Solution of biharmonic problems with circular boundaries using null-field integral equations

九十四年電子計算機於土木水利工程應用研討會

1

Solution of biharmonic problems with circular boundaries using null-field integral equations

Name: Chia-Chun HsiaoDate: 2005/9/2Place: NCKU


2

Outlines

Introduction Formulation Numerical examples Conclusions


3

Outlines



4

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)


5

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


6

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


7

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


8

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)


9

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.


10

Outlines



11

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


12

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


13

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


14

Null-field integral equation

0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C

B


0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C

B


0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), Cm m m m

B


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


15

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




, ( )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)


16

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


17

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


18

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


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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


1

0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ),j

HC

j j j j jj B


H: number of circular boundaries

Collocation circle index

Routing circle index


21

Numerical

Analytical

0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C

B


0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C

B



Potential

Flowchart of the present method

Degenerate kernels

Fourier series

Fourier coefficients

BIE for domain point

Adaptive observer system

Collocation method

Matching B.C.


22

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)


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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


0 ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( ), C

B


Unknown constant

Given

Unknown

c


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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:


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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


26

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


27

Outlines



28

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)


29

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)





30

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


Bird and Steele (1991)

FEM (ABAQUS)FEM mesh

(No. of nodes=3,462, No. of elements=6,606)


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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

0.2

0.4

0.6

0.8

1

Pa

rse

val s

um

1v1m


0

2

4

6

8

Pa

rse

val s

um


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


32


0

200

400

600

Pa

rse

val s

um


40

60

80

100

120

140

Pa

rse

val s

um

3v3m


0

50

100

150

200

250

Pa

rse

val s

um


10

20

30

40

50

60

70

Pa

rse

val s

um

4v4m

Parseval sum for convergence


33

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)


34

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


35

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)(∞)


36

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


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


37

-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


-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


Contour plot of Streamline for 0.8

Kelmanson (Q=0.0740, n=160)


38

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


39

-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)


40

Outlines



41

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.


42

Thank you for your kind attention!

The end


43

Direct BIEM Indirect BIEM

8 ( ) ( , ) ( ) ( , ) ( )

( , ) ( ) ( , ) ( ) ( ),B

u x U s x v s s x m s


0 ( , ) ( ) ( , ) ( )

( , ) ( ) ( , ) ( ) ( ),

B

C

U s x v s s x m s


( ) ( , ) ( ) ( , ) ( )} ( )

,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 !


44

?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


45

?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


46

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

Documents

Solution of biharmonic problems with circular boundaries using null-field integral equations