Aco

usti

cs a

t Ren

ssel

aer

The Boundary Element Method(and Barrier Designs)

Architectural Acoustics II

March 31, 2008

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

Barrier Designs

Aco

usti

cs a

t Ren

ssel

aer

BEM: Outline

• Review Complex Exponentials Wave equation

• Huygens’ Principle

• Fresnel’s Obliquity Factor

• Helmholtz-Kirchhoff Integral

• Boundary Element Method

• Relationship to Wave-Field Synthesis

Aco

usti

cs a

t Ren

ssel

aer

References• Encyclopedia of Acoustics, M. Crocker (Ed.), Chapter 15,

“Acoustic Modeling: Boundary Element Methods”, 1997.

• Acoustic Properties of Hanging Panel Arrays in Performance Spaces, T. Gulsrud, Master’s Thesis, Univ. of Colorado, Boulder, 1999.

• Boundary Elements X Vol. 4: Geomechanics, Wave Propagation, and Vibrations, C. Brebbia (Ed.), 1988.

• Boundary Element Fundamentals, G. Gipson, 1987.

• “Assessing the accuracy of auralizations computed using a hybrid geometrical-acoustics and wave-acoustics method,” J. Summers, K. Takahashi, Y. Shimizu, and T. Yamakawa J. Acoust. Soc. Am. 115, 2514 (2004).

Aco

usti

cs a

t Ren

ssel

aer

Complex Exponentials

sincos je j

krjkre jkr sincos

krjkr sincos

In general:

For the upcoming derivation:

tjte tj sincos

Aco

usti

cs a

t Ren

ssel

aer

Wave Equation

01

2

2

22

t

p

cp

zzyyxx ()()()()2

• Hyperbolic partial differential equation

• Partial derivatives with respect to time (t) and space ( )

• Can be derived using equations for the conservation of mass and momentum, and an equation of state

Aco

usti

cs a

t Ren

ssel

aer

Huygens’ Principle

(From 1690): Consider a source from which (light) waves radiate, and an isolated wavefront created by the source. Each element on such a wavefront can be considered as a secondary source of spherical waves, and the position of the original wavefront at a later time is the envelope of the secondary waves.

Christiaan Huygens (1629 – 1695)

Aco

usti

cs a

t Ren

ssel

aer

Huygens’ Principle

S

Point source S emitting spherical waves.

Aco

usti

cs a

t Ren

ssel

aer

Huygens’ Principle

S

Secondary sources on an isolated wavefront.

Aco

usti

cs a

t Ren

ssel

aer

Huygens’ Principle

S

Spherical wavelets from secondary sources.

Aco

usti

cs a

t Ren

ssel

aer

Huygens’ Principle

S

Envelope of wavelets: outward inward

This is the problem with the original Huygens’ Principle.

Aco

usti

cs a

t Ren

ssel

aer

Huygens’ Principle

S

Envelope of wavelets, outward only.

Aco

usti

cs a

t Ren

ssel

aer

Fresnel

Huygens-Fresnel Principle (1818): Fresnel added the concept of wave interference to Huygens’ principle and showed that it could be used to explain diffraction. He also added the idea of a direction-dependent obliquity factor: secondary sources do not radiate spherically.

Augustin Fresnel (1788 – 1827)

Aco

usti

cs a

t Ren

ssel

aer

Kirchhoff

Gustav Kirchhoff (1824 - 1887)

Kirchhoff showed that the Huygens-Fresnel Principle is a non-rigorous form of an integral equation that expresses the solution to the wave equation at an arbitrary point within the field created by a source. He also explicitly derived the obliquity factor for the secondary sources.

Aco

usti

cs a

t Ren

ssel

aer

Helmholtz

Hermann von Helmholtz (1821 - 1894)

Namesake of the Helmholtz equation and a huge contributor to the science of acoustics.

Aco

usti

cs a

t Ren

ssel

aer

Fresnel’s Non-Spherical Secondary Sources

Secondary sources have cardioid pattern:

2

cos1 r

S

θ

θ

Aco

usti

cs a

t Ren

ssel

aer

Fresnel’s Non-Spherical Secondary Sources

Secondary sources have cardioid pattern:

2

cos1 r

S

Aco

usti

cs a

t Ren

ssel

aer

Fresnel’s Secondary Sources

Secondary sources have cardioid pattern:

2

cos1 r

1r cosr

Monopole Dipole

+-

Cardioid- =

Aco

usti

cs a

t Ren

ssel

aer

• Start with the wave equation

• Assume p is time harmonic, i.e.

• Then the wave equation becomes the Helmholtz Equation:

• k = ω/c is the wave number

Helmholtz Equation

022 pkp

tjep

01

2

2

22

t

p

cp

Aco

usti

cs a

t Ren

ssel

aer

Green’s Functions

• To represent free-field radiation, we need the function

• G is called a “Green’s Function” (after George Green (1793-1841))

• A Green’s Function is a fundamental solution to a differential equation, i.e. where L is a linear differential operator

• In this case (the Helmholtz equation),

r

ePQG

jkr

),(

)'()',( xxxxLG

)( 22 kL

r = dist. between Q and P

Aco

usti

cs a

t Ren

ssel

aer

Two ApplicationsExterior Problem

(Object Scattering)

Source

V

nS

r

Q

Interior Problem (Room Modeling)

SourceV

Q

n

Sr

S = surrounding surface

V = volume

n = surface normal

Q = receiver

r = distance from Q to a point on S

Aco

usti

cs a

t Ren

ssel

aer

• Start with these equations

• Multiply (1) by G and (2) by p

• Subtract (3) from (4)

Helmholtz-Kirchhoff Integral

022 pkp

)(,, 002

02 rrrrGkrrG

022 pGkpG

)( 022 rrpGpkGp

)()( 02222 rrppGkGpkpGGp

0

(1)

(2)

(3)

(4)

(5)

Aco

usti

cs a

t Ren

ssel

aer

• From the previous slide

• Integrate over the volume V

• Apply Green’s Second Identity

• The result is the Helmholtz-Kirchhoff Integral

Helmholtz-Kirchhoff Integral

)( 022 rrppGGp

VV

pdVrrpdVpGGp 4)( 022

V S

dSnn

dV 22

dSn

pG

n

Gp

eQp

S

ss

tj

4)(

Aco

usti

cs a

t Ren

ssel

aer

• From the previous slide

• Recall

• So

Helmholtz-Kirchhoff Integral

dSn

pG

n

Gp

eQp

S

ss

tj

4)(

r

eG

jkr

dSn

p

r

e

r

e

np

eQp

S

sjkrjkr

s

tj

4)(

Aco

usti

cs a

t Ren

ssel

aer

Helmholtz-Kirchhoff Integral

dSn

p

r

e

r

e

np

eQp

S

sjkrjkr

s

tj

4)(

p(Q) = sound pressure at receiver point Q

= 2f = frequency of sound

pS = sound pressure on the surface S

n = surface normal

r = distance from point on S to Q

k = /c = wave number

Rec. (Q)

Src.

dSSurface S

rn

(f = frequency in Hz)

c = speed of sound

Aco

usti

cs a

t Ren

ssel

aer

Helmholtz-Kirchhoff Integral

The Helmholtz-Kirchhoff integral describes the (frequency domain) acoustic pressure at a point Q in terms of the pressure and its normal derivative on the surrounding surface(s).

dSn

p

r

e

r

e

np

eQp

S

sjkrjkr

s

tj

4)(

The normal derivative of the pressure is proportional to the particle velocity.

Aco

usti

cs a

t Ren

ssel

aer

Helmholtz-Kirchhoff Integral →Boundary Element Method

• HK Integral gives us the (acoustic) pressure at a point Q in space if we know the pressure p and normal velocity δp/δn everywhere on a surrounding closed surface

• For the BEM, we 1) Discretize the boundary surface into small pieces

over which p and δp/δn are constant

2) Calculate p and δp/δn for each patch

3) Use the patch values to calculate p(Q)

Aco

usti

cs a

t Ren

ssel

aer

BEM Details

• Discretization changes the integral to a summation over patches

• Patches can be rectangular, triangular, etc.• Each patch can be defined by multiple nodes (e.g. for a

triangle at the three corners and the center) or just one at the center Multiple nodes per patch: interpolate p and δp/δn between them One node per patch: p and δp/δn are assumed to be constant over the

patch• Patches/node spacing must be smaller than a wavelength so p

and δp/δn don’t vary much over the patch• Typically at least 6 per wavelength, so high-frequency

calculations are prohibitively expensive computation-wise• There are several methods to find p and δp/δn

Aco

usti

cs a

t Ren

ssel

aer

Simplest Solution: The Kirchhoff Approximation

• At each patch, let p = RRefl ·PInc

RRefl = surface reflection coeff.

PInc = incident pressure

• Surface velocity found in a similar way

• Surface conditions are due to source only. No patch-to-patch interaction!

• Useful only for the exterior problem

Aco

usti

cs a

t Ren

ssel

aer

Proper BEM

• To make this easier, we’ll make two assumptions The surface is rigid, so δp/δn = 0 We have one node per patch (at the center) A surface with N patches and N nodes

• So, we have

N

ii

i

jkr

i

i

tj

Ar

e

np

eQp

i

14)(

Image from “Sounds Good to Me!”, Funkhouser, Jot, and Tsingos, Siggraph 2002 Course Notes

Aco

usti

cs a

t Ren

ssel

aer

BEM

• Create N new receivers and place one at each node on the surface

• So for receiver j we have

• And a set of N linear equations in matrix form

N

ii

ij

jkr

i

i

tj

jDirj Ar

e

np

epp

ij

1, 42

1

surfDirrec pFpp 2

1i

ij

jkr

i

jwt

ji Ar

e

n

eF

ij

4,where

Direct sound at receiver j Influence of other patches on j

Aco

usti

cs a

t Ren

ssel

aer

BEM

• But since each receiver is on the surface

• So

surfDirrec pFpp 2

1surfDirsurf pFpp

2

1

Dirsurf pFIp1

2

1

where I is the identity matrix

This is why BEM is only useful at low frequencies and/or for small spaces. F is an n x n matrix, and matrix inversion is ~O(n2.4)!

Aco

usti

cs a

t Ren

ssel

aer

BEM

• Now we have the pressure at each node/patch, specifically the N-element vector

• Use the values in psurf to find p(Q) using our original equation

1

2

1

FIpsurf

N

ii

i

jkr

i

i

tj

Ar

e

np

eQp

i

14)(

Aco

usti

cs a

t Ren

ssel

aer

Results

A new analysis method of sound fields by boundary integral equation and its applications, Tadahira and Hamada.

Aco

usti

cs a

t Ren

ssel

aer

Results

A new analysis method of sound fields by boundary integral equation and its applications, Tadahira and Hamada.

Aco

usti

cs a

t Ren

ssel

aer

Results

Prediction and evaluation of the scattering from quadratic residue diffusers, Cox and Lam, JASA 1994.

Aco

usti

cs a

t Ren

ssel

aer

Hybrid BEM/GA Modeling

IFFT M

+

M

thHF

fH LF

th fH LF *

CATT-Acoustic

Sysnoise BEM

100 Hz

100 Hz

J. Summers, K. Takahashi, Y. Shimizu, and T. Yamakawa, “Assessing the accuracy of auralizations computed using a hybrid geometrical-acoustics and wave-acoustics method,” 147th ASA Meeting, New York, NY, May 2004.

Aco

usti

cs a

t Ren

ssel

aer

Test Case: Assembly Hall at Yamaha

X

X

X

X

Hz 50

s 51

m 2400

Sch

3

f

.T

V

Summers et al. 2004

Aco

usti

cs a

t Ren

ssel

aer

Test Case: Assembly Hall at Yamaha

• Why this space? Reasonable size allows for tractable BEM Easy access for measurements and surface impedance

measurement Existing computer model

• Model details 11180 linear triangular elements Δl = 0.64 m f = 10 – 100 Hz elements / λ ≥ 5 for all frequencies

Summers et al. 2004

Aco

usti

cs a

t Ren

ssel

aer

Results: Time Domain

63 Hz octave band

GA+BEM

GA

Measured

Summers et al. 2004

Aco

usti

cs a

t Ren

ssel

aer

Results: Frequency Domain

Summers et al. 2004

Aco

usti

cs a

t Ren

ssel

aer

Results: Energy-Time

T20: solidEDT: dashed

ts: dotted

Summers et al. 2004

Aco

usti

cs a

t Ren

ssel

aer

Overall Results

• Hybrid GA / WA techniques can model full-scale auditoria

• Uncertainties in input parameters limit accuracy of low-frequency computations

• Use of WA-based models at low frequencies affects audible variations

• Substantially larger data set required to assess classification schemes (6 subjects, 10 tests per subject, convolution with organ music)

Summers et al. 2004

Aco

usti

cs a

t Ren

ssel

aer

Barrier Analysis with BEM

Aco

usti

cs a

t Ren

ssel

aer

Barrier Analysis with BEM

Aco

usti

cs a

t Ren

ssel

aer

Helmholtz-Kirchhoff Integral and Wave-Field Synthesis

• Pressure on surface can be represented with a monopole

• Velocity on the surface can be represented with a dipole

• Reconstruct the surface (boundary) conditions with speakers to synthesize the interior sound field

S

Sjkrjkr

S dSn

p

r

e

r

e

npQp

41

)(

Aco

usti

cs a

t Ren

ssel

aer

Helmholtz-Kirchhoff Integral and Wave-Field Synthesis

http://recherche.ircam.fr/equipes/salles/WFS_WEBSITE/Index_wfs_site.htm