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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.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
IntroductionSound waves
00
000
0 '
ρρ pp' u +ρ ρ+p' pp
<<<<===
rrρ
5
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
∇=∂∂
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
§ 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
8
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
Finite difference methodConclusion
12
§ 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
13
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
Multi-level methodApplication to the Helmholtz equation
222 kkk yx ≈+
17
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation 19
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.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
Multi-level methodConclusion
222 kkk yx ≈+
20
§ 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
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).
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
Wave-Ray algorithmIntroduction
23
§ For a numerical method the number of rays should be limited.§ Wave-ray method uses 8 rays.§ 90% correction.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
Wave-Ray algorithmRay-directions
2
6
5
4
3
7
1
0
θ2
θ1
24
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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:
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
Wave-Ray algorithmBoundary conditions
30
Wave boundaries
Ray boundaries
§ 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
31
0100
200300
400500
600
0
200
400
600-15
-10
-5
0
5
10
15
20
ij
Re(
u)
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
ResultsSound-sources outside the domain
§ Solution of problem with 16 plane wave sources outside domain.
32
Grid: 512x512
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
ResultsConvergence speed
§ 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
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
ResultsConvergence speed
§ 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)
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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
§ 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
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.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
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.
23-06-2010Multi-Level Wave-Ray Method for 2D Helmholtz Equation
Conclusions and recommendationsRecommendations
39
Thank you for your attention