1. The Boundary Element Method (and Barrier Designs)
Architectural Acoustics II
March 31, 2008

2. Barrier Designs

3. Barrier Designs

4. Barrier Designs

5. Barrier Designs

6. Barrier Designs

7. Barrier Designs

8. Barrier Designs

9. Barrier Designs

10. Barrier Designs

11. Barrier Designs

12. Barrier Designs

13. BEM: Outline Review Huygens’ Principle Fresnel’s Obliquity Factor
Complex Exponentials
Wave equation
Huygens’ Principle
Fresnel’s Obliquity Factor
Helmholtz-Kirchhoff Integral
Boundary Element Method
Relationship to Wave-Field Synthesis

14. 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).

15. Complex Exponentials
In general:
For the upcoming derivation:

16. 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

17. Christiaan Huygens (1629 – 1695)
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)

18. Point source S emitting spherical waves.
Huygens’ Principle S
Point source S emitting spherical waves.

19. Secondary sources on an isolated wavefront.
Huygens’ Principle S
Secondary sources on an isolated wavefront.

20. Spherical wavelets from secondary sources.
Huygens’ Principle S
Spherical wavelets from secondary sources.

21. Envelope of wavelets: outward inward
Huygens’ Principle S
This is the problem with the original Huygens’ Principle.
Envelope of wavelets: outward inward

22. Envelope of wavelets, outward only.
Huygens’ Principle S
Envelope of wavelets, outward only.

23. 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)

24. 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 ( )

25. Hermann von Helmholtz (1821 - 1894)
Namesake of the Helmholtz equation and a huge contributor to the science of acoustics.
Hermann von Helmholtz ( )

26. Fresnel’s Non-Spherical Secondary Sourcesθ S θ
Secondary sources have cardioid pattern:

27. Fresnel’s Non-Spherical Secondary Sources
Secondary sources have cardioid pattern:

28. Fresnel’s Secondary Sources
Secondary sources have cardioid pattern: - +
Monopole -
Dipole =
Cardioid

29. Helmholtz Equation 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

30. Green’s Functions
To represent free-field radiation, we need the function
G is called a “Green’s Function” (after George Green ( ))
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 = dist. between Q and P

31. Two Applications Interior Problem (Room Modeling)
Source V Q n S r
Exterior Problem (Object Scattering)
Source V n S r Q
V = volume
S = surrounding surface
n = surface normal
Q = receiver
r = distance from Q to a point on S

32. Helmholtz-Kirchhoff Integral
Start with these equations
Multiply (1) by G and (2) by p
Subtract (3) from (4)
(1)
(2)
(3)
(4)
(5)

33. Helmholtz-Kirchhoff Integral
From the previous slide
Integrate over the volume V
Apply Green’s Second Identity
The result is the Helmholtz-Kirchhoff Integral

34. Helmholtz-Kirchhoff Integral
From the previous slide
Recall So

35. Helmholtz-Kirchhoff Integral
p(Q) = sound pressure at receiver point Q
 = 2f = frequency of sound
(f = frequency in Hz)
Rec. (Q)
Src. dS
Surface S r
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

36. 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.

37. 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
Discretize the boundary surface into small pieces over which p and δp/δn are constant
Calculate p and δp/δn for each patch
Use the patch values to calculate p(Q)

38. 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

39. 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

40. Proper BEM To make this easier, we’ll make two assumptions So, we have
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
Image from “Sounds Good to Me!”, Funkhouser, Jot, and Tsingos, Siggraph 2002 Course Notes

41. 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

42. 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)!

43. 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

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

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

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

47. Hybrid BEM/GA Modeling
CATT-Acoustic
100 Hz +
Sysnoise BEM M
100 Hz M
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.

48. Test Case: Assembly Hall at YamahaX
Summers et al. 2004

49. 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

50. Results: Time Domain GA+BEM GA Measured 63 Hz octave band
Summers et al. 2004

51. Results: Frequency Domain
Summers et al. 2004

52. Results: Energy-Time T20: solid EDT: dashed ts: dotted
Summers et al. 2004

53. 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

54. Barrier Analysis with BEM

55. Barrier Analysis with BEM

56. 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

57. Helmholtz-Kirchhoff Integral and Wave-Field Synthesis