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

Application Finite Elements Vibrations in stoch. layers Boundary Elements Noise Barriers Vibrations in Tunnels FMM-BEM Calc. of HRTFs Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 2 / 13

FEM Example: Laplace Equation, weighted residual 2 u = 0, 2 uωdω = 0 Gauss-Green theorem 2 uωdx = Ω Γ Ω u n ωdγ u ωdω Ω Discretize Ω with a grid of simple geometric elements and approximation of u with basis u(x) = u i ψ i (x) Choose weighting function ω, f.e. Galerkin: ψ m (x) Linear system of equations Ku = f Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 3 / 13

BEM Lot of possibilities to choose ω Fundamental solution: Solution of 2 ω = δ(ξ x) Second time Gauss-Green theorem 2 u uωdx = Ω Γ n ωdγ u ωdω Ω u = n ωdγ u w n dγ + Ω Γ u 2 ωdω = uδ(ξ x)dω = Ω Γ Ω u 2 ωdω u(ξ) ξ Ω 1 2 u(ξ) ξ Γ 0 ξ / Ω Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 4 / 13

Boundary Integral Equation κu(ξ) + u ω Γ n dγ = u Γ n ωdγ 2D: ω = 1 2π log r, r = ξ x 3D: ω = 1 4πr Discretization linear system of equations Only necessary for points on boundary Once values for boundary are calculated, results for ξ / Γ are easy to get Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 5 / 13

FEM vs BEM FEM (large) sparse sym. matrix mesh for entire domain simple integrals widely applicable BEM (smaller) nonsym. fully pop. matrix mesh only for boundary singular integrals restricted to some problems What if there is no fundamental solution? What if the system gets too big FMM Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 6 / 13

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 = δ F LĜ = 1 Calculation of approximation for G in the Fourier domain on some grid Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 7 / 13

Example Vibrations in tunnels immerged in orthotropic layered soil Propagation of waves in soil without tunnel with pointload (δ functional) at different depths z Deformation and stresses at different depths z After Fourier backtransformation w.r.t. y, results from above are taken for BEM-formulation of the tunnel κu(ξ) + u G Γ n dγ = u Γ n GdΓ Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 8 / 13

Orthotropic layers No singularity in the Fourier domain Problems with backtransformation No FFT possible Interpolation with αe β y, αe βy2??? Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 9 / 13

HRTFs Localization of sound sources dependent on the form of the pinna Calcualtion of acoustic pressure on the head Model has about 30.000 nodes and over 65.000 elements Too big for BEM Fast Multipole Method Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 10 / 13

FMM Originally developed for N-body problems Man-in-the-middle principle Near field classical BEM Far field fast mulitipole methode Single or multilevel Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 11 / 13

Helmholtz and FMM Helmholtz equation 2 Φ(x) + kφ(x) = 0 Fundamental solution: G(x, y) = eikr 4πr, r = x y expansion of G(x,y) possible e i D+d D + d = ik 4π (2l + 1)i l h l (kd) l S e iskd P l (sˆd)ds with ˆD = D D, h l(x) Hankel functions, P l (x) Legendre polynomials (h l for l ) Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 12 / 13

Helmholtz and FMM Helmholtz equation 2 Φ(x) + kφ(x) = 0 Fundamental solution: G(x, y) = eikr 4πr, r = x y expansion of G(x,y) possible Φ(x) = ik 4π S e ik(x z 2)s M L (s, z 2 z 1 ) A e ik(z 1 y a)s q a ds a=1 Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 12 / 13

Acknowledgments/Literature Vibrations in orthotropic layers: BMVIT/FFG Pr. 809089 HRTFs: FWF Pr. P-18401B15 Peter Hunter: FEM/BEM Notes Matthias Fischer: The Fast Multipole Boundary Element Method and its Application to Structured-Acoustic Field Interaction H. Waubke: Boundary Element Method for Isotropic Media with Random Shear Moduli, J. Comput. Acoust., 13(1) Z. Chen et al: A Formulation of the Fast Multipole Boundary Element Method (FMBEM) for Acoustic Radiation and Scattering from Three-Dimensional Structures, to appear Kreuzer, Chen, Waubke (ARI) FEM-BEM-FMM ARI meets NuHAG 13 / 13