Acoustical scattering from N spheres using

a multilevel fast multipole method

Nail A. Gumerov Ramani Duraiswami

Institute for Advanced Computer StudiesUniversity of Maryland at College Parkwww.umiacs.umd.edu/~gumerovwww.umiacs.umd.edu/~ramani

This study has been supported by NSF

Outline! Introduction! Problem Formulation! Method of Solution

! Multipole Reexpansion (T-matrix) Method! Iterative Methods! Fast Multipole Method

! Results of Computations! Conclusion

Introduction

Multiple Scattering Problems! Sound propagation in disperse media

(particles, bubbles, etc.)! Modeling of scattering from environment

(humans, animals, fish, etc.)! Electromagnetic scattering problems

(microwaves, optics, etc.)! Efficient parametrization in inverse problems

(tomography, etc.)

Introduction

Why Multipole Methods?

Introduction

BEM Mesh5402 nodes10800 elementsDiscretization # Dn=30

Can be usedto compute the fieldonly for ka < 25 (for human head < 16.5 kHz)

Run Time for one frequencyon Dual Processor1 GHz Pentium III ~ 1 day.

Required Maximum Frequency to Compare with ExperimentalHRIR (22 or 44 kHz), 200 frequencies

Formal Requirement:ka << Dn

Scattering computation with BEM for a single object

Why Multipole Methods?

! Meshless for spherical scatterers

! Fast

! Needs Mesh! Relatively Slow

Introduction

Multipole Methods BEM

Problem Formulation

Equations and Boundary Conditions

Helmholtz Equation

Impedance Boundary Conditions

Field Decomposition

Sommerfield Radiation Condition

4

2

1

6

5

3

Incident Wave

Formulation

Wave EquationFourierTransform

Multipole Reexpansion(T-Matrix) Method

Scattered Field Decomposition

T-Matrix Method

Expansion Coefficients

Singular Basis Functions Hankel Functions

Spherical Harmonics

Vector Form:

dot product

Incident Field Decompositionand T-matrix for a Single Sphere

T-Matrix Method

Regular Basis Functions Bessel Functions

Analytical Solution of the Problem:

T-matrix

Solution of Multiple Scattering Problem

T-Matrix Method

4

2

1

6

53

Incident Wave

Scattered Wave

Coupled System of Equations:

(S|R)-TranslationMatrix

�Effective� Incident Field

Reexpansions/Translations

T-Matrix Method

q

p

M

O

rprq

r�pr�q

r�pq

r q

p

M

O

rprq

r�pr�q

r�pq

r

Two Spheres: Convergence with Respect to Truncation Number

T-matrix Method

-30

-20

-10

0

10

0 10 20 30 40 50Truncation Number

HR

TF (d

B)

ka1=30

5110

20

Two spheres,θ1 = 60o, φ1 = 0o,rmin/a1 = 2.3253.

Three Spheres Comparisons ofBEM & MultisphereHelmholtz

T-matrix Method

BEM: 5184 triangular elementsMH: Ntrunc = 9 (100 coefficients for each sphere)

-12

-9

-6

-3

0

3

6

9

12

-180 -90 0 90 180

Angle φ1 (deg)

HR

TF (d

B)

BEMMultisphereHelmholtz

θ1 = 0o

30o

60o

90o

60o

30o

90o

120o

120o

150o

150o

180o

Three Spheres, ka1 =3.0255.

Conclusions on T-matrix Method! We used recursive computation of translation

matrices (Chew, 1992; Gumerov & Duraiswami, 2001).

! In some cases speed up of computations 103-104

times compared to BEM.! But� Computational Complexity is O(N3P3)= O(N3

Nt6), where P= Nt

2 is the total length of the vector of expansion coefficients. Method is not suitable for large N and ka.

! Details can be found in our paper JASA 112(6), 2002, 2688-2701.

T-matrix Method

Iterative Methods

Reflection Method & Krylov Subspace Method (GMRES)

Reflection (Simple Iteration) Method:

General Formulation (used in GMRES)

Iterative Methods

Convergence of Reflection Iteration Method

Iterative Methods

Exponential Convergence

Conclusions on Iterative Methods

! Both the Reflection Method (RM) and the GMRES converge well, while the RM is simpler and faster;

! Some problems in convergence were found for larger ka and regular spacing of the scatterers;

! In iterative methods fast translation algorithms can be used (weused O(Nt

3)=O(P3/2) fast translation based on sparse matrix decomposition of translation operators). This cost potentially can be reduced further (we are working on O(PlogP) methods).

! Complexity of Iterative Methods in this case O(N2Niter Nt3);

! Savings in complexity compared to straightforward T-matrix are O(Nt

3 N/Niter)! For N~200, Niter~20, Nt~10 (P~100) this yields of order 104

times savings.

Iterative Methods

Multilevel Fast Multipole Method

Some Facts on the Fast Multipole Methods (FMM)

! Introduced by Rokhlin & Greengard (1987,1988) for computation of 2D and 3D fields for Laplace Equation;

! Reduces complexity of matrix-vector product from O(N2) to O(N) or O(NlogN) (depends on data structure);

! Hundreds of publications for various 1D, 2D, and 3D problems (Laplace, Helmholtz, Maxwell, Yukawa Potentials, etc.);

! Application to acoustical scattering problems (Koc & Chew, 1998; JASA);

! We taught the first in the country course on FMM fundamentals & application at the University of Maryland (2002);

! Our recent report on fundamentals and data structures is available online (visit our web pages).

MLFMM

Far and Near Fields

MLFMM

Neighborhood(Near Field)

Far Field

Computation of the Far Field (1)

MLFMM

xixc

(n,L)

y

1). Set Data Structure (hierarchically subdivide space with an oct-tree)

2). (S|S)-translate S-expansions for all scatterers in a box at the finest level to the center of the box and sum up (determine contribution to Far Field for each box at the finest level).

3). Recursively (S|S)-translate S-expansions to the center of the parent box and sum up (determine contribution to Far Field for each box at all courser levels).

UpwardPass

(From the finestto the coarsest

level)

Computation of the Far Field (2)

MLFMM

4). (S|R)-translate S-expansions for boxes which are inside the parent neighborhood but outside the box neighborhood to the center of the box (convert S-expansion to R-expansion).

5). (R|R)-translate R-expansions from the center of the box to the center of its child boxes (determine Far Field for each box at all levels).

DownwardPass

(From the coarsest

to the finest level)

Steps 4 and 5 performed one after the other recursively

6). (R|R)-translate R-expansions from the center of the boxes at the finest level to the centers of the spheres.

Complexity of MLFMM

MLFMM

! Savings in complexity compared to the iterative T-matrix method are O(NNt/(logN)).

! For N~104, Nt~10 this yields about 104 times speed up (109

speed up compared to straightforward T-matrix).

(For translation cost O(Nt3)= O(P3/2) without optimization):

(with optimization of the number of scatterers inside the finest level box):

Computable Problems on Desktop PC

MLFMM

Met

hod

Number of Scatterers101 102 103100 104 105

BEM

Multipole Straightforward

Multipole Iterative

MLFMM

Also strongly depends on ka !

Results of Computations

Range of Parameters! Number of Spheres: 1-104;! ka: 10-3-50;! Random and regularly spaced grids of

spheres;! Polydispersity: 0.5-1.5 (ratio to the

mean radius);! Volume fractions: 0.01-0.1

Results

4 spheres (T-matrix straightforward)

Results

Vector of the incidentplane wave

Imaging plane

Scatterers ka=15.2

Incident Field Total Field Scattered Field

100 random spheres (MLFMM)

Results

ka=1.6 ka=4.8ka=2.8

R G B

Convergence for 100 spheres (MLFMM)

Results

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

0 5 10 15 20 25 30 35Iteration #

Max

Abs

olut

e Er

ror

Iterations with Reflection Method

3D Helmholtz Equation,MLFMM100 Spheres

ka = 4.8

2.8 1.6

1000 random sphereska=1

Conclusions

! We developed, implemented, and tested the Multilevel Fast Multipole Method for computation of multiple scattering problems.

! Performance of the method depends on a number of controlling parameters. At proper selection of these parameters fast and accurate results can be achieved.

! Some convergence problems in iterative methods were observed for short wave propagation in regularly spaced sphere grids. This may be due to some internal resonances, which should be investigated.

Future work! Development of faster translation algorithms;! Extension for non-spherical scaterrers;! Comparisons with continuum (averaging)

theories and theories of wave propagation in random media;

! Computations of acoustic fields in disperse systems (bubbly liquids, particulate systems);

! Comparisons with experimental data.

THANK YOU !