Finite and Boundary Element Methods in Acoustics€¦ · Finite and Boundary Element Methods in Acoustics W. Kreuzer, Z. Chen, H. Waubke Austrian Academy of Sciences, Acoustics Research

Finite and Boundary Element Methods inAcoustics

W. Kreuzer, Z. Chen, H. Waubke

Austrian Academy of Sciences, Acoustics Research Institute

ARI meets NuHAG

Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 1 / 13

Application

Finite ElementsVibrations in stoch. layers

Boundary ElementsNoise BarriersVibrations in Tunnels

FMM-BEMCalc. of HRTFs


FEM

Example: Laplace Equation, weighted residual

∇2u = 0,∫

Ω∇2uωdΩ = 0

Gauss-Green theorem∫Ω∇2uωdx =

∫Γ

∂u∂nωdΓ−

∫Ω∇u∇ωdΩ

Discretize Ω with a grid of simple geometric elements andapproximation of u with basis u(x) =

∑uiψi(x)

Choose weighting function ω, f.e. Galerkin: ψm(x)Linear system of equations Ku = f


BEM

Lot of possibilities to choose ωFundamental solution: Solution of ∇2ω = −δ(ξ − x)Second time Gauss-Green theorem∫

Ω∇2uωdx =

∫Γ

∂u∂nωdΓ−

∫Ω∇u∇ωdΩ

=∫

Γ

∂u∂nωdΓ−

∫Γ

u∂w∂n

dΓ +∫

Ωu∇2ωdΩ

∫Ω

u∇2ωdΩ = −∫

Ωuδ(ξ − x)dΩ =

−u(ξ) ξ ∈ Ω−1

2 u(ξ) ξ ∈ Γ0 ξ /∈ Ω


Boundary Integral Equation

κu(ξ) +∫

Γu∂ω

∂ndΓ =

∫Γ

∂u∂nωdΓ

2D: ω = − 12π log r, r = ||ξ − x||

3D: ω = 14πr

Discretization → linear system of equationsOnly necessary for points on boundaryOnce values for boundary are calculated, results for ξ /∈ Γare easy to get


FEM vs BEM

FEM BEM(large) sparse sym. matrix (smaller) nonsym. fully pop. matrix

mesh for entire domain mesh only for boundary“simple” integrals singular integralswidely applicable “restricted” to some problems

What if there is no fundamental solution ?What if the system gets too big → FMM


No fundamental solution ?

“No BEM without fundamental solution G(ξ, x)”“Solution of the problem Lu = 0 with a singularity at ξ”Fourier transformation F

LG = δF→ LG = 1

Calculation of approximation for G in the Fourier domain onsome grid


Example

Vibrations in tunnels immerged in orthotropiclayered soil

Propagation of waves insoil without tunnel withpointload (δ functional) atdifferent depths z

Deformation and stressesat different depths z

After Fourierbacktransformation w.r.t. y,results from above aretaken for BEM-formulationof the tunnel

κu(ξ) +∫

Γu∂G∂n

dΓ =∫

Γ

∂u∂n

GdΓ


Example





κu(ξ) +∫

Γu∂G∂n

dΓ =∫

Γ

∂u∂n

GdΓ


Example





κu(ξ) +∫

Γu∂G∂n

dΓ =∫

Γ

∂u∂n

GdΓ


Example





κu(ξ) +∫

Γu∂G∂n

dΓ =∫

Γ

∂u∂n

GdΓ


Example





κu(ξ) +∫

Γu∂G∂n

dΓ =∫

Γ

∂u∂n

GdΓ


Example





κu(ξ) +∫

Γu∂G∂n

dΓ =∫

Γ

∂u∂n

GdΓ


Example





κu(ξ) +∫

Γu∂G∂n

dΓ =∫

Γ

∂u∂n

GdΓ


Orthotropic layers

No singularity in theFourier domainProblems withbacktransformationNo FFT possibleInterpolation withαeβ|y|, αeβy2

???


Orthotropic layers


???


Orthotropic layers


???


HRTFs

Localization of soundsources dependent on theform of the pinnaCalcualtion of acousticpressure on the headModel has about 30.000nodes and over 65.000elementsToo big for BEM → FastMultipole Method


HRTFs

Localization of soundsources dependent on theform of the pinnaCalcualtion of acousticpressure on the headModel has about 30.000nodes and over 65.000elementsToo big for BEM → FastMultipole Method


FMM

Originally developed forN-body problemsMan-in-the-middleprincipleNear field → classical BEMFar field → fast mulitipolemethodeSingle or multilevel


FMM



FMM



FMM



FMM



Helmholtz and FMM

Helmholtz equation

∆2Φ(x) + kΦ(x) = 0

Fundamental solution:

G(x, y) =eikr

4πr, r = ||x− y||

expansion of G(x,y) possible

ei|D+d|

|D + d|=

ik4π

∑`

(2`+ 1)i`h`(kD)∫

SeiskdP`(sD)ds

with D = D||D|| , h`(x) Hankel functions, P`(x) Legendre

polynomials (h` →∞ for `→∞)


Helmholtz and FMM

Helmholtz equation

∆2Φ(x) + kΦ(x) = 0

Fundamental solution:

G(x, y) =eikr

4πr, r = ||x− y||

expansion of G(x,y) possible

Φ(x) =ik4π

∫S

eik(x−z2)sML(s, z2 − z1)A∑

a=1

eik(z1−ya)sqads


Acknowledgments/Literature

Vibrations in orthotropic layers: BMVIT/FFG Pr. 809089HRTFs: FWF Pr. P-18401B15Peter Hunter: FEM/BEM NotesMatthias Fischer: The Fast Multipole Boundary ElementMethod and its Application to Structured-Acoustic FieldInteractionH. Waubke: Boundary Element Method for Isotropic Mediawith Random Shear Moduli, J. Comput. Acoust., 13(1)Z. Chen et al: A Formulation of the Fast MultipoleBoundary Element Method (FMBEM) for AcousticRadiation and Scattering from Three-DimensionalStructures, to appear


Documents

Finite and Boundary Element Methods in Acoustics€¦ · Finite and Boundary Element Methods in Acoustics W. Kreuzer, Z. Chen, H. Waubke Austrian Academy of Sciences, Acoustics Research