Fast Multipole Boundary Element Method of ... - Semantic Scholar
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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 © 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.1.108-114
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 threedimensional 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 : 2u 0 (in )
(1)
Boundary conditions: The basic boundary conditions: u u (in u ) The natural boundary conditions: q
u q (in q ) n
ci u i ( 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 , c i shape coefficient, u* ( x, y) and q* ( x, y) are the basic solutions of threedimensional potential problems, usually expressed as 1 4 R
(5)
u* ( x, y ) n
(6)
u* ( x, y)
q* ( x, y)
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.
(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) © 2014 ACADEMY PUBLISHER
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
ci
R i j
( Rij is distance between
center. Then first sum term of the formula (3.1) can be written as the follows,
ij
q
any two different particle charge, ci 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 c formula i ( Rij is distance between source and R i j ij integral point, constant ci 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 FMBEM is that derive for BEM solutions and related form of kernel function for FMM, get formula related to FMBEM. Here, the corresponding FM-BEM solutions and related kernel function formula is derived for the potential problem, also derive first-order derivatives c formula of i and the corresponding Cartesian R i j ij coordinate calculation formula in spherical coordinates
k ,s
c u ( x ) q * x , y ( ) u ( ) J y( ) q
q
s
kl
l
s
s
k ,l , s
u * x q , y( s ) u kl l ( s ) J y( s ) s 0
k ,s
f ( x q ) Cks R k ,s
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 =4
(8)
l ,s
for every unit area, the integral at all boundary points is invariant. If solving the equation by iteration method, u kl and q kl are assignment before the iteration, then in each k ,l , s
kl
l ( s ) J y( s ) s
of each unit is a fixed value, similarly, the product qkl l ( s ) J y( s ) s also has the same properties. k ,l , s
The first sum term of the formula (8), Let Cks u kl l ( s ) J y( s ) s k ,l , s
For q x q , y( s ) , If it can be written as the function 1 R , i. e. *
q* x q , y( s ) f ( x q ) 1 R
Among them, x q is the source point, R xq yc is the distance between source point and the multipole
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1 1 1 1 R n1 2 R n2 3 R n3
(10)
So, formula (8) can be rewritten as
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 ) J y( s ) s
u
u* ( x, y ) 1 = n n 4 R
q * ( x, y )
s
k ,l , s
iteration step, the product term
(9)
f ( x q )(Cks R)
At the point x q , formula (2. 4) numerical integrals i
x q , y ( s ) u kl l ( s ) J y ( s ) s
f ( x q )(1 R)Cks
A. The Basic Solution of Cartesian Coordinates
i
*
k ,l , s
ci u i ( x q )
k ,l , s
1 m 4
u (
1
kl
x yc
1 4
q
k ,l , s
1 x yc q
l
s
)nm y ( s ) J y ( s ) s
q ( kl
l
s
(11)
) J y ( s ) s 0
Because m 1, 2,3 , 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 1 applied to BEM is and its derivatives. In the rij specific implementation process, mainly manifests the form of partial derivatives for first order, two order in the local expansion
j
( X ) Lkj Y jk ( , )r j j 0 k j
(12)
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In the formula (12), Lkj is the expansion coefficients for the local, which is equivalent to the constant, so the problem is derivative for Yjk ( , )r j . Let V Ynm ( , )r n
between multipole moments (M2M), multipole moments to local moments (M2L) and the conversion between local moment (L2L). A
B
2n 1 (n m)! m Pn (cos )eim r n 4 (n m)! Cnm
y y0
M 2M
y1
T M 2L
x
x1 L2L
2n 1 (n m)! 4 (n m)!
(13)
C
x0
Figure 2. Key operation for fast multipole algorithm
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 V V V grda(V ) (vr , v , v ) , , r r r sin
A. Creating Adaptive Tree Structure The operation mode of adaptive tree structure for fast multipole boundary element method includes the following steps:
vr nr n 1Ynm ( , )
v r n 1Cnm eim sin m cos (sin 2 )1 Pnm (cos ) sin
1
Pn
m 1
(cos )
v r n 1Cnm sin Pn (cos )m sin(m ),cos(m ) The rectangular coordinate form of the first order partial derivatives: i (V ) vi (i 1, 2,3) , in the Cartesian coordinate system, the first order partial derivative for V respectively as follows. 1 (V ) v1 vr sin cos v cos cos v sin -1
m
2 (V ) v2 vr sin sin v cos sin v cos
3 (V ) v3 vr cos v sin 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 © 2014 ACADEMY PUBLISHER
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
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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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V.
NUMERICAL RESULTS
Examples 1: Laplace equation 2u 2u 2 u 2 2 0 , x, y . x y Set is a square area: ( x, y) 0 x 6,0 y 6 The following four straight line segments consists of the boundary ( x, y) x 0,6,0 y 6; y 0,6,0 x 6 The known boundary conditions: ( x, y) x 0, 0 y 6 300 u ( x, y ) ( x, y) x 6, 0 y 6 0 u ( x, y) 0, ( x, y) y 0,6,0 x 6 n 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).
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. 1.2 Fast Muitiople 1.0
The relative error
complexity of the whole process of solving the fast multipole boundary element method is O (N).
113
Conventional
BEM BEM
0.8 0.6 0.4 0.2 0.0 2
4
6
8
10
DEGREE of FREEDOM N 10 3
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. Fast Muitiople
1200
Conventional
BEM BEM
1000 800 600 400 200
2
4
6
Freedom degree N/ 10
8
10
3
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. Figure 5. Heat conduction in a cub
ACKNOWLEDGMENT
As you can see from Figure 3, when the number of degrees of freedom N 2000 , the calculation results of FM-BEM and BEM compared with the analytical
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
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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.
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[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 threedimensional 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. 4956(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.
JOURNAL OF NETWORKS, VOL. 9, NO. 1, JANUARY 2014
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 © 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.1.108-114
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 threedimensional 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 : 2u 0 (in )
(1)
Boundary conditions: The basic boundary conditions: u u (in u ) The natural boundary conditions: q
u q (in q ) n
ci u i ( 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 , c i shape coefficient, u* ( x, y) and q* ( x, y) are the basic solutions of threedimensional potential problems, usually expressed as 1 4 R
(5)
u* ( x, y ) n
(6)
u* ( x, y)
q* ( x, y)
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.
(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) © 2014 ACADEMY PUBLISHER
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
ci
R i j
( Rij is distance between
center. Then first sum term of the formula (3.1) can be written as the follows,
ij
q
any two different particle charge, ci 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 c formula i ( Rij is distance between source and R i j ij integral point, constant ci 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 FMBEM is that derive for BEM solutions and related form of kernel function for FMM, get formula related to FMBEM. Here, the corresponding FM-BEM solutions and related kernel function formula is derived for the potential problem, also derive first-order derivatives c formula of i and the corresponding Cartesian R i j ij coordinate calculation formula in spherical coordinates
k ,s
c u ( x ) q * x , y ( ) u ( ) J y( ) q
q
s
kl
l
s
s
k ,l , s
u * x q , y( s ) u kl l ( s ) J y( s ) s 0
k ,s
f ( x q ) Cks R k ,s
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 =4
(8)
l ,s
for every unit area, the integral at all boundary points is invariant. If solving the equation by iteration method, u kl and q kl are assignment before the iteration, then in each k ,l , s
kl
l ( s ) J y( s ) s
of each unit is a fixed value, similarly, the product qkl l ( s ) J y( s ) s also has the same properties. k ,l , s
The first sum term of the formula (8), Let Cks u kl l ( s ) J y( s ) s k ,l , s
For q x q , y( s ) , If it can be written as the function 1 R , i. e. *
q* x q , y( s ) f ( x q ) 1 R
Among them, x q is the source point, R xq yc is the distance between source point and the multipole
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1 1 1 1 R n1 2 R n2 3 R n3
(10)
So, formula (8) can be rewritten as
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 ) J y( s ) s
u
u* ( x, y ) 1 = n n 4 R
q * ( x, y )
s
k ,l , s
iteration step, the product term
(9)
f ( x q )(Cks R)
At the point x q , formula (2. 4) numerical integrals i
x q , y ( s ) u kl l ( s ) J y ( s ) s
f ( x q )(1 R)Cks
A. The Basic Solution of Cartesian Coordinates
i
*
k ,l , s
ci u i ( x q )
k ,l , s
1 m 4
u (
1
kl
x yc
1 4
q
k ,l , s
1 x yc q
l
s
)nm y ( s ) J y ( s ) s
q ( kl
l
s
(11)
) J y ( s ) s 0
Because m 1, 2,3 , 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 1 applied to BEM is and its derivatives. In the rij specific implementation process, mainly manifests the form of partial derivatives for first order, two order in the local expansion
j
( X ) Lkj Y jk ( , )r j j 0 k j
(12)
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In the formula (12), Lkj is the expansion coefficients for the local, which is equivalent to the constant, so the problem is derivative for Yjk ( , )r j . Let V Ynm ( , )r n
between multipole moments (M2M), multipole moments to local moments (M2L) and the conversion between local moment (L2L). A
B
2n 1 (n m)! m Pn (cos )eim r n 4 (n m)! Cnm
y y0
M 2M
y1
T M 2L
x
x1 L2L
2n 1 (n m)! 4 (n m)!
(13)
C
x0
Figure 2. Key operation for fast multipole algorithm
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 V V V grda(V ) (vr , v , v ) , , r r r sin
A. Creating Adaptive Tree Structure The operation mode of adaptive tree structure for fast multipole boundary element method includes the following steps:
vr nr n 1Ynm ( , )
v r n 1Cnm eim sin m cos (sin 2 )1 Pnm (cos ) sin
1
Pn
m 1
(cos )
v r n 1Cnm sin Pn (cos )m sin(m ),cos(m ) The rectangular coordinate form of the first order partial derivatives: i (V ) vi (i 1, 2,3) , in the Cartesian coordinate system, the first order partial derivative for V respectively as follows. 1 (V ) v1 vr sin cos v cos cos v sin -1
m
2 (V ) v2 vr sin sin v cos sin v cos
3 (V ) v3 vr cos v sin 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 © 2014 ACADEMY PUBLISHER
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
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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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V.
NUMERICAL RESULTS
Examples 1: Laplace equation 2u 2u 2 u 2 2 0 , x, y . x y Set is a square area: ( x, y) 0 x 6,0 y 6 The following four straight line segments consists of the boundary ( x, y) x 0,6,0 y 6; y 0,6,0 x 6 The known boundary conditions: ( x, y) x 0, 0 y 6 300 u ( x, y ) ( x, y) x 6, 0 y 6 0 u ( x, y) 0, ( x, y) y 0,6,0 x 6 n 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).
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. 1.2 Fast Muitiople 1.0
The relative error
complexity of the whole process of solving the fast multipole boundary element method is O (N).
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Conventional
BEM BEM
0.8 0.6 0.4 0.2 0.0 2
4
6
8
10
DEGREE of FREEDOM N 10 3
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. Fast Muitiople
1200
Conventional
BEM BEM
1000 800 600 400 200
2
4
6
Freedom degree N/ 10
8
10
3
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. Figure 5. Heat conduction in a cub
ACKNOWLEDGMENT
As you can see from Figure 3, when the number of degrees of freedom N 2000 , the calculation results of FM-BEM and BEM compared with the analytical
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
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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.
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[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 threedimensional 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. 4956(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.
