particular solutions for some engineering problems

1

Particular solutions for some engineering problems

Chia-Cheng Tasi 蔡加正

Department of Information Technology Toko University, Chia-Yi County, Taiwan

2008 NTOU

2

Motivation

Method of Particular Solutions (MPS)

Particular solutions of polyharmonic spline

Numerical example I

Particular solutions of Chebyshev polynomials

Numerical example II

Conclusions

Overview

3

Motivation

BEM has evolved as a popular numerical technique for solving linear, constant coefficient partial differential equations.

Other boundary type numerical methods: Treffz method, MFS…

Advantage: Reduction of dimensionalities (3D->2D, 2D->1D)

Disadvantage: domain integration for nonhomogeneous problem

For inhomogeneous equations, the method of particular solution (MPS) is needed.

In BEM, it is called the dual reciprocity boundary element method (DRBEM) (Partridge, et al., 1992).

4

Motivation and Literature review

5

Motivation

RBF

Golberg (1995)

Chebyshev

MFSMPS with

Chebyshev Polynomial

s

spectral convergence

Golberg, M.A.; Muleshkov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003)

6

Motivation

1 21 2( ) ( ) ( )

L

7

Motivation

We note that the polyharmonic and the poly-Helmholtz equations are encountered in certain engineering problems, such as high order plate theory, and systems involving the coupling of a set of second order elliptic equations, such as a multilayered aquifer system, or a multiple porosity system.

These coupled systems can be reduced to a single partial differential equation by using the Hörmander operator decomposition technique. The resultant partial differential equations usually involve the polyharmonic or the products of Helmholtz operators.

Hence My study is to fill an important gap in the application of boundary methods to these engineering problems.

8

Method of particular solutions

( ) ( );u f L x x

( ) ( );u g B x x

p hu u u

( ) ( );pu f L x x

( ) 0;hu L x

( ) ( ) ( );h pu g u B Bx x


Method of fundamental solutions, Trefftz method, boundary element metho

d, et al.

9

( ) ( );pu f L x x

1

( ) ( )n

i ii

f

x x

( ) ( )i i L x

1

( ).n

p i ii

u

x

1

( ).n

p i ii

u

B B x


10

1

( ) ( )n

i ii

f

x x

2 2

2 1 2

1

: ( )

( ( )) : ( ) (3 ), ln (2 )

: ( ) (1 ) ( )

nn n

i ii n

MQ r r c

RBF r APS r r D r r D

CSRBF r r g r

,

, , ,,

,

: ( ) sin sin

( ( )) : ( )

: ( ) ( ) ( )

m n

m nm n m n m n

m n

m n m n

Tri m x n y

Global Polynomials x y

Chebyshev T x T y

x

x x

x

1 , 1(1 )

0, 1

r rr

r

( (?) )iL x

Method of particular solutions (basis functions)

11

1 21 2( ) ( ) ( )

L

1 2, , ,

1 2, ,...,

( ) ( )i i L x

Particular solutions for

the engineering problems

Method of particular solutions (Hörmander Operator Decomposition technique)

12

Example

11 12 13

21 22 23

1 2 3

0 0

0 0

0 0

u u u

u u u

p p p

L

2 2

2 2

( ) 0

0 ( )

0

x

Ly

x y

2 22 2

2

2 22 2

2

2 2 2 2 2 2 2 2

( )

( )

( ) ( ) ( )

adj

y x y x

x y x y

x y

L

2 2 2( )adj LL I

13

Example

11 12 13

21 22 23

1 2 3

0 01

0 0

0 0

adj

u u u

u u u

p p p

L

0 0 0 01

( ) 0 0 0 0

0 0 0 0

adj

L L11 12 13

21 22 23

1 2 3

0 0

0 0

0 0

u u u

u u u

p p p

L

2 2 2( )

14

Other examples

2

11 12 132

21 22 23

1 2 3

00 0

0 0 0

0 0

0

x u u u

u u uy

p p p

x y

2

1 * * *11 12 13

2 * * *21 22 23

2 * * *2 1 2 3

* * *1 2 3

1 2

0 0

0 0 000 0 0 00

0 0 000 0 0

0 0 0 000 0

T

xu u u

u u ux

T T Tk

p p p

x x

Stokes flow

Thermal Stokes flow

15

Other examples

* * *11 12 13

* * *21 22 23

* * *31 32 33

0 0

0 0

0 0

u u u

L u u u

u u u

22 2(1 ) (1 )

( )2 2ij ij

i j

D D vL

x x

2

3 3

(1 )

2i ii

DL L

x

22

33

(1 )

2

DL

Thick plate

2 22

2 * *1 1 2 11 12

* *2 22 21 22

21 2 2

( ) ( )0

0( ) ( )

x x x u u

u u

x x x

Solid deformation

16

Remark

Particular solutions for engineering

problems

Particular solutions for product operator

1 21 2( ) ( ) ( )

L

Hörmander operator decomposition techn

ique

17

Particular solutions for

( )

L

L

L

Partial fraction decomposition

Particular solutions for product operator

1 21 2( ) ( ) ( )

L

Method of particular solutions (Partial fraction decomposition)

18

Partial fraction

decomposition (Theorem)

19


(Proof 1)

,( ) ( , , ) ( , , )m

m

lm m lA x y z a x y z

1 21 2( ) ( ) ( ) ( , , ) ( , , )A x y z a x y z

1 21 2 ,( ) ( ) ( ) ( , , ) ( ) ( , , )m

m

lm m lA x y z A x y z

,1

( , , ) ( ) ( , , )i

mm l ii

A x y z A x y z

,m

i i

l

i m

20


(Proof 2)

1

,1 0 1

1 ( )m

im l i

m l i

C

1

,1 0 1

( , , ) ( ) ( , , )m

im l i

m l i

A x y z C A x y z

,

1

( ) ( , , ) ( , , )i

mi m li

A x y z A x y z

1

, ,0 0

( , , ) ( , , )m

mm l m lm l

A x y z C A x y z

1,

1 01

1

( ) ( )

m

i m

m ll

m li i m

C

21

Example

(1)

2 2( 4)( 9) ( , ) m nx y x y

2 2 2 2

1 1 1 1 1 35

( 4)( 9) 324 11664 2704( 4) 1053( 9) 123201( 9)

(2) (1) (1) (2) (1)2L 2L 2M 2H 2H35

324 11664 2704 1053 123201

2 (2)2L

m nx y (1)2L

m nx y (1)2M( 4) m nx y

2 (2)2H( 9) m nx y (1)

2H( 9) m nx y

2 2 2 2

1 1 1 1 1 35

( 4)( 9) 324 11664 2704( 4) 1053( 9) 123201( 9)

2 (2)2L

m nx y (1)2L

m nx y (1)2M( 4) m nx y

2 (2)2H( 9) m nx y (1)

2H( 9) m nx y

22

Example (2)

2 2 4 ( , )0 0 0( 2 cos 2 ) ( )m n m nr r r x y

2 2 ( , )1 2( )( ) ( )m n m nr x y

0

0

1 0

2 0

r e

r e

i

i

( , ) ( , )( , ) 1 2

2 2 2 21 2 2 1

( ) ( )( )

m n m nm n r r

r

2 ( , )1 1( ) ( )m n m nr x y

2 ( , )2 2( ) ( )m n m nr x y

23

Remark

( , , , )( ) ( , , ) ( ) ( ) ( )L L l m n l m nl m nP x y z x y z orT x T y T z

( , , , )0 ( , , ) ( ) ( ) ( )L L l m n l m n

l m nP x y z x y z orT x T y T z

1 21 2( ) ( ) ( ) ( ) ( ) ( )l m n

l m nP x y z orT x T y T z


24

Particular solutions of polyharmonic spline (APS)

( ) ( );u q L x x

25

Particular solutions of polyharmonic spline (APS)

( )

( ) ( )

j jP p

F f r

L

L

26

Particular solutions of polyharmonic spline (Definition)

27

Particular solutions of polyharmonic spline (Generating Theorem)

28


29


30

Particular solutions of polyharmonic spline (2D Poly-Helmholtz Operator)

31


Generating Theorem

proof

32


33


34


35


Generating Theorem

proof

36

Particular solutions of polyharmonic spline (Limit Behavior)

37

Particular solutions of polyharmonic spline (Limit Behavior)

38

2 2 2 ( ) in ( ) ( ) ( )pD G k qu u u xx x x

Numerical example I

1 1

2 2

( ) ( ) on

( ) ( ) on

B u u

B u u

x x

x x

39

Numerical example I

1 2 1 2 2 2 1 1

1 1 1

( )( ) ( )( ) ( )( )

40

Numerical example I (BC)

( ) 1u K

1 1

2 2

( ) ( ) on

( ) ( ) on

B u u

B u u

x x

x x

( )( )

x

Kn

22

2

( )( ) ( ) (1 )m

xx

Kn

2 2( ) ( )( ) (1 )v

x

x x x x

Kn t n t

( ) 1u K1 2

1 2( ) ( )

( )x x

n n

K

2 2 2

1 2 32 21 21 2

( ) ( ) ( )( )m x xx x

K

3 3 3 3

1 2 3 43 2 2 31 1 2 1 2 2

( ) ( ) ( ) ( )( )v x x x x x x

K

2 21 21 Dn Dn

1 22 2(1 )Dn n 2 2

2 13 Dn Dn 2 2

1 2 1 21 (1 )Dn n Dn n

2 3 22 1 2 1 22 (1 ) 2(1 )Dn n Dn n nD

2 3 21 2 1 2 13 (1 ) 2(1 )Dn n Dn n nD

2 24 2 1 2 1(1 )Dn n Dn n

41

Numerical example I (BC)

42

Numerical example I (MFS)

1 1 1

2 2 2

( ) ( ) ( )

( ) ( ) ( )

on

on

h p

h p

B u u B u

B u u B u

x x x

x x x

1 2 1 1 2 21 1

( ; , , ) ( , ) ( , )L L

j j j jh j j j

j j

u G G

x s x s x s

1 0 1

2 0 2

( , ) ( )

( , ) ( )

j j

j j

G K r

G K r

x s

x s

21

22

21

22

2 ( )

2 ( )

( )( )

G

G

x s

x s

1 2 1 1 2 21 1

( ; , , ) ( , ) ( , )L L

j j j jh j j j

j j

u G G

x s x s x s

2 0 1 0 2

1 0 1 0 2

( , ) ( ) ( )

( , ) [ ( ) ( )]

G K r K r

G K r K r

x s

x s i

2 2 2 21 2( )( ) ( ) ( )u qD x x

43

Numerical example I (results)

44

Particular solutions of Chebyshev polynomials (why orthogonal polynomials)Fourier series: exponential convergence but Gibb’s phenomena

Lagrange Polynomials: Runge phenomena

Jacobi Polynomials (orthogonal polynomials): exponential convergence

45

Particular solutions of Chebyshev polynomials (why Chebyshev)

FFT

46

Chebyshev interpolation (1)

2 2 2( , , ) ( ) ( ) ( )

l m nb a b a b a

ijk i j ki j k b a b a b a

x x x y y y z z zf x y z a T T T

x x y y z z

, , , , , ,

( , , )8cos cos cos

l m ni j k

ijki j kl i m j n k l i m j n k

f x y z i i j j k ka

lmnc c c c c c l m n

no matrix inverse

47

Chebyshev

interpolation (2)

( , , ) ( ) ( ) ( )l m n

ijk i j ki j k

f x y z a T x T y T z

[ / 2]( ) 2

0

( )n

n n kn k

k

T x c x

( ) 2 1

(2 )

( 1)!( 1) 2 , 2

!( 2 )!

( 1) , 0

n k n kk

k kk

n n kc n k

k n k

c k

( , , )l m n

i j kijk

i j k

f x y z b x y z

no book keeping

by multiple loops

48

Particular solutions of Chebyshev polynomials

49

Particular solutions of Chebyshev polynomials (poly-Helmholtz)

proof

Generating Theorem

Golberg, M.A.; Muleshkov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003)

50

Particular solutions of Chebyshev polynomials (polyharmonic)

51

Particular solutions of Chebyshev polynomials (polyharmonic)

52

Method of fundamental solutions

1

( ) 0iAi h

i

u

1( )

1 0 1

( ) ( )i

i

i

A KA j

h ijk ki j k

u G

x x s

( ) ( ) ( )L LG x x Fig. 2: Geometry configuration of the MFS.

53

2 2( 4)( 9) ( ) 0hu x

2 (2)2LG

(1)2LG (1)

2M( 4)G

2 (2)2H( 9) G (1)

2H( 9)G

(2) (1) (1) (2) (1)1 2L 2 2L 3 2M 4 2H 5 2H

1

( ) ( ( ) ( ) ( ) ( ) ( ))K

h k k k k k k k k k kk

u G r G r G r G r G r

x

Method of fundamental

solutions (example)

54

Example (2D modified Helmholtz)

( 900) 899(e e )x yu

e ex yu

RMSEs 2.16E-02 1.38E-06 4.18E-12 3.07E-13 2.41E-12

Table I: The RMSEs for Example 3

4l m 8l m 12l m 16l m 20l m


55

Example (2D Laplace)

e ex yu e ex yu

RMSEs 4.77E-05 2.94E-10 1.92E-10 1.92E-10 1.76E-10

Table II: The RMSEs for Example 4

4l m 8l m 12l m 16l m 20l m


56

Example (3D modified Helmholtz)

( 900) 899(e e e )x y zu

e e ex y zu

RMSEs 1.48E-01 5.45E-06 8.33E-12 4.15E-12 1.77E-11

Table III: The RMSEs for Example 5

4l m n 8l m n 12l m n 16l m n 20l m n


57

Example (3D Laplace)

e e ex y zu

e e ex y zu

RMSEs 4.18E-05 2.65E-10 4.17E-11 4.15E-11 2.89E-10

Table IV: The RMSEs for Example 6

4l m n 8l m n 12l m n 16l m n 20l m n


58

Example (2D polyharmonic)

e ex yu 4 e ex yu

2 3

2 31

T

n n n

B

RMSEs 4.75E-10 2.98E-12 2.98E-12 2.98E-12 2.98E-12

Table V: The RMSEs for Example 7

4l m 8l m 12l m 16l m 20l m

6 4 21 2 3 4

1

( ) ( ln ln ln ln )K

h k k k k k k k k k k kk

u r r r r r r r

x


59

Example (2D product operator)

e ex yu 2 3

2 31

T

n n n

B

2 ( 900)( 100) 89001(e e )x yu

21 0 2 0 3 4

1

( ) ( (30 ) (10 ) ln ln )K

h k k k k k k k k kk

u K r K r r r r

x

RMSEs 7.29E-06 2.61E-10 3.29E-10 3.29E-10 3.23E-10

Table VI: The RMSEs for Example 8

4l m 8l m 12l m 16l m 20l m


60

1. MFS+APS => scattered data in right-hand sides

2. MFS+Chebyshev => spectral convergence

3. Hörmander operator decomposition technique

4. Partial fraction decomposition

5. polyHelmholtz & Polyharmonic particular solutions

6. MFS for the product operator

Conclusion

61

Thank you

particular solutions for some engineering problems

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