multi-level wave-ray method for 2d helmholtz equation

Multi-Level Wave-Ray Method for 2D Helmholtz Equation

P.C. Verburg June 23 2010

§ Introduction§ Finite difference method§ Multi-level method§ Wave-Ray algorithm§ Results§ Conclusions and recommendations

23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation

Contents

2

§ Helmholtz equation describes waves.§ Applications in acoustics, mechanics, thermodynamics and electrical

engineering.§ This research: acoustics.

23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation 3

IntroductionHelmholtz equation

§ Noise pollution is an important problem in daily life.§ Tool is needed to evaluate sound-mitigation measures.§ Generally no analytical solution is available.§ Therefore numerical methods are required.


IntroductionNumerical methods

4

Source: Rijkswaterstaat

§ Sound waves cause small-amplitude perturbations:

§ No external force fields/heat sources.§ No viscosity and heat conduction.§ Isentropic flow.§ Calorically perfect gas.

§ Governing equations for mass, momentum and energy conservation are applied.


IntroductionSound waves

00

000

0 '

ρρ pp' u +ρ ρ+p' pp

<<<<===

rrρ

5


IntroductionSeparation of variables

( ) ( ) ιωtexutxu ˆ, rr=

Space Time

§ Helmholtz equation describes the space-dependence of sound waves.

6

Wave equation:Standing wave

Separation of variables:

uctu 222

2

∇=∂∂


IntroductionHelmholtz equation

fuku =+∇ 22§ Helmholtz equation (elliptic PDE):

§ Describes space-dependence of pressure/density/velocity perturbations of standing waves.

§ Wavenumber k: frequency of the waves in the solution (this research: constant k).

§ Difficult to solve numerically for many wave-periods in domain.

7

λπ2

=k



Contents

8


Finite difference methodIntroduction

uku 22 +∇

hji

y

hji

hji

x

hji

hji

hji

jih uk

huuu

huuu

uL ,2

21,,

h1-ji,

2,1,,1

,

22+

+−+

+−= ++−

§ Left-hand side of Helmholtz equation:

§ Discrete operator:

§ Second-order accurate.

9


Finite difference methodSingle grid iterative solver

§ Residual:

§ Difference between right-hand side of equation and discrete operator applied to approximate solution.

§ Converged solution:

§ Iterative solver: adjust solution point by point such that the residual is reduced or eliminated.

§ Sweep across all points: relaxation.

ji

hhhji

hji uLfr

,,,~−=

Ω∈∀= jir hji , 0,

10


Finite difference methodSingle grid iterative solver - 1D demonstration

02

2

=dx

ud

Computation time for 2D situations:

( )2NΟ

Laplaceequation:

11

§ Single grid iterative solvers are unable to remove components with a low frequency/large wavelength relative to the grid.

§ Such components are insensitive to local relaxations because only a small part of the wave is taken into account by the discrete operator.


Finite difference methodConclusion

12



Contents

13


Multi-level methodIntroduction

§ Multi-level principle: represent every frequency component in the error on a scale at which it can be accurately described and efficiently solved.

§ Multi-level solvers have been successfully applied to several ellipticPDE’s such as the Laplace equation.

§ Ideally, computation time: ( )NΟ

14


Multi-level methodV-cycle

Mesh size

h=H

h=2H

h=4H

Level

3

2

1

§ Highest level should represent solution with the desired accuracy.

§ Lower levels are a means to accelerate the convergence speed.

§ Components with a low frequency relative to a fine grid have a high frequency relative to a coarse grid.

15


Multi-level methodMulti-level solver - 1D demonstration

02

2

=dx

ud

Computation time for 2D situations:

( )NΟ

Laplaceequation:

16

§ A multi-level solver should be able to efficiently remove all frequency components in the error to obtain required performance.

§ However, standard multi-level methods fail to remove errors with a frequency close to the solution for the Helmholtz equation.

§ Characteristic components


Multi-level methodApplication to the Helmholtz equation

222 kkk yx ≈+

17


Multi-level methodRemoving characteristic components on fine grids

§ Locally the solution satisfies the discrete operator.§ Low frequency relative to the grid.

Ratio between error before/after relaxation:

222 kkk yx ≈+

18

§ A phase error always occurs when discretising a continuous problem.§ Phase error accumulates with every wave-period in the domain.§ Phase error becomes larger on coarse grids.

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Multi-level methodRemoving characteristic components on coarse grids

§ Characteristic components in the error with a frequency close to the solution cannot be removed on both fine and coarse grids.

§ To make the multi-level approach succeed for the Helmholtz equation, a separate treatment is necessary for the characteristic components.

§ Multi-level cycle for Helmholtz equation: wave-cycle.


Multi-level methodConclusion

222 kkk yx ≈+

20



Contents

21

§ Wave-cycle shows good performance except for characteristic frequencies.

§ Performance of wave-cycle should be retained for other frequencies.

§ Return to principle behind multi-level method: describe all components in the error in a way that allows efficient removal.

§ Brandt and Livshits: characteristic components are described separately for each wave-propagation direction (wave-ray method).


Wave-Ray algorithmIntroduction

22

§ Frequency of characteristic components is known.§ Wave propagation direction (ray) is fixed.§ Therefore only the amplitude has to be described.§ Amplitude changes very gradually, coarse grid is sufficient, no phase

errors.


Wave-Ray algorithmIntroduction

23

§ For a numerical method the number of rays should be limited.§ Wave-ray method uses 8 rays.§ 90% correction.


Wave-Ray algorithmRay-directions

2

6

5

4

3

7

1

0

θ2

θ1

24


Wave-Ray algorithmRay equations

rfuku −=+∇ ~~ 22

rvkv =+∇ 22

vuu += ~

with r the residual

25

§ However, is just as difficult to solve.§ Therefore a new description of v and r is needed.§ Represent characteristic components separately for each wave

propagation direction:

§ ξ: wave propagation direction

§ Characteristic components are represented by smooth envelope functions, therefore no phase error.


Wave-Ray algorithmRay equations

rvkv =+∇ 22

( )∑=

=8

1

,i

ki eav ξιηξ ( )∑

=

=8

1 ,

i

ki err ξιηξ

( ) ( ) ( ) ( )ηξηξξ

ιηξη

ηξξ

,,2,, 2

2

2

2

iiii raaa =∂∂

+∂∂

+∂∂

26


Wave-Ray algorithmOutline of the solver

Run standard multi-level solver (wave-cycle)

Split residual into different wave propagation directions

Solve ray equationsCorrect wave-cycle solution

27

§ Separation procedure for the wave-cycle residual:

§ Solve ray equations:

§ Merge corrections:


Wave-Ray algorithmSeparation and merging

( )x,yr

( ) ( ) ( ) ( )ηξηξξ

ιηξη

ηξξ

,,2,, 2

2

2

2

iiii raaa =∂∂

+∂∂

+∂∂

( )∑=

8

1i

ιkξi eξ,ηr

( ) ( ) ( )∑=

+=8

1 ,,~,

i

ki eayxuyxu ξιηξ

28


Wave-Ray algorithmSeparation and merging

Rotate Add correction

Separation/ solve ray equationResidual Rotate

29

§ Ray equations are solved on a larger domain than the original problem.§ Allows Sommerfeld condition to be introduced: waves that propagate

through the domain without obstructions.§ Near-field solution can be obtained without computing far-field solution.


Wave-Ray algorithmBoundary conditions

30

Wave boundaries

Ray boundaries



Contents

31

0100

200300

400500

600

0

200

400

600-15

-10

-5

0

5

10

15

20

ij

Re(

u)


ResultsSound-sources outside the domain

§ Solution of problem with 16 plane wave sources outside domain.

32

Grid: 512x512


ResultsConvergence speed

§ Convergence speed remains constant up to machine accuracy.§ About 25 wave-ray cycles required for full convergence.

0 10 20 30 40 5010-15

10-10

10-5

100

Number of cycles

L2 N

orm

of t

he re

sidu

al

33

Standardmulti-level

Wave-Rayalgorithm



§ Convergence speed independent of number of grid points.§ computation time to solve discrete Helmholtz problem.( )NΟ

0 5 10 15 20 25 30 35 40 45 5010-14

10-12

10-10

10-8

10-6

10-4

10-2

Number of Wave-Ray cycles

L2 N

orm

of t

he re

sidu

al

N=512x512N=1024x1024N=2048x2048

34

0 10 20 30 40 5010-15

10-10

10-5

100

Number of Wave-Ray cycles

L2 N

orm

of t

he re

sidu

al

kd=32kd=64kd=128



§ Convergence speed independent of number of wave-periods in domain.§ kd: product of wavenumber and domain size.

35

0100

200300

400500

600

0

200

400

600-1

-0.5

0

0.5

1

1.5

ij

Re(

u)


ResultsSound-sources inside the domain

§ Solution of problem with a line source inside the domain.§ Identical performance compared with sources outside the domain.

36

Line source



Contents

37

§ A very efficient multi-level solver has been developed for the 2D Helmholtz equation.

§ Several types of boundary conditions have been introduced.

§ It is expected that the wave-ray method can be developed into a solver suitable for applications in engineering.


Conclusions and recommendationsConclusions

38

§ Investigate how the wave-ray solver can deal with additional boundary conditions.

§ Extend the solver from 2D to 3D.

§ Adapt method to problems with strongly varying wavenumbers.

§ Further investigate accuracy of the wave-ray method.


Conclusions and recommendationsRecommendations

39

Thank you for your attention

multi-level wave-ray method for 2d helmholtz equation

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