solution of biharmonic problems with circular boundaries using null-field integral equations
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九十四年電子計算機於土木水利工程應用研討會
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Solution of biharmonic problems with circular boundaries using null-field integral equations
Name: Chia-Chun HsiaoDate: 2005/9/2Place: NCKU
九十四年電子計算機於土木水利工程應用研討會
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Outlines
Introduction Formulation Numerical examples Conclusions
九十四年電子計算機於土木水利工程應用研討會
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Outlines
Introduction Formulation Numerical examples Conclusions
九十四年電子計算機於土木水利工程應用研討會
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Engineering problems with arbitrary boundaries
Circular boundary
Circular boundary
Degenerate boundary
Straight boundary
Degenerate boundary
(Legendre polynomials)
(Chebyshev polynomials)
(Fourier series)
Elliptic boundary(Mathieu
function)
九十四年電子計算機於土木水利工程應用研討會
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MotivationBEM/BIEM
Improper integral
Contour Fictitious boundary method
: collocation point
B BB
C
C
Direct Indirect (Interior)
Singular Desingular (Regular)
Limiting process
xxB B
Null-field approach
Fictitious boundary
九十四年電子計算機於土木水利工程應用研討會
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MotivationBEM/BIEM
Improper integral
CPV & HPV
Contour Fictitious boundary method
B BB CC
ill-posed
Direct Indirect (Exterior)
Singular Desingular (Regular)
Limiting process
xxB B
Degenerate kernel
( , )IK s x
C ( , )EK s x
Field point
Present approach
: collocation point
Null-field
Fictitious boundary
九十四年電子計算機於土木水利工程應用研討會
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Literature review
Engineering
problems
Laplace problems
Helmholtz problems
Biharmonic problems
Torsion bar with circular holesSteady state heat conduction of tube
Electromagnetic wave
Membrane vibration
Water wave and Acoustic problems
Plane elasticity ： Airy stress functionSolid mechanics ： plate problemFluid mechanics ： Stokes flow
2 0u
2 2( ) 0k u
4 0u
九十四年電子計算機於土木水利工程應用研討會
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Literature review1. Plane elasticity:
3. Viscous flow (Stokes Flow):
2. Solid mechanics (Plate problem):
4 0, :u u Airy stress function
4 0, :u u stream function
ntdisplacemelateraluu :,04
Jeffery (1921), Howland and Knight (1939),Green (1940) and Ling (1948)
Kamal (1966), DiPrima and Stuart (1972), Mills (1977) and Ingham and Kelmanson (1984)
Bird and Steele (1991)
九十四年電子計算機於土木水利工程應用研討會
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Purpose
A semi-analytical approach in conjunction with Fourier series, degenerate kernels and adaptive observer system is applied to biharmonic problems.
Advantages ： 1. Mesh free. 2. Accurate. 3. Free of CPV and HPV.
九十四年電子計算機於土木水利工程應用研討會
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Outlines
Introduction Formulation Numerical examples Conclusions
九十四年電子計算機於土木水利工程應用研討會
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Problem statement
Governing equation:
Essential boundary condition:
4 ( ) 0,u x x ( ), ( ),u x x x B
:lateral displacement,( )u x ( )x
Bu
u
:slope
Natural boundary condition:( ), ( ),m x v x x B( )m x ( )v x: moment, : shear
force
v
m
九十四年電子計算機於土木水利工程應用研討會
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Boundary integral equations
BIEs are derived from the Rayleigh-Green identity ：
8 ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ),B
u x U s x v s s x m s M s x s V s x u s dB s x
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C
B
U s x v s s x m s M s x s V s x u s dB s x
xW
x
Interior problem
BCW
BIE for the domain point
Null-field integral equation
九十四年電子計算機於土木水利工程應用研討會
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Boundary integral equation for the domain point
8 ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ),B
u x U s x v s s x m s M s x s V s x u s dB s x
8 ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ),B
x U s x v s s x m s M s x s V s x u s dB s x
8 ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ),m m m m
B
m x U s x v s s x m s M s x s V s x u s dB s x
8 ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ),v v v v
B
v x U s x v s s x m s M s x s V s x u s dB s x : Poisson ratio
,
( )( )x
x
u xK
n
22
, 2
( )( ) ( ) (1 )m x
x
u xK u x
n
2
,
( ) ( )( ) (1 )v x
x x x x
u x u xK
n t n t
Displacement
Slope
Displacement
Moment
Displacement
Shear force
九十四年電子計算機於土木水利工程應用研討會
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Null-field integral equation
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C
B
U s x v s s x m s M s x s V s x u s dB s x
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C
B
U s x v s s x m s M s x s V s x u s dB s x
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), Cm m m m
B
U s x v s s x m s M s x s V s x u s dB s x
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), Cv v v v
B
U s x v s s x m s M s x s V s x u s dB s x : Poisson ratio
,
( )( )x
x
u xK
n
22
, 2
( )( ) ( ) (1 )m x
x
u xK u x
n
2
,
( ) ( )( ) (1 )v x
x x x x
u x u xK
n t n t
Displacement
Slope
Displacement
Moment
Displacement
Shear force
九十四年電子計算機於土木水利工程應用研討會
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is the fundamental solution, which satisfies
Relation among the kernels
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
m m m m
v v v v
U s x s x M s x V s x
U s x s x M s x V s x
U s x s x M s x V s x
U s x s x M s x V s x
, ( )m sK , ( )v sK , ( )sK
, ( )xK
, ( )m xK
, ( )v xK
2( , ) lnU s x r r 4 ( , ) 8 ( )U s x s x
Continuous (Separable form of degenerate kernel)
九十四年電子計算機於土木水利工程應用研討會
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Degenerate kernels
( , )x r
( , )s R
32 2
2
222
32 2
2
22
1( , ) (1 ln ) ln [ (1 2ln ) ]cos( )
2
1 1[ ]cos[ ( )],
( 1) ( 1)( , ) ln
1( , ) (1 ln ) ln [ (1 2ln ) ]cos( )
2
1 1[ ]co
( 1) ( 1)
I
m m
m mm
E
m m
m mm
U s x R R R R RR
m Rm m R m m R
U s x r rR
U s x R R
R R
m m m m
s[ ( )],m R
R
EUO
IU
q f
s
r
xx
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Fourier series
01
( ) ( cos sin ),M
n nn
u s g g n h n s B
01
( ) ( cos sin ),M
n nn
s c c n d n s B
01
( ) ( cos sin ),M
n nn
m s a a n b n s B
01
( ) ( cos sin ),M
n nn
v s p p n q n s B
The boundary densities are expanded in terms of Fourier series:
M: truncating terms of Fourier series
九十四年電子計算機於土木水利工程應用研討會
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Adaptive observer system
ja
22R
1R
1O2O
3O
1B
2B3B
: Collocation point
: Radius of the jth circle
: Origin of the jth circle
: Boundary of the jth circle
( , )x
jR
jO
jB
1
33R
1x
2x
3x
2 1Mx
2Mx
kx
九十四年電子計算機於土木水利工程應用研討會
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Vector decomposition for normal derivative
i j j
e
ie
iB
jOjB
j
iiO
je
Tangential direction
True normal direction
Radial direction
( , ) ( , )( , ) cos( ) cos( )
2nx x
U s x U s xU s x
n t
: normal derivative
xt xn
( , ) ( , )( , ) cos( ) cos( )
2nx x
s x s xs x
n t
( , ) ( , )( , ) cos( ) cos( )
2nx x
M s x M s xM s x
n t
( , ) ( , )
( , ) cos( ) cos( )2n
x x
V s x V s xV s x
n t
: tangential derivative
x( , )x
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Linear algebraic system
Null-field integral equations for and formulations( )u x ( )xq
1
1
2
2
11 11 12 12 1 1
11 11 12 12 1 1
21 21 22 22 2 2
21 21 22 22 2 2
1 1 2 2
1 1 2 2H
H
H H
H H
H H
H H
H H H H HH HH
H H H H HH HH
θ θ θ θ θ θ
θ θ θ θ θ θ
θ θ θ θ θ θ
U Θ U Θ U Θ v
U Θ U Θ U Θ m
U Θ U Θ U Θ v
U Θ U Θ U Θ m
U Θ U Θ U Θ v
U Θ U Θ U Θ m
1
1
2
2
11 11 12 12 1 1
11 11 12 12 1 1
21 21 22 22 2 2
21 21 22 22 2 2
1 1 2 2
1 1 2 2H
H
H H
H H
H H
H H
H H H H HH HH
H H H H HH HH
θ θ θ θ θ θ
θ θ θ θ θ θ
θ θ θ θ θ θ
M V M V M V θ
M V M V M V u
M V M V M V θ
= M V M V M V u
M V M V M V θ
M V M V M V u
1
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ),j
HC
j j j j jj B
U s x v s s x m s M s x s V s x u s dB s x
1
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ),j
HC
j j j j jj B
U s x v s s x m s M s x s V s x u s dB s x
H: number of circular boundaries
Collocation circle index
Routing circle index
九十四年電子計算機於土木水利工程應用研討會
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Numerical
Analytical
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C
B
U s x v s s x m s M s x s V s x u s dB s x
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C
B
U s x v s s x m s M s x s V s x u s dB s x
Linear algebraic system
Potential
Flowchart of the present method
Degenerate kernels
Fourier series
Fourier coefficients
BIE for domain point
Adaptive observer system
Collocation method
Matching B.C.
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Stokes flow problems (Eccentric case)
2R
e1B
2B
1R
1
1 1 1 1,u c R
2 20, 0u
Governing equation:
Essential boundary condition:
4 ( ) 0,u x x
on
on
1B
2B
u
: stream function
: normal derivative of stream function
/u n
(Stationary)
九十四年電子計算機於土木水利工程應用研討會
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Linear algebraic system
1 1
1 1
2 2
2 2
11 11 12 12 11 11 12 12
11 11 12 12 11 11 12 12
21 21 22 22 21 21 22 22
21 21 22 22 21 21 22 22
θ θ θ θ θ θ θ θ
θ θ θ θ θ θ θ θ
U Θ U Θ M V M Vv θ
U Θ U Θ M V M Vm u=
U Θ U Θ M V M Vv θ
U Θ U Θ M V M Vm u
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C
B
U s x v s s x m s M s x s V s x u s dB s x
0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C
B
U s x v s s x m s M s x s V s x u s dB s x
Unknown constant
Given
Unknown
c
九十四年電子計算機於土木水利工程應用研討會
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Constraint equation
2 21
2 2
2
, ,1
1, ,
{ [ ( , ) ( ) ( , ) ( )
( , ) ( ) ( , ) ( )] ( )} ( ) 0,
jj jn nB B
j
j j jn n
U s x v s s x m s
M s x s V s x u s dB s dB x x
1 11 1 0nB B
dB dBn
2 2
2 2
2
1
1( ) { [ ( , ) ( ) ( , ) ( )
8
( , ) ( ) ( , ) ( )] ( )},
jj jB
j
j j j
x U s x v s s x m s
M s x s V s x u s dB s x
2( ) ( )x u x
x
1B
2B
e
Vorticity:
Constraint:
九十四年電子計算機於土木水利工程應用研討會
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Inner circle
Trapezoid integral
2
01
2( ) ( )
N
kk
f d fN
2 21
2 2
2
, ,1
1, ,
{ [ ( , ) ( ) ( , ) ( )
( , ) ( ) ( , ) ( )] ( )} ( ) 0
jj jn nB B
j
j j jn n
U s x v s s x m s
M s x s V s x u s dB s dB x
( 2)j
Analytical
Numerical
( 1)j
x
1B2B
eSeries sum Trapezoid integral
1 11 1( , ) ( ) ( )
B Bf s x dB s dB x 1 2
2 1( , ) ( ) ( )B B
f s x dB s dB xVector
decomposition
Outer circle
九十四年電子計算機於土木水利工程應用研討會
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Linear algebraic augmented system
1
1
1 12
2
1
11 11 12 12 11 11
11 11 12 12 11 11
21 21 22 22 21 21
21 21 22 22 21 21
11 11 12 12 11 11
R
u
θ θ θ θ θ θ
θ θ θ θ θ θ
U Θ U Θ V Mv
U Θ U Θ V Mm
U Θ U Θ V Mv
U Θ U Θ V Mm
U Θ U Θ V M2 2 2 2 2 2,n ,n ,n ,n ,n ,n
Unknown
九十四年電子計算機於土木水利工程應用研討會
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Outlines
Introduction Formulation Numerical examples Conclusions
九十四年電子計算機於土木水利工程應用研討會
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Plate problems
1B
4B
3B
2B1O
4O
3O
2O
Geometric data:
1 20;R 2 5;R
( ) 0u s 1B( ) 0s
1 (0,0),O 2 ( 14,0),O
3 (5,3),O 4 (5,10),O 3 2;R 4 4.R
( ) sinu s
( ) 1u s
( ) 1u s
( ) 0s
( ) 0s
( ) 0s
2B
3B
4B
and
and
and
and
on
on
on
on
Essential boundary conditions:
(Bird & Steele, 1991)
九十四年電子計算機於土木水利工程應用研討會
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Contour plot of displacement
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
Present method (N=21)
Present method (N=61)
Present method (N=41)
Present method (N=81)
九十四年電子計算機於土木水利工程應用研討會
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Contour plot of displacement
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
Present method (N=101)
Bird and Steele (1991)
FEM (ABAQUS)FEM mesh
(No. of nodes=3,462, No. of elements=6,606)
九十四年電子計算機於土木水利工程應用研討會
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Parseval sum for convergence ( )
0 10 20 30 40Te rm s o f Fo u rie r se rie s
0
0.4
0.8
1.2
1.6
Pa
rse
val s
um
0 10 20 30 40Te rm s o f Fo u rie r se rie s
0
0.2
0.4
0.6
0.8
1
Pa
rse
val s
um
1v1m
0 10 20 30 40Te rm s o f Fo u rie r se rie s
0
2
4
6
8
Pa
rse
val s
um
0 10 20 30 40Te rm s o f Fo u rie r se rie s
0
1
2
3
4
Pa
rse
val s
um
2v2m
2 2 2 2 200
1
( ) 2 ( )n nn
f d a a b
九十四年電子計算機於土木水利工程應用研討會
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0 10 20 30 40Te rm s o f Fo u rie r se rie s
0
200
400
600
Pa
rse
val s
um
0 10 20 30 40Te rm s o f Fo u rie r se rie s
40
60
80
100
120
140
Pa
rse
val s
um
3v3m
0 10 20 30 40Te rm s o f Fo u rie r se rie s
0
50
100
150
200
250
Pa
rse
val s
um
0 10 20 30 40Te rm s o f Fo u rie r se rie s
10
20
30
40
50
60
70
Pa
rse
val s
um
4v4m
Parseval sum for convergence
九十四年電子計算機於土木水利工程應用研討會
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Stokes flow problems
1
2 1R
e
1 0.5R
1B
Governing equation:
4 ( ) 0,u x x
Boundary conditions:
1( )u s u and ( ) 0.5s on 1B
( ) 0u s and ( ) 0s on 2B
2 1( )
e
R R
Eccentricity:
Angular velocity:
1 1
2B
(Stationary)
九十四年電子計算機於土木水利工程應用研討會
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Comparison of stream function
Kelmanson & Ingham (BIE)Analytical solution
Present method
n=80 n=160 n=320Limitn→∞
0.0 0.1066 0.1062 0.1061 0.1061 0.1060 0.1060 (N=5)
0.1 0.1052 0.1048 0.1047 0.1047 0.1046 0.1046 (N=7)
0.2 0.1011 0.1006 0.1005 0.1005 0.1005 0.1005 (N=7)
0.3 0.0944 0.0939 0.0938 0.0938 0.0938 0.0938 (N=7)
0.4 0.0854 0.0850 0.0848 0.0846 0.0848 0.0848 (N=9)
0.5 0.0748 0.0740 0.0739 0.0739 0.0738 0.0738 (N=11)
0.6 0.0622 0.0615 0.0613 0.0612 0.0611 0.0611 (N=17)
0.7 0.0484 0.0477 0.0474 0.0472 0.0472 0.0472 (N=17)
0.8 0.0347 0.0332 0.0326 0.0322 0.0322 0.0322 (N=21)
0.9 0.0191 0.0175 0.0168 0.0163 0.0164 0.0164 (N=31)
n: number of boundary nodes
1u
N: number of collocation points
九十四年電子計算機於土木水利工程應用研討會
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0 80 160 240 320 400 480 560 640
0.0736
0.074
0.0744
0.0748
0 80 160 240 320
Comparison for 0.5
DOF of BIE (Kelmanson)
DOF of present method
BIE (Kelmanson) Present method Analytical solution
(160)
(320)
(640)
u1
(28)
(36)
(44)(∞)
九十四年電子計算機於土木水利工程應用研討會
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Contour plot of Streamline for
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Present method (N=81)
Kelmanson (Q=0.0740, n=160)
Kamal (Q=0.0738)
e
Q/2
Q
Q/5
Q/20-Q/90
-Q/30
0.5
0
Q/2
Q
Q/5
Q/20-Q/90
-Q/30
0
九十四年電子計算機於土木水利工程應用研討會
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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Present method (N=21)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Present method (N=81)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Present method (N=41)
Contour plot of Streamline for 0.8
Kelmanson (Q=0.0740, n=160)
九十四年電子計算機於土木水利工程應用研討會
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Contour plot of vorticity for
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Present method (N=21) Present method (N=41)
Kelmanson (n=160)
0.5
九十四年電子計算機於土木水利工程應用研討會
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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Present method (N=21) Present method (N=41)
Contour plot of vorticity for 0.8
Kelmanson (n=160)
九十四年電子計算機於土木水利工程應用研討會
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Outlines
Introduction Formulation Numerical examples Conclusions
九十四年電子計算機於土木水利工程應用研討會
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Conclusions
Successful applied to biharmonic problems with circular boundaries by using the present method.
Good agreement was obtained after compared with previous results, exact solution and ABAQUS data.
Stream function and vorticity were found to be independent of Poisson ratio as we predicted.
Once engineering problems satisfy the biharmonic equation with circular boundaries, our present method can be used.
九十四年電子計算機於土木水利工程應用研討會
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Thank you for your kind attention!
The end
九十四年電子計算機於土木水利工程應用研討會
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Direct BIEM Indirect BIEM
8 ( ) ( , ) ( ) ( , ) ( )
( , ) ( ) ( , ) ( ) ( ),B
u x U s x v s s x m s
M s x s V s x u s dB s x
0 ( , ) ( ) ( , ) ( )
( , ) ( ) ( , ) ( ) ( ),
B
C
U s x v s s x m s
M s x s V s x u s dB s x
( ) ( , ) ( ) ( , ) ( )} ( )
,B
u x U s x s s x s dB s
x
( ) ( , ) ( ) ( , ) ( )} ( )
,
B
C
u x U s x s s x s dB s
x
Conclusions
Null-field integral equation available?
Null-field ! No !
九十四年電子計算機於土木水利工程應用研討會
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?Further research
Viewpoint Finished Further research
Direct BIEM
& formulation & formulation
Indirect BIEM
Single & double layer potentials
Triple & Quadruple layer potentials
Post processin
gLateral displacement
Stress or moment diagram
Boundary condition
Essential boundary condition
Natural boundary condition...etc.
( )u x ( )x ( )m x ( )v x
九十四年電子計算機於土木水利工程應用研討會
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?Further research
Viewpoint Finished Further research
Degenerate scale
Simply-connected problem had finished by Wu (2004)
Doubly-connected & multiply-connected problems
Shape of domain
Circular domain Arbitrary domain
九十四年電子計算機於土木水利工程應用研討會
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uniform pressure
a
B
w=constant
0)(,0)( xxu
: flexure rigidity
: deflection of the circular plate
: uniform distributed load
: domain of interest
)(xu
D
)(xw
Governing equation: xD
xwxu ,
)()(4
Boundary condition: 0)(,0)( xxu
Splitting method
xxu ,0)(*4Governing equation:
Boundary condition:D
wax
D
waxu
16)(,
64)(
3*
4*
General form
Governing equation: xxu ,0)(*4
Boundary condition: )()(),()(**** xxxuxu
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