3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/...

3rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004

Plane Wave Basis Boundary Elements and Finite Elements for Wave Scattering

Problems

Peter Bettess

Emmanuel Perrey-Debain

Omar Laghrouche

Jon Trevelyan

Joe Shirron

School of Engineering

University of Durham


ResearchersCurrent Affiliations

Peter Bettess University of Durham

Omar Laghrouche Heriot-Watt University, Edinburgh

Emmanuel Perrey-Debain UMIST, Manchester

Joe Shirron Metron, Inc, Virginia, U.S.A.

Jon Trevelyan University of Durham


Durham Cathedral


Background

I do not propose to survey the extensive literature.

Two recent volumes give an introduction to the field.


1. Introduction

2. Mathematical formulation for the 2D Helmholtz problem

3. Conditioning and Singular Value Decomposition

4. Numerical results, convergence and accuracy analysis

5. The 3D Helmholtz problem

6. The 2D elastodynamic problem

7. Conclusions and prospects

Presentation topics


1. Introduction

Motivation


1. Introduction

Volume discretization scheme

• Partition of Unity Method (Babuška and Melenk 1997, Laghrouche et al. 2002)

• Least-Squares (Stojek 1998, Monk and Wang 1999)

• Ultra Weak Formulation (Cessenat and Després 1998, Huttunen et al. 2002)

• Discontinuous Enrichment Method (Farhat et al. 2002)

What about the use of plane waves ?What about the use of plane waves ?

Surface discretization scheme

• Micro-local discretization (de La Bourdonnaye 1994)

• Wave Boundary Elements (Perrey-Debain et al. 2002, 2003)

• Specific use of plane waves for high frequency scattering problems in

(Abboud et al. 1995, Bruno et al. 2003, Chandler-Wilde et al. 2003)


2. Mathematical formulation…

Problem: we consider a two dimensional obstacle of general shape in an infinite propagative medium impinged by a incident time-harmonic wave inc

We seek the potential in as the solution of the Helmholtz equation:

The integral formulation reads

where is the free-space Green function

(1)



Geometry

Boundary conditions

( )

Incident wave field



Plane wave basis function

We can write the solution of (1) in the compact form:

where the Q plane waves propagate in various directions evenly distributed over the unit circle:

Note: Continuity requirement leads to N=2nQ degrees of freedom



Numerical implementation

Matrix system

Plane wave coefficients

Plane wave interpolation matrix (sparse)

Boundary matrix (full)


3. Conditioning and SVD

Ideal case, plane wave interpolation on the real line

The interpolation matrix reads

where denotes the discretization level (DOF per wavelength)

and is the sampling rate

Computed case, we define the average discretization level by

FFT matrix when =2



Condition number (2-norm) vs discretization level

Computed

(: unit circle, =0, N=2nQ=64)

Ideal case

FFT matrix



Consider the unit circle

and a regular subdivision

with and

Then the analytical solution can be expanded as

where



Test case: scattering by the unit circle with =100 and =-100i

The QR solver is used for all except these two most ill-conditioned cases for which SVD is used with =10-12

Q=5

Q=10

Q=25

Quadratic



12

1 2



Incident plane wave at 45o

Water wave-structure interaction (a=1.7).

Discretization: n=2 elements and Q=16 wave directions



Test case: scattering by a 50-width boomerang-shaped obstacle



Problem statement and notation


Finite element discretization

Plane wave basis

kK

k

1



8 triangular patches and 6 vertices are sufficient to describe the scatterer




Scattering of a vertical plane wave by the unit sphere

Integral formulation

Finite element formulation

(From O. Laghrouche)


(0,0,1)

(1,0,1)/2

Iluminated zone Shadow zone

Re ()


Scattering by a thin plate (=3.1)


Ellipsoid (20 x 4 x 4) – Far Field Pattern

Convergence reached with N=1308 variables (=2.65)



Distorted scatterer geometry (=2.9)

(0,0,1)(1,0,1)/2

(1,0,0)

||



6. The 2D elastodynamic problem


13. Conclusions and prospects

Positive aspects of the `wave boundary element method’

• Not restricted to a specific problem

• Possible extension of the method to other wave problems

• Only approximately 2.5 - 3 variables per wavelength are required

• Can provide extremely accurate results

Drawbacks

• Ill-conditioned matrices require careful integration procedure and the use

of appropriate solvers like truncated SVD

• Matrix coefficients evaluation is time-consuming

In Prospects

• Speed up the numerical integration

• Investigate the Galerkin formulation for BE

• Find good preconditioners for iterative algorithms


Motivation: Solve short wave problems where L/ >> 1

Applications: geophysics (hydro-carbon exploration), coastal and earthquake waves, acoustic and electromagnetic scattering, …

7. Finite elements for Short Wave Modelling


Aim: Develop wave finite elements capable of containing many wavelengthsper nodal spacing

Applications: Problems involving large boundaries and/or short wavelengths

frequency

Conventionalmethods

Ray theory& SEA?

Objective - Applications


Idea: Include the wave character of the wave field in the element formulation

Conventional Plane wave basis

9. Formulation of the plane wave finite elements


Potential around the cylinder, ka=24

10. Wave scattering in 2D


Reduction in dof ~ 1/15

10. Wave scattering in 2D


Finite elementsBoundary elements

Reduction in dof ~ 1/50

11. Wave scattering in 3D (parallel coding)


12. Wave scattering in heterogeneous media


k1=10, k2=2, = 4.2, 2=0.07% k1=10, k2=6, = 3.4, 2=0.4%

12. Wave scattering in heterogeneous media


Wave refraction due to wave speed changes


Special numerical integration schemefor plane wave basis finite elements

tetrahedra


Special integration scheme for plane wave basis f.e.

Evaluate integrals of the form:

Typically p(x) is of the following form:

Scheme:


Gordon’s integration scheme

Applying twice the divergence theorem

1

2

3


Gordon’s integration scheme

Evaluate a surface integral of the form

Applying again the divergence theorem


Gordon’s integration scheme (cont.)

1

2

3

S

The integral becomes

Linear parametric representation of the nth edge of S

with w* is obtained by rotating w by 90o and x’(t) means the derivative of x(t) with respect to t.



The contribution of the nth edge of S



Finally

The method above explains the integration of the plane wave itself.The higher order polynomial terms can be evaluated using Shirron’s Method


Shirron’s integration scheme

Consider that we have already evaluated the integral

Derivation of W000 with respect to k1

Generalization for any terms


Singular cases

Three singular cases could arise:

1. Wave normal to an edge (w . a = 0)then the term (sinx / x) is replaced with the series approximationsbut still integrated using Gordon’s formula.

2. Wave normal to a face (wj = 0)

a - if the local wave number is very close zero (wave almost normalto a face), the Gordon’s formula is replaced with series approximations.

b - the local wave number is equal to zero, the term eiw.x is replaced with series approximations.

2. Case of no wave (k = 0)there is no more trigonometric functions to integrate but only polynomials.Gauss-Legendre integration scheme is used.


Testing the procedure


Timing results

Tetrahedron element with 4 directions at each node


Timing results


Special numerical integration schemefor plane wave basis finite elements

rectangles


Computing the weights - Shirron

We expand the integrand in Legendre polynomials then the integration weightsWould have the form

Using the following

The result is


Integration routine in 2D


Integration routine in 2D (cont.)


Routine: spherical Bessel functions


Problems with a simple square domain


Results from cylinder diffraction problem

Results using analytically integrated wave finite elements


Results from cylinder diffraction problem

Results using numerically integrated wave finite elements


Adaptivity scheme forplane wave basis

boundary elements


How many plane waves per node?

We have found that the accuracy and efficiency are dependent on the number of plane waves (more so than their directions).

It is generally better to accumulate degrees of freedom by using:

few elements with lots of plane waves per node

than

many elements with few plane waves per node


Prospects for adaptivity

But can the plane wave directions be chosen more selectively?

Adaptivity

Use of an error indicator in combination with model improvement and reanalysis to convergence.

In this context we can start with a ‘coarse’ model and progressively add plane waves where they are most needed to gain accuracy.

Apart from some special cases, using fewer than 2 DOF per wavelength causes numerical instability, so this defines the lower bound for our coarse starting point.


The error indicator

The BEM is a collocation method forming equations by setting the error in the governing integral equation to zero at discrete points.

)(d)(

),(d)(),(

2

)(x

yyxy

yxxyy in

Gn

G

This provides a matrix equation that can be solved, in this case for the plane wave amplitudes.

The error indicator we use is a consideration of the same integral equation by considering new collocation points not in the initial set.

Model of a circular scatterer with boundary .


The error indicator

The error indicator R(x’) is simply a measure of how well the integral equation is satisfied at a ‘collocation’ point x’.

)(d)(

),(d)(),(

2

)(x

yyxy

yxxyy in

Gn

G

)(d)(

),(

d)(),(

2

)(

1)(

x'y

yx'

yyx'x'

x'

y

y

inG

n

G

AR

We normalise and non-dimensionalise by dividing by A, the amplitude of the incident wave.


Scattering from circular cylinder

Now plot the error indicator R(x’) against angle and observe form of errors.

Incident wave



Now plot the error indicator R(x’) against angle and observe form of errors.

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

90 95 100 105 110 115 120 125 130 135

Angle (deg)

R( x

')

For the case

ka = 50

8 elements

10 plane waves/node

giving efficiency

= 3.2 DOF/wavelength

Notice how R(x’) becomes very small at the collocation points in the original set.


Error indicator – properties

The error indicator R(x’) exhibits useful properties:

It seems to act as a good global error indicator. If it generally takes the value of order 10-3 or less then accuracy should be at least a good engineering accuracy.

It tends to be noticeably higher towards the element ends than in the middle of the element.

It has useful local properties that we can use as a guide to the adaptive model improvement.



Now we look at a simple adaptive analysis – scattering from circular cylinder.

For the case

ka = 150

20 elements

8 plane waves/node

giving efficiency

= 2.13 DOF/wavelength1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 18 36 54 72 90 108 126 144 162 180

Angle (deg)

R(x

')

Decide to add an extra plane wave here

Initial L2 error in potential around the boundary = 7.7%.



Results of the first adaptive step.

For the case

ka = 150

20 elements

1st adaptive step

with efficiency


1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 18 36 54 72 90 108 126 144 162 180

Angle (deg)

R(x

')




Jumping forward to the third adaptive step….

For the case

ka = 150

20 elements

3rd adaptive step

with efficiency


1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 18 36 54 72 90 108 126 144 162 180

Angle (deg)

R(x

')




Jumping forward to the fifth and final adaptive step….

For the case

ka = 150

20 elements

5th adaptive step

with efficiency


1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 18 36 54 72 90 108 126 144 162 180

Angle (deg)

R(x

' )

With error indicator largely < 10-3, decide to stop at this point. L2 error around boundary in potential results = 0.40%.



As a comparison, run a single analysis with uniform 9 plane waves/node giving the same = 2.40.

For the case

ka = 150

20 elements

9 plane waves/node

giving efficiency


1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 18 36 54 72 90 108 126 144 162 180

Angle (deg)

R(x

' )

Notice how the errors are significantly greater than the final adaptive step with same DOF. L2 error around boundary in potential results = 1.8%.


Conclusions

An adaptive scheme is proposed based on error indicator R(x’) taking the form of the residual of the governing integral equation at a ‘new’ collocation point.

The error indicator has useful global properties allowing it to be used as an effective stopping criterion.

Initial tests show the scheme to converge to a set of plane waves that offer more efficiency than a uniform distribution.

It also has useful local properties that we can use as a guide to the adaptive model improvement.


The problem

In 3D analysis using a plane wave basis we wish to define a set of wave

We feel that this is likely to give a more efficient solution (though there may be

directions that are as evenly spaced as possible.

cases in which we want to cluster wave directions).


Existing algorithm

Currently we define the wave directions using a cube with a boundary ‘mesh’

uniformly defined on the cube surface.


Existing algorithm



The vectors from the cube centre to the points give a reasonably distributed set.


Existing algorithm



The vectors from the cube centre to the points give a reasonably distributed set.

Disadvantage:

Limited to a few numbers of directions of the form:

n3 (n – 2)3


New algorithm

A new algorithm is presented which allows rapid determination of ‘almost’ evenly spaced wave directions for arbitrary numbers of waves.

Based on repulsion of charged particles.

Coulomb forces between charged particles are of the form:

F =

where q1 and q2 are the two charges and r is the distance between them.

We consider each wave direction to be represented by a particle of unit mass and unit charge on the surface of a unit sphere, and find a static equilibrium state.

r2

q1q2


New algorithm

For each particle we…

Sum to find the resultant of the Coulomb force vectors

Find the projection of that force vector in the sphere

Include a damping term to derive a net force on the particle

Determine its acceleration, velocity and new position

Relocate it back onto the spherical surface

Repeat in an explicit time-stepping scheme until convergence


New algorithm

Mathematically:

n

jtj

ti

tj

tiijt

i AF1

3

))(1(

ti

ti

ti ucfu

tuuu ti

ti

tti

tu tti

ti

tti ̂

Position of particle i at time t

Force scaling parameter

Equivalent damping coefficient

A set of parameters that works well:

A = 100

c = 10

t = 0.01 (n < 100)

= 0.001 (n > 100)NB: no stiffness term is included

tti

ttitt

i

ˆ

ˆ

ti

ti

ti

ti Ff )(


Graphical display

On the surface of a sphere we display in blue circles the directions on the near hemisphere and in orange the directions on the hidden hemisphere.

Example: 74 directions


Evaluation of algorithm

A convenient measure of the algorithm is the minimum angle between any two directions in the set.

jinjniji

ji ;1,..., ;1,..., .

cosmin 1


Comparison with earlier algorithm

5 x 5 grid of points (as shown) on each face = 98 directions:

Cube method:

Min. angle = 15.8º

Charged particle method:

Min. angle = 20.4º

4 x 4 grid = 56 directions:

Cube method:

Min. angle = 22.0º

Charged particle method:

Min. angle = 26.6º


Biasing of directions

By placing a large positive or negative charge on or near the sphere we can attract or repel the particles.

Point charge of strength -0.4 x sum of particle charges.

NB. Negative attracts

Positive repels

Possible use in far field FE to cluster directions around the radial direction, or in adaptive scheme.

Example : 30 directions


Conclusions

A new scheme is proposed that efficiently determines a set of wave directions approximately evenly spaced on the unit sphere.

Waves are considered as particles of unit mass and unit charge on the surface of a unit sphere

Tests show the new scheme to give a more even spread of directions than the existing algorithm, and importantly applies for an arbitrary number of directions

A static equilibrium position is found through a simple time-stepping scheme

Wave directions can be clustered around a dominant wave direction


Personal comments in conclusion

• I retire at the end of September, 2004

• This is my last presentation

• In my last conference

• If any part of my talk was of interest, you can e-mail me for reprints, lists of publications etc. at: [email protected]


I plan to spend a lot more time in the mountains


I plan to spend a lot more time on my bike


I plan to relax more


I plan to do a lot less of this

3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/...

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