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A discontinuous Galerkin method with plane waves for sound-absorbing materials. (English) Zbl 1352.74145
Summary: Poro-elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications, including the aerospace, automotive and energy industries. More efficient computational models are required to further optimise the use of such materials. In this paper, we present a discontinuous Galerkin method (DGM) with plane waves for poro-elastic materials using the Biot theory solved in the frequency domain. This approach offers significant gains in computational efficiency and is simple to implement (costly numerical quadratures of highly oscillatory integrals are not needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave-based DGM. A key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media. This is particularly important when modelling porous materials as they are generally coupled with other media, such as the surround fluid or an elastic structure. The validation of the method is described first for a simple wave propagating through a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational cost of the method are assessed, and comparison with the standard finite element method is included. It is found that the benefits of the wave-based DGM are fully realised for the Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material.

74J15 Surface waves in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Full Text: DOI
[1] Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range, The Journal of the Acoustical Society of America 28 (2) pp 168– (1956)
[2] Johnson, Theory of dynamic permeability and tortuosity in fluid-saturated porous media, Journal of Fluid Mechanics 176 pp 379– (1987) · Zbl 0612.76101
[3] Champoux, Dynamic tortuosity and bulk modulus in air-saturated porous media, Journal of Applied Physics 70 (4) pp 1975– (1991)
[4] Allard, Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials (2009)
[5] Hesthaven, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (2007)
[6] Melenk, The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099
[7] Cessenat, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem, SIAM Journal in Numerical Analysis 35 pp 255– (1998) · Zbl 0955.65081
[8] Farhat, The discontinuous enrichment method, Computer Methods in Applied Mechanics and Engineering 190 (48) pp 6455– (2001) · Zbl 1002.76065
[9] Farhat, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime, Computer Methods in Applied Mechanics and Engineering 192 (11) pp 1389– (2003) · Zbl 1027.76028
[10] Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems, Journal of Computational Physics 225 pp 1961– (2007) · Zbl 1123.65102
[11] Gittelson, Plane wave discontinuous Galerkin methods: analysis of the h-version, ESAIM: Mathematical Modelling and Numerical Analysis 43 pp 297– (2009) · Zbl 1165.65076
[12] Hiptmair, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM Journal of Numerical Analysis 49 (1) pp 264– (2011) · Zbl 1229.65215
[13] Hiptmair, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations, Mathematics of Computation 82 (281) pp 247– (2013) · Zbl 1269.78013
[14] Gittelson, Dispersion analysis of plane wave discontinuous Galerkin methods, International Journal for Numerical Methods in Engineering 98 (5) pp 313– (2014) · Zbl 1352.65503
[15] Huttunen, Solving Maxwell’s equations using the ultra weak variational formulation, Journal of Computational Physics 223 (2) pp 731– (2007) · Zbl 1117.78011
[16] Gabard, A comparison of wave-based discontinuous Galerkin, ultra-weak and least-square methods for wave problems, International Journal for Numerical Methods in Engineering 85 pp 380– (2011) · Zbl 1217.76047
[17] Monk, A least-squares method for the Helmholtz equation, Computer Methods in Applied Mechanics and Engineering 175 pp 121– (1999) · Zbl 0943.65127
[18] Deckers, Efficient treatment of stress singularities in poroelastic wave based models using special purpose enrichment functions, Computers & Structures 89 (11) pp 1117– (2011)
[19] Deckers, A wave based method for the efficient solution of the 2D poroelastic Biot equations, Computer Methods in Applied Mechanics and Engineering 201 pp 245– (2012) · Zbl 1239.74023
[20] Lähivaara, A non-uniform basis order for the discontinuous Galerkin method of the 3D dissipative wave equation with perfectly matched layer, Journal of Computational Physics 229 (13) pp 5144– (2010) · Zbl 1192.65129
[21] Chazot, Performances of the partition of unity finite element method for the analysis of two-dimensional interior sound fields with absorbing materials, Journal of Sound and Vibration 332 (8) pp 1918– (2013)
[22] Chazot, The partition of unity finite element method for the simulation of waves in air and porous media, Journal of the Acoustical Society of America 135 (2) pp 724– (2014)
[23] Dazel, An alternative Biot’s displacement formulation for porous materials, Journal of the Acoustical Society of America 121 (6) pp 3509– (2007)
[24] Whitham, Linear and Nonlinear Waves (1999)
[25] Leveque, Finite Volume Methods for Hyperbolic Problems (2002) · Zbl 1010.65040
[26] Kreiss, Initial boundary value problems for hyperbolic systems, Communications on Pure and Applied Mathematics 23 pp 277– (1970) · Zbl 0188.41102
[27] Higdon, Initial-boundary value problems for linear hyperbolic systems, SIAM Review 28 (2) pp 177– (1986) · Zbl 0603.35061
[28] Moiola A Approximation properties of plane wave spaces and application to the analysis of the plane wave discontinuous Galerkin method Technical Report 2009-06 2009
[29] Atalla, A mixed displacement-pressure formulation for poroelastic materials, Journal of the Acoustical Society of America 104 (3) pp 1444– (1998)
[30] Atalla, Enhanced weak integral formulation for the mixed (u,p) poroelastic equations, Journal of the Acoustical Society of America 109 (6) pp 3065– (2001)
[31] Göransson, A 3D, symmetric finite element formulation of the Biot equations with application to acoustic wave propagation through an elastic porous medium, International Journal for Numerical Methods in Engineering 41 pp 167– (1998) · Zbl 0909.76048
[32] Hörlin, A 3-D hierarchical FE formulation of Biot’s equations for elasto-acoustic modelling of porous media, Journal of Sound and Vibration 243 (4) pp 633– (2001)
[33] Huttunen, The ultra-weak variational formulation for elastic wave problems, SIAM Journal in Scientific Computing 25 (5) pp 1717– (2004) · Zbl 1093.74028
[34] El Kacimi, Improvement of PUFEM for the numerical solution of high-frequency elastic wave scattering on unstructured triangular mesh grids, International Journal for Numerical Methods in Engineering 84 (3) pp 330– (2010) · Zbl 1202.74166
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