲
⳵u
共 y 兲 dS⫺
⳵n
冕
⳵ G 共 x⫺y兲
u 共 y 兲 dS, (13)
⳵ny
⳵D
where G is the fundamental solution of the Laplace equation
given by
G 共 x⫺y兲 ª
1
,
4 ␲ 兩 x⫺y兩
and D is the domain under consideration. One shows that the
following expansion holds true
⬘
1
1
ជ 兲 , (14)
⫽
R
Oy 兲 S n ⬘ ,m ⬘ 共 Ox
共ជ
4 ␲ 兩 x⫺y兩 4 ␲ n ⬘ ⫽0 m ⬘ ⫽⫺n ⬘ n ⬘ ,m ⬘
⬁
n
兺 兺
Oy 兩 ⬍ 兩 ជ
Ox 兩 兲
共兩ជ
where ā stands for the complex conjugate of a, and the
functions R n,m and S n,m are defined in terms of the polar
coordinate (r, ␪ , ␾ ) of the point x viewed from the origin O
and the associated Legendre functions P m
n by
ជ 兲⫽
R n,m 共 Ox
Fig. 3
Neighboring cells and far cells
1
P m 共 cos ␪ 兲 e im ␾ r n ,
共 n⫹m 兲 ! n
ជ 兲 ⫽ 共 n⫺m 兲 ! P mn 共 cos ␪ 兲 e im ␾
S n,m 共 Ox
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1
r n⫹1
(15)
.
304
Nishimura: Fast multipole accelerated BIEMs
Appl Mech Rev vol 55, no 4, July 2002
Therefore, for a part of ⳵ D denoted by S 0 and a point x
which is away from S 0 , we have the multipole expansion
given by
冕冉
S0
冊
⬁
⫽
2.4.2 Helmholtz’ equation in 2D [27]
The solution of the 2D Helmholtz equation
⳵u
⳵ G 共 x⫺y兲
u 共 y 兲 dS
共 y 兲⫺
⳵n
⳵ny
G 共 x⫺y兲
共 ⌬⫹k 2 兲 u⫽0
n
兺 兺
n⫽0 m⫽⫺n
ជ 兲 M n,m 共 O 兲 ,
S n,m 共 Ox
(16)
where M n,m (O) stands for the multipole moment centered at
O, given by 关22兴
M n,m 共 O 兲 ⫽
冕冉
S0
冊
ជ兲
⳵u
⳵ R 共 Oy
ជ 兲 共 y 兲 ⫺ n,m
R n,m 共 Oy
u 共 y 兲 dS.
⳵n
⳵n
(17)
In 共16兲, we have assumed that the point x is sufficiently far
ជ 兩 ⬎max兩Oy
ជ兩 holds.
from S 0 that the inequality 兩 Ox
y苸S0
As the origin is shifted from O to O ⬘ we obtain the following M2M formula:
M n ⬘ ,m ⬘ 共 O ⬘ 兲 ⫽
n⬘
兺 兺
n⫽0 m⫽⫺n
ជ
R n,m 共 O
⬘ O 兲 M n ⬘ ⫺n,m ⬘ ⫺m 共 O 兲 .
Also, the local expansion is given by
S0
⬁
冊
n
兺 兺
n⫽0 m⫽⫺n
R n,m 共 ជ
x 0 x 兲 L n,m 共 x 0 兲
(19)
for points x in the neighborhood of a certain point x 0 , where
the coefficient of the local expansion L n,m is given by the
following M2L formula:
L n,m 共 x 0 兲 ⫽
兺 兺
n ⫽0 m ⫽⫺n
⬘
⳵D
G 共 x⫺y兲
⬘
⬘
n⬘
兺 兺
冊
⳵u
⳵ G 共 x⫺y兲
u 共 y 兲 dS
共 y 兲⫺
⳵n
⳵n
in D
(23)
where G is the fundamental solution of Helmholtz’ equation
given by
i
G 共 x⫺y兲 ⫽ H (1)
共 k 兩 x⫺y兩 兲 .
4 0
(24)
In this equation, k and H (1)
0 stand for the wave number and
the Hankel function of the first kind and 0th order, respectively. Graf’s addition theorem 关28兴 gives
⬁
G 共 x⫺y兲 ⫽
i
ជ 兲 I ⫺n 共 Oy
ជ 兲,
O n 共 Ox
4 n⫽⫺⬁
兺
(25)
where the functions O n and I n are defined by
I n 共 x兲 ⫽ 共 ⫺i 兲 n J n 共 kr 兲 e in ␪ . (26)
In these expressions, J n stands for the Bessel function and
(r, ␪ ) indicates the polar coordinate of a vector x. The ⫾p
term truncation of the series in the RHS of 共25兲 gives a good
ជ 兩.
approximation for the LHS as one takes p larger than k 兩 Oy
The RHS of 共23兲, integrated over a subset S 0 of ⳵ D, gives
冕冉
S0
G 共 x⫺y兲
⫽
共 ⫺1 兲 n
(20)
Ox 0 兩 ⬎ 兩 ជ
x 0 x 兩 has been assumed.
and an inequality given by 兩 ជ
The L2L takes the following form
L n,m 共 x 1 兲 ⫽
冕冉
冊
⳵u
⳵ G 共 x⫺y兲
u 共 y 兲 dS
共 y 兲⫺
⳵n
⳵n
⬁
ជ0 兲 M n ,m 共 O 兲 ,
⫻S n ⬘ ⫹n,m ⬘ ⫹m 共 Ox
⬘ ⬘
⬁
u共 x 兲⫽
n⬘
⬁
(22)
in ␪
O n 共 x兲 ⫽i n H (1)
,
n 共 kr 兲 e
⳵u
⳵ G 共 x⫺y兲
G 共 x⫺y兲 共 y 兲 ⫺
u 共 y 兲 dS
⳵n
⳵ny
⫽
in D
is known to have a potential representation given by
n
(18)
冕冉
sion in 共14兲 could have been obtained with the Taylor expansion of the kernel function together with a careful arrangement of terms.
n ⬘ ⫽n m ⬘ ⫽⫺n ⬘
R n ⬘ ⫺n,m ⬘ ⫺m 共 ជ
x 0 x 1 兲 L n ⬘ ,m ⬘ 共 x 0 兲
(21)
as one shifts the center of expansion from x 0 to x 1 .
Notice that the formulation presented here is applicable
regardless of the types of the boundary conditions, as long as
the direct BIEM is used. FMM formulations for other integral equations such as those for indirect BIEM can also be
obtained similarly.
Finally, we point out that the simplicity of the FMM formulation achieved by the use of solid harmonics in 共15兲 is
noticed in 关24 –26兴, etc. As a matter of fact, the function R n,m
is a polynomial of the cartesian coordinate, and the expan-
i
ជ 兲 M ⫺n 共 O 兲 ,
O n 共 Ox
4 n⫽⫺⬁
兺
(27)
where x is a point which is far from S 0 , and M n is the
multipole moment defined by
M n共 O 兲 ⫽
冕冉
S0
ជ兲
I n 共 Oy
冊
ជ兲
⳵u
⳵ I n 共 Oy
u 共 y 兲 dS.
共 y 兲⫺
⳵n
⳵n
(28)
In this formula the origin O is assumed to be located close to
S 0 so that 兩 ជ
Ox 兩 ⬎max兩ជ
Oy兩 holds. The M2M formula is given
y苸S0
by
⬁
M n共 O 兲 ⫽
兺
␯ ⫽⫺⬁
ជ⬘ 兲 M ␯ 共 O ⬘ 兲 .
I n⫺ ␯ 共 OO
(29)
The expression in 共27兲 allows a local expansion given by
冕冉
S0
G 共 x⫺y兲
冊
⳵u
⳵ G 共 x⫺y兲
u 共 y 兲 dS
共 y 兲⫺
⳵n
⳵n
⬁
⫽
i
ជ
共 ⫺1 兲 n I n 共 O
⬘ x 兲 L ⫺n 共 O ⬘ 兲 ,
4 n⫽⫺⬁
兺
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(30)
Appl Mech Rev vol 55, no 4, July 2002
Nishimura: Fast multipole accelerated BIEMs
where O ⬘ is located near x. The coefficient of the local expansion L n is related to M n by the following M2L formula
⬁
L n共 O ⬘ 兲 ⫽
兺
␯ ⫽⫺⬁
ជ⬘ 兲 M ␯ 共 O 兲 .
O n⫺ ␯ 共 OO
(31)
Finally, the L2L formula is given by
⬁
L n共 O ⬘ 兲 ⫽
兺
␯ ⫽⫺⬁
ជ
I n⫺ ␯ 共 O
⬘O 兲 L ␯共 O 兲 .
(32)
These formulas are derived from results in Rokhlin 关27兴.
The corresponding 3D formulas 关22兴 are complicated, and
are given in terms of the Wigner-3j symbols 关29兴.
3 FMM FOR LAPLACE AND
HELMHOLTZ EQUATIONS
3.1
Original FMM and related approaches
3.1.1 Fundamental algorithms
The fast multipole method was first introduced by Rokhlin
关2兴 as an order N numerical method of solving integral equations for the Laplace equation in 2D. He uses indirect BIEMs
(2)
(k (3)
with the function k (1,2)
n
n ⫽k n ) in 共2兲 taken as solutions
of the Laplace equation. The formulation uses essentially a
real variable approach, although the multipole expansion is
obtained with the help of complex variables. Nyström’s
method is used to discretize the integral equation. Because of
the use of Nyström, his algorithm can be interpreted as a fast
method of computing gravitational forces produced by many
particles with given masses.
This work of Rokhlin is really amazing in that it includes
almost all the essence of what is now called the original
FMM, except that the proposed algorithm uses a binary tree
structure of unknowns, rather than the quad-tree explained in
the previous section. Indeed, it is quite interesting to compare Rokhlin’s FMM with related methods to compute mutual forces in particle systems by Appel 关30兴 共which is another amazing paper considering the fact that it stems from
the author’s undergraduate thesis written in 1981兲 published
in 1985 and Barnes & Hut 关10兴 published in 1986, both of
which propose O(N log N) methods for computing gravitation forces between many particles. Interpreted from the
point of view of FMM, these papers use only the upward
pass and evaluate multipole expansions at each point of observation, considering only the moments of the zeroth and
first 共always zero in their approaches兲 order in the expansions. In their approaches, M2M simply reduces to the computation of the total mass and the center of gravitation, and
they have no M2L or L2L. Appel’s approach uses a binary
tree data structure, while Barnes and Hut introduce quad-tree
and oct-tree structures in 2D and 3D problems, respectively.
Rokhlin’s approach went beyond these methods by introducing the local expansion which made the complexity of the
algorithm O(N). 共Notice, however, that this does not necessarily mean that Rokhlin’s FMM is faster than O(N log N)
approaches in problems of any size. For example, FMM is
not considered to be the best choice in astrophysical N body
problems 关9,31兴.兲 The most outstanding part of Rokhlin’s
305
approach is found in the downward pass which combines
M2L and L2L. Unfortunately, this paper 关2兴 is somewhat
hard to read because of sporadic misprints and errors. Therefore, the present author recommends this paper only to those
who already have some knowledge of FMM.
The original FMM was further developed and made
famous by Greengard 关3,4兴 as he applied this method to
many body problems. The name Fast Multipole Algorithm
seems to have appeared for the first time in published materials in 关4兴. The ingenious use of quad-tree in 2D 共or oct-tree
in 3D兲 characterizes these works. Greengard and Rokhlin 关4兴
present the non-adaptive version of FMM, where all the
leaves have the same level. The thesis of Greengard 关3兴 also
includes adaptive FMM where leaves may have varying levels, and details of the FMM formulation in 3D, which was
already hinted in Rokhlin 关2兴.
Hackbusch and Nowak’s panel clustering method 关11兴,
developed independently of FMM, provides another fast algorithm for solving integral equations using tree structures of
elements. Panel clustering uses a tree structure of elements,
or panels, together with an expansion of the kernel function
of the following form:
K 共 x,y 兲 ⫽
(2)
兺n k ⬘ (1)
n 共 x,y 0 兲 k ⬘ n 共 y 兲 .
Notice that the function k ⬘ (2)
n (y) is independent of the center
of expansion y 0 . An expansion of this form is obtained, for
example, by expanding K(x,y) into a Taylor series with respect to y at y 0 , followed by expansion of polynomials. A
tree structure called admissible covering 共of the discretized
boundary兲 is constructed for each point of evaluation x, in
order to obtain multipole moments, or far field coefficients in
their terminology. These authors assume no particular structure for this tree in their paper. Their approach in 关11兴 is of
the Barnes & Hut type, and the complexity of the algorithm
is
O 共 N logd⫹2 N 兲 ,
(33)
where d is the dimensionality of the problem. Obviously the
M2M in this approach reduces to an identity operator. Further details on the panel clustering algorithm are found in
Sauter 关32兴.
It would be of interest to consider the reason why the
complexity estimates of the FMM and the panel clustering
differ. An obvious reason for the appearance of the log N
factor in 共33兲 is that Hackbusch and Nowak’s approach is of
the Barnes and Hut type. Also, they seem to have the Taylor
expansion of kernel functions in mind when they make assumptions on the number of certain operations in their algorithm, and the use of other expansions may change the estimate. Another reason for the difference is that the estimate in
共33兲 gives the complexity when one solves an integral equation with a series of increasingly refined meshes. Indeed,
Hackbusch and Nowak derive 共33兲 assuming that the number
of terms in the multipole expansion increases proportional to
log N as N increases, in order to be consistent with the increased accuracy of the discretized integral equation with the
mesh refinement. Hence, their estimate is relevant when one
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306
Nishimura: Fast multipole accelerated BIEMs
is interested in obtaining highly accurate numerical results
for an integral equation with very fine meshes. On the other
hand the order N estimate given in Section 2.3 assumes that
the number of terms in the multipole expansion stays the
same as N increases. This can be interpreted as estimating
the complexities of a series of different problems with increasing number of unknowns solved with almost the same
mesh size 共linear dimension兲 and accuracy. This fact becomes more evident in Helmholtz’ equation, as we shall see
later. We thus see that the estimate in 共33兲 cannot be compared directly with the standard O(N) estimate for the FMM.
The original panel clustering method by Hackbusch and
Nowak, which uses collocation, has been extended to Galerkin in Sauter 关33兴. His approach can be interpreted as using
both multipole and local expansions. Sauter’s improved
panel clustering 关34兴 uses expansions of the kernel with variable order 共number of terms兲 depending on the level, in a
Galerkin discretized integral equation of the second kind
with the double layer kernel. This approach is much closer to
the original FMM by Rokhlin and Greengard than Hackbusch and Nowak’s version of panel clustering is except for
the expansion used, and the complexity of the algorithm is
O(N).
3.1.2 Integral equation solvers
Several authors have attempted to develop solvers of general
boundary value problems for Laplace equations using BIEM
and FMM.
Greenbaum et al 关35兴 considered both Dirichlet and Neumann problems for Laplace’s equation in 2D. They express
solutions of the Dirichlet problem with double layer potentials, and those of the Neumann problem with single layer
potentials. When the domain under consideration has holes,
they either modify the kernel function or add a logarithmic
term for each hole in the double layer formulation. The use
of FMM enhances the efficiency of the numerical methods,
as expected.
Nabors et al 关21,36 –38兴 investigated the use of the first
kind integral equation in Dirichlet problems for Laplace’s
equation in 3D, expressing the solution with a single layer
potential. They were interested in obtaining the integral of
the single layer density when the Dirichlet data on each piece
of boundary is constant 共capacitance extraction兲. Their approach, however, is applicable to the general Dirichlet problems. They use basis functions, rather than Nyström’s
method, and trees having leaves with uniform depths. Also,
these authors exclude cells in the second neighborhood
共neighbors of neighbors兲 of a cell C from the definition of
cells far from C. In 关36兴, a 3D implementation using a nonadaptive algorithm is presented together with some numerical examples of the size of about 6000 unknowns. In 关37兴,
they propose an adaptive FMM 共notice that Greengard’s
adaptive algorithm uses trees having leaves with variable
depths 关3兴 and, hence, is different from Nabors et al’s兲,
which economizes the calculation in the original FMM by
skipping some of M2M, M2L, and/or L2L conversions when
they are inefficient. These authors also propose to precondition the linear equation using essentially the inverse of the
Appl Mech Rev vol 55, no 4, July 2002
part of the coefficient matrix computed directly in FMM, ie,
contributions from nearby cells. In 关38兴, they extended their
approach to the cases including multiple dielectrics, which is
equivalent to inclusion problems with interface conditions of
continuous potential and flux, in addition to Dirichlet boundaries. In 关21兴, these authors also make efforts to rigorously
prove that the complexity of their algorithm is really O(N).
Interestingly, the suspicion raised by Aluru 关39兴 concerning
the order N property of the original FMM by Greengard 关3兴
seems to have already been answered in this paper before the
question was asked! Some numerical examples for capacitance extraction are given in these papers 关21,36 –38兴.
Japanese researchers started investigations of fast BIEM
in 1994, as one can see in the work of Watanabe and Hayami
关40兴 which uses FMM 共what they actually did, however, is
closer to Hackbusch and Nowak’s panel clustering 关11兴 or
Barnes and Hut’s method 关10兴 since they did not implement
the downward pass兲 in Laplace’s equation in 2D. Subsequent
developments include works by Nishida and Hayami 关41兴 for
Laplace’s equation in 3D and Fukui et al 关42,43兴 for 2D
problems. Both of them use the original FMM. Nishida and
Hayami 关41兴 proposed the use of the block diagonal matrix
corresponding to leaves as the preconditioner.
Gáspár 关44兴 presents some detail 共eg, explicit formulas for
the moments兲 of the multiple expansion in BIEM in
Laplace’s equation in 2D. The formulation is of the Barnes
and Hut type with no downward pass, and no numerical results are shown.
Grama et al 关45兴 propose a variable p implementation in
either Barnes and Hut’s approach or FMM for Laplace’s
equation to enhance accuracy. Their paper includes examples
of solutions to Laplace’s equation in 3D obtained with BIEM
and Barnes and Hut’s treecode.
3.1.3 Solvers of specialized problems
Efforts are also aimed at applications of FMM in specialized
problems. In McKenney et al 关46兴 these authors consider application of FMM to the solution to Poisson’s equation in 2D
⌬u⫽⫺ f
in D
(34)
subject to the homogeneous Dirichlet boundary condition,
where the function f is nonzero only in a bounded set. These
authors state that the direct evaluation of the volume potential
冕
D
G 共 x⫺y兲 f 共 y 兲 dV
(35)
with FMM will not compete with other Poisson solvers of
the domain type. Hence, they use the finite difference
method, rather than the direct evaluation of 共35兲, to obtain
a particular solution for Poisson’s equation, and add a solution of Laplace’s equation to satisfy the boundary condition.
FMM is applied to the boundary integral equation obtained
from the double layer representation of the solution of the
latter problem. Additional logarithmic terms have to be considered when the domain under consideration has holes.
Greengard and Lee 关47兴 considered a related problem of
solving Poisson’s equation in the whole plane without
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Appl Mech Rev vol 55, no 4, July 2002
boundary. The exact solution is given by 共35兲, but, again, the
authors avoid the direct evaluation because of the same reason. They take a square which includes the support of f in its
interior and construct a quad-tree of cells as in Section 2.3. A
particular solution of Poisson’s equation is obtained in each
of these leaves thus constructed with the help of Chebyshev
polynomials. However, one cannot patch these solutions directly, since they are not continuous across the boundaries of
the leaf cells. This discontinuity is resolved with the aid of
single and double layer potentials having appropriate discontinuities of the particular solution as the density. These authors use FMM in the computation of these layer potentials
with known densities. They succeeded in obtaining a fast and
very accurate Poisson solver in this way.
Nishida and Hayami 关48兴 discussed the use of FMM in
the analysis of electron guns. This paper deals with a special
case of Poisson’s equation where the source term 共f in 共34兲兲
is composed of many concentrated charges. The motions of
the charges are determined in interaction with other charges
and walls. These authors use FMM in the direct BIEM to
deal with the boundary conditions, as well as in computing
the interactions of charges. This paper also discusses methods to compute multipole moments for planar elements using
a recursive formula.
In a series of papers, Greengard and his group investigated inclusion problems for Laplace’s equation. Analyses of
this type are useful in obtaining the effective conductivity of
composites, etc. Their problem is to solve Laplace’s equation, in a planar domain 共matrix兲 having many inclusions,
under the interface conditions of continuous potential and
flux 共normal derivative of the potential multiplied by the
conductivity constant兲. The whole system is subjected to either remote uniform field, or periodic boundary conditions.
The conductivity varies, but stays constant within an inclusion or in the matrix. In Greengard and Moura 关49兴, they use
ordinary single layer potential distributed on the interfaces to
represent the solution. They could solve large problems fast
with FMM. But the distance between inclusions could not be
made too small. To overcome this difficulty, Cheng and
Greengard developed a modified integral equation approach
关50兴 for circular inclusions which use single layer potential
with a modified kernel function, considering possibility of
very dense arrangements of the inclusions. As a matter of
fact, they start from the observation that the use of Fourier
expansion for the single layer density reduces the single
layer potential on a circle with the ordinary fundamental solution into a multipole series located at the center of the
circle 共Rayleigh’s method兲. To this series, they add image
multipoles considering effects of nearby circles. These authors thus actually deal with the problem of determining the
coefficients of the multipole series thus obtained, rather than
to attempt to discretize an integral equation with an explicitly
written modified kernel function. In 关51兴 they could deal
with realistic problems of the size of about 16,000 inclusions
共each having 11 DOF兲 with efficiency and accuracy. See also
Cheng 关52兴 for a 3D version. One may also mention the
paper by Helsing 关53兴 where similar problems with various
shapes of inclusions are solved mainly with ordinary single
layer potentials, sometimes with the help of FMM.
Nishimura: Fast multipole accelerated BIEMs
307
Nishimura et al 关23兴 used the original FMM for Laplace’s
equation in 3D crack problems using collocation. The regularized hypersingular integral equation for crack problems
was discretized with piecewise constant shape functions.
Pan et al 关54,55兴 consider the problems of determining
the capacitance of conductors above stratified dielectric media, or Dirichlet problems for a half space with holes located
above strata. Their approach is of interest in that it includes a
treatment of the images of the multipole moments in problems where the Green function is expressed as a sum of the
source and its mirror images.
3.1.4 Helmholtz equation
We have so far seen applications of FMM in Laplace’s equation. Similar approach leads to an O(N) algorithm in Helmholtz equation when the frequency is low. Such an approach
is sometimes referred to as low-frequency FMM. One may
mention Fukui 关56,57兴, Hoyler and Unbehauen 关58兴 共according to 关59兴—This paper was unavailable to the present author兲, Zhao and Chew 关60兴, etc, as attempts of this type in
2D. Similar attempts in 3D are found in Zhao and Chew
关59兴, Giebermann 关61兴, and Fukui and Kozuka 关62兴. The paper by Giebermann 关61兴 includes 3D numerical examples
solved with a new version of panel clustering which uses
spherical harmonics rather than the Taylor series to expand
kernel functions. Fukui and Kozuka 关62兴 present numerical
examples of acoustic half-space problems solved with a lowfrequency formulation of the Barnes and Hut type. Zhao and
Chew 关59,60兴 introduce a scaling to keep the magnitudes of
all the 共spherical兲 Bessel and Hankel functions moderate
since Bessel 共Hankel兲 functions can otherwise become very
small 共large兲 in the quasi static cases. These scaling techniques are related to overflow and underflow problems in the
evaluation of Bessel and Hankel functions, and should not be
confused with instabilities of diagonal forms where loss of
information is the issue. See Section 3.2.
For high frequency problems, however, approaches of this
type will lead to O(N 2 ) algorithms 关63兴 since one cannot
take the number of terms p in the multipole expansion independent of the level. This is why the use of diagonal forms,
or the high-frequency FMM discussed in the next section is
considered imperative in the FMM for the Helmholtz equation when the frequency is high. As will be made clear later,
however, the high-frequency FMM does not work in low
frequency problems, making low-frequency FMM worth the
investigation.
3.2 Diagonal forms
One problem of the original FMM is that the M2L is rather
expensive. In 2D, for example, the M2L gives an operation
of the convolution type 共see 共31兲 for example兲, whose complexity is of the order of O(p 2 ) if one truncates the infinite
series with p terms. This operation has to be used 27 times in
the worst case in order to evaluate the contributions from all
the cells in the interaction list. Even worse is the 3D case
where the M2L 共see 共20兲 for example兲 is an O(p 4 ) operation,
which has to be repeated 189 times in the worst case.
This problem becomes quite serious in Helmholtz equation for high frequency. Consider the 2D case, for example.
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Appl Mech Rev vol 55, no 4, July 2002
One takes a few boundary elements per wavelength in the
discretization to maintain accuracy. To express potential
fields produced by an aggregate of such elements, one would
have to take p proportional to the size of the cell multiplied
by the wave number. With such p the total complexity of the
original FMM becomes O(N 2 ) 关63兴, thus making a naive use
of FMM equally or possibly more expensive than the conventional O(N 2 ) approaches.
However, the operations in 共29兲, 共31兲, and 共32兲 are multiplications of matrices to vectors. If one can diagonalize the
matrices, the complexities could be reduced. Rokhlin 关27兴
noticed that one can achieve this diagonalization by using
finite Fourier transform of the formulas in 共29兲, 共31兲, and
共32兲, since they are of convolutional form. Indeed, he introduces an auxiliary variable q and defines
F 共 q;O 兲 ⫽
兺n e ⫺inq M n共 O 兲 ,
H 共 q;O ⬘ 兲 ⫽
兺n e ⫺inq L n共 O ⬘ 兲 ,
(36)
p
␯ 共 q,x兲 ⫽
兺
e ⫺inq O n 共 x兲
n⫽⫺p
to rewrite M2M, M2L, and L2L formulas as
ជ
F 共 q;O 兲 ⫽e ⫺ikk̂•OO ⬘ F 共 q;O ⬘ 兲 ,
ជ⬘ 兲 F 共 q;O 兲 ,
H 共 q;O ⬘ 兲 ⫽ ␯ 共 q,OO
(37)
ˆ ជ
H 共 q;O ⬘ 兲 ⫽e ikk•OO ⬘ H 共 q;O 兲 ,
respectively, where
k̂共 q 兲 ⫽ 共 cos q,sin q 兲 .
(38)
The derivation of these formulas uses the following identity
p⫽kD⫹c log共 kD⫹ ␲ 兲 ,
兺n e ⫺inq I n共 x兲 ⫽e ⫺ikk̂•x.
Also, in the present context with 共23兲, one has
F 共 q;O 兲 ⫽
冕
ជ
e ⫺ikk̂•Oy
S0
冉
冊
⳵u
共 y 兲 ⫹ikk̂•nu 共 y 兲 dS.
⳵n
(39)
This is nothing other than the scattering amplitude of the
potential function given in 共23兲 共with ⳵ D replaced by S 0 兲. In
practice one computes F in 共39兲 for several 共say N q ⫽2 p兲
directions q j ( j⫽1,...,N q ) in each source cluster of elements, shifts F(q j ;•) with 共37兲 to obtain H(q j ;•) in clusters
away from sources and evaluates the potential via
i
8␲
冖
ˆ ជ
e ikk•O ⬘ x H 共 q;O ⬘ 兲 dq
i
⬃
8␲
Nq
兺
j⫽1
e
ជx
ikkˆ j •O
⬘
H 共 q j ;O ⬘ 兲 w j ,
called Rokhlin’s diagonal form. Note that the computations
in 共37兲 are done with O(N q )⬃O( p) operations. Rokhlin
thus showed that this approach gives an O(N 3/2) algorithm in
two level approaches, and an O(N 4/3) algorithm with one
more hierarchy of clusters. He states that the introduction of
quad-tree will further reduce the complexity to O(N log N).
These estimates for the complexity are established with an
assumption that the number of unknowns per wavelength
stays constant as N increases. This means that one is considering a series of problems with increasing N in which either
the domain sizes or the wave numbers increase, but the solutions are obtained with almost the same accuracy.
Notice that the series in the definition of ␯ (q,x) in 共36兲 is
truncated at ⫾p terms. This truncation is essential since this
series is divergent as p→⬁. This subtlety came from the
derivation of 共37兲 from 共31兲 where a summation and an integration has been interchanged; an operation which can be
justified only if the series is finite. This number p is typically
chosen proportional to kD where D is the diameter of the
source region 关27兴.
Engheta et al 关64兴 present the same formulation as in 关27兴
using what these authors consider easy words for nonmathematical readers. This paper also includes attempts of
avoiding irregular frequencies 共fictitious eigenfrequencies
关65兴兲 by using complexification, or addition of a small artificial imaginary part to the wave number. Rokhlin extended
his formulation to 3D in a mathematical paper 关66兴, and in a
companion paper by Coifman et al for physicists and engineers 关67兴. This paper by Coifman et al 关67兴 points out the
possibility of the instability, or subwavelength breakdown
关68兴 of this formulation when the parameter of truncation p
in 共36兲 is taken too large. Empirical formulas for recommended values of p in terms of the wave number k and the
diameter D of the domain 共source cell兲 are given in the following form:
where c is a number dependent on the precision of the arithmetics. See also Song, Lu, and Chew 关69兴 for a similar result
and Darve 关70兴 for a mathematical justification. We shall
further discuss this problem in the next section. Epton and
Dembart 关22兴 relate translation operators in the original and
diagonal form FMMs in Helmholtz’ equation in 3D.
Lu and Chew 关63兴 provided another point of view to the
diagonal forms for Helmholtz’ equation. They started from
the following plane wave expansion of the interior solution
I n (x) 共See 共26兲兲:
I n 共 x兲 ⫽
1
共 ⫺1 兲 n
2␲
冖
ˆ
e ikk•x⫹inq dq
to obtain
(40)
where w j is the weight of a certain numerical integration
formula. Effects from nearby elements are evaluated in the
conventional manner and added to 共40兲. This formulation is
G 共 x⫺y兲 ⫽
⫽
i
4
兺␯ 共 ⫺1 兲 ␯ I ⫺ ␯共 x⫺X⫹Y⫺y兲 O ␯共 X⫺Y兲
i
8␲
冖
ˆ
e ikk•(x⫺X) ␯ 共 q;X⫺Y兲 e ⫺ikk̂•(y⫺Y) dq,
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(41)
Appl Mech Rev vol 55, no 4, July 2002
where X and Y are points close to x and y, respectively. As
one uses this expansion with 共23兲 one shows that the LHS in
共40兲 is obtained. One thus sees that the plane wave point of
view of Lu and Chew gives the same formulation as Rokhlin’s diagonal form. Lu and Chew 关63兴 present numerical
examples obtained with a two level implementation whose
complexity is O(N 3/2).
Implementation of multilevel FMM is done with the help
of the quad-tree or oct-tree structures of cells as in the case
of the original FMM. One complication, however, comes
from the number of directions N q used for q in 共40兲, etc,
which is chosen basically proportional to the diameter of
cells, as stated above. Hence, the number of directions for
level l, denoted by N q,l , approximately doubles as one goes
up 共with decreased l兲 the tree of cells by one level (N q,l⫺1
⬃2N q,l ). Since the quantity F for N q,l⫺1 directions in a
parent cell is computed from the Fs for the children known
only in N q,l directions, the Fs in the children have to be
interpolated into N q,l⫺1 directions before they are shifted to
the parent cell by (37) 1 . One also needs a reverse operation
共filtering, or sometimes called anterpolation兲 when one computes H by tracing the tree downward. See Lu and Chew
关71兴, Song and Chew 关72兴, Song, Lu, and Chew 关69兴, and
Gyure and Stalzer 关73兴 for details.
Numerical implementations of the formulations for Helmholtz’ equation discussed above have been investigated in
many publications. Particularly impressive, both qualitatively and quantitatively, are the works by Chew’s group.
The two level implementation in 关63兴 was extended by Lu
and Chew 关71兴 to a 2D multilevel one whose complexity is
O(N log2 N). Wagner and Chew 关74兴 further proposed to use
only those directions k̂ close to X⫺Y in 共41兲 with the help of
a certain window function to reduce the computational complexity to O(N 4/3) even in the two level formulation in 2D.
This approach is named RPFMA 共Ray Propagation Fast Multipole Algorithm兲 关74兴. See also Burkholder and Kwon 关75兴
where the RPFMA idea is more clearly defined, and Rokhlin
关76兴 where the definition of ␯ (q;X) is modified so that approaches similar to RPFMA are justified mathematically.
RPFMA is extended to 3D electromagnetic scattering problem by Song and Chew 关77兴, who implemented a two level
algorithm with curved boundary elements. The multilevel implementation 共MLFMA, Multi Level Fast Multipole
Algorithm兲 of this approach with O(N log N) complexity is
presented by Song and Chew 关78兴, which developed into a
code called FISC 共Fast Illinois Solver Code兲 whose requirements and scaling properties as of 1998 are presented in 关72兴.
A remarkable piece of information from this paper is that
the number of iterations needed with their code with CG
method scales approximately as O(N 1/4) for closed targets
solved with CFIE 共to be discussed later兲. For open targets
they report a slower O(N 1/2) convergence. Song, Lu, and
Chew 关69兴 discuss the preconditioning, truncation of multipole series, and initial guess related to their code. The update of their status with their parallel code named ScaleME
is seen in their homepage 关79兴 where they proudly announce
that they could solve 3D electromagnetic scattering problems with about 10.2 million unknowns in a few hours 共or,
Nishimura: Fast multipole accelerated BIEMs
309
eight times faster than with the previous fastest algorithm,
in their expression兲 on the 128-processor Silicon Graphics
Origin2000 computer at the University of Illinois’s National
Center for Supercomputing Applications. Figure 4 is taken
from Chew’s home page 关79兴.
Their more recent developments include coupling with
FEM 关80兴, use of a higher order basis with Galerkin 关81兴, a
fast method of evaluating the 3D counterpart of ␯ (q;X) in
共36兲 using a 1D FMM 关82兴 and an application of FMM to a
2D electromagnetic inverse scattering problem of determining the shapes of unknown conductors from the scattering
data 关83兴.
Other groups also implemented FMM for Helmholtz’s
equations. In Gyure and Stalzer 关73兴, these authors present
some details of their O(N log2 N) implementation. Specifically, they discuss the issues of interpolation and filtering
共anterpolation兲 of the multipole moments in the upward and
downward processes. Dembart and Yip 关68兴 pay particular
attention to the problem of subwavelength breakdown to be
discussed in the next section. The paper by Darve 关84兴 provides details in the implementation of FMM codes for the
Maxwell equation, which are useful in Helmholtz’ equation
as well.
In high frequency problems for exterior problems, an
ordinary BIEM easily breaks down because of the fictitious
eigenfrequencies 关65兴. The most standard technique in electromagnetic analysis to avoid fictitious eigenfrequencies
is CFIE 共combined field integral equation兲, which is conceptually close to Burton and Miller’s method 关85兴 for direct
BIEMs in acoustics or in elastodynamics. Examples of the
use of Burton and Miller’s method with FMM in Helmholtz’
equation in 2D are found in Fukui et al 关62,86,87兴. Fukui and
Kozuka 关62兴 include discussion and numerical examples of
acoustic half-plane problems solved with the diagonal form.
Rokhlin’s diagonal form can also be used in Laplace’s
equation 关22兴. Or equivalently, one may use discrete Fourier
transform to diagonalize the M2M, M2L, and L2L operations
in the original FMM 关22兴. A numerical implementation of
this type in 3D is found in Elliot and Board 关88兴, where FFT
is used to accelerate computation of some convolutions. Here
Fig. 4 The computation of the current distribution on the VFY218
plane at 2 GHz. The plane wave is incident 30 degrees from the
nose, and is vertically polarized. At 2 GHz, the VFY218 with inlet
sealed is 155 wavelengths long, and is refined to 2,032,518 unknowns. The problem can be solved on the Origin 2000 with eight
processors, 6.6 GB of memory, and 13 hours of CPU time.
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310
Nishimura: Fast multipole accelerated BIEMs
Appl Mech Rev vol 55, no 4, July 2002
again the instability related to the truncation of the divergent
series similar to the one in (36) 3 is observed. Elliot and
Board propose to avoid this problem by dividing the series
similar to the one in (36) 3 into blocks, and applying FFT to
each block followed by a block-level convolution. Finally,
we mention the paper by Epton and Dembart 关22兴 once again
which is valuable in that the tools used in the original and
diagonal form FMMs for both Laplace and Helmholtz equations in 3D are summarized.
3.3 New FMMs
Rokhlin’s diagonal form provides a very tricky fast alternative to the original FMM, as we have seen in the previous
section. It is instructive to see how the information needed in
the original FMM is stored in the formulation using the diagonal form. Consider, for example, the special case of 共41兲
with x⫽X
G 共 x⫺y兲 ⫽
i
4
the expression of the fundamental solution in terms of a Fourier integral 共Fourier inversion with respect to ␰ 3 has been
carried out兲
1
冑共 x 1 ⫺y 1 兲
⫽
冕
e i ␰ ␣ (x ␣ ⫺y ␣ )⫺(x 3 ⫺y 3 ) 兩 ␰兩
d ␰ 1d ␰ 2 ,
兩 ␰兩
R2
1
冑共 x 1 ⫺y 1 兲
2
⫹ 共 x 2 ⫺y 2 兲 2 ⫹ 共 x 3 ⫺y 3 兲 2
s(␧) M (k)
⫽
兺 兺
k⫽1 j⫽1
␻k
e ⫺(␭ k /d)(x 3 ⫺y 3 )
M 共 k 兲d
⫻e i(␭ k /d)((x 1 ⫺y 1 )cos ␣ j ⫹(x 2 ⫺y 2 )sin ␣ j ) ⫹␧ 共 x 3 ⬎y 3 兲 ,
(44)
兺j ␯ 共 q j ;x⫺Y兲 F 共 q j ;Y 兲 w j
(42)
where
M ␯ 共 Y 兲 ⫽I ␯ 共 y⫺Y兲 ,
F 共 q j ;Y 兲 ⫽e ⫺ikk̂(q j )•(y⫺Y) ⫽
兺n e ⫺iq n M n共 Y 兲 ,
j
(43)
and q j and w j are the abscissa and weight of the quadrature
rule used to approximate the integral in 共41兲. It would be
obvious that one has to be able to recover good approximations for the numbers O ␯ (x⫺Y)M ⫺ ␯ (Y ) from ␯ (q j ;x⫺Y)
and F(q j ;Y ) for the diagonal form to work. For moderate
values of 兩 x⫺Y兩 , 兩 y⫺Y兩 , and a small k, however, one estimates
冉
ek 兩 y⫺Y兩
兩 M ␯共 Y 兲兩 ⬃
冑2 ␲ 兩 ␯ 兩 2 兩 ␯ 兩
1
兩 O ␯ 共 x⫺Y兲 兩 ⬃
⫹ 共 x 2 ⫺y 2 兲 2 ⫹ 共 x 3 ⫺y 3 兲 2
the RHS of which can be evaluated with certain numerical
integration formulas as 关90,91兴
兺␯ O ␯共 x⫺Y兲 M ⫺ ␯共 Y 兲
i
⬃
8␲
1
2␲
2
冉
冊
ek 兩 y⫺Y兩
冑␲ 兩 ␯ 兩 2 兩 ␯ 兩
2
where ␣ j is a number given by ␣ j ⫽2 ␲ j/M (k), ␧ is the error
term, and d is a scale factor, respectively. Also, the number
of integration points s(␧) and M (k), together with the abscissa of the Gauss points ␭ k and their weights w k are tabulated in 关89–91兴. In deriving this formula we have assumed
that x 3 ⬎y 3 holds. One thus obtains an expansion of the type
in 共2兲 for the fundamental solution from the Fourier transform of the fundamental solution, which is readily available
for any PDE with constant coefficients. One thus sees that
the single layer potential for Laplace’s equation in 3D, for
example, allows an expansion of the following form
冕
s(␧) M (k)
␾共 y 兲
dS y ⬃ 兺
k⫽1
S 0 兩 x⫺y兩
冊
with appropriately chosen origin and d, where E k j (O) is the
coefficient of exponential expansion defined by
⫺兩␯兩
as 兩␯兩→⬁. Hence 兩 M ␯ (Y ) 兩 converges to 0 and 兩 O ␯ (x⫺Y) 兩
diverges, both very rapidly, as 兩␯兩→⬁. This means that the
information on M ␯ (Y ) (O ␯ (x⫺Y)) for large 共small兲 兩␯兩 is
easily lost as one tries to extract it from F(q j ;Y ) in 共43兲
( ␯ (q j ;x⫺Y) in 共36兲兲 with floating point arithmetics, while
these numbers O ␯ (x⫺Y) and M ␯ (Y ) are needed in order to
evaluate the multipole expansion accurately. Thus the diagonal form of Rokhlin suffers from instability in Helmholtz’
equation for small wave number or in Laplace’s equation.
The new versions of FMM are introduced to resolve this
problem 共Laplace 关89–91兴, Helmholtz 关92兴兲. These formulations are designed to remain diagonal while suffering from
no instability problems. In the new FMM for Laplace’s equation in 3D by Greengard and Rokhlin 关90兴, they start from
␻k
e ⫺(␭ k /d)x 3
M 共 k 兲d
⫻e i(␭ k /d)(x 1 cos ␣ j ⫹x 2 sin ␣ j ) E k j 共 O,d 兲 (45)
兩␯兩
,
兺
j⫽1
E k j 共 O,d 兲 ⫽
冕
S0
e (␭ k /d)y 3 e ⫺i(␭ k /d)(y 1 cos ␣ j ⫹y 2 sin ␣ j ) ␾ 共 y 兲 dS.
(46)
It is easily seen that the shift of the origin of the coefficient
of exponential expansion 共this corresponds to M2L in the
original FMM兲 reduces to a simple multiplication of an exponential function, thus making the shift operation diagonal.
Also, since the exponentials in 共45兲 and 共46兲 are harmonic,
they can be expanded with the solid harmonics R N,M 共see
共15兲兲 in a bounded set. This implies that the exponential
expansion in 共45兲 共coefficients of exponential expansion in
共46兲兲 can easily be converted into the local expansion 共multipole moments兲 for the original FMM 关90兴. One thus sees
that the following algorithm is possible:
1兲 Introduce the tree structure of elements.
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Appl Mech Rev vol 55, no 4, July 2002
2兲 Compute multipole moments in each cell. Up to here
the procedure is exactly the same as that in the original
FMM.
3兲 Convert multipole expansion into exponential expansion in each cell.
4兲 Shift the coefficients of exponential expansion from
source cells to target cells. This operation replaces the M2L
in the original FMM. The shifted exponential expansion is
added in each target cell to form an exponential expansion
with the origin taken in the target cell. This expansion represents the sum of contributions from far cells.
5兲 Convert the exponential expansions in cells into local
expansion, and proceed as in the original FMM.
Notice, however, that the expression in 共44兲 is valid only
for x 3 ⫺y 3 ⬎0. This is why Greengard and Rokhlin introduced 6 kinds of exponential expansions in 关90兴, ie, upward,
downward, northward, southward, eastward, and westward
共or z⫹, z⫺, y⫹, y⫺, x⫹, x⫺兲 exponential expansions,
taking the direction of exponential decay in the exponential
functions as the plus or minus the coordinate directions, respectively. The expression in 共45兲, for example, gives the
upward exponential expansion 共the word upward expansion
is created by the present author for the purpose of explanation兲. Also, the conversion from these exponential expansions to expansions in the original FMM needs the rotation
of the coordinate 关90兴 using formulas given in 关29兴, for example, since Legendre’s functions are not isotropic. In spite
of these complications the new FMM was found to be faster
than the original FMM 关89–91兴, since this approach reduces
the computational complexity needed for M2L from O(p 4 )
to O( p 3 ) 关90兴. The use of rotation in M2M and L2L is another reason for the speedup since the complexities of M2M
and L2L are also reduced from O(p 4 ) to O(p 3 ) 关90兴. In
addition, this approach is shown to be devoid of the instability seen in the original diagonal form by Rokhlin.
This new FMM was formulated by Hrycak and Rokhlin
关89兴 for Laplace’s equation in 2D. This paper includes comparison of the efficiencies of the new and original FMMs in
the analysis of particle systems, according to which the new
FMM is faster than the original one in all the examples presented. The reduction of the CPU time varies from approximately 10% to 76% depending on the problem. This formulation was tested favorably in an engineering problem by
Nishimura et al 关93兴. Indeed, the new FMM was concluded
to be about 1.8 times faster than the original FMM for an
array of 10 times 10 straight cracks (N⫽20,000). The 3D
version of the new FMM presented above is due to Greengard and Rokhlin 关90兴, further details of which, related to
numerical implementation, are elaborated in Cheng et al
关91兴. Yoshida et al 关94兴 investigated the use of the new FMM
in crack problems for Laplace’s equation in 3D. They found
that the new FMM is about 30% faster than the original
FMM in problems where the cracks are distributed densely
in the domain. The extension of the new FMM to Helmholtz’
equation is made in 关92兴, but the complete details are not
available yet.
In passing, we note that another O(p 3 ) M2L algorithm for
Nishimura: Fast multipole accelerated BIEMs
311
Laplace’s equation in 3D can be obtained simply by rotating
the coordinate axes in the original FMM 关25兴. Zhao and
Chew state that the rotation technique in Helmholtz’ equation
in 3D does not reduce the number of operations, but saves
the storage 关95兴.
In Helmholtz’ equation in 2D, Hu et al 关96兴 proposed a
faster alternative to FMM called FIPWA 共Fast Inhomogeneous Plane Wave Algorithm兲. This approach is related to the
new FMM by Greengard and Rokhlin in that an expansion of
the fundamental solution is obtained by writing it in the form
of a Fourier integral, followed by discretization using a certain numerical integration. Indeed, Hu et al started from an
observation that the fundamental solution of Helmholtz’
equation has an integral representation
H (1)
0 共 k 兩 x⫺y兩 兲 ⫽
1
␲
冕
⌫
ˆ
e ikk•(x⫺y) dq,
(47)
where k̂ is defined in 共38兲, and ⌫ is an infinite path in the
complex plane which runs from the north-west to the southeast of ␾ x⫺y⫽tan⫺1((x2⫺y2)/(x1⫺y1)) 共See 关96兴. The ⌫ in
共47兲 is identical with the one in 关96兴, although the definition
of the angle q in 共47兲 is different from the one in 关96兴兲. As a
matter of fact, 共47兲 has been utilized by Burkholder and
Kwon 关75兴 and by Michielssen and Chew 关97兴 in different
contexts. Suppose now that one computes the single layer
potential
冕
S0
H (1)
0 共 k 兩 x⫺y兩 兲 ␾ 共 y 兲 dS y
for x in a ball having the radius of R centered at x o . The
surface S 0 is also assumed to be included in a ball having the
radius of R centered at y s . In this case, the path ⌫ is chosen
to pass the interval ( ␾ xo ⫺ys ⫺ ␺ 0 , ␾ xo ⫺ys ⫹ ␺ 0 ), where ␺ 0
⫽sin⫺1(2R/兩xo ⫺ys 兩 ). The above integral is then evaluated as
冕
S0
H (1)
0 共 k 兩 x⫺y兩 兲 ␾ 共 y 兲 dS y ⫽
1
␲
兺␣ e ikk(q )•(x⫺x )
ˆ
⫻e ikk(q ␣ )•(xo ⫺ys ) F 共 q ␣ ;y s 兲 w ␣
ˆ
␣
o
(48)
where q ␣ and w ␣ are the abscissa and weight for the numerical integration used for the q integration on ⌫ and
F 共 q;y s 兲 ⫽
冕
S0
e ⫺ikk̂(q)•(y⫺ys ) ␾ 共 y 兲 dS y .
(49)
Hu et al 关96兴 propose to sample F(q;y s ) at a finite number
of points on the real axis in the interval of ( ␾ xo ⫺ys
⫺ ␺ 0 , ␾ xo ⫺ys ⫹ ␺ 0 ), and use interpolation and extrapolation
共into the complex plane兲 to compute the sum in 共48兲. The
overall complexity of the algorithm is O(N 4/3) in the two
level scheme and O(N log N) for the multilevel implementation. In 关96兴 the authors conclude that FIPWA outperforms
ordinary FMM and RPFMA. They also point out that FIPWA
is more stable than ordinary FMM, as expected. Hu and
Chew 关98兴 extends the FIPWA formulation to the case of
layered media in 2D.
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3.4
Related approaches
Phillips and White 关99兴 investigated a precorrected-FFT approach in the analysis of Dirichlet problems for the Laplace
equation in 3D using single layer potentials. This approach
introduces a uniform grid with cubic cells over a right parallelepiped which includes the boundary of the domain in its
interior. Each of the cells is a cube with the edge length of a
few grid intervals. In this approach the contribution from the
elements within a cell to the single layer potential is approximated by a set of sources acting on the grid points associated
with the cell, ie, those grid points on the boundary or in the
interior of the cell. This approximation is valid away from
the cell, of course. The magnitude of the equivalent sources
is determined by actually equating the single layer potential
in the cell and the potential by the sources at several points
away from the cell. The single layer potential at the grid
points can now be evaluated easily with the help of FFT
since the kernel function depends on x 共point of evaluation兲
and y 共integration point兲 via x⫺y. The estimated values of
the single layer potential at grid points are now interpolated
to give the values of the single layer potential at the collocation points. However, a careless use of this procedure will
estimate contribution from nearby elements inaccurately
since the equivalent source approximation is valid only in the
far field. Therefore one has to precorrect the estimation by
subtracting the inaccurate contribution from the nearby cells
and adding back accurate values obtained by direct computation. This approach gives an O(N log N) algorithm, and
will be inferior to FMM in very large problems with sparse
element distribution, but the authors concluded that this approach is faster than an FMM counterpart in all examples
shown in this paper.
An advantage of this approach is that one may use
Green’s function in place of the fundamental solution without difficulty. Indeed, an application of this approach is attempted by Kring et al 关100兴 in their analysis of water wave
problems using Green’s function satisfying linearized boundary condition on the free water surface. This paper also includes examples of the time domain analysis 共with Laplace’s
equation兲 with nonlinear boundary condition for the free water surface. See also Korsmeyer et al 关101兴 for more applications to offshore structure problems.
Analysis of this type with FFT and a regular grid is applicable also to Helmholtz’ equation. Indeed, Bleszynski
et al 关102兴 propose an Adaptive Integral Method 共AIM兲 for
electromagnetic scattering problems, whose complexity is
estimated to be O(N 3/2 log N) for the case of boundary integral equations. A related formulation for Helmholtz’ equation
in 3D is proposed by Bespalov 关103兴. This formulation splits
the source at the grid points into the sum of a singular function with a compact support and a smooth part, and uses FFT
to take the sum of only the smooth part of the kernel. Also,
the 2D nature of the boundary surface is taken into account
so that the amount of computation is reduced. The complexity of this approach is also O(N 3/2 log N).
The formulation by Bruno and Kunyansky 关104兴 also
uses a regular grid and FFT in the analysis of Helmholtz’
equation in 3D, but differs from the precorrected-FFT or
Appl Mech Rev vol 55, no 4, July 2002
AIM approaches in that the equivalent sources are distributed two dimensionally on planes containing the faces of a
cell, rather than three-dimensionally. Also, the equivalent
sources considered in 关104兴 include both monopoles and dipoles. Their approach is characterized by higher accuracy
and the complexity of O(N 6/5 log N) to O(N 4/3 log N), the
former being for highly complex surfaces and the latter for
smooth surfaces.
Kapur and Long 关105兴 propose an approach called IES3
which decomposes hierarchically the coefficient matrix into
submatrices which allow singular value decompositions
共SVD兲 with small ranks, thus facilitating multiplication with
vectors. The SVDs of the submatrices are obtained with an
interpolation idea based on sampling of some rows and columns of the submatrices. The complexity of their approach is
O(N log N), and the formulation is applicable to any kernel
function as long as they are smooth in the far field. Their
formulation is applied to both Laplace’s equation and Maxwell’s equation for low frequency. From numerical examples
with O(104 ) unknowns, they concluded their formulation to
be faster and to yield more accurate results than an FMM
based code.
Lu and Chew 关106兴 propose what they call fast far-field
approximation which resembles FMM, but differs in that
it uses the far field approximation instead of the multipole
expansion.
Michielssen and Chew 关97兴 proposed an approach called
Fast Steepest Descent Path Algorithm 共FSDPA兲. This is
considered to be a cross between FMM and Matrix Decomposition Algorithm 共MDA兲 关107兴, which is a multilevel
fast method of calculating a matrix-vector product having
some resemblance to FFT. FSDPA is an O(N 4/3) method, but
is considered far from being optimum 关96兴 since it is not
diagonal; an observation which led to the development of
FIPWA 关96兴.
3.5
Introductory and review articles
The introductory article by Ramaswamy et al 关108兴 covers
FMM for Laplace and Helmholtz equations including the
new FMM, precorrected approaches, wavelets, and Stokes
flow problems. This paper puts more stress on explaining
basic ideas of the methods covered than to present a list of
existing papers.
The paper by Seberino and Bertram 关26兴, although written
as a research paper, can be recommended also as an introductory paper to FMM for Laplace’s equation in 3D because
it is very clearly written and covers various topics from
the formulation to parallelization. Also recommendable is
the paper by Sun and Pitsianis 关109兴 where the original
FMM in Laplace’s equation in 3D is written neatly with
matrix notations.
Chew et al 关110兴 present various fast methods of solving
electromagnetic problems not restricted to boundary integral
formulations, but including domain methods, wavelets, etc.
This article is particularly useful for those wishing to obtain
information on various descendents of FMM and related
techniques in equations of the Helmholtz type proposed before 1997.
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Appl Mech Rev vol 55, no 4, July 2002
4
Nishimura: Fast multipole accelerated BIEMs
FMM IN TIME DOMAIN
4.1 Wave equation
We consider the initial boundary value problem for the wave
equation in 3D:
⌬u⫽
1
ü
c2
in D⫻ 共 t⬎0 兲
subject to standard initial and boundary conditions, where D
is a domain. The solution to this problem is written as
u 共 x,t 兲 ⫽
⫺
冕
⳵D
G 共 x⫺y,t 兲 *
⳵u
共 y,t 兲 dS y
⳵n
冕
⳵ G 共 x⫺y,t 兲
* u 共 y,t 兲 dS y
⳵ny
⳵D
x苸D
(50)
where G is the fundamental solution of the wave equation
given by
␦ 共 t⫺ 兩 x兩 /c 兲
G 共 x,t 兲 ⫽
4 ␲ 兩 x兩
and * indicates the convolution with respect to time t. Also,
we have assumed that the initial data u 兩 t⫽0 and u̇ 兩 t⫽0 vanish
for the purpose of simplicity. The representation of the solution in 共50兲 and the boundary condition yield an integral
equation of the following form
冕
⳵D
K 共 x,y 兲共 unknown quantities for t⫽t 兲共 y 兲 dS y
⫽
冕
⳵D
K ⬘ 共 x,y,t 兲 * 共 known quantities before t 兲共 y 兲 dS y
(51)
where K and K ⬘ are certain kernel functions. The LHS is
discretized into a sparse matrix, and solving the resulting
linear equations is not expensive once the RHS is obtained.
Hence the question is how one computes rapidly the RHS of
the above equation, or the terms representing the effects of
the past history.
It would be a natural question to ask if the developments
presented so far for the Helmholtz equation carry over to
wave equation as one considers the Fourier inverse transform
with respect to the frequency ␻. This approach, although
nobody seems to have tried it yet, appears to be out of the
question. Indeed, the Fourier inverse transform of the expansion for the fundamental solution 共25兲 for the Helmholtz
equation in 2D given in 共24兲 reduces to
1
2 ␲ 冑t ⫺ 兩 x⫺y兩 2 /c 2 ⫹
2
冉 冊
冉 冊
ct in ␪
ct ⫺in ␾
T 兩n兩
e
T 兩n兩
e
⬁
1
兩
x
兩
兩
y兩
⫽ 2
共 ⫺1 兲 n 2
2 ␲ n⫽⫺⬁
冑t ⫺ 兩 x兩 2 /c 2 ⫹ * 冑兩 y兩 2 /c 2 ⫺t 2 ⫹
兺
where T n is the Chebyshev polynomial and the angular variables of x and y are denoted by ␪ and ␾, respectively. Also,
313
the suffix ⫹ indicates that the corresponding term vanishes if
the quantity within the radical symbol is negative. With this
expression, one would have to express the solution for large
t in terms of functions behaving like polynomials of order
n⫺1. It would be obvious that such an approach would yield
a disastrous result as time grows. We therefore have to find
more sophisticated treatments than this.
As we have noticed, Lu and Chew’s point of view 关63兴 for
the diagonal form for the Helmholtz equation is that the diagonal form can be interpreted as a plane wave expansion of
the fundamental solution. It would therefore be natural to
seek a plane wave expansion of the fundamental solution for
the wave equation. To derive one, one starts from the Fourier
transform of the fundamental solution of the wave equation
in 3D.
1
兩 ␰兩 ⫺ ␻ 2 /c 2
(52)
2
where ␰ and ␻ are Fourier parameters for the spatial variable
x and t. The fundamental solution satisfying causality 共anticausality兲 is obtained as one computes the Fourier inverse
transform of 共52兲 assuming the imaginary part of ␻ to be
positive 共negative兲, as the limiting absorption principle tells.
Indeed, by computing the inverse transform with respect to
␰ 3 first, assuming that x 3 ⬎0, and by using a subsequent substitution given by ␰ ␣ ⫽ ␻ ␩ ␣ /c ( ␣ ⫽1,2) one has
␦ 共 t⫾r/c 兲
1
⫽
4␲r
2共 2 ␲ 兲3c
⫻
冕冕冕
e i ␻ ((x ␣ ␩ ␣ ⫹i sgn ␻
冑␳
␳ ⬎1
⫻d ␳ d ␪ d ␻ ⫿
⳵t
8 ␲ 2c
2
冑␳ 2 ⫺1x 3 )/c⫺t)
␳
⫺1
冕
␦ 共 t⫺k̂•x/c 兲 dS kˆ ,
S 0 艚(k̂ 3 ⭵0)
(53)
where k̂ indicates the unit outward normal vector to the unit
sphere S 0 and ␳ ⫽ 冑␩ ␣ ␩ ␣ . Also, the second integral in 共53兲 is
taken on the upper half (k̂ 3 ⬎0) or lower half (k̂ 3 ⬍0) of S 0 ,
respectively, when one chooses upper or lower signs in the
same formula. The first integral in 共53兲 for ␳⬎1 is called the
evanescent part, while the integral on the unit sphere is interpreted as a superposition of plane waves. One may eliminate the evanescent parts from these formulas to have
⳵t
␦ 共 t⫺r/c 兲 ␦ 共 t⫹r/c 兲
⫺
⫽⫺
4␲r
4␲r
8 ␲ 2c
冕␦
S0
共 t⫺k̂•x/c 兲 dS kˆ
(54)
This formula holds for an arbitrary r⫽ 兩 x兩 (⫽0). The anticausal fundamental solution 共the second term in the LHS兲
vanishes for a positive t, thus will never be mixed with the
physical signal produced by the causal fundamental solution
G which arrives at a positive t(⫽r/c). One thus sees that the
RHS of 共54兲 does give a plane wave expansion for the fundamental solution G satisfying causality 共the physically
meaningful one兲 to within a ghost which vanishes when the
true signal is nonzero. This formulation is called the
Whittacker-type. As a matter of fact, one may take only a cap
shaped part of the unit sphere S 0 visible from the point of
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314
Nishimura: Fast multipole accelerated BIEMs
Appl Mech Rev vol 55, no 4, July 2002
observation x in 共54兲 as the domain of integration to obtain a
plane wave representation of the fundamental solution valid
to within a ghost. We shall call this the finite-cone formulation after Ergin, Shanker, and Michielssen 关111兴. Of course,
the form of the ghost depends on the shape of the cap. When
this cap covers the whole unit sphere, the ghost takes the
form of the anticausal fundamental solution, as we have seen
in 共54兲.
We now consider the single layer potential for the wave
equation in 3D with the density function ␾, given by
u 共 x,t 兲 ⫽
冕␦
S
冕␦
S
⬇⫺
⳵t
8 ␲ 2c
冕 冕␦
S0
S
共 t⫺k̂• 共 x⫺y兲 /c 兲 * ␾ 共 y,t 兲 dS y dS kˆ
(55)
and the symbol ⬇ indicates that the equality holds to within
the ghost.
Now, let x be a point in a spherical region R o centered at
x oc having the radius of R. Also, let the surface S be included in a spherical region R s centered at x sc having the
radius of R. It is seen from Fig. 5 that the signal from R s
reaches R o after T 2 if
R c ⬎ 共 T 2 ⫺T 1 兲 c⫹2R
(56)
holds. Also, this condition guarantees that the ghost produced by ␾ in R s vanishes in R o before the arrival of the
signal from R s . Hence the condition in 共56兲 is useful in
constructing an algorithm for evaluating potentials.
We now rewrite 共55兲 into
S0
S
共 t⫺k̂• 共 x⫺xoc 兲 /c 兲
* ␦ 共 t⫹k̂• 共 y⫺xsc 兲 /c 兲 * ␾ 共 y,t 兲 dS y dS kˆ
⬇⫺
*
共 t⫺ 兩 x⫺y兩 /c 兲
␾ 共 y,t 兲 dS y
4 ␲ 兩 x⫺y兩 *
冕 冕␦
1
8 ␲ 2c
⳵
* ⳵ t ␦ 共 t⫺k̂• 共 xoc ⫺xsc 兲 /c 兲
共 t⫺ 兩 x⫺y兩 /c 兲
␾ 共 y,t 兲 dS
4 ␲ 兩 x⫺y兩 *
where S is a surface, x is a point located away from S, and *
denotes the convolution with respect to time. We assume, for
a moment, that the density ␾ is nonzero only in the time
interval given by 0⬍T 1 ⬍t⬍T 2 . This assumption will be
removed later. Equation 共54兲 shows that the single layer potential can be rewritten as follows 关111–114兴:
u 共 x,t 兲 ⫽
u 共 x,t 兲 ⫽⫺
1
8 ␲ 2c
冕␾
S
兺i ␦ 共 t⫺k̂i • 共 x⫺xoc 兲 /c 兲 * Ti共 t,xoc ⫺xsc 兲
共 y,t⫹k̂i • 共 y⫺xsc 兲 /c 兲 dS y w i
where Ti (t,xoc ⫺xsc ) is 共essentially兲 equal to
⫺
1 ⳵
␦ 共 t⫺k̂i • 共 xoc ⫺xsc 兲 /c 兲
8 ␲ 2c ⳵ t
(57)
with k̂i and w i indicating integration points and the corresponding weight of a certain quadrature formula for the integral on S 0 . The integral
M i 共 t;x sc 兲 ⫽
冕␾
S
共 y,t⫹k̂i • 共 y⫺xsc 兲 /c 兲 dS y ,
(58)
R
R
T 1 ⫺ ⬍t⬍T 2 ⫹
c
c
is called SST 共Slant Stack Transform兲 of ␾, or the outgoing
ray. As the notation implies, this quantity corresponds to the
moment in the FMM in frequency domain. We then translate
M i (t;x sc ) to the observation sphere by
L i 共 t;x oc 兲 ⫽Ti 共 t⫺k̂i • 共 xoc ⫺xsc 兲 /c 兲 * M i 共 t;x sc 兲
(59)
for all k̂i . The quantity L i , called the incoming ray, corresponds to the coefficient of the local expansion in the FMM
in frequency domain. Finally, at a point x o we compute u via
u共 t 兲⫽
兺i ␦ 共 t⫺k̂i • 共 xo ⫺xoc 兲 /c 兲 * L i共 t;x oc 兲 w i .
(60)
We now use the scheme discussed above and the two
level implementation to compute the single layer potential
u 共 x,t 兲 ⫽
冕
␦ 共 t⫺ 兩 x⫺y兩 /c 兲
␾ 共 y,t 兲 dS.
4 ␲ 兩 x⫺y兩 *
⳵D
(61)
We introduce N g disjoint groups of elements, each of which
is contained in a sphere centered at x i and having a radius of
R. Each group is assumed to contain O(M s ) elements. In
general, the source ␾ (•,t) may not have a finite duration.
Therefore we split ␾ (•,t) into sub-sources ␾ l (•,t), each of
which vanishes outside of an interval 关 T l1 ,T l2 兴 (l
⫽1,2, . . . ) and satisfies
␾ 共 •,t 兲 ⫽ 兺 ␾ l 共 •,t 兲 .
l
A typical choice for T l1 and T l2 are
Fig. 5
Signal and ghost
T l1 ⫽ 共 M 共 l⫺1 兲 ⫺ p 兲 ⌬t,
T l2 ⫽ 共 M l⫹ p 兲 ⌬t
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(62)
Appl Mech Rev vol 55, no 4, July 2002
Nishimura: Fast multipole accelerated BIEMs
where M and p are integers, and ⌬t is used as the time step
in a time marching analysis. The number p represents the
overlap of the time intervals. With 共62兲, the calculation goes
as follows
1兲 Set l⫽1.
2兲 Let t be such that t⭐T l2 holds.
a兲 For each point x in a sphere S o , compute directly the
contributions from sources ␾ (y,s) for s⬍t in spheres
S s close to S o , where a sphere S s is said to be close to
S o if the following inequality holds 共see 共56兲兲:
兩xoc ⫺xsc 兩 ⭐c 共 T l2 ⫺T l1 兲 ⫹2R.
Notice that this part of the calculation is identical with the
one used in the conventional approach.
b兲 For each point x in a sphere S o , use 共60兲 to compute
contributions from sources ␾ m (y,s) for m⭐l⫺1 in
far spheres S s which satisfy
兩xoc ⫺xsc 兩 ⬎c 共 T l2 ⫺T l1 兲 ⫹2R.
(63)
It is easily seen that contributions from such spheres for
m⫽l do not reach S o unless t⬎T l2 .
3兲 At t⫽T l2 , we compute the outgoing ray associated with
␾ l given by 共58兲 at the centroid of each sphere S s , and
propagate it via 共59兲 to spheres S o which satisfy 共63兲. The
incoming ray at S o from many far spheres S s are added
together for each propagation direction k̂i to yield L li .
Update l by l⫹1.
The above description of the method presents the basis of
the fast multipole BIEM for the wave equation in time domain. Here are some more details.
1兲 In practice the source function ␾ is rather smooth as a
function of time 共or band-limited兲. For such ␾, it is permissible to use a smooth approximation to the derivative
of Dirac’s delta as the function Ti in 共57兲 关111,113兴. Indeed, Ergin et al uses
K
Ti ⫽⫺
⳵t
共 2k⫹1 兲 P k 共 ct/ 兩 xoc ⫺xsc 兩 兲
2
16␲ 兩 xoc ⫺xsc 兩 k⫽0
by N s discrete sources, they found that the computational
complexity for the two level algorithm is of the order of
O(N t N s3/2 log Ns), and that for the multilevel approach is
O(N t N s log2 Ns), where N t is the number of time steps.
These estimations may further be improved to
O(N t N s4/3 log Ns) and O(N t N s log Ns) with treatments similar
to RPFMA 关74兴. In BIE applications one may take N s to be
the number of unknowns at a certain time.
These authors tried similar analyses with the Whittackertype formulation in 关111兴. One of the advantages of using the
Whittacker-type formulation is the simplicity of the formulation. Again, in this paper numerical examples are limited to
simple wave fields in 3D produced by a few sources with
known time dependence.
More interesting numerical examples are found in 关114兴,
where the same authors consider the scattering of acoustic
waves by a 3D hard object. The direct BIE formulation is
used. As is known, the ordinary frequency domain integral
equation for exterior problems may fail to give a unique
solution at the so called fictitious eigenfrequencies 关65兴.
These authors have shown, however, that phenomena similar
to the fictitious eigenfrequency also exist in the time domain
where the solution to the integral equation is unique, but its
discretized equation tends to amplify the error having the
frequency close to one of the fictitious eigenfrequencies
关117兴. Based on this observation, they use a formulation of
the Burton-Miller type to obtain a stable algorithm. The
analyses presented include scattering from cavities, screens,
and a submarine shaped object 共about 15,000 DOF兲. 共In
passing, we point out that the existence of numerical problems related to the fictitious eigenfrequency in the time domain has been observed by Abboud and Sayah 关118兴 before
this paper.兲 The corresponding multilevel implementation
has been presented in 关116兴, in which the size of the the
problem was increased to as large as 107,500 DOF with 500
time steps.
4.2 Heat equation
The solution of the initial boundary value problem for the
heat equation
兺
⫻ P k 共 cos共xoc ⫺xsc ,k̂i 兲兲
315
(64)
for ct/ 兩 xoc ⫺xsc 兩 ⭐1, where P k is the Legendre function
and K is a certain number.
2兲 The 2D equivalent of 共55兲 is somewhat more complicated
including the Hilbert transform instead of the time derivative. See Lu et al 关115兴.
3兲 Here again, one has interpolation and anterpolation issues
in multilevel implementations 关116兴.
The time domain approach of the type discussed above
was first presented by Ergin, Shanker, and Michielssen 关113兴,
where the authors developed what they call PWTD 共Plane
Wave Time Domain兲 algorithm for the wave equation in 3D
based on the finite-cone formulation. They present simple
numerical examples of computing wave fields produced by a
few sources. They also provide estimates of computational
complexities of both two level and multilevel PWTD algorithms. In the context of estimating the wave fields produced
⳵
u⫽⌬u
⳵t
in D⫻ 共 t⬎0 兲
(65)
is well known to have a potential expression given by
u 共 x,t 兲 ⫽
冕冉
G 共 x⫺y,t 兲 *
⳵D
⳵u
共 y,t 兲
⳵n
冊
⫺
⳵ G 共 x⫺y,t 兲
* u 共 y,t 兲 dS
⳵n
⫹
冕
D
G 共 x⫺y,t 兲 u 0 共 y 兲 dV,
(66)
where G is the fundamental solution for the heat equation
given by
G 共 x,t 兲 ⫽
冉 冊
1
2 冑␲ t ⫹
d
e ⫺ 兩 x⫺y兩
2 /4t
,
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(67)
316
Nishimura: Fast multipole accelerated BIEMs
Appl Mech Rev vol 55, no 4, July 2002
where d stands for the dimensionality of the problem and u 0
is the initial data. As in the case of the wave equation, the
boundary value problem under consideration reduces to a
solution of a time dependent boundary integral equation of
the form given in 共51兲, with an essentially sparse matrix K,
since the fundamental solution in 共67兲 is almost zero outside
a small ball centered at y for small t. Therefore, our interest
is to find a fast method of evaluating contributions to the
integrals in 共66兲 from the past. Consider, for example, the
contributions from the single layer potential or the volume
potential in 共66兲, which may be discretized as
q i, j G 共 x⫺yi ,t⫺ ␶ j 兲 ,
兺
i, j
where q i, j is a certain number, and y i and ␶ j are spatial and
time integration points, respectively. We may further simplify the expression by extracting the contribution from a
certain time ␶ j , followed by multiplication of (t⫺ ␶ j ) d/2 to
consider a sum of the following form:
N
兺i q i e ⫺兩x⫺y 兩 / ␦
i
2
(68)
where ␦ ⫽4(t⫺ ␶ j )⬎0.
Greengard and Strain 关119兴 proposed a fast method of
computing the sum in 共68兲 for N sources at M points x with
O(M ⫹N) operations, improving the performance over the
naive O(M N) approach. The mathematical tools used by
Greengard and Strain 关119兴 are rather simple, ie, the Taylor
2
expansions of e ⫺ 兩 x⫺y兩 / ␦ which take the following forms:
e
e
⫺ 兩 x⫺y兩 2 / ␦
⫺ 兩 x⫺y兩 2 / ␦
⫽
⫽
冉 冊冉 冊
冉 冊 冉 冊
1
x⫺y0
h
␣ ! ␣ 冑␦
兺
兩 ␣ 兩 ⭓0
兺 共 ⫺1 兲
兩 ␣ 兩 ⭓0
兩␣兩
y⫺y0
␣
,
冑␦
1 x⫺x0
␣ ! 冑␦
␣
h␣
x0 ⫺y
冑␦
(69)
,
(70)
where we have used the following multi-index notation:
兩 ␣ 兩 ⫽ ␣ 1 ⫹¯⫹ ␣ d ,
␣
␣
␣ !⫽ ␣ 1 !¯ ␣ d !,
␣
t ␣ ⫽t 1 1 ¯t d d ,
␣
D ␣⫽ ⳵ 1 1¯ ⳵ d d,
h ␣ 共 x兲 ⫽h ␣ 1 共 x 1 兲 ¯h ␣ d 共 x d 兲 .
Also,
冉 冊
h n共 t 兲 ⫽ ⫺
d
dt
2
i
冑␦
M ␣共 y 0 兲 ,
1
␣!
兺i
qi
冉 冊
yi ⫺y0
冑␦
␣
.
(72)
3兲 共Taylor series兲 Use 共70兲 to evaluate 共68兲 as
兺
0⭐␣i⭐p
冉 冊
x⫺x0
冑␦
␣
L ␣共 x 0 兲 ,
(73)
where
L␤共x0兲⫽
1
␤!
兺i qi共⫺1兲兩␤兩h␤
冉 冊
x0 ⫺yi
冑␦
.
(74)
4兲 共Hermite⫹Taylor兲 Combine the above two methods to
evaluate 共68兲 as in 共73兲 with
L ␤共 x 0 兲 ⫽
冉 冊
x0 ⫺y0
共 ⫺1 兲 兩 ␤ 兩
M ␣共 y 0 兲 h ␣⫹␤
.
␤ ! 0⭐ ␣ i ⭐p
冑␦
兺
(75)
In order to evaluate 共68兲 for many points x i (i⫽1,...,M ),
Greengard and Strain 关119兴 introduce a scaling with which
all the points x i (i⫽1,...,M ) and y j ( j⫽1,...,N) are included in a unit cube. They then subdivide this cube into
small boxes whose side lengths are equal and of the order of
冑␦ . In evaluating the sum in 共68兲 for x i s in a certain box, one
may consider only a finite number of nearby boxes as the
source boxes, since contributions from farther boxes can be
neglected due to the exponential decay of the exponential
function in 共68兲. Greengard and Strain then set thresholds
M L and N F to the numbers of points x i and y j in target and
source boxes, respectively. If the number of source points N B
in a box B satisfies N B ⭓N F (N B ⬍N F ), then one sends out
Hermite expansion 共Gaussian兲 from B with y 0 set at the centroid of B. If the number of observation points M C in a box
C satisfies M C ⭓M L (M C ⬍M L ), one transforms the fields
sent to C into Taylor’s series setting x 0 at the centroid of C
共evaluates the fields sent to C immediately兲. These authors
concluded that the algorithm thus obtained can be executed
with O(M ⫹N) operations for a fixed error bound for the
expansion and the numbers ␦ and p. Using this approach,
Greengard and Strain 关119兴 could speed up a summation of
the form in 共68兲 involving about 100,000 source points 3000
times compared to a naive approach. Greengard and Strain
关119兴 also state that the 共continuous兲 single layer potential
冕
e ⫺t .
冉 冊
x⫺y0
M ␣共y0兲⫽
S0
n
G 共 x⫺y,t 兲 ␾ 共 y 兲 dS
could be evaluated effectively with 共71兲 as one replaces 共72兲
with
The following four methods are available for evaluating the
sum in 共68兲 共Greengard and Strain 关119兴. See Strain 关120兴 for
some corrections.兲:
1兲 共Gaussian兲 Use 共68兲 as it is.
2兲 共Hermite expansion兲 Use 共69兲 to evaluate 共68兲 as 共p: integer兲
兺 h␣
0⭐␣ ⭐p
where
(71)
1
M ␣共 y 0 兲 ⫽
␣!
冕冉
S0
yi ⫺y0
冑␦
冊
␣
␾ 共 y 兲 dS y .
(76)
This idea was further investigated and tested numerically by
Strain 关121兴. Unfortunately, the proposed approach is somewhat complicated including adaptive mesh refinement, and
the numerical test for single layer potential in 2D was concluded to be expensive but accurate.
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Appl Mech Rev vol 55, no 4, July 2002
In 关120兴 Strain extended the approach in 关119兴 to the case
where the number ␦ depends either on the source or target
locations. Greengard 关122兴 presents a simplified approach for
evaluating 共68兲 by using only the Hermite expansion, whose
complexity is also O(M ⫹N).
The mathematical tools used in the fast method for the
heat equation, presented above, are relatively simple, but do
not seem to have been tested much in engineering applications related to BIEM. An attempt of using tools for FMM in
BIEM for heat equations is found in Takahashi et al 关123兴,
where the authors used an expansion of the form 共75兲 in 2D
panel clustering BIEM. They use a hierarchical quad-tree
structure of boundary elements, whose cell size does not depend on time, thus leaving room for further improvement.
One advantage of this approach over the conventional BIEM
is that one does not have to store all the boundary data from
the past, since one may store just their moments instead.
5 APPLICATIONS IN
COMPUTATIONAL MECHANICS
5.1
Linear elasticity
5.1.1 Elastostatics
In this section we review fast methods applied to linear elasticity. We consider piecewise homogeneous and isotropic
materials, unless stated otherwise.
In 2D, Yamada and Hayami 关124兴 formulated a multipole BIEM based on the real variable formulation and the
standard direct BIEM. Their approach is close to panel clustering, since their formulation does not include downward
pass. Complex variables are used in an auxiliary manner to
formulate a multipole expansion of the fundamental solution.
They use four 共8兲 types of moments for the single 共double兲
layer potential. Since they considered rather small problems
with less than 800 unknowns, they ended up concluding
that the proposed approach is slower than the conventional
method in analyses using GCR or Bi-CGSTAB as the
solvers. In larger problems the conclusion might have been
different.
Greengard et al 关125兴 proposed an FMM based on the
complex variable formulation for elastostatic boundary value
problems. The Sherman-Lauricella integral equation is
solved with Nyström’s method and GMRES. Since the integral equation is essentially of the Cauchy integral type, the
method developed for Laplace’s equation in 2D can be applied with some modification. This paper includes numerical
examples of 100 inclusions.
Fukui et al 关126兴 presented an FMM formulation for elastostatics in 2D based on the real variable formulation and
the standard direct BIEM, using the complex variable in an
auxiliary manner. This approach is considered natural since
it uses two types of moments 共M n and N n in their notation兲,
as is expected from the number of unknown functions. This
paper includes numerical examples of the size of 105
unknowns.
Greengard and Helsing 关127兴 use Sherman’s complex
variable formulation to solve periodic inclusion problems in
2D. Their primal interest was to find a fast and robust nu2D.
Nishimura: Fast multipole accelerated BIEMs
317
merical method to obtain the elastic field and effective
moduli of composite materials. The periodicity is introduced
with the help of lattice sums. Their integral equation is a
singular equation of the Cauchy type. The accuracy of the
analysis is maintained with the help of an adaptive Gaussian
quadrature and a posteriori mesh refinement. This paper includes various numerical examples, the largest being of the
size of about 200,000 unknowns.
Helsing 关128兴 uses a complex variable formulation for
plane crack problems developed by Helsing and Peters 共see
references cited in 关128兴兲 to solve a periodic boundary value
problem in which the unit cell includes 10,000 randomly
oriented cracks. His integral equation is of Fredholm’s second kind with a singular kernel. He claims that the result is
accurate to nine digits.
Helsing and Jonsson 关129兴 present another complex variable formulation for stress and strain problems for a perforated plate. This formulation is simpler to use than the
classical Sherman-Lauricella equation, and is quite stable
numerically yielding good results even in problems with
closely spaced boundaries.
Peirce and Napier 关130兴 propose a spectral multipole
method for 2D linear elasticity. This approach is related to
the precorrected-FFT method proposed later by Phillips and
White 关99兴. Peirce and Napier use multipole expansion of the
kernel functions obtained by Taylor’s expansion 共with considerable simplification using biharmonic property of some
functions兲 to shift element integrals to grid points. With the
use of FFT and local correction, they could evaluate the
boundary integrals at the grid points, the interpolation of
which gives the values of the boundary integrals at collocation points. Both single and double layer potentials are considered. The use of multipole expansion provides a systematic approach to shift boundary integrals to grid points, but
the convolution using FFT has to be applied to each term of
the multipole expansion. This is in contrast to the precorrected-FFT where only the convolution of the kernel
function and equivalent source needs to be computed. The
complexity of the matrix vector product computation with
this approach is estimated to be O(N log N). These authors
present numerical examples of a punch applied to a granular
assembly whose intergranular boundaries are modeled as
cracks whose strength is governed by the Mohr-Coulomb
law. Richardson et al 关131兴 present a similar method in elastostatics in 2D using both ordinary and traction regularized
direct BIEs.
An FMM formulation for 2D anisotropic elastostatics is
included in the paper by Akaiwa et al 关132兴 which proposes
a simulation method for the Ostwald ripening. They use the
method proposed in 关35兴 for the simulation of the mass diffusion, and the FMM for anisotropic elastostatics is used in
evaluating potentials with known densities.
3D.
The first attempt at using fast methods in linear elasticity in 3D is found in Hayami and Sauter 关133,134兴, where
they give a sketch of a panel clustering formulation using
Taylor expansion and direct BIEM. They state that a fast
method for a potential having 兩 x⫺y兩 as the kernel will solve
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318
Nishimura: Fast multipole accelerated BIEMs
Appl Mech Rev vol 55, no 4, July 2002
elasticity problems in 3D as well. The complexity of such
formulation is estimated to be N log5 N in 关135兴.
In Hayami and Sauter 关136,137兴, they formulated a panel
clustering method based on an expansion of 兩 x⫺y兩 in terms
of spherical harmonics. They derived a multipole expansion
formula for the elastic single layer potential in terms of five
types of moments. Their investigations do not include numerical examples.
Fu et al 关138兴 developed an FMM formulation for linear
elasticity in 3D based on the observation that the elastostatic
fundamental solution ⌫ i j in 3D can be written as
冉 冊
冉 冊
1
yj
⫹Q i 共 x 兲
,
⌫ i j 共 x⫺y兲 ⫽ P i j 共 x 兲
兩 x⫺y兩
兩 x⫺y兩
(77)
where P i j (x) and Q i (x) are operators defined in terms of the
Lamé constants 共␭, ␮兲 by
冉
冊
P i j共 x 兲⫽
␭⫹3 ␮
1
␭⫹ ␮
␦i j⫺
x⳵ ,
8 ␲ ␮ ␭⫹2 ␮
␭⫹2 ␮ j i
Q i共 x 兲 ⫽
1 ␭⫹ ␮
⳵
8 ␲ ␮ ␭⫹2 ␮ i
S0
⌫ i j 共 x⫺y兲 ␾ j 共 y 兲 dS y
冕
冕
(1)2
M N,M
共 O 兲⫽
ជ 兲 ␾ j 共 y 兲 dS y
R N,M 共 Oy
(78)
y j R N,M 共 ជ
Oy 兲 ␾ j 共 y 兲 dS y
(79)
S0
S0
S0
C jknp
⳵
⌫ 共 x⫺y兲 n k 共 y 兲 ␸ j 共 y 兲 dS y
⳵ y n ip
⬁
(1)2
Notice that M (1)1
j,N,M has three components and M N,M is a
scalar for a given pair of (N,M ), hence one has four types of
moments in this formulation. In the same manner Fu et al
obtained a multipole expansion of elastostatic double layer
potential in terms of 12 types of moments. Their formulation
allows a black box use of FMM for Laplace’s equation. Indeed, they compute single and double layer potentials for
elastostatics by invoking FMM for Laplace’s equation 16
times. Fu et al 关138兴 presents numerical examples of many
spherical inclusions 共not periodic兲 embedded in an infinite
elastic space, the largest being of the size of about 400,000
unknowns. They do not present the final numerical results
because they performed only one matrix-vector product operation due to a limitation for computer access. Fu et al 关139兴
include discussion on the use of the new FMM in anisotropic
elastostatics.
Independently of Fu et al’s developments, Yoshida et al
关140兴 derived related, but different multipole expansions for
N
1
⫽
8 ␲ ␮ N⫽0
S
ជ 兲 M (2)1
共 F ip,N,M
共 Ox
兺 M兺
p,N,M 共 O 兲
⫽⫺N
S
(2)2
⫹G i,N,M
Ox 兲 M N,M
共ជ
共 O 兲兲 ,
(80)
where C i jkl ⫽␭ ␦ i j ␦ kl ⫹ ␮ ( ␦ ik ␦ jl ⫹ ␦ il ␦ jk ) is the elasticity tensor,
␭⫹3 ␮
ជ兲
␦ S 共 Ox
␭⫹2 ␮ i j N,M
⫺
S
Ox 兲 ⫽
G i,N,M
共ជ
(2)2
M N,M
共 O 兲⫽
␭⫹ ␮
ជ 兲 j ⳵ xi S N,M 共 Ox
ជ 兲,
共 Ox
␭⫹2 ␮
␭⫹ ␮ x
ជ 兲,
⳵ S 共 Ox
␭⫹2 ␮ i N,M
冕
冕
M (2)1
p,N,M 共 O 兲 ⫽
is given as a sum of single layer potentials for Laplacian with
densities given by ␾ j and ␾ j y j , postprocessed by P and Q,
respectively. Hence one may use the FMM for Laplace’s
equation to formulate an FMM for elastostatics. It is obvious
that the multipole expansion thus obtained is given in terms
of moments defined by 共See 共15兲兲
M (1)1
j,N,M 共 O 兲 ⫽
冕
ជ 兲⫽
F Si j,N,M 共 Ox
and ⳵ i is the partial differential operator with respect to x i . It
would be obvious from 共77兲 that the elastic single layer potential
冕
elastostatic potentials in 3D. They were primarily interested
in hypersingular integral equations for crack problems formulated with double layer potential. They tried both the 12
moment expression for the derivative of the double layer
potential obtained with regularization, and the four moment
expression for the same quantity derived from the following
multipole expansion for the double layer potential
S0
S0
C jknp
C jknp
⳵
ជ 兲 ␸ j n k dS y ,
R 共 Oy
⳵ y n N,M
(81)
(82)
(83)
⳵
ជ 兲 p R N,M 共 Oy
ជ 兲兲 ␸ j n k dS y .
共共 Oy
⳵yn
(84)
共See 共15兲兲 They found, in crack problems, that the four moment formulation without regularization is computationally
more efficient than the other with 12 moment expression
based on regularization. This result, however, is to be expected, since regularization is devised to deal with singularities while FMM is for evaluating far fields to which singularity has little relevance. Their approach does not allow the
black box use of FMM for Laplace’s equation, but the forms
of M2M, M2L, and L2L formulas in this formulation differ
only slightly from the counterparts in the Laplace case. This
paper includes numerical examples for crack problems having about 66,000 unknowns. Continuing efforts made in
关140兴, Yoshida et al 关87,141兴 developed a Galerkin formulation for the same integral equations for crack problems.
Thanks to improved accuracy of Galerkin’s method and reasonable computational loads for the double integration
achieved with the local expansion, they could solve large
problems with a small desk top computer. In 关87兴 for example, they present an example of 1000 interacting cracks
with about 470,000 unknowns, solved with GMRES. Takahashi et al 关142兴 utilized the same formulation as in 关140兴 to
solve ordinary 共not restricted to cracks兲 boundary value
problems with direct collocation BIEM. Since
ជ 兲⫽
P i j 共 x 兲 S N,M 共 Ox
1
ជ 兲,
FS
共 Ox
8 ␲ ␮ i j,N,M
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Appl Mech Rev vol 55, no 4, July 2002
Q i 共 x 兲 S N,M 共 ជ
Ox 兲 ⫽
1
GS
Ox 兲
共ជ
8 ␲ ␮ i,N,M
holds 共see 共77兲兲, the single layer potential also admits a mulS
.
tipole expansion in terms of functions F Si j,N,M and G i,N,M
Because the multipole moments are the coefficients of these
functions, one may combine the moments for single and
double layer potentials to obtain a four moment multipole
expansion for the general direct BIEM. Hence, the number of
the types of multiple moments is always four.
Fukui et al 关143兴 also obtained the same FMM formulation as in Yoshida et al 关87,140–142兴 for elastostatics using a
different derivation.
In Yoshida et al 关144兴 these authors modify the formulation presented in 关140兴 by using the new version of FMM
proposed by Greengard and Rokhlin 关90兴. They present numerical examples of many crack problems similar to the one
considered in 关87兴 solved with a desktop computer, with the
number of unknowns increased to 1.2 million.
Popov and Power 关145兴 presented an FMM using the Taylor series and the direct BIEM for elastostatics in 3D. Their
formulation is closely related to the one proposed earlier by
Hayami and Sauter 关133,134兴, but these authors succeeded in
actually implementing the formulation. Their paper includes
numerical examples of the size of N⫽O(104 ).
5.1.2 Elastodynamics
In 2D elastodynamics, Chen et al 关146兴 presented what
might be called a straightforward extension of their approach
in Helmholtz’ equation using the diagonal form. In this formulation, they work with 16 components of plane waves per
directions 共P and S wave components for the two components of single and double layer potentials with densities
having two components. This number 16 can be reduced to
eight since two of them always appear as a sum兲. This paper
includes a numerical example of the scattering from a rough
interface between two elastic materials obtained with a two
level RPFMA 关74兴.
Fukui 关147兴 proposed an FMM of the original type for
elastodynamics for low frequency using Galerkin’s vector.
This formulation uses four types of moments. This paper
includes examples of scattering of a plane wave by many
holes. Fujiwara 关148兴 also proposes an FMM of the original
type for elastodynamics for low frequency. This formulation
is in terms of eight types of moments. But this author considers cases where four of them are zero because of the
boundary condition. This paper includes examples of scattering by many cavities or cracks.
In 3D, Fujiwara 关149兴 presents a direct extension of the
diagonal form for Helmholtz’ equation to elastodynamics.
This formulation is in terms of 12 components of plane
waves per direction, and the multilevel implementation is
combined with the CGS 共Conjugate Gradients Squared兲
solver. The interest in this paper is directed to low frequency
problems related to earthquakes. The instability of the diagonal form in low frequency is avoided by taking less terms
共p in ␯ (q;•) in 共36兲兲 than required by the accuracy. He presents numerical examples of a concave halfspace 共topography model兲 subjected to an earthquake motion. He also pre-
Nishimura: Fast multipole accelerated BIEMs
319
sents an analysis of a basin model 共topography model with
the concave part filled with a different material兲, solved with
a special preconditioner.
Yoshida et al 关150兴 considered low frequency crack problems in 3D. A formulation of the original FMM type with
Wigner-3j symbols are used. They utilize the fact that the
fundamental solution of elastodynamics can be written as
Ui j⫽
1
4 ␲ ␮ k T2
冉
⳵ i⳵ j
e ik L r
e ik T r
⫹e ips e jqs ⳵ p ⳵ q
r
r
冊
(85)
modulo Dirac’s delta to show that the number of the type of
moments in isotropic elastodynamics is always four 共two in
2D兲, as in elastostatics, where k L and k T are wave numbers
of P and S waves. This number four is equal to the number of
components of scalar and vector potentials.
Takahashi et al 关151兴 present a fast time domain BIEM
for elastodynamics in 2D by combining the decomposition of
the fundamental solution similar to 共85兲 and the formulation
given in 关115兴. This paper presents numerical examples of
the size of O(104 ) spatial DOF and 100–240 time steps.
5.2 Stokes flow and other topics in fluid mechanics
The Stokes flow problem is considered to be the limiting
case of ␭→⬁ in linear elasticity. However, this problem differs from the elastic one in several aspects, and has been
investigated separately in the literature.
Greengard et al 关125兴 considered 2D problems using
complex variables. They have also shown equivalence of
their approach with the so called completed double layer
formulation for velocity 共Dirichlet兲 boundary value problems
for domains having holes. Their formulation requires introduction of additional logarithmic functions with unknown
multipliers, thus distorting the form of the matrix equation to
be solved. Greengard et al utilized a special preconditioner
to deal with this problem. These authors present numerical
examples for domains exterior to many holes, half plane
problems, etc, of the size of thousands of unknowns.
Gómez and Power 关152兴 proposed real variable approaches based on both direct and indirect BIEs. Their indirect BIEM uses the completed double layer formulation for
velocity 共Dirichlet兲 boundary value problems in 2D. The
name completed comes from the addition of terms which are
deficient in the conventional double layer formulation when
the domain has holes. They use an expansion of kernel functions with a real valued Taylor series. Their approach is not
the genuine FMM, but has some relevance to Hackbusch’s
panel clustering 关11兴 or to Barnes and Hut’s method 关10兴.
Indeed, their approach is an O(N log N) one, and does not
use the downward pass. These authors say that it was not
possible to formulate the downward pass in a way similar to
the counterpart in Laplace’s equation. Instead, they evaluate
the coefficients of the local expansion in the leaves using
M2L directly with cells of each level without using the tree
structure or L2L. They concluded that the indirect BIEM is
superior to the direct one in terms of convergence. They also
discuss parallel implementation of their approach 关153兴, and
present numerical examples of the size of several thousands
of unknowns.
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Nishimura: Fast multipole accelerated BIEMs
Mammoli and Ingber 关154兴 applied Gómez and Power’s
approach 关152兴 to the analysis of viscous flow containing
floating particles. They present examples of the size of
O(10,000) unknowns. These authors also considered parallelization of the same formulation in 关155兴, which includes
simulation results of 2D suspension flow with as many as
999 particles. Numerical results are seen to agree qualitatively with experimental results obtained with MRI.
Fu and Rodin 关156兴 present a formulation for 3D Stokes
flow as an extension of their approach in elastostatics. As in
the elastic case, their method invokes the standard FMM
routine for Laplace’s equation 16 times and postprocesses to
obtain layer potentials for Stokes flow. Their approach is an
O(N) one, and should have had no problem with the downward pass. They do not show numerical examples, though.
Takahashi et al 关157兴 proposed a four moment formulation for Stokes flow problems in 3D, based on their approach
in elastostatics in 3D. This original FMM approach shows no
problem with the downward pass, and the complexity of the
algorithm is O(N). They present numerical examples based
on the direct BIEM in which they did not see convergence
problems. The numerical examples include one with more
than 1,000,000 unknowns and applications to the determination of permeability constants in underground water flow
problems.
Zinchenko and Davis 关158兴 utilize a different FMM accelerated boundary integral formulation in their 3D analysis of
periodic deformable drops within a viscous medium. Both
drops and the medium are viscous fluids governed by Stokes’
equation. Their implementation is considered to be a single
stage FMM. The periodic kernel functions can be expressed
as a sum of the free space fundamental solution plus an additional term expressing the effect form outside the unit cell.
The multipole moments associated with the free space fundamental solution are called the near field multipole moments, and those associated with the other term of the kernel
function are named far field moments. It is easy to see that
the pair (u i ,p) satisfying
u i 共 x 兲 ⫽e ipq x q ␺ ,p ⫹ ␾ ,i ⫹
⫻
冉
1
共 2n⫹3 兲共 n⫹1 兲
共 n⫹3 兲
x p x p p ,i ⫺npx i
2
冊
gives a solution of the Stokes equation, where ␾, ␺ are harmonic functions and p is a homogeneous harmonic function
of order n. The solutions obtained by substituting appropriate singular 共regular兲 solid harmonics into ␾, ␺, and p in the
above expression are known as Lamb’s singular 共regular兲
forms. Their multipole expansion uses Lamb’s singular
in 共2兲, and their local expansion is made in
forms as k (1)
n
terms of Lamb’s regular forms. Hence, their formulation is in
terms of three types of near field moments, namely the coefficients of solid harmonics appearing in ␾, ␺, and p. They
could solve problems including 125 deformable drops with
each drop discretized into O(103 ) elements. One could also
mention the FMM formulation for Stokes’ flow by Sangani
Appl Mech Rev vol 55, no 4, July 2002
and Mo 关159兴, which is not based on an integral equation,
but is related to formulations by Cheng and Greengard 关50兴,
for example.
Ly et al 关160兴 consider an application of FMM 共the authors say that their approach is an O(N log N) one兲 to a simulation of particle dynamics in magnetorheological fluids in
2D. This analysis is aimed at simulating a smart controllable
material which may be transformed from liquid to solid
states under the action of a magnetic field. From an analytical point of view, this paper deals with essentially the same
integral equation as considered in 关49兴 except for the double
layer potential on the outer boundary of the domain.
The paper by Greengard and Kropinski 关161兴 deals with
the incompressible steady and unsteady Navier-Stokes equations in 2D in the special case of a circular domain. The
authors state that their approach for the steady state can be
extended to the general domain case with techniques developed in 关46,125兴.
See also references 关100,101兴, discussed in Section 3.4,
which deal with water wave problems.
5.3
Corrosion problems
Finally, we mention the paper by Nakayama et al 关162兴
which discusses corrosion problems. Their problems are formulated into a boundary value problem for Laplace’s equation with a boundary condition given in terms of a non-linear
relation between the potential and flux 共polarization curve兲
on the boundary. They use the original FMM for Laplace’s
equation in 3D and a modified Bi-CGSTAB which takes the
non-linearity of the boundary condition into consideration.
They apply their technique to a problem of corroded pipes in
the soil and conclude that their FMM results agree with another obtained with special pipe elements.
6
CONCLUDING REMARKS
1兲 We have reviewed efforts made so far to enhance the
performance of BIEM with the help of FMM and related
techniques. These methods combine fast methods of matrix
vector multiplication and iterative solvers for a matrix equation to yield O(N 1⫹ ␣ log␤ N) 共␣,␤⭓0, ␣⬍1. Typically, O(N)
in statics兲 solvers of integral equations.
2兲 As we have seen, the FMM accelerated BIEM is already
quite a mature technique in electrical engineering, although it
is still in an incipient stage in applied mechanics. Indeed,
even the FMM formulation in 3D elastostatics is not completely established yet. Some other areas of applied mechanics where further investigation of FMM is needed include
elastodynamics, heat equation, etc. Even in areas where
FMM formulations are already available, it would be necessary to accumulate further experiences by solving larger
problems.
3兲 We can mention the following as other areas of research
requiring further investigation:
• BIEM has been considered weak in problems where one
has to evaluate volume integrals. Such problems include
problems with inhomogeneity 共source terms兲, eigenvalue
problems, nonlinear problems, etc. With the new machinery for evaluating volume integrals of the Newton poten-
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Appl Mech Rev vol 55, no 4, July 2002
tial type provided by the FMM, the performance of BIEM
in these problems may improve. Investigations of FMM in
problems of this type, however, seem to remain open except in a few cases including Poisson’s equation. Use of
FMM with integral equations defined in a domain 共hence
not BIEs兲 is also a possibility.
• As Chew et al 关110兴 point out, the number of iterations
needed in achieving convergence remains unpredictable.
One may possibly remedy this problem numerically by
using good preconditioners. Alternatively, one may possibly improve the conditioning of the linear equation by using better integral equations 共See Sections 3.1.3 and 5.1.1
for examples of such attempts兲. Investigations along these
lines are considered important.
• FMM is not the only fast method for integral equations
which uses the tree data structure of elements. Indeed,
methods of the Barnes and Hut type use quad- and oct-tree
structures of elements, as we have seen. In astrophysics,
however, various treecodes based on different tree data
structures are in use 共See Anderson 关31兴 for example兲. Use
of data structures other than the spatially balanced quadand oct-trees in integral equations can also be an interesting research subject.
4兲 In this article, we did not discuss numerical issues such as
estimation of the errors, convergence, etc. As a matter of
fact, estimation of errors introduced by truncating various
series in the FMM for Laplace’s equation has been a concern
since the early paper by Rokhlin 关2兴. The panel clustering
community has also been investigating this issue since the
Hackbusch and Nowak 关11兴 paper. Related estimates in
Helmholtz’ equation are found in Rahola 关163兴 共3D兲, Song
et al 关69兴 共3D兲, Koc et al 关164兴 共3D兲, Amini and Profit 关165兴
共2D兲, etc. Labreuche 关166兴 addresses convergence of FMM
in Helmholtz’ equation in 2D.
5兲 Parallel implementation of FMM is another big issue
which we did not discuss very much in this article. One finds
many publications on this subject in particle simulations, but
we here cite the paper by Grama et al 关167兴 which considers
specifically the parallelization of the Barnes and Hut method
applied to BIEM. The papers by Gómez and Power 关153兴,
Mammoli and Ingber 关155兴, and Fu et al 关156兴 also include
discussions on parallel implementations.
6兲 There are innumerable web sites providing information
on FMM. It would be impossible to list all of them, but it
should not be very difficult to find them with the help of
search engines. We here present just a few URLs containing
source codes 关168 –170兴, and bibliography on FMM
关171,172兴.
7兲 Finally, we cite an important reference book on FMM by
Chew et al 关173兴, which became available to present author
recently.
ACKNOWLEDGMENT
The author wishes to express his gratitude to Profs/Drs B
Alpert, L Greengard, K Hayami, J Helsing, GJ Rodin, SA
Sauter, J Strain, S Wandzura, and J White for providing him
with information on existing publications, and to Prof WC
Nishimura: Fast multipole accelerated BIEMs
321
Chew for permitting the use and providing the original of
Fig. 4. Finally, the author would like to thank Drs K Yoshida
of Kyoko University and T Takahashi of the Institute of
Physical and Chemical Research for their help in preparing
the manuscript.
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N Nishimura has been an associate professor of civil engineering at Kyoto University since 2002.
He received the B Eng, M Eng, and Dr Eng degrees in civil engineering from Kyoto University in
1977, 1979, and 1988. He joined the faculty of the Department of Civil Engineering at Kyoto
University in 1979, moved to the Department of Global Environment Engineering in 1991, and
returned to the Department of Civil Engineering in 2002. His collaborations with other institutions
include his stays in the Department of Civil Engineering at Northwestern University during 1983–
1985 and in Centre de Mathématiques Appliquées at Ecole Polytechnique in 1988–1989. He conducts research on fundamentals and applications of boundary integral equation methods in continuum mechanics with special attention to wave propagation, fracture, inverse, and large scale
problems. He has over 100 publications in archival journals and conference proceedings, and is an
author of chapters in 15 books.
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