NATO Undersea Research Centre Partnering for Maritime Innovation Presented at the COMSOL Conference 008 Hannover Acoustics Session Wed 5 November 008, 13:00 15:40. Imroved Perfectly Matched Layers for Acoustic Radiation and Scattering Problems Mario Zamolli, *,1 Nils Malm, Alessandra Tesei 1 1 NURC NATO Research Centre, La Sezia (Italy) COMSOL AB, Stockholm (Sweden)
Overview The Sommerfeld Radiation Condition and Perfectly Matched Layers (PML s). Problems caused by the stee decay of evanescent waves at low frequencies. Imroved PML formulation: (i) Imroved accuracy in the resence of evanescent waves at low frequencies. (ii) Stability of the mesh with resect to frequency. Real-life alication: loudseaker design. Conclusions and further develoment.
The Sommerfeld Radiation Condition In an unbounded medium [ex(+i ω t), k= ω /c] : r + + k ik = = 0 1 o r Helmholtz equation describing the acoustic ressure., r In a comuter model, r must be finite. Sommerfeld radiation condition r source(s) and/or scatterer(s) exterior fluid The Sommerfeld condition must be aroximated numerically.
Perfectly Matched Layer (PML) An (efficient!) technique for aroximating the Sommerfeld BC. J.-P. Bérenger, A erfectly matched layer for the absortion of electromagnetic waves, Journal of Comutational Physics, Vol. 114,. 185 00 (1994). PML s Can be easily adated to CONVEX geometries:. Straightforward imlementation via comlex coordinate scaling. F. Collino, P. Monk, The erfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comut., Vol. 19(6),. 061 090 (1998). F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Sringer-Verlag (1998). M. Zamolli, A. Tesei, F.B. Jensen, N. Malm, J.B. Blottman, J.Acoust.Soc.Am. 1, 147 1485 (007). z Fluid Domain (hysical) x PML (non-hysical) Perturbed Helmholtz Equation (PML): iω iω + + k = 0 iω + σ ( x) x ( ) iω + σ x x z Reflections are negligible. ~ x + z + k Comlex coordinate Daming only in the x-direction. = 0
PML Formulation in the Current Imlementation Fluid z (hysical) PML x PDE in the PML: ~ x + z + k = 0 D PML scaled coordinate: ~ x = (1 i) λ / Normalizing the scaled coordinate w.r.t. the wavelength λ no need to adjust the mesh density in the PML in x-direction as the frequency varies Mesh stability. The scaled coordinate is a olynomial, with equal ower n in the real art and in the imaginary art. ( x D) n The real and the imaginary art of the scaled coordinate each have different effects, deending on whether the incident wave is roagating or evanescent.
Effect of the Real and Imaginary Parts of the Scaling Wave Tye PML scaled coordinate Real Part Imaginary Part Proagating Evanescent Resolution of the oscillatory comonents in the PML, no daming. Daming (Decay) of the evanescent wave in the PML. Daming in the PML. Raidly growing imaginary art good daming. Surious anti-causal waves, which are absorbed by the PML. Problems at very low frequencies, where the evanescent field decays steely: the PML over-dams an already steely decaying wave PML accuracy roblems.
Imroved PML Scaling Fluid z (hysical) PML D x ( ) n ( x / D) r + i log 1 ( x / D ) i ~ x = Aλ n arameter ~0.5 sensitivity convergence n i ka 1 log, = 10 10 1, ka ka < 10 10 At high frequencies ka>10 : raidly growing log (1-x/D) scaling of the imaginary art good daming of roagating waves. At low frequencies: log (1-(x/D) ni ) with exonent of n i >1 in the imaginary art imaginary coordinate grows more slowly near the interface with the hysical domain, element size comressed near the interface, good resolution of the hase of the decaying wave.
Imroved PML Scaling Fluid z (hysical) PML D x ( ) n ( x / D) r + i log 1 ( x / D ) i ~ x = Aλ n n r 1 log10 = 1, ka, ka < 1 ka 1 At low frequencies : real art grows slowly near the boundary element size comressed near the interface, raidly decaying evanescent waves are better resolved by the PML near the interface with the hysical domain, imroved accuracy.
Examle: Circular Piston Radiation Sound-ressure level on axis, at 0.1a. PML COMSOL: 1 quadratic elements in the PML, old PML scaling. air rigid baffle rescribed constant normal velocity. Piston radius = a. Errors at low frequency in the solution with the old PML scaling.
Comarison: old vs Imroved Scaling Relative error vs. ka
Plane-wave Scattering from a Hard Shere Scattered ressure level on surface, backscatter. Relative error vs. ka
Helmholtz resonator Evanescent field at the oening. Performance of the Modified PML Cylindrical Sub-woofer Accurate solution with 4-layer modified PML, comared to 16-layer standard PML.
Audio Loudseaker Design SPL (db) Black: 8 Layer standard PML in Comsol Red: 1 Layer new PML. Freq. (Hz)
Conclusions and Further Develoment The real and imaginary arts of the PML coordinate scaling affect evanescent and roagating wave comonents in different ways. Modified scaling strategy roosed: (i) imroves the erformance at low frequencies, where evanescent waves dominate. (ii) mesh stability with resect to frequency, from ka = 1/100 to ka = 100. Increasing error at high frequencies observed in radiation roblems To be addressed. What is a in ka? Problem deendence? A measure of the smallest wave scales of the roblem?