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Department of Civil Engineering and Hydrotech Research Institute National Taiwan University. The Application of H ypersingular M eshless M ethod for 3D Potential and Exterior Acoustics Problems. Reporter : Professor D. L. Young 2008/01/03. - PowerPoint PPT Presentation
Citation preview
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
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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
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Brief detail of MFS
FDM FEM BEM
M esh method
M Q M FS
M eshless method
Numerical method
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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.
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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
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Brief detail of MFS From the principles of method of fundamental solutio
ns, 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 t
he field point, and is the position of the source point.
2 ( ) ( )G x x
( ) ln ijG x r
)(
x
x
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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
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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.
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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
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Introduction"
B fB.C.
G.E.
G x s L
1
,n
j jj
x G x s
L 0Time-independent
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Introduction
(n+1)dt
y
t
(n)dt
(n-)dt
Field pointSource pointTime-dependent
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Numerical methodsfor 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 & Tosaka1998 Hon
2002 Li, Hon & Chen
Introduction (Burgers’ equation)
2000 Lin & Atluri
Modified Helmhotz fundamental solution
Domain-type method
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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.
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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. 2006 Eng. Anal. Bound. Elem.
Multiply-connected-domain Laplace problem in 2D.
Young et al. 2006 J. Acoust. Soc. Am. Exterior acoustics problems in 2D.
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Introduction (3/4)Source point location
MFS HMM
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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
rnyxsB 5
2 3,
ij
lklkijkkiji
rnnyyrnn
xsB
ij
iji
rxsA 1, 3,
ij
kkiji
rnyxsA
3
1,
ij
kkikr
ijije
rnyeikr
xsAij
5
22 131,
ij
kkllikr
ijijklikr
ijkkllikr
ije
rnynyeikrrnneikrnynyek
xsBijijij
ij
ikrije
rexsA
ij
,
3
1,
ij
kkikr
ijije
rnyeikr
xsBij
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
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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
rnyxsA
5
2 3,
ij
lklkijkkiji
rnnyyrnn
xsB
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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
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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
lmmlmmkhkjlm
mmkhkjm
Nika
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
lmmlmmkhkjlm
mmkhkjm
Nikb
mm
N
l mm
mmmmii
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Case 1-1:Dirichlet boundary case:
1
0 0
22
22
1 1 sinh
sinh)1(1)1(1)sin()sin(4nm
znmnm
ynxmnm
m n
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Point Distribution with Normal Vectors
24
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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)
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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 solu tion
HM M (1350 nodes)
M FS (1350 nodes)
FE M (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 solutionY
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 solutionY
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Absolute Error Distribution Map at x=0.5MFS
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Absolute Error Distribution Map at x=0.5HMM
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Case 1-2:Dirichlet and Neumann mixed boundary case:
1
0 0
22
22
1 1 cosh
cosh)1(1)1(1)sin()sin(4nm
znmnm
ynxmnm
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)
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Case 2:
Analytical solution:
: Bessel function : The root of
10
1
)(12n
nzz
nn
rJeeJee
nn
nn
n 0Jthn0J
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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)
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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
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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 0.37647 0.1428 0.28718 0.1250 0.21319 0.1111 0.151010 0.1000 9.84E-211 0.0909 5.44E-212 0.0833 2.41E-213 0.0769 3.39E-214 0.0714 6.10E-215 0.0666 8.73E-2
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Sensitive Test of Number of nodesNodes RMSE
234 6.870E-2420 3.070E-2684 2.726E-2
1050 2.481E-21518 2.507E-21800 2.412E-22106 2.319E-22842 2.299E-23720 2.253E-24240 2.172E-2
0 1000 2000 3000 4000 5000N u m be r o f no de s
2.00E -002
3.00E -002
4.00E -002
5.00E -002
6.00E -002
7.00E -002
R M SE
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Case 3-1:
Inside radius: 1Outside radius: 2
Height: 11
0
0
0
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Point Distribution with Normal Vectors
1991 nodes
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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
zye x 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
rnyeikr
xsAij
5
22 131,
ij
kkllikr
ijijklikr
ijkkllikr
ije
rnynyeikrrnneikrnynyek
xsBijijij
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 rnyxsAxsA
ji
ikr
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
mm
N
N
mm
i
aaa
aaaa
aaaa
,2,1,
,21
2,2,21,2
,12,11
1,1,1
NNNN
N
N
mm
N
N
mm
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
lmmlmmkhkjlm
mmkhkjm
Nika
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
lmmlmmkhkjlm
mmkhkjm
Nikb
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
cossinsincossin
rzryrx
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
A na lytical so lution
M FS
HM 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
Ana lytica l solu 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
11
ka
Governing equation:
Sommerfeld radiation condition:
Neumann boundary condition:
4
04 sincoscos,
n
n nnP
y
z
a
x
:4nP Associated Lengendre polynomial
cossinsincossin
rzryrx
56
Analytical Solution
nBnAPkrhr nnm
n
sincoscos,, 444
4
04
,1
1
44
r
n
rkrh
A 1
44
1
r
n
rkrh
B
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 solution
M FS
HM 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:
RzRzyRzR
Rzx
2
22
cos4.0164.0cos3.0
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 21
ji )()()1(4
1 2
1 12
N
pq
N
qp
ikr
j
ij
WWikr
er
nrJH
1 2
0
1 1
-ikr
( ) ( ) i j4
-1 (e -1) i j2
N N ikr
j p qp q
ij
eJ W WrG
ik
70
The Diagonal Coefficient of Influence Matrices for HMM
3,,limij
kkijeije
sx rnyxsAxsA
ji
ikr
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
mm
N
N
mm
i
aaa
aaaa
aaaa
,2,1,
,21
2,2,21,2
,12,11
1,1,1
NNNN
N
N
mm
N
N
mm
i
bbb
bbbb
bbbb
,2,1,
,21
2,2,21,2
,12,11
1,1,1