The Boundary Element Method (and Barrier Designs)

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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
• Fresnels 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, Masters 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
In general
For the upcoming derivation
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Wave Equation

• 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
Point source S emitting spherical waves.
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Huygens Principle
Secondary sources on an isolated wavefront.
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Huygens Principle
Spherical wavelets from secondary sources.
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Huygens Principle
This is the problem with the original Huygens
Principle.
Envelope of wavelets outward
inward
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Huygens Principle
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
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.
Gustav Kirchhoff (1824 - 1887)
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Helmholtz
Namesake of the Helmholtz equation and a huge
contributor to the science of acoustics.
Hermann von Helmholtz (1821 - 1894)
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Fresnels Non-Spherical Secondary Sources
?
?
S
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Fresnels Non-Spherical Secondary Sources
S
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Fresnels Secondary Sources

-
Monopole
Dipole
Cardioid
-

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Helmholtz Equation

• Start with the wave equation
• Assume p is time harmonic, i.e.
• Then the wave equation becomes the Helmholtz
  Equation
• k ?/c is the wave number

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Greens Functions

• To represent free-field radiation, we need the
  function
• G is called a Greens Function (after George
  Green (1793-1841))
• A Greens 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 dist. between Q and P
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Two Applications
V volume
S surrounding surface
n surface normal
Q receiver
r distance from Q to a point on S
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Helmholtz-Kirchhoff Integral

• Start with these equations
• Multiply (1) by G and (2) by p
• Subtract (3) from (4)

(1)
(2)
(3)
(4)
(5)
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Helmholtz-Kirchhoff Integral

• From the previous slide
  
• Integrate over the volume V
• Apply Greens Second Identity
• The result is the Helmholtz-Kirchhoff Integral

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Helmholtz-Kirchhoff Integral

• From the previous slide
• Recall
• So

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Helmholtz-Kirchhoff Integral
p(Q) sound pressure at receiver point Q
? 2?f frequency of sound
(f frequency in Hz)
pS sound pressure on the surface S
n surface normal
k ?/c wave number
c speed of sound
r distance from point on S to Q
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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).

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 dp/dn everywhere on a surrounding
  closed surface
• For the BEM, we
• Discretize the boundary surface into small pieces
  over which p and dp/dn are constant
• Calculate p and dp/dn for each patch
• 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 dp/dn
  between them
• One node per patch p and dp/dn are assumed to be
  constant over the patch
• Patches/node spacing must be smaller than a
  wavelength so p and dp/dn dont 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 dp/dn

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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, well make two assumptions
• The surface is rigid, so dp/dn 0
• We have one node per patch (at the center)
• A surface with N patches and N nodes
• So, we have

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

Direct sound at receiver j
Influence of other patches on j
where
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BEM

• But since each receiver is on the surface
• So

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

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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
CATT-Acoustic
100 Hz
Sysnoise BEM
100 Hz
IFFT
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
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
• ?l 0.64 m
• f 10 100 Hz
• elements / ? 5 for all frequencies

Summers et al. 2004
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Results Time Domain
GABEM
GA
Measured
63 Hz octave band
Summers et al. 2004
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Results Frequency Domain
Summers et al. 2004
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Results Energy-Time
T20 solid EDT 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

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Helmholtz-Kirchhoff Integral and Wave-Field
Synthesis
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