Validation of open source code BEM++ for simulation 

of acoustic problems

P. Lukashin (ISP RAS, Russia), S. Strijhak, (ISP RAS, Russia), G. Shcheglov(BMSTU, Russia)

Jet’s engines of Launch abort vehicle

A rotating 7-bladed propeller on submarine Compressor stator test rig

Car’s outside rear-view mirror

Aeroacoustic sources

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Sound‐pressure levels

C. Wagner, T. Hüttl, and P. Sagaut, eds. Large‐Eddy Simulation for Acoustics.Cambridge Univ. Press, New York, 2007. 3

Definition of acoustic’s problem and boundary conditions

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The main approaches for numerical modelling in aeroacoustics

Approach Numerical method Accuracy Required resources / Applicability

Solution of «full» Navier – Stokes

equations

Direct Numerical Simulation (DNS) Fine Very much / Impossible

Large Eddy Simulation (LES)

Good Very much / Impossible

Unsteady Reynolds-Averaged Navier –Stokes (URANS)

Low Few / Possible but not reliable

CFD-CAACFD-ALECFD-APE

Linearized Euler Equation (LEE) or Acoustic Perturbation Equation (APE)

Good Needs many resources due to grid resolution for LES and acoustics.Applicable for small-scale cases.

CFDSurface Integral

Analytic Methods (Lighthill, Kirchhoff,

FW-H)

Hybrid methods(LES/RANS) and analytical methods for

far-field modelling

Sufficient Can not used in cases withreflection of acoustic waves

somewhere in considered region

CFD-BEM Connection of the CFD solution (LES/RANS) in the near-field and the

solution of Helmholtz equation by BEM (Boundary Element Method) in the far-

field

Good Good for large cases to avoid big grids

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The BEM++ Library

• Core library in C++, complete interface via

Python

• Support for Laplace, Helmholtz, Maxwell

• Shared-Memory parallelization

• H-matrix assembly

• Open source license

• Mac and Linux supported6

Introduction to BEM++

BEM++ project: www.bempp.orgUniversity College London

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Classification of acoustic problems2

22 2

1 0, ( , )uu u f x tc t

( , ) ( ) i tu x t U x e

2 2 0, /U k U k c wavenumber

U i U hn

0 1

0h

0h

Boundary conditions on S:

(Neumann BCs) (Dirichlet BCs)

1) Radiation problem:2) Scattering problem:

Acoustic wave equation:

Helmholtz equation:

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Formulation of the test problem

2 2

0

0

0

lim [ ] 0

S I

I ikxa

SS

R

U k UU i Un

U U U

U U e

UR ikUR

| 0SU

| 0SUn

0, , ,k U R a

Source data:

(Neumann problem)

(Dirichlet problem)

Scattering from the unit sphere of plane wave:

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BEM problem

*1( ) ( ) |2

II

n SUI T i S U i Un

*1( ( )) ( ) |

2

II

SUH i I K U i Un

* *nU and U

[ ]( ) ( , ) ( ) ( )S

S U x G x y U y dS у ( , )[ ]( ) ( ) ( )( )S

G x yK U x U y dS уn y

( , )[ ]( ) ( ) ( )( )S

G x yT U x U y dS уn x

( , )[ ]( ) ( ) ( )

( ) ( )S

G x yH U x U y dS уn x n y

[ ]( ) ( ) ( )S

I U x U x dS x

Boundary Integral equation (BIE) for

Neumann problem:Dirichlet problem:

are surface potentials, which need to be calculated.

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BEM solution

*'[ ]InU U S U

*'[ ]IU U K U

'[ ]( ) : ( , ) ( ) ( )S

S U x G x y U y dS у

( , )'[ ]( ) : ( ) ( )( )S

G x yK U x U y dS уn y

| |

( , )4 | |

ik x yeG x yx y

Integral equation to derive the solution for

Neumann problem:Dirichlet problem:

Single-layer potential operator:

Double-layer potential operator:

Green’s function:

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Discretization of the BIE and solving

1,( )

0,k

kk

xx

x

1,( )

0,j i

i jx

i j

kix

Triangulatedsurface with elementsand nodes

Piecewise constant functionsfor Neumann data:

Continuous piecewise linear functions for

Dirichlet data:

Solving the system by GMRES or CG

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Interface basics of BEM++ 

Implementation in BEM++Geometry grid = bempp.api.shapes.regular_sphere (n)

Space function space = bempp.api.function_space(grid, "P", 1)BCs gr_fun = bempp.api.GridFunction(space, fun=fun)

Identity operator

I = bempp.api.operators.boundary.sparse.identity(space, space, space)

Boundary Integral

Operators

K = bempp.api.operators.boundary.helmholtz.double_layer (space, space, space, k)H = bempp.api.operators.boundary.helmholtz.hypersingular (space, space, space, k)

Potential Operators

S = bempp.api.operators.potential.helmholtz.single_layer(space, points, k)K = bempp.api.operators.potential.helmholtz.double_layer(space, points, k)

Solving the System

func,info=bempp.api.linalg.gmres (-H+1j *k*(0.5*I-K))=gr_fun)

Deriving the results

res = np.absolute(u_inc + K.evaluate(func)+1j*k*S.evaluate(b*func))

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Analytical Solution

I SU U U

00

(2 1) ( ) (cos )n

I ikxa ll l

l

U U e l i j kr P

0( ) (cos )

nS

l l ll

U Ah kr P

0,I SU U at r ar r

0,I SU U at r a

0( )(2 1)( )

l ll

l

j kaA U l ih ka

1 10

1 1

( ) ( 1) ( )(2 1)( ) ( 1) ( )

l l ll

l l

lj ka l j kaA U l ilh ka l h ka

lj

lP

lh

Dirichlet problem:

Spherical Bessel functionLegendre functionSpherical Hankel function

Neumann problem:

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Compare the results for Dirichlet problem

010, 1, 1, {1;0;0}k U R a

OXY planeAnalytical solutionBEM solution

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Compare the results for Neumann problem

OXY plane

010, 1, 1, {1;0;0}k U R a

Analytical solutionBEM solution

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Polar plots 

OXY plane

010, 1, 1, {1;0;0}k U R a

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Pressure range for a rigid sphere

Point APoint B

A

B

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Simulation for prototype of Launch Vehicle

| 0SUn

100f Hz

{1;1; 1}a

Mesh size: 56500 elements

Pressure plotCPU Time

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Conclusions

•Successful validation of the BEM++ open source library on test case «Sphere»•The pressure error is less than 5 % compared to analytical solution•Shared memory technology restricts efficiency of solver: sufficiently time of computation for industrials problems with large meshes is possible for low frequencies ( 100 Hz)

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