fmm accelerated bem for 3d laplace & helmholtz...

FMM accelerated BEMfor 3D Laplace & Helmholtz equations

Nail A. Gumerov and Ramani DuraiswamiInstitute for Advanced Computer Studies

University of Maryland, U.S.A.

Presentation for the International Conference on Boundary Element Technique BETEQ-7, September 2006, Paris, France

© Gumerov & Duraiswami, 2006

Outline

• Introduction• Problem formulation• BEM speed up with the FMM• Performance tests

• Laplace equation• Helmholtz equation

• Conclusions


Introduction

• BEM is a great method

• Drawbacks of the standard method• Slow: O(N3)-direct, O(NiterN2)-iterative• Memory expensive: O(N2)-full matrix storage• For modern PC’s are not suitable for solution

of large problems, N > 104

• FMM acceleration/memory reduction• Fast: O(NiterN) (or O(NiterNlogN))• Memory inexpensive: O(N) (or O(NlogN))• Modern PC’s can solve large problems, N ~ 106


Large problems: Example 1Introduction


Mesh: 30000 elements15002 vertices

Acoustic scattering from a sphere(model of human head)

Mesh: 33082 elements16543 vertices

Large problems: Example 1/1Introduction


KEMAR: 405002 elements, 202503 vertices

Large problems: Example 2Introduction


Multiparticle system (e.g. blood flow)

Large mesh (hundreds of thousands elements)

Problem formulation


Laplace: Helmholtz:

Boundary conditions:

For external problems: For external problems:

where

n

S

Layer potentials


Laplace: Helmholtz:

Problem formulation

Normal derivatives:

Boundary integral equations


Problem formulation

Green’s identity (direct) formulation:

Layer potential (indirect) formulation:

BEM speed up with the FMM


• Use iterative methods (we use GMRES)• The system matrix can be decomposed as

A = Asparse+Adense .• The sparse part includes diagonal

elements+neighbor element influence• Integrals can be computed using standard BEM

techniques, O(N) memory/operation complexity• The dense part is related to far element

influence• Most memory/computationally expensive• Can be done with the FMM• Low order quadratures over elements can be used

BEM and FMM data structures



FMM neighborhood

BEM neighborhood

This element is close to red one in all senses(high order quadrature, influence directly)

This element is far to red one in all senses(low order quadrature, influence via expansion)

This element is far to red one in the sense of BEM, but close in the sense of the FMM(low order quadrature, influence directly)

rmin

Level lmax

Computation of normal derivatives



Differential operators in the space of expansion coefficients:

Detailed theory and expressions can be found in bookN.A. Gumerov and R. Duraiswami, Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, Elsevier, Oxford UK, 2004.

sparse matrix

Expansion of boundary integrals



Greens function expansion:

Quadratures:

Expansion coefficients:

Particulars of the FMM employed



• Two or four BEM matrix-vector multiplications are performed at “one shot” using sparse matrix representation of differential operators

• O(p3) translation methods based on the RCR*

-decomposition are used both for the Laplace and Helmholtz equation

• Details can be found in our technical reports and FMM papers, 2003-2006.

* RCR-decomposition is the Rotation-Coaxial translation-Rotation decomposition of an arbitrary translation operator in three dimensions

Basic FMM flow chart



1. Get S-expansion coefficients(directly)

2. Get S-expansion coefficients from children

(S|S translation)

Level lmax

3. Get R-expansion coefficients from far neighbors

(S|R translation)

Level 2


(S|R translation)

6. Evaluate R-expansions (directly)

7. Sum sourcesin close neighborhood

(directly)

Start

End

Level lmaxLevel lmax

5. Get R-expansion coefficients from parents

(R|R translation)

Levels lmax-1,…, 2 Levels 3,…,lmax

1. Get S-expansion coefficients(directly)

2. Get S-expansion coefficients from children

(S|S translation)

Level lmax


(S|R translation)

Level 2


(S|R translation)

6. Evaluate R-expansions (directly)

7. Sum sourcesin close neighborhood

(directly)

Start

End

Level lmaxLevel lmax

5. Get R-expansion coefficients from parents

(R|R translation)

Levels lmax-1,…, 2 Levels 3,…,lmax

Performance tests


• Flat triangular surface discretization• Accuracy of the FMM selected to be

consistent with the accuracy of the BEM (relative errors normaly within the range 10-2 -10-4)

• Problem sizes 500-3,000,000 boundary elements

• Where possible compared with direct BEM, iterative BEM, and low memory direct BEM

• Hardware: PC: 3.2 GHz, Intel Xeon processor, 3.5 GB RAM

Laplace equation


Performance tests

1000 randomly oriented ellipsoids488000 vertices and 972000 elements

Potential

Typical test configuration: for external Dirichletand Neumann problems

Laplace equation


Performance tests

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06Number of Vertices, N

Tota

l CP

U ti

me

(s)

GMRES+FMM

GMRES+Low Mem Direct

GMRES+High Mem Direct

LU-decomposition

O(N2) Memory Threshold

y=ax

y=bx2

y=cx3

BEMDirichlet Problemfor Ellipsoids(3D Laplace)

FMM: p = 8

Laplace equation


Performance tests

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06Number of Vertices, N

Sing

le M

atrix

-Vec

tor M

ultip

licat

ion

CPU

Tim

e (s

)

y=ax

y=bx2Number of GMRES Iterations

FMM

Direct+Matrix EntriesComputation

Multiplicationof Stored Matrix

3D Laplace

FMM: p = 8

Helmholtz equation


Performance tests

Incident wavePlane wave scattering from a sphere, ka=30BEM/FMM, 240002 vertices, 480000 elements

(comparison with analytical solution)

GMRES: 115 iterations, ε =10-5,FMM: lmax= 6, pmax= 22

Helmholtz equation


Performance tests

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+03 1.E+04 1.E+05 1.E+06Number of Vertices, N

Tota

l CP

U ti

me

(s)

GMRES+FMM

GMRES+Low Mem Direct

GMRES+High Mem Direct

LU-decomposition

O(N2) Memory Threshold

y=ax

y=bx2

y=cx3

BEMNeumann Problemfor Sphere (3D Helmholtz)

kD=34.64

Helmholtz equation


Performance tests

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+03 1.E+04 1.E+05 1.E+06Number of Vertices, N

Sing

le M

atrix

-Vec

tor M

ultip

licat

ion

CPU

Tim

e (s

)

y=ax

y=bx2Number of GMRES Iterations

FMM

Low MemDirect

Multiplicationof Stored Matrix

3D Helmholtz, kD=33.64

Helmholtz equation


Performance tests

Mesh: 249856 vertices/497664 elements

kD=29, Neumann problem kD=144, Robin problem(impedance, sigma=1)

(some other scattering problems were solved)

Helmholtz equation


Performance tests


Incident plane wave

Mesh: 132072 elements 65539 vertices

exp(ikz)

Helmholtz equation


Performance tests


kD=0.96 kD=9.6 kD=96(250 Hz)

Sound pressure

(2.5 kHz) (25 kHz)

Conclusions


• Developed method (BEM with GMRES+FMM) shows a good performance and can be used for solution of large boundary value problems for the Laplace and Helmholtz equtaions (up to million size) on a single PC

• Future research: • Parallel versions of the method should bring additional

speedups• BEM for different equations (e.g. biharmonic and Stokes

equations) can be also speeded up in the same way • Efficient iterative methods for BEM are of high demand,

as the matrix-vector products can be accelerated via the FMM

Software Availability:


• FMM-based software is licensed from the University of Maryland to Fantalgo, LLC

• Includes matrix-vector product for 3D Laplace, biharmonic, Helmholtz (Maxwell is coming) equations and Gaussian kernel

• BEM and function fitting via iterative solvers• Contact

[email protected] further information

THANK YOU !

fmm accelerated bem for 3d laplace & helmholtz...

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