Post on 20-Dec-2015
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
NNational ational TTaiwan aiwan UUniversityniversity
Brief detail of MFS
FDM FEM BEM
M esh method
M Q M FS
M eshless method
Numerical method
5
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
Introduction
"
B fB.C.
G.E.
G x s
L
1
,n
j jj
x G x s
L 0Time-independent
12
NNational ational TTaiwan aiwan UUniversityniversity
Introduction
(n+1)dt
y
t
(n)dt
(n-)dt
Field pointSource pointTime-dependent
13
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
Absolute Error Distribution Map at x=0.5MFS
27
NNational ational TTaiwan aiwan UUniversityniversity
Absolute Error Distribution Map at x=0.5HMM
28
NNational ational TTaiwan aiwan UUniversityniversity
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
( analytical solution, numerical result)
30
NNational ational TTaiwan aiwan UUniversityniversity
Case 2:
Analytical solution:
: Bessel function
: The root of
10
1
)(12
nn
zz
nn
rJeeJee
nn
nn
n 0Jthn0J
31
NNational ational TTaiwan aiwan UUniversityniversity
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
( analytical solution, numerical result)
32
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
Case 3-1:
Inside radius: 1Outside radius: 2
Height: 11
0
0
0
36
NNational ational TTaiwan aiwan UUniversityniversity
Point Distribution with Normal Vectors
1991 nodes
37
NNational ational TTaiwan aiwan UUniversityniversity
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
Point Distribution with Normal Vectors
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
( analytical solution, numerical result)
- 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
Point Distribution with Normal Vectors
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
Department of Civil Engineering and Hydrotech Research Institute Department of Civil Engineering and Hydrotech Research Institute
National Taiwan UniversityNational Taiwan University
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
RMSE: 1.26E-5 RMSE: 9.69E-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
55
Scattering of a Plane Wave by a Rigid Sphere
0),,(22 rk
1
1
k
a
Governing equation:
Sommerfeld radiation condition:
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
RMSE: 1.34E-4 RMSE: 1.47E-2 1866 nodes 1866 nodes MFS HMM
),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
RMSE: 5.98E-3 RMSE: 5.76E-2 1866 nodes 1866 nodes MFS HMM
),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:
Sommerfeld radiation condition:
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
NNational ational TTaiwan aiwan UUniversityniversity
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
NNational ational TTaiwan aiwan UUniversityniversity
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
Department of Civil Engineering and Hydrotech Research Institute Department of Civil Engineering and Hydrotech Research Institute
National Taiwan UniversityNational Taiwan University
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