1 the application of hypersingular meshless method for 3d potential and exterior acoustics problems...

1

The Application of Hypersingular Meshless Method

for 3D Potential and Exterior Acoustics Problems

Reporter ： Professor D. L. Young 2008/01/03

Department of Civil Engineering and Hydrotech Research Institute National Taiwan University

Scientific Computing & Visualization Lab

2

含超強奇異性無網格法於三維勢能及外域聲學問題之應用

楊德良教授 2008/01/03

Department of Civil Engineering and Hydrotech Research Institute National Taiwan University

Scientific Computing & Visualization Lab

3

NNational ational TTaiwan aiwan UUniversityniversity

Outline: Introduction Potential problems

Formulation The diagonal coefficient of influence matrices Numerical results

Cube Cylinder Arbitrary shape

Exterior acoustics problems Formulation The diagonal coefficient of influence matrices Numerical results

Scattering by a soft sphere Scattering by a rigid sphere Scattering by a bean shape obstacle

Conclusions Further researches

4


Brief detail of MFS

FDM FEM BEM

M esh method

M Q M FS

M eshless method

Numerical method

5


Brief detail of MFS

Method of fundamental solutions （ MFS ） is involved through the combination of meshless and the concept of indirect boundary element method.

The MFS considers an artificial boundary outside the computational domain, to locate the source points and some field points locate on the boundary. Using these points and boundary conditions can solve the coefficients used in the fundamental solution.

6


Brief detail of MFS

V1

V2

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frame 001 29 Jul 2002 Frame 001 29 Jul 2002

X

Y

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frame 001 27 Oct 2003 MESH TEST

-40.00 0.00 40.00 80.00 120.00 160.00

-40.00

0.00

40.00

80.00

120.00

160.00Field point

Source point

Domain method

MFS

7


Brief detail of MFS

From the principles of method of fundamental solutions, for the given governing equation, the free space Green’s function has to be satisfied.

For example of the Laplace equation as follows the free space Green’s function can be written

where is the fundamental solutions

is the Dirac delta function, is the position of the field point, and is the position of the source point.

2 ( ) ( )G x x

( ) ln ijG x r

)(

x

x

8


Brief detail of MFS

Method of Fundamental Solutions (MFS)

0 L

N

D

'G x x

- L

1'

N

j jjG x x

A B

9


Brief detail of MFS Using the above expression, the approximate solution

can be obtained as

1

,N

ji i ijj

x y G r

And the field points located on the boundary and com

bined with boundary condition that can solve coefficients and advance to solve any region in the solution domain.

10


Singular Value Decomposition

SVD is the technique for dealing with sets of equations or matrices are either singular or else numerically very close to singular.

1

2

n

TA U V

Orthogonal matrix

Matrix of the singular values

11


Introduction

"

B fB.C.

G.E.

G x s

L

1

,n

j jj

x G x s

L 0Time-independent

12


Introduction

(n+1)dt

y

t

(n)dt

(n-)dt

Field pointSource pointTime-dependent

13


Numerical methods

for Burgers’ eq.

Mesh method

Meshless method

FDM

FEM

MQ

Mesh-reduction method

BEM

MFS-DRM

MLPG

1980 Varoglu & Finn 1981 Caldwell & Wanless 1982 Nguyen & Reynen 2004 Dogan

1984 Evans & Abdullah

1990 Kakuda & Tosaka

1998 Hon

2002 Li, Hon & Chen

Introduction (Burgers’ equation)

2000 Lin & Atluri

Modified Helmhotz fundamental solution

Domain-type method

14


Introduction (1/4)

The Method of Fundamental Solutions proposed by Kupradze and Aleksidze, 1964.

The MFS has been generally applied to solve some engineering problems. It is a kind of meshless methods, since only boundary nodes are distributed.

However because of the controversial artificial boundary (off-set boundary) outside the physical domain, the MFS has not become a popular numerical method.

MFS only works well in regular geometry with the Dirichlet and Neumann boundary conditions.

15


Introduction (2/4)

This research extends the Hypersingular Meshless Method to solve the 3D potential and exterior acoustics problems.

Young et al. 2005 J. Comput. Phys. Potential problems in 2D.

Chen et al. 2006Eng. Anal. Bound. Elem.

Multiply-connected-domain Laplace problem in 2D.

Young et al. 2006 J. Acoust. Soc. Am. Exterior acoustics problems in 2D.

16


Introduction (3/4)Source point location

MFS HMM

17

NNational ational TTaiwan aiwan UUniversityniversityIntroduction (4/4) Comparison of HMM and MFS

HMM MFSMeshless features Yes Yes

Source point location Real boundary Fiction boundaryAccuracy

Acceptable Better

PotentialDouble layer Single layer

Kernel functions for 3D potential

problems

Kernel functions for 3D exterior acoustics

3

,ij

kkiji

r

nyxsB

5

2 3,

ij

lklkijkkiji

r

nnyyrnnxsB

ij

iji

rxsA

1,

3,

ij

kkiji

r

nyxsA

3

1,

ij

kkikr

ijije

r

nyeikrxsA

ij

5

22 131,

ij

kkllikr

ijijklikr

ijkkllikr

ije

r

nynyeikrrnneikrnynyekxsB

ijijij

ij

ikrije

r

exsA

ij

,

3

1,

ij

kkikr

ijije

r

nyeikrxsB

ij

1818

Potential ProblemsPotential Problems

Department of Civil Engineering and Hydrotech Research Institute Department of Civil Engineering and Hydrotech Research Institute

National Taiwan UniversityNational Taiwan University

19


Formulation Governing equation: ,

the representation of the solution for interior problem can be approximated as:

Kernel functions:

023 x Dx

( )

1

( ) ( , )N

i i j i j

j

x A s x

( )

1

( ) ( , ) ,N

i i j i j

j

x B s x

3

,ij

kkiji

r

nyxsA

5

2 3,

ij

lklkijkkiji

r

nnyyrnnxsB

20


The Diagonal Coefficient of Influence Matrices

1, 1,1 1,2 1,1

2,1 2, 2,2 2,1

,1 ,2 , ,1

,

N

m Nm

N

m Ni jm

N

N N N m N Nm

a a a a

a a a a

a a a a

1, 1,1 1,2 1,1

2,1 2, 2,2 2,1

,1 ,2 , ,1

( )

( ).

( )

N

m Nm

N

m Ni jm

N

N N N m N Nm

b b b b

b b b b

b b b b

21


Analytical derivation of diagonal

coefficients Analytical solution:

:kwhere

:Nwave number

number of nodes

radius of sphere:

}))!(()!(4

)!22()!2()()()124(

)!(4

))!2(()()()14(2{

2

22)12()1(

)12(

1

1 0

')2(

042

2)1(

)2('

)2(

2

lmm

lmmkhkjlm

m

mkhkjm

N

ika

mm

N

l mm

mmmmii

}))!(()!(4

)!22()!2()()()124(

)!(4

))!2(()()()14(2{

2

22)12()1(

)12(

1

1 0

')12(

042

2)1(

)2('

)2(

3

lmm

lmmkhkjlm

m

mkhkjm

N

ikb

mm

N

l mm

mmmmii

22


Case 1-1:Dirichlet boundary case:

1

0 0

22

22

1 1 sinh

sinh)1(1)1(1)sin()sin(4

nm

znm

nmynxm

nm

m n

23


Point Distribution with Normal Vectors

24

NNational ational TTaiwan aiwan UUniversityniversityResultsCross-section at x=0.5

MFS HMM FEM RMSE: 6.26E-5 RMSE: 2.01E-3 RMSE: 4.10E-3 1350 NODES 1350 NODES 3375 NODES (13720 Elements)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

( analytical solution, numerical result)

25


Comparison of Three MethodsNumerical values at x=0.5, z=0.5

0 0.2 0.4 0.6 0.8 1

0

0.04

0.08

0.12

0.16

0.2

0 0.2 0.4 0.6 0.8 1

0

0.04

0.08

0.12

0.16

0.2

0 0.2 0.4 0.6 0.8 1

0

0.04

0.08

0.12

0.16

0.2

0 0.2 0.4 0.6 0.8 1Y

0

0.04

0.08

0.12

0.16

0.2

Ph a i

Exact solution

HM M (1350 nodes)

M FS (1350 nodes)

FEM (1000 nodes)

00001

0.040190.040150.039430.040150.92857

0.077330.077340.077510.077340.85714

0.10910.109230.109420.109230.78571

0.134140.134440.134640.134440.71429

0.151970.152420.152620.152420.64286

0.162570.163120.163320.163120.57143

0.166080.166670.166860.166670.5

0.162570.163120.163320.163120.42857

0.151970.152420.152620.152420.35714

0.134140.134440.134640.134440.28571

0.10910.109230.109420.109230.21429

0.077330.077340.077510.077340.14286

0.040190.040150.039430.040150.07143

00000

FEMMFSHMMAnalytical solution

Y

00001

0.040190.040150.039430.040150.92857

0.077330.077340.077510.077340.85714

0.10910.109230.109420.109230.78571

0.134140.134440.134640.134440.71429

0.151970.152420.152620.152420.64286

0.162570.163120.163320.163120.57143

0.166080.166670.166860.166670.5

0.162570.163120.163320.163120.42857

0.151970.152420.152620.152420.35714

0.134140.134440.134640.134440.28571

0.10910.109230.109420.109230.21429

0.077330.077340.077510.077340.14286

0.040190.040150.039430.040150.07143

00000

FEMMFSHMMAnalytical solution

Y

26


Absolute Error Distribution Map at x=0.5MFS

27


Absolute Error Distribution Map at x=0.5HMM

28


Case 1-2:Dirichlet and Neumann mixed boundary case:

1

0 0

22

22

1 1 cosh

cosh)1(1)1(1)sin()sin(4

nm

znm

nmynxm

nm

m n

29

NNational ational TTaiwan aiwan UUniversityniversityResultsCross-section at x=0.5

MFS HMM FEM RMSE: 8.23E-5 RMSE: 2.02E-3 RMSE: 4.10E-3 1350 NODES 1350 NODES 3375 NODES (13720 Elements)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


30


Case 2:

Analytical solution:

: Bessel function

: The root of

10

1

)(12

nn

zz

nn

rJeeJee

nn

nn

n 0Jthn0J

31


ResultsCross-section at z=0.5

MFS HMM 1800 nodes 1800 nodes RMSE: 3.80E-3 RMSE: 2.41E-2

-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8


32


Point Distribution Comparison

RMSE : 2.41E-2 RMSE : 0.1523

RMSE : 8.89E-2 RMSE : 0.1738

-1 .2 -0 .8 -0 .4 0 0.4 0.8 1.2

-1 .2

-0 .8

-0 .4

0

0.4

0.8

1.2

-1 .2 -0 .8 -0 .4 0 0.4 0.8 1.2

-1 .2

-0 .8

-0 .4

0

0.4

0.8

1.2

-1 .2 -0 .8 -0 .4 0 0.4 0.8 1.2

-1 .2

-0 .8

-0 .4

0

0.4

0.8

1.2

-1 .2 -0 .8 -0 .4 0 0.4 0.8 1.2

-1 .2

-0 .8

-0 .4

0

0.4

0.8

1.2

33


Sensitivity Test of Point Distribution

The distance of the nodes on the top surface is fixed at 0.0833

Number of nodes on Z axis

Distance of the nodes on Z axis

RMSE

6 0.1666 0.3764

7 0.1428 0.2871

8 0.1250 0.2131

9 0.1111 0.1510

10 0.1000 9.84E-2

11 0.0909 5.44E-2

12 0.0833 2.41E-2

13 0.0769 3.39E-2

14 0.0714 6.10E-2

15 0.0666 8.73E-2

34


Sensitive Test of Number of nodes

Nodes RMSE

234 6.870E-2

420 3.070E-2

684 2.726E-2

1050 2.481E-2

1518 2.507E-2

1800 2.412E-2

2106 2.319E-2

2842 2.299E-2

3720 2.253E-2

4240 2.172E-2

0 1000 2000 3000 4000 5000N u m be r o f n o d e s

2.00E-002

3.00E-002

4.00E-002

5.00E-002

6.00E-002

7.00E-002

R M SE

35


Case 3-1:

Inside radius: 1Outside radius: 2

Height: 11

0

0

0

36



1991 nodes

37


ResultsCross-section at z=0

MFS HMM FEM 1991 nodes 1991 nodes 1320 nodes (5000 elements)

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

38

Case 3-2:

BC:

Analytical solution:

zyex cos

02

zyex cos

39


40

ResultsCross-section at x=0

MFS (d=0.5) MFS (d=1) HMM 2826 nodes 2826 nodes 2826 nodes RMSE: 9.63E22 RMSE: 1.26E-4 RMSE: 4.12E-2


- 1 - 0 . 5 0 0 . 5 1- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

2 . 5

41

Case 3-3:

02

1

0

42

ResultsCross-section at x=0

2826 nodes 2981 nodes 2826 nodes

(a) MFS (d=1) (b) LDQ (c) HMM

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

1.5

2

2.5

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

1.5

2

2.5

43

Case 3-4

02 1

0

44


2261 nodes

45

ResultsCross-section at z=0

2826 nodes 2981 nodes 2826 nodes

(a) MFS (d=2) (b) LDQ (c) HMM

-1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

-1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

4646

Exterior Acoustics Exterior Acoustics ProblemsProblems



47

Formulation Governing equation:

Sommerfeld radiation condition:

the representation of the solution for exterior problem can be approximated as:

Kernel functions:

,0223 xkx eDx

3

1,

ij

kkikr

ijije

r

nyeikrxsA

ij

5

22 131,

ij

kkllikr

ijijklikr

ijkkllikr

ije

r

nynyeikrrnneikrnynyekxsB

ijijij

1

( ) ( , ) , N

i e j i j e

j

x A s x x D

1

( ) ( , ) , N

i e j i j e

j

x B s x x D

1( 1)

2lim ( ) 0, d

rr ik r

r

48

The Diagonal Coefficient of Influence Matrices

3

,,limij

kkijeije

sx r

nyxsAxsA

ji

ik

r

nynyrnnxsBxsB

ij

llkkijkkijeije

sx ji4

3,,

2

5

2

lim

The kernel function will be approximated by:

The diagonal coefficients for the exterior problem can be extracted out as:

NNNN

N

N

m

m

N

N

m

m

i

aaa

aaaa

aaaa

,2,1,

,21

2,2,21,2

,12,11

1,1,1

NNNN

N

N

m

m

N

N

m

m

i

bbb

bbbb

bbbb

,2,1,

,21

2,2,21,2

,12,11

1,1,1

49

Analytical derivation of diagonal

coefficients Analytical solution:

:kwhere

:Nwave number

number of nodes

radius of sphere:

}))!(()!(4

)!22()!2()()()124(

)!(4

))!2(()()()14(2{

2

22)12(

)1()12(

'1

1 0)12(

042

2)1(

)2('

)2(

2

lmm

lmmkhkjlm

m

mkhkjm

N

ika

mm

N

l mm

mmmmii

}))!(()!(4

)!22()!2()()()124(

)!(4

))!2(()()()14(2{

2

22)12(

)1()12(

'1

1 0

')12(

042

2)1(

)2(''

)2(

3

lmm

lmmkhkjlm

m

mkhkjm

N

ikb

mm

N

l mm

mmmmii

50

Scattering of a Plane Wave by a Soft Sphere

0),,(22 rk

1,1 ka

Governing equation:

Plane wave incidence:y

z

a

x

cos

sinsin

cossin

rz

ry

rx

ikzi eAnalytical solution of total field:

0

1 cos12,,n

nnnnnt Pkrhakrjnir

J. J. Bowman,T. B. A. Senior, P. L. E.Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes, Hemisphere publishing Corp., 1987.

Analytical solution of the scattered field:it

51

Point Distribution and Normal Vectors

1866 nodes

52

ResultsValues on the y=0, z=0 line

0.8 1.2 1.6 2 2.4 2.8 3.2

-1

-0 .8

-0 .6

-0 .4

-0 .2

0

0.2

0.8 1.2 1.6 2 2.4 2.8 3.2

-1

-0 .8

-0 .6

-0 .4

-0 .2

0

0.2

0.8 1.2 1.6 2 2.4 2.8 3.2r

-1

-0 .8

-0 .6

-0 .4

-0 .2

0

0.2

Analytica l so lu tion

M FS

H M M

Real part Imaginary part

0.8 1.2 1.6 2 2.4 2.8 3.2

-0 .4

-0 .3

-0 .2

-0 .1

0

0.8 1.2 1.6 2 2.4 2.8 3.2

-0 .4

-0 .3

-0 .2

-0 .1

0

0.8 1.2 1.6 2 2.4 2.8 3.2r

-0 .4

-0 .3

-0 .2

-0 .1

0

Analytica l so lu tion

M FS

H M M

53

ResultsCross-section of y=0 plan for real part

RMSE: 8.73E-5 RMSE: 4.73E-3 1866 nodes 1866 nodes MFS HMM

Exact solution

Numerical solution

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

54

ResultsCross-section of y=0 plan for imaginary part


Exact solution

Numerical solution

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

55

Scattering of a Plane Wave by a Rigid Sphere

0),,(22 rk

1

1

k

a

Governing equation:


Neumann boundary condition:

4

04 sincoscos,

n

n nnP

y

z

a

x

:4nP Associated Lengendre polynomial

cos

sinsin

cossin

rz

ry

rx

56

Analytical Solution

nBnAPkrhr nnm

n

sincoscos,, 444

4

04

,1

1

44

r

n

rkrh

A 1

44

1

r

n

r

krhB

where

K. Gerdes, L. Demkowicz, Solution of 3D-Laplace and Helmholtz equations in exterior domains using hp-infinite elements. Comput. Meth. Appl. Mech. Eng. 137 1996 239–273.

:4h Spherical Hankel function of the first kind

57

Results

Real part Imaginary part

,

2,2

,

2,2

0 100 200 300 400

-2

-1

0

1

2

0 100 200 300 400

-2

-1

0

1

2

0 100 200 300 400

-2

-1

0

1

2Exact so lution

M FS

H M M

58

ResultsCross-section of


),2

,(Re r

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

Exact solution

Numerical solution

59

ResultsCross-section of


),2

,(Im r

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

Exact solution

Numerical solution

60

Scattering of a Plane Wave by a Soft Bean Shape Obstacle

,0),,(22 rk

Governing equation:


Plane wave incidence:

ikxi e

M. Ganesh and I.G. Graham, A high-order algorithm for obstacle scattering in three dimensions, J. Comput. Phys. 198 2004 211–242.

Radius parameter R:

RzRz

yRzR

Rz

x

222

cos4.0164.0

cos3.0

cos1.0164.0

1k

61

Mesh of the Bean Shape Obstacle

2600 elements

62

Point Distribution and Normal Vectors

2500 nodes

63

Results

real part imaginary part

64

Results of Real Part by MFS

d=-0.1 d=-0.2

d=-0.15 d=-0.25

65

Results of Imaginary Part by MFS

d=-0.1 d=-0.2

d=-0.15 d=-0.25

66


Conclusions

The controversy of the artificial (off-set) boundary outside the physical domain by using the MFS no longer exists.

From the series cases of the complex irregular shape, MFS required a lot of time to adjust the distance of source points, HMM has figured out acceptable answers immediately.

From the sensitivity test of point distribution, we can know that to obtain the high accuracy of HMM, improving the point seeding is necessary. HMM required the uniform point distribution to obtain the good results.

67


Further Researches

For the next step, to solve the Helmholtz problem in vector field which relate to the electromagnetic problem in three dimensions is we are going to do.

The combination of other numerical methods such as method of particular solutions (MPS) or domain decomposition method (DDM) and HMM to solve Poisson, Helmholtz, modified Helmholtz equation would be interesting topics to research.

6868



Thank YouThank YouScientific Computing & Visualization Lab

69

The Diagonal Coefficient of Influence Matrices for BEM

ji 2

1

ji )()()1(4

1 2

1 12

N

pq

N

qp

ikr

j

ij

WWikr

e

r

nrJ

H

1 2

0

1 1

-ikr

( ) ( ) i j4

-1(e -1) i j

2

N N ikr

j p qp q

ij

eJ W W

rG

ik

70

The Diagonal Coefficient of Influence Matrices for HMM

3

,,limij

kkijeije

sx r

nyxsAxsA

ji

ik

r

nynyrnnxsBxsB

ij

llkkijkkijeije

sx ji4

3,,

2

5

2

lim

The kernel function will be approximated by:

The diagonal coefficients for the exterior problem can be extracted out as:

NNNN

N

N

m

m

N

N

m

m

i

aaa

aaaa

aaaa

,2,1,

,21

2,2,21,2

,12,11

1,1,1

NNNN

N

N

m

m

N

N

m

m

i

bbb

bbbb

bbbb

,2,1,

,21

2,2,21,2

,12,11

1,1,1

1 the application of hypersingular meshless method for 3d potential and exterior acoustics problems...

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