1 the numerical methods for helmholtz equation 報告人：陳義麟...

1

The numerical methods for Helmholtz equation

報告人：陳義麟國立高雄海洋技術學院造船系副教授

於海洋大學河海工程系

基隆 2003/10/23

Outline

Helmholtz equation

Engineering applications

Numerical methods

Boundary integral equation method

BEM for interior and exterior problems

The Trefftz method

The MFS method

Concluding remarks

2

Helmholtz equation


Numerical methods



The Trefftz method

The MFS method

Concluding remarks

3

Helmholtz equation

Time domain

Wave equation

Dxt

txu

ctxu

,),(1

),(2

2

22

Fourier transformation

Dxxuk ,0)()( 22

Frequency domain

Helmholtz equation

4

Helmholtz equation


Numerical methods



The Trefftz method

The MFS method

Concluding remarks

5


1. Waveguide problem

2. Vibration of membranes

3. Water wave diffraction problem

4. Exterior acoustic problem

5. Elastic wave problem

6

Two-dimensional Helmholtz problem with a circular domain:

Dxxukxu ,0)()( 22G.E. :

B

D

a

is the angle along the circular domain

a is the radius of the circular domain

)(xu is the potential function

2 denotes the Laplacian operator

7

aD

B

Interior : Exterior :

Problem statement

k is the wave number

Wave equation


Numerical methods



The Trefftz method

The MFS method

Concluding remarks

8

Numerical Methods

Finite Element Method

Finite Difference Method

Boundary Element Method

mesh Methods

Meshless Methods

9

Helmholtz equation


Numerical methods



The Trefftz method

The MFS method

Concluding remarks

10

11

Dxxuk ,0)()( 22

Original system

Reciprocal work theorem

)(),()( 22 xsxsUk

Auxiliary system

e

BB

BB

Dx

Bx

Dx

Dfor

sdBstxsLsdBsuxsMxt

sdBstxsUsdBsuxsTxu

,0

,

,2

,2

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

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

UT (singular) formulation

LM (hypersingular) formulation

)(2

),( )1(0 krH

ixsU

Kernel function


12

Discrete the boundary integral equation (BEM)

}]{[}]{[ tUuT

4

3

2

1

1

23

41

2

1

2

1

)(),(11

B

BsdBxsUa

x2

x1

11

][

a

U

3 2

12a

4

3

13a

4

14a

24232221 aaaa

x3

34333231 aaaa

x4

44434241 aaaa

Influence matrix

13

Helmholtz equationEngineering applicationsNumerical methodsBoundary integral equation methodBEM for interior and exterior problems The Trefftz method The MFS methodConcluding remarks

Interior problem (Eigenproblem)

0 .00 1 .00 2 .00 3 .00 4 .00 5 .00

0 .00

0 .00

0 .00

0 .01

0 .10

1 .00

10 .00

100 .00

01J

11J

]det[ eU

k

}0{}]{[}]{[ uTtU

0||det U

For the Dirichlet B.C., u=0

To obtain the nontrivial solution

14

Field solution

}{2

1)(

)()(),()(2

tUxu

sdBstxsUxuB

Field integral equation

x

s

<U> : Influence row vector

-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

0J

-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

-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

1J

15

Exterior problem (radiation or scattering)

}]{[}]{[ tUuT

For the Neumann B.C., tt

}]{[][}{

}]{[}]{[1 tUTu

tUuT

1),( at0),( at

Drruk ),( ,0),()( 22

9

1),( at0),( at

Drruk ),( ,0),()( 22

9

Field solution}{}{)(2 tUuTxu

u(a,0)

-1 .50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

16

Fictitious frequency

0 2 4 6 8

-0.8

-0.4

0 .0

0 .4

0 .8

1 .2UT method

LM method

Burton & Miller method

Analytical solution

ka

1),( at0),( at

Drruk ),( ,0),()( 22

9

1),( at0),( at

Drruk ),( ,0),()( 22

9

For the Neumann B.C., tt

}]{[][}{

}]{[}]{[1 tUTu

tUuT

0||det T , fail at the fictitious frequency

17

Helmholtz equation


Numerical methods



The Trefftz method

The MFS method

Concluding remarks

18

Trefftz method

12

1

)()(TN

iii xuzxuField solution :

izwhere is complex type

19

where 12 TN is the number of complete functions

iz is the unknown coefficient

iu is the T-complete function which satisfies

the Helmholtz equation

T-complete set functions :

20

T-complete set

The superscripts “I” and “E” denote the interior and exterior problems, respectively.

Interior:

Exterior:

)sin()(),cos()(),(0 mkJmkJkJ mm

)sin()(),cos()(),(0 mkHmkHkH mm

N

mmmm

N

mmmm

I

mkJihg

mkJifekJifeu

1

1000

)sin()()(

)cos()()()()(),(

N

mmmm

N

mmmm

E

mkHihg

mkHifekHifeu

1

)1(

1

)1()1(000

)sin()()(

)cos()()()()(),(

For the Dirichlet B.C. u=0

By matching the boundary condition at a

21

Derivation of unknown coefficients

Field solution:Interior :

N

mmmm

N

mmmm

I

mkJihg

mkJifekJifeu

1

1000

)sin()()(

)cos()()()()(),(

N

mmmm

N

mmmm

mkaJihg

mkaJifekaJife

1

1000

)sin()()(

)cos()()()()(0

}.0{}]{[ zA

Find the eigenvalue and eigenvector

22


Field solution:Exterior :

N

mmmm

N

mmmm

E

mkHihg

mkHifekHifeu

1

)1(

1

)1()1(000

)sin()()(

)cos()()()()(),(

N

mmmm

N

mmmmm

mkaHihg

mkaHifekaHifeau

1

)1(

1

)1()1(000

)sin()()(

)cos()()()()(),(

uu For the Dirichlet B.C.

By matching the boundary condition at a

}.{][}{

},{}]{[1 uAz

uzA

12

1

)()(N

ii xuzxu

Helmholtz equation


Numerical methods



The Trefftz method

The MFS method

Concluding remarks

23

ej

N

jjj DssxUcxu

M

,),()(1

Field solution :

MN is the number of source points in the MFS

jc is the unknown coefficient

),( jsxU is the fundamental solution

eD is the complementary domain

s is the source point

x is the collocation point

Method of Fundamental Solutions (MFS)

24

B

B’

),( ax ),( Rs

R

a

D

R

B

B’

D

a

),( ax ),( Rs

Interior: Exterior:

)(2

),( )1(0 krH

ixsU

rr

Symmetry property for kernel :

),(),( xsUsxU jj .,),()(1

ej

N

jjj DsxsUzxu

M

25

Degenerate kernel :

Separable property of kernel function

RnkRJkYkiJRU

RnkJkRYkRiJRUkrH

ixsU

nnn

ne

nnn

ni

)),(cos()()]()([2

),;,(

)),(cos()()]()([2

),;,()(

2),( )1(

0

26

By matching the boundary condition for interior problem with Dirichlet B. C., u=0.

)()()()(

))(cos()(2

),(1

kRJakRYbikRJbkRYa

mkJru

mjmjmjmj

N

j

m

mm

M


0

)()()()(

))(cos()(2

),(1

kaJakaYbikaJbkaYa

mkaJu

mjmjmjmj

N

j

m

mm

M

}.0{}]{[ zA Found the igenvalue and eigenvector

27

By matching the boundary condition for exterior problem with Dirichlet B. C.

)()()()(

))(cos()(2

),(1

kJakYbikJbkYa

mkRJru

mjmjmjmj

N

j

m

mm

M


u

kJakYbikJbkYa

mkJu

mjmjmjmj

N

j

m

mm

M

)()()()(

))(cos()(2

),(1

}.{][}{

},]{[}{1 uAz

zAu

Failed at det|T|=0.

Numerical example(interior problem)

B

D

a

0 .00 1 .00 2 .00 3 .00 4 .00 5 .00

0 .00

0 .00

0 .00

0 .01

0 .10

1 .00

10 .00

100 .00

01J

11J

]det[ eU

k

BEM

1 2 3 4 5

k

-24

-20

-16

-12

-8

-4

0

|d e t[U ]|

Tre fftz m e thod fo r in te rio r p rob lem(1 3 POIN TS)

2.405

3.832

Trefftz -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

0J

-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

-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

1J

281 2 3 4 5

k

-36

-34

-32

-30

-28

|d e t[U ]|

M FS fo r in te rio r p ro b le m

2.408 3.838

Numerical example(exterior problem)

0 2 4 6 8 10

-12

-8

-4

0

4

8

BEM MFS

k

t

Fig.3 The contour plot for the real-part solutions.

Drruk ),(,0),()( 22

4cos),( au

Γ

Radiator

29

?Trefftz

Multiply domain ?

Degenerate boundary ?Degenerate boundary ?Degenerate scale ?

Further research

30

Helmholtz equation


Numerical methods



The Trefftz method

The MFS method

Concluding remarks

31

Concluding Remarks

1. The three numerical methods have been demonstrated.

2. The BEM has the mesh concept, the others are meshless.

3. The BEM and MFS adopted the fundamental solution as a kernel function and basis function, respectively.

4. The Trefftz method adopted the T-complete set as a basis function.

5. The drawbacks of those numerical methods are the objective of research.

32

The EndThanks for your

attention

33

Degenerate kernel (step1)

34

Step 1

~~ln)ln(),( xsrxsU

S

x

rx: variable

s: fixed

Rm

R

mRU

m

me

1)),(cos()(

1)ln(),,,(

Rm

RmRRU

m

mi

1)),(cos()(

1)ln(),,,(

Degenerate kernel (step2, step3)

35

x

s

A

B

Step 2

RA

iUeU

AStep 3

x

B

A

s

RB

iU

eUB

1 the numerical methods for helmholtz equation 報告人：陳義麟...

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