fast multipole boundary element method of potential …€¦ · · 2016-01-09fast multipole...

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

108 JOURNAL OF NETWORKS, VOL. 9, NO. 1, JANUARY 2014

© 2014 ACADEMY PUBLISHERdoi:10.4304/jnw.9.1.108-114

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

JOURNAL OF NETWORKS, VOL. 9, NO. 1, JANUARY 2014 109

© 2014 ACADEMY PUBLISHER

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)



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.



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



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



Qinggong College Hebei United University (Grant No.

qz201205), all support is gratefully acknowledged.

REFERENCES

[1] Shen Guangxian, Xiao Hong, Chen Yiming. Boundary element method. Beijing: Mechanical industry press, 1998 pp. 26-129.

[2] Rokhlin V. Rapid solution of integral equations of classical potential theory. Journal of Computational Physics, 1985, 60(2) pp. 187-207.

[3] Yoshida K. Application s of fast multipole method to boundary integral equation method. Kyoto: Kyoto University, 2001.

[4] Chew W C, Chao H Y, Cui T J, et al. Fast integral equation solvers in computational electromagnetics of complex structures. Engineering Analysis with Boundary Element, 2003, 27(8) pp. 803-823.

[5] Liu Y J. A new fast multipole boundary element method for solving large-scale two-dimensional elastostatic problems. International Journal for Numerical Methods in Engineering, 2006, 65(6) pp. 863-881.

[6] Li Huijian, Shen Guangxian, Liu Deyi. Fretting damage

mechanism occurred sleeve of oil-film bearing in rolling mill and multipole boundary element method. Chinese Journal of Mechanical Engineering, 2007, 43(1) pp. 95-99.

[7] Nishimura N, Yoshida K, Kobayashi S. A fast multipole boundary integral equation method for crack problems in 3D. Engineering Analysis with Boundary Element, 1999, 23 pp. 97-105.

[8] Yoshida K, Nishimura K, Kobayashi S. Application of new fast multipole boundary integral equation method for crack problems in 3D. Engineering Analysis with Boundary Element, 2001, 25 pp. 239-247.

[9] Shen Guangxian, Liu Deyi, Yu Chunxiao. Fast multipole boundary element method and Rolling engineering. Beijing: Science press, 2005 pp. 81-86, 185-193.

[10] Wang Haitao, Yao Zhenhan. A new fast multipole boundary element method for large scale analysis of mechanical properties in 3D particle-reinforced composites. Computer Modeling in Engineering & Sciences, 2005, 7(1) pp. 85-95.

[11] Ling Nengxiang. Convergence rate of E·B estimation for location parameter function of one-side truncated fami1y under NA samples. China Quartly Journal of Mathmatics,

2003, 18(4) pp. 400~405.

[12] WU Chengjun, CHEN Hualing, HUANG Xieqing. Theoretical prediction of sound radiation from a heavy

fluid-loaded cylindrical coated shell. Chinese Journal of Mechanical Engineering, 2008, 21(3) pp. 26-30.

[13] LIU Y J. A new fast multipole boundary element method for solving large-scale two-dimensional elastostatic problems. International Journal for Numerical Methods in Engineering, 2006, 65(6) pp. 863-881.

[14] Chen K, Harris P J. Efficient preconditioners for iterative solution of the boundary element equations for the three-dimensional Helmholtz equation. Applied Numerical Mathematics, 2001, 36(4) pp. 475-489.

[15] Cui Yuhuan, Qu Jingguo, Chen Yiming, Yang Aimin.

Boundary element method for solving a kind of biharmonic equation, Mathematica Numerica Sinica, 34(1), pp. 49-56(2012. 2)

[16] Yuhuan cui, Jingguo Qu, Yamian Peng and Qiuna Zhang. Wavelet boundary element method for numerical solution of Laplace equation, International Conference on Green Communications and Networks, 2011/1/15-2011/1/17

[17] Lei Ting, Yao Zhenhan, Wang Haitao. Parallel computation of 2D elastic solid using fast multipole boundary element method. Engineering Mechanics, 2004, 21(Supp.) pp. 305~308.

[18] Lei Ting, Yao Zhenhan, Wang Haitao. The comparision of

parallel computation between fast multipole and conventional BEM on PC CLUSTER. Engineering Mechanics, 2006, 23(11) pp. 28-32

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.



fast multipole boundary element method of potential …€¦ · · 2016-01-09fast multipole...

Documents