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![Page 1: Acoustics at Rensselaer The Boundary Element Method (and Barrier Designs) Architectural Acoustics II March 31, 2008.](https://reader037.fdocuments.in/reader037/viewer/2022103022/56649f535503460f94c78301/html5/thumbnails/1.jpg)
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The Boundary Element Method(and Barrier Designs)
Architectural Acoustics II
March 31, 2008
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Barrier Designs
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Barrier Designs
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Barrier Designs
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Barrier Designs
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Barrier Designs
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Barrier Designs
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Barrier Designs
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Barrier Designs
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Barrier Designs
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Barrier Designs
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Barrier Designs
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BEM: Outline
• Review Complex Exponentials Wave equation
• Huygens’ Principle
• Fresnel’s Obliquity Factor
• Helmholtz-Kirchhoff Integral
• Boundary Element Method
• Relationship to Wave-Field Synthesis
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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).
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Complex Exponentials
sincos je j
krjkre jkr sincos
krjkr sincos
In general:
For the upcoming derivation:
tjte tj sincos
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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
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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)
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Huygens’ Principle
S
Point source S emitting spherical waves.
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Huygens’ Principle
S
Secondary sources on an isolated wavefront.
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Huygens’ Principle
S
Spherical wavelets from secondary sources.
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Huygens’ Principle
S
Envelope of wavelets: outward inward
This is the problem with the original Huygens’ Principle.
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Huygens’ Principle
S
Envelope of wavelets, outward only.
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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)
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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.
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Helmholtz
Hermann von Helmholtz (1821 - 1894)
Namesake of the Helmholtz equation and a huge contributor to the science of acoustics.
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Fresnel’s Non-Spherical Secondary Sources
Secondary sources have cardioid pattern:
2
cos1 r
S
θ
θ
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Fresnel’s Non-Spherical Secondary Sources
Secondary sources have cardioid pattern:
2
cos1 r
S
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Fresnel’s Secondary Sources
Secondary sources have cardioid pattern:
2
cos1 r
1r cosr
Monopole Dipole
+-
Cardioid- =
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• 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
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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
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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
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• 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)
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• 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)(
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• 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)(
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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
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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.
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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)
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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
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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
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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
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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
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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)!
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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)(
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Results
A new analysis method of sound fields by boundary integral equation and its applications, Tadahira and Hamada.
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Results
A new analysis method of sound fields by boundary integral equation and its applications, Tadahira and Hamada.
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Results
Prediction and evaluation of the scattering from quadratic residue diffusers, Cox and Lam, JASA 1994.
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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.
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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
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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
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Results: Time Domain
63 Hz octave band
GA+BEM
GA
Measured
Summers et al. 2004
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Results: Frequency Domain
Summers et al. 2004
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Results: Energy-Time
T20: solidEDT: dashed
ts: dotted
Summers et al. 2004
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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
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Barrier Analysis with BEM
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Barrier Analysis with BEM
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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
)(
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Helmholtz-Kirchhoff Integral and Wave-Field Synthesis
http://recherche.ircam.fr/equipes/salles/WFS_WEBSITE/Index_wfs_site.htm