1. 3 rd 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

2. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Researchers Current 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 TrevelyanUniversity of Durham

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

4. Durham Cathedral

5. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004

7. Background I do not propose to survey the extensive literature. Two recent volumes give an introduction to the field.

8. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004

13. 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

14. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 1. Introduction Motivation

15. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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 ? 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)

16. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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)

17. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 2. Mathematical formulation… Geometry Boundary conditions ( ) Incident wave field

18. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 2. Mathematical formulation… 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

19. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 2. Mathematical formulation… Numerical implementation Matrix system Plane wave coefficients Plane wave interpolation matrix (sparse) Boundary matrix (full)

20. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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

21. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 3. Conditioning and SVD Condition number (2-norm) vs discretization level  Computed (  : unit circle,  =0, N=2nQ=64) Ideal case FFT matrix

22. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 3. Conditioning and SVD

23. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 3. Conditioning and SVD

24. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 4. Numerical results, convergence and accuracy analysis Consider the unit circle and a regular subdivision with and Then the analytical solution can be expanded as where

25. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 4. Numerical results, convergence and accuracy analysis 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

26. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 4. Numerical results, convergence and accuracy analysis 1212 1 2

27. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 4. Numerical results, convergence and accuracy analysis 

28. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 4. Numerical results, convergence and accuracy analysis Incident plane wave at 45 o Water wave-structure interaction (  a=1.7). Discretization: n=2 elements and Q=16 wave directions

29. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 4. Numerical results, convergence and accuracy analysis Test case: scattering by a 50 -width boomerang-shaped obstacle

30. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 5. The 3D Helmholtz problem Problem statement and notation

31. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Finite element discretization Plane wave basis 5. The 3D Helmholtz problem

32. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 8 triangular patches and 6 vertices are sufficient to describe the scatterer 5. The 3D Helmholtz problem

33. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 5. The 3D Helmholtz problem Scattering of a vertical plane wave by the unit sphere Integral formulation Finite element formulation (From O. Laghrouche)

34. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 (0,0,1) (1,0,1)/  2 Iluminated zone Shadow zone Re (  ) 5. The 3D Helmholtz problem Scattering by a thin plate (  =3.1)

35. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Ellipsoid (20 x 4 x 4 ) – Far Field Pattern Convergence reached with N=1308 variables (  =2.65) 5. The 3D Helmholtz problem

36. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Distorted scatterer geometry (  =2.9) (0,0,1) (1,0,1)/  2 (1,0,0) |||| 5. The 3D Helmholtz problem

37. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 6. The 2D elastodynamic problem

38. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 6. The 2D elastodynamic problem

39. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 6. The 2D elastodynamic problem

40. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 6. The 2D elastodynamic problem

41. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 6. The 2D elastodynamic problem

42. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 6. The 2D elastodynamic problem

43. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 6. The 2D elastodynamic problem

44. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 6. The 2D elastodynamic problem

45. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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

46. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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

47. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Aim: Develop wave finite elements capable of containing many wavelengths per nodal spacing Applications: Problems involving large boundaries and/or short wavelengths frequency Conventional methods Ray theory & SEA ? Objective - Applications

48. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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

49. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Potential around the cylinder, ka=24 10. Wave scattering in 2D

50. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Reduction in dof ~ 1/15 10. Wave scattering in 2D

51. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Finite elements Boundary elements Reduction in dof ~ 1/50 11. Wave scattering in 3D (parallel coding)

52. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 12. Wave scattering in heterogeneous media

53. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 k 1 =10 , k 2 =2 ,  = 4.2,  2 =0.07%k 1 =10 , k 2 =6 ,  = 3.4,  2 =0.4% 12. Wave scattering in heterogeneous media

54. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Wave refraction due to wave speed changes

55. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Special numerical integration scheme for plane wave basis finite elements tetrahedra

56. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Special integration scheme for plane wave basis f.e. Evaluate integrals of the form: Typically p(x) is of the following form: Scheme:

57. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Gordon’s integration scheme Applying twice the divergence theorem 1 2 3

58. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Gordon’s integration scheme Evaluate a surface integral of the form Applying again the divergence theorem

59. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Gordon’s integration scheme (cont.) 1 2 3 S The integral becomes Linear parametric representation of the n th edge of S with w* is obtained by rotating w by 90 o and x’(t) means the derivative of x(t) with respect to t.

60. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Gordon’s integration scheme (cont.) The contribution of the n th edge of S

61. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Gordon’s integration scheme (cont.) Finally The method above explains the integration of the plane wave itself. The higher order polynomial terms can be evaluated using Shirron’s Method

62. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Shirron’s integration scheme Consider that we have already evaluated the integral Derivation of W 000 with respect to k 1 Generalization for any terms

63. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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 approximations but still integrated using Gordon’s formula. 2.Wave normal to a face (w j = 0) a - if the local wave number is very close zero (wave almost normal to a face), the Gordon’s formula is replaced with series approximations. b - the local wave number is equal to zero, the term e iw.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.

64. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Testing the procedure

65. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Timing results Tetrahedron element with 4 directions at each node

66. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Timing results

67. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Special numerical integration scheme for plane wave basis finite elements rectangles

68. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Computing the weights - Shirron We expand the integrand in Legendre polynomials then the integration weights Would have the form Using the following The result is

69. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Integration routine in 2D

70. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Integration routine in 2D (cont.)

71. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Routine: spherical Bessel functions

72. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Problems with a simple square domain

73. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Results from cylinder diffraction problem Results using analytically integrated wave finite elements

74. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Results from cylinder diffraction problem Results using numerically integrated wave finite elements

75. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Adaptivity scheme for plane wave basis boundary elements

76. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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

77. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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.

78. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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. 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 .

79. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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’. We normalise and non-dimensionalise by dividing by A, the amplitude of the incident wave.

80. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Scattering from circular cylinder Now plot the error indicator R(x’) against angle  and observe form of errors.  Incident wave

81. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Scattering from circular cylinder 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 9095100105110115120125130135 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.

82. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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.

83. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Scattering from circular cylinder 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/wavelength 1.00E-03 1.00E-02 1.00E-01 1.00E+00 01836547290108126144162180 Angle  (deg) R ( x' ) Decide to add an extra plane wave here Initial L 2 error in potential around the boundary = 7.7%.

84. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Scattering from circular cylinder Results of the first adaptive step. For the case ka = 150 20 elements 1 st adaptive step with efficiency  = 2.20 DOF/wavelength 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 01836547290108126144162180 Angle  (deg) R ( x' ) Decide to add an extra plane wave here

85. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Scattering from circular cylinder Jumping forward to the third adaptive step…. For the case ka = 150 20 elements 3 rd adaptive step with efficiency  = 2.29 DOF/wavelength 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 01836547290108126144162180 Angle  (deg) R ( x' ) Decide to add an extra plane wave here

86. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Scattering from circular cylinder Jumping forward to the fifth and final adaptive step…. For the case ka = 150 20 elements 5 th adaptive step with efficiency  = 2.40 DOF/wavelength 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 01836547290108126144162180 Angle  (deg) R ( x' ) With error indicator largely < 10 -3, decide to stop at this point. L 2 error around boundary in potential results = 0.40%.

87. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Scattering from circular cylinder 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  = 2.40 DOF/wavelength 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 01836547290108126144162180 Angle  (deg) R ( x' ) Notice how the errors are significantly greater than the final adaptive step with same DOF. L 2 error around boundary in potential results = 1.8%.

88. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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.

89. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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).

90. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Existing algorithm Currently we define the wave directions using a cube with a boundary ‘mesh’ uniformly defined on the cube surface.

91. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Existing algorithm Currently we define the wave directions using a cube with a boundary ‘mesh’ uniformly defined on the cube surface. The vectors from the cube centre to the points give a reasonably distributed set.

92. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Existing algorithm Currently we define the wave directions using a cube with a boundary ‘mesh’ uniformly defined on the cube surface. 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: n 3  (n – 2) 3

93. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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 q 1 and q 2 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. r2r2 q1q2q1q2

94. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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

95. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 New algorithm Mathematically: 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

96. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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

97. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 Evaluation of algorithm A convenient measure of the algorithm is the minimum angle between any two directions in the set.

98. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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

99. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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

100. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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

101. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 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: peter.bettess@durham.ac.uk

102. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 I plan to spend a lot more time in the mountains

103. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 I plan to spend a lot more time on my bike

104. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 I plan to relax more

105. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004 I plan to do a lot less of this

106. 3 rd International Conference On Boundary Integral Methods: Theory and Applications – LMA/ University of Reading – September 2004