Fast Multipole Boundary Element Method of
Potential Problems
Yuhuan Cui and Jingguo Qu Qinggong College, Heibei United University, Tangshan, China
Email: [email protected]
Aimin Yang and Yamian Peng College of Science, Heibei United University, Tangshan, China
Abstract—In order to overcome the difficulties of low
computational efficiency and high memory requirement in
the conventional boundary element method for solving
large-scale potential problems, a fast multipole boundary
element method for the problems of Laplace equation is
presented. through the multipole expansion and local
expansion for the basic solution of the kernel function of the
Laplace equation, we get the boundary integral equation of
Laplace equation with the fast multipole boundary element
method; and gives the calculating program of the fast
multipole boundary element method and processing
technology; finally, a numerical example is given to verify
the accuracy and high efficiency of the fast multipole
boundary element method.
Index Terms—Boundary Element Method; Fast Multipole
Methods; Fast Multipole Boundary Element Method;
Potential Problems; Laplace Equation
I. INTRODUCTION
The boundary element method (BEM) [1] is a
numerical method for solving the field problem based on
the boundary integral equation, but there is very limited
in the solving large scale. The conventional boundary element is difficult to deal with large-scale computing
problems in engineering. This is because the coefficient
matrix of the linear algebraic equations is a full matrix
formatted by the boundary element method, but also
shows the properties of asymmetric in the treatment of
some special problems. The matrix operation requires a
large amount of computer resources, such as direct Gauss
elimination method requires O(N2) storage and
O(N3)CPU time, N is the degree of freedom. Therefore,
computing power has become a bottleneck restricting the
development and the application of the boundary element
method. From the late 1970s to now, the boundary element
method has been applied to the fluid mechanics, wave
theory, electromagnetism, and heat conduction problems
and unsteady issues problem of composite materials
axisymmetric and so on. In recent years, the boundary
element method began to be used during material
processing, in order to obtain numerical solutions. There
are more and more engineering examples, in the engineering examples we use boundary element method
to solve nonlinear problems and dynamic problems. For
potential non-linear problems, such as proliferation issues
we can do some transformation, so that control
differential equations is linear, and solving the problem of
heterogeneity.
In 1985, ROKHLIN [2] first put forward the fast
multipole algorithm (Fast Multipole Methods) (FMM),
Amount of the potential problem calculation for N particles interact with each other is reduced to O(N). The
essence of fast multipole algorithm is that multipole
expansion of node clusters to approximate shows
boundary integral of kernel function and boundary
variable product, the amount of calculation and storage is
reduced from the original O(N2) to O(N). This algorithm
is little demand for computer memory, and with the
expansion of problems, the increased memory demand is also slow. it create a sufficient condition for the computer
to large-scale operations. Computing efficiency, reduced
memory usage and high accuracy will greatly strengthen
the advantages of boundary element method and expand
the application range of the boundary element method.
this is a breakthrough. Therefore, the fast multipole
algorithm is suitable for large-scale computing problems.
Using FMM to accelerate the process of solving algebraic equations in boundary element, based on
iterative algorithm, using fast algorithm of FMM and
recursive operations of product tree structure to replace
the matrix and the unknown vector algebra equations, it is
no need to form the dominant. it effectively overcomes
the disadvantages of traditional boundary element
calculation, so it is suitable for solving large-scale
problems. In recent years, research of FMM is used to solve the
acceleration of the traditional boundary element method
[3-6], namely the establishment of the fast multipole
boundary element method (Fast multipole boundary
element method, FM-BEM), it is successful
implementation of large-scale complex engineering
problems on a personal computer for million degrees of
freedom, such as electromagnetism problem [4], mechanics [5-6]. For example, Nishimura N and Yoshida
K of University of Tokyo in Japan used to solve three-
dimensional fracture problems [7-8]; research group of
Yanshan University professor Shen Guangxian apply
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multipole boundary element to mechanical engineering
elasto-plastic contact problems [9]. In mechanical
engineering, computational mechanics, computational
mathematics and other fields, FM-BEM has high
efficiency for numerical calculation, and it has very broad
application prospects. This new concept of "Fast
multipole BEM" has generated, it is bound with the "finite difference method," "finite element method",
"boundary element method", as an important numerical
analysis in 21st century, and will be further development
and promotion.
Based on the fundamental solution of Laplace equation,
the multipole boundary element should be used for 2D,
3D potential problems, the boundary integral equations of
potential problems and the basic solution is presented, then gives the basic solution of FM-BEM; then gives the
calculation program of FM-BEM and processing
technology; finally, numerical calculation example verify
the accuracy of FM-BEM, indicating that the FM-BEM
computational efficiency compared with the traditional
BEM has the order of magnitude improvement, it can
effectively solve large-scale complex problems. This
paper belongs to computational mathematics, boundary element method, potential problems, elasticity problems,
and rolling theoretical research. the research is
interdisciplinary with significant academic and practical
significance, and it has broad application prospects in
engineering.
II. BOUNDARY INTEGRAL EQUATIONS OF POTENTIAL
PROBLEMS
In the 3D domain , control equations and boundary
conditions for the potential u and potential gradient q :
2 0u (in ) (1)
Boundary conditions:
The basic boundary conditions: u u (in u )
The natural boundary conditions:
u
q qn
(in q ) (2)
2 is the Laplace operator; u is called the potential,
and it is usually said temperature, concentration, pressure,
potential in specific issues. Along the boundary q is the
normal derivative of u , a source body; u q ,
u is the given boundary of potential (known as the
essential boundary conditions), q is the gradient of
potential for the given boundary (known as natural
boundary condition), n is outside the normal of
boundaries , as shown in figure 1.
For complex boundary conditions, use the combination
of the above two parameters to be marked as
u q (3)
In the formula, and are the correlation
coefficients. The boundary integral equation form of
formula (2. 1)
*
*
( ) ( , ) ( )
( , ) ( )
i ic u x q x y u y d
u x y q y d
(4)
Among them, x is source point, y is the arbitrary
boundary point on the boundary , ic shape coefficient, *( , )u x y and *( , )q x y are the basic solutions of three-
dimensional potential problems, usually expressed as
* 1( , )
4u x y
R (5)
*
* ( , )( , )
u x yq x y
n
(6)
Among them, R is distance between the source point
and observation point, n is the outside normal vector of
boundary .
The solution domain boundary is divided into
boundary element, discrete boundary integral equations
form linear algebraic equations, Introducing boundary
conditions, rearrange the equations to format the final
equation:
AX B (7)
A is a symmetric matrix; X is an unknown vector,
B is a known variable. Solving the equation (7), we can
obtain the boundary unknown variable.
When using the fast multipole algorithm, all the
elements of the coefficient matrix A do not need to be
calculated. For a fixed source, the contribution of the unit
far away from the source to the source point, we can use
fast multipole algorithm through the steps of polymerization, transfer, configuration to achieve. Only a
small number of units adjacent to the source point, we
should use the conventional boundary element to
calculate.
Figure 1. Schematic diagram of two-dimensional ordinary potential
problems
III. THE BASIC SOLUTION OF FM-BEM
Multipole boundary element method can be set up, and
the fundamental reason is that FMM can be applied to
rapid calculating of BEM remote effects coefficient.
Cross point lies in the basic solution decomposition.
Through the research, we can discover that the formula of
FMM to calculate the interaction potential in the set of a large number of particles can be abstracted as
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mathematical formula i
i j ij
c
R
(ijR is distance between
any two different particle charge, ic is constant of
particle charge electric), and the use of BEM to solve the
potential problems of the boundary integral equation of
discrete occasions, can be abstracted as mathematical
formula i
i j ij
c
R
(ijR is distance between source and
integral point, constant ic is the influence coefficient
between the source and the integral point) and its
derivatives form. In BEM the core part of the boundary
integral equation and its discrete form is the basic
solution and related function, therefore, FMM will be
applied to BEM and the key for the establishment of FM-
BEM is that derive for BEM solutions and related form of
kernel function for FMM, get formula related to FM-BEM.
Here, the corresponding FM-BEM solutions and
related kernel function formula is derived for the
potential problem, also derive first-order derivatives
formula of i
i j ij
c
R
and the corresponding Cartesian
coordinate calculation formula in spherical coordinates
A. The Basic Solution of Cartesian Coordinates
At the point qx , formula (2. 4) numerical integrals
, ,
, ,
( ) * , ( ) ( ) ( )
* , ( ) ( ) ( ) 0
i i q q s kl l s s s
k l s
q s kl l s s s
k l s
c u x q x y u J y
u x y u J y
(8)
Among them, s is the integral point of unit, s is
integral weight function at s , ( )J y is the Jacobi
determinant.
By formula (8), the operation ,
( ) ( )l s s s
l s
J y
for every unit area, the integral at all boundary points is
invariant. If solving the equation by iteration method, klu
and klq are assignment before the iteration, then in each
iteration step, the product term , ,
( ) ( )kl l s s s
k l s
u J y
of each unit is a fixed value, similarly, the product
, ,
( ) ( )kl l s s s
k l s
q J y also has the same properties.
The first sum term of the formula (8), Let
, ,
( ) ( )s kl l s s s
k
k l s
C u J y
For * , ( )q sq x y , If it can be written as the function
1 R , i. e.
* , ( ) ( ) 1q s qq x y f x R
Among them, qx is the source point, q
cR x y is
the distance between source point and the multipole
center. Then first sum term of the formula (3.1) can be
written as the follows,
*
, ,
,
,
,
, ( ) ( ) ( )
( )(1 )
( )( )
( )
q s kl l s s s
k l s
q s
k
k s
q s
k
k s
q s
k
k s
q x y u J y
f x R C
f x C R
f x C R
(9)
Obviously, the sum term in formula (9) can be
computed quickly by FMM. The second sum term in
formula (8) the function 1 R , without the need for
decomposition, it can fast calculation by FMM. So, FMM
can be used to solve the boundary integral equation with
potential problems.
In order to make the fundamental solution *( , )q x y
adapt to the the multipole expansion method, *( , )q x y
will be decomposed into
**
1 1 2 2 3 3
( , ) 1( , ) =
4
1 1 1 1=-
4
u x yq x y
n n R
n n nR R R
(10)
So, formula (8) can be rewritten as
, ,
, ,
1( )
4
1( ) ( ) ( )
1 1( ) ( ) 0
4
i i q
m
kl l s s s s
mqk l s c
kl l s s s
qk l s c
c u x
u n y J yx y
q J yx y
(11)
Because 1,2,3m , so the first sum term of the
formula (8) call for multipole expansion 3 times, the
second sum term of the formula (8) call for multipole
expansion 1 times, for a total of 4 times.
x is column vector construct by the unknown potential
and its derivatives.
B. The Basic Solution of Spherical Coordinate System.
FM-BEM theory is based on spherical coordinate
system, therefore, we discuss basic solution form in the spherical coordinates. The core problem for the FMM
applied to BEM is 1
ijr and its derivatives. In the
specific implementation process, mainly manifests the
form of partial derivatives for first order, two order in the
local expansion
0
( ) ( , )j
k j
j jk
j k j
X L Y r
(12)
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In the formula (12), k
jL is the expansion coefficients
for the local, which is equivalent to the constant, so the
problem is derivative for ( , ) j
jkY r . Let
( , )
2 1 ( )!(cos )
4 ( )!
n
nm
m im n
n
V Y r
n n mP e r
n m
2 1 ( )!
4 ( )!
m
n
n n mC
n m
(13)
After the strict derivation, and obtained the following
results.
A fundamental solution for the potential problem
relates to first order partial derivative of function 1/ R, in
the first order partial derivatives, we first solve the
gradient, and converts it into a rectangular coordinate form.
Firstly, solve the gradient for V
( ) ( , , ) , ,sin
r
V V Vgrda V v v v
r r r
1 ( , )n
r nmv nr Y 1 sinn m im
nv r C e
1 12 1cos (sin ) (cos ) sin (cos )m m
n nm P P
-11 sin (cos ) sin( ),cos( )
mn m
n nv r C P m m m
The rectangular coordinate form of the first order partial
derivatives: ( ) ( 1,2,3)i iV v i , in the Cartesian
coordinate system, the first order partial derivative for V
respectively as follows.
1 1( ) sin cos cos cos sinrV v v v v
2 2( ) sin sin cos sin cosrV v v v v
3 3( ) cos sinrV v v v
IV. THE BASIC PRINCIPLE AND PROCESS OF FM-BEM
Using the traditional boundary element method to solve, such as direct method and iterative method, the
coefficient matrix of formula (7) needs to be stored and
arithmetic formula. The basic principle of the fast
multipole boundary element method is:
1) Using adaptive tree structure instead of the
traditional matrix, A does not need to be explicitly stored,
the information is hidden in a tree structure;
2) Based on the iterative method, the multiplication between coefficient matrix A and iteration vector X is
instead by multipole expansion of basic solution
approximation and tree structure in iterative process;
3) Computing and storage capacity of tree structure is
O (N). Therefore if the iteration can converge rapidly, the
amount of computation and storage of fast multipole
boundary element method are approaching O(N).
The multipole expansion method and adaptive tree structure for basic solution of fast multipole boundary
element method potential problems are explained below.
The fast multipole algorithm involves 4 steps key
operation (Fig. 2) the multipole expansion, conversion
between multipole moments (M2M), multipole moments
to local moments (M2L) and the conversion between
local moment (L2L).
y
1y
T
2M L
0y
2M MB
A
0xC
1xx
2L L
Figure 2. Key operation for fast multipole algorithm
A. Creating Adaptive Tree Structure
The operation mode of adaptive tree structure for fast
multipole boundary element method includes the
following steps:
Figure 3. Three-dimensional tree structure
1) generating tree structure. Based on the 3D tree
structure as an example, the solution domain is contained
in a cube, on behalf of the root node of the tree structure;
a large cube is decomposed into 8 sub - cube; cube further decomposition into smaller cubes, if the number
of boundary element contained in cube less than a
predetermined value, then stop decomposition.
2) upward traversal tree structure. The steps is used to
calculate the multipole expansion coefficient of boundary
element method for each leaf node.
3) downward traversal tree structure. The multipole
expansion using the steps to calculate the coefficient of local expansion coefficient, interested readers see article
[10].
4) using the tree structure to calculate the integral.
5) the iterative calculation. In each iteration step, by
the tree structure to complete the operational equivalence
matrix - vector product. Figure 3 is an example of 3D tree
structure.
The fast multipole boundary element method use the storage structure of tree, each tree node includes adjacent
boundary element. Generally speaking, the tree structure
is the data structure of nonlinear and unbalanced, the task
division of the tree structure and communication
operations between the division tasks is complex than
conventional matrix [11], this is the the main difficulties
that fast multipole boundary element parallel format is
different from the conventional boundary element.
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The whole area is divided into NP task, where NP is
usually a number of required processors for parallel
calculation, division of the unit is a node of tree structure,
division of the goal is to ensure each task contains a
roughly the same number of units, but the classification
method is that firstly sort boundary nodes on the tree
structure and the unit in the boundary element method in accordance with the same spatial order, Then divided
according to node, figure 4 shows the automatic
generation diagram that of 3D problem into 4 task.
Task0
Task1
Task2
Task3
Figure 4. Task partition scheme of the tree structure
B. Calculation Procedures for Fast Multipole Boundary
Element Method
The literature [11, 12] detailed description of the fast
multipole boundary element method algorithm procedures, basic calculation process of this algorithm are
as follows.
(1) the model boundary discrete unit by the traditional
boundary element method.
(2) spanning tree structure. Contains all the boundary
element model with a cube, on behalf of the root node of
the tree structure; a large cube is decomposed into eight sub - cube; cube further decomposition into smaller cubes,
if the number of boundary element contained in cube less
than a predetermined value, then stop decomposition.
This set contains the hierarchical tree structure of all
boundary element, and it is used to store the multipole
expansion coefficients and local expansion coefficient.
(3) Calculate multipole expansion coefficient with
upward traversal. The steps is used to calculate the multipole expansion coefficient of boundary element
method for each leaf node, start from the leaf nodes of the
tree structure, recursive computation form a layer to a
layer, until the root node.
(4) Calculate the local expansion coefficient with
downward traversal. The multipole expansion using the
steps to calculate the coefficient of local expansion
coefficient, start from the root node of the tree structure, down recursive computation form a layer to a layer, until
a leaf node.
(5) Calculate the integral using the tree structure.
(6) Iterative solution. In each iteration step, product of
equation coefficient matrix and the unknown vector
equivalence complete by the tree structure. The iterative
process is repeated until the solution of the unknown
variable converges to the reasonable accuracy, then the process ends, and we get the solution of the problem.
C. The Pretreatment Technology of Fast Multipole
Boundary Element Method
Sometimes state of the coefficient matrix A formed by boundary element methods is not good, the coefficient
matrix that the state is not good will lead to iterative
convergence inefficient, even fail to converge, so the
pretreatment of the coefficient matrix is crucial. The fast
multipole boundary element algorithm uses the GMRES
solver after pretreatment proposed by CHEN [13] in this
paper, and combined with block diagonalization
pretreatment technology for iterative solution of linear systems equations. this pretreatment technology has
advantages, namely the generalized minimum residual
method solver after pretreatment can store the used
coefficients, and can be directly reused in the calculation
of the near field contributions, so we do not have to
repeat the computation of these coefficients in each
iteration step, and it can accelerate the speed of iteration
algorithm and improve the convergence efficiency. For large scale problems of N degrees of freedom, if the
truncated term number p of multipole expansion
boundary element method and number of units in each
leaf node keep the set for a constant, the computational
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complexity of the whole process of solving the fast
multipole boundary element method is O (N).
V. NUMERICAL RESULTS
Examples 1: Laplace equation 2 2
2
2 20
u uu
x y
, ,x y .
Set is a square area:
( , ) 0 6,0 6x y x y
The following four straight line segments consists of
the boundary
( , ) 0,6,0 6; 0,6,0 6x y x y y x
The known boundary conditions:
300 ( , ) 0,0 6( , )
0 ( , ) 6,0 6
x y x yu x y
x y x y
( , ) 0, ( , ) 0,6,0 6u
x y x y y xn
Solve the potential on the boundary point and domain
point value.
Results solved by the traditional BEM and FM-BEM
by two methods are given below (see table 1).
TABLE I. POTENTIAL VALUE OF POINT ( , )x y
coordinates ( , )x y BEM FM-BEM Exact solutions
(1, 0) 252. 25 250. 02 250. 00
(2, 4) 200. 28 199. 98 200. 00
(3, 6) 150. 02 150. 01 150. 00
(4, 2) 99. 74 100. 01 100. 00
(5, 0) 47. 75 49. 99 50. 00
(5, 6) 47. 75 50. 02 50. 00
Seen from table 4.1, using FM-BEM to solve the
potential problems, solutions are more accurate. Below
we through examples show FM-BEM efficient. Example 2: cube regional heat conduction. Cube side
length is 2m, as shown in figure 4.1. the lower bottom
surface temperature is 100 ℃, the upper bottom surface
temperature is 0 ℃, the normal heat flux of other 4 sides
is 0. Divided into 24 by 8 node quadrilateral quadratic
unit, every square surface is divided into 4 units.
Calculate interior point potential of angular point A
(temperature).
Figure 5. Heat conduction in a cub
As you can see from Figure 3, when the number of
degrees of freedom 2000N , the calculation results of
FM-BEM and BEM compared with the analytical
solution, the relative error is 0.2% , with the number
of degrees of freedom increases, further reduced, thus,
the truncation error introduced by multipole expansion
and local expansion is very small. Figure 3clearly show
that the accuracy of FM-BEM and BEM, are equivalent,
demonstrated that the fast multipole boundary element
method in this paper has high accuracy, meet the
requirements of engineering calculations.
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
DEGREE of FREEDOM N 103
The
rela
tive
erro
r
Conventional BEM
Fast Muitiople BEM
Figure 6. Calculation accuracy of FM - BEM and BEM
As you can see from figure 4, when the free degree
reached 1000, computing speed of FM-BEM is faster
than BEM; as the number of degrees of freedom increases,
computational advantage of FM-BEM fully reflected, the
calculation speed is much higher than that of BEM, effectiveness demonstrated that the computational
efficiency of the FM-BEM is efficient.
2 4 6 8 10
200
400
600
800
1000
1200
Conventional BEM
Fast Muitiople BEM
Freedom degree N/ 310
Figure 7. Calculation efficiency of FM - BEM and BEM
VI. CONCLUSIONS
In this paper, the fast multipole boundary element
method is applied to solve the 3D potential problems, and
do the numerical calculation. In the premise of ensuring
high calculation accuracy, compared computation time
and memory requirement with the conventional boundary
element method. Numerical examples show that, the fast multipole boundary element method has the advantages
of accuracy and efficiency, it is suitable to large scale
numerical computing in the engineering.
ACKNOWLEDGMENT
This research was supported by the National Nature
Foundation of China (Grant No. 61170317), the National
Science Foundation for Hebei Province (Grant No.
A2012209043) and Natural science foundation of
JOURNAL OF NETWORKS, VOL. 9, NO. 1, JANUARY 2014 113
© 2014 ACADEMY PUBLISHER
Qinggong College Hebei United University (Grant No.
qz201205), all support is gratefully acknowledged.
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Yuhuan Cui Female, born in 1981, Master degree candidate. Now he acts as the Math teacher in Qinggong College, Hebei United University. She graduated from Yanshan University, majoring computational mathematics. Her research directions include mathematical modeling and computer simulation, the
design and analysis of parallel computation, elastic problems numerical simulation, and etc.
Jingguo Qu Male, born in 1981, Master degree candidate. Now he acts as the Math teacher in Qinggong College, Hebei United University. He graduated from Harbin University of Scince and Technology, majoring fundamental mathematics. His research directions include mathematical modeling and computer simulation, the design and analysis of parallel computation, elastic problems numerical simulation, and etc.
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