×

zbMATH — the first resource for mathematics

The discontinuous enrichment method for medium-frequency Helmholtz problems with a spatially variable wavenumber. (English) Zbl 1295.76030
Summary: Numerical dispersion, or what is often referred to as the pollution effect, presents a challenge to an efficient finite element discretization of the Helmholtz equation in the medium frequency regime. To alleviate this effect and improve the unsatisfactory pre-asymptotic convergence of the classical Galerkin finite element method based on piecewise polynomial basis functions, several discretization methods based on plane wave bases have been proposed. Among them is the discontinuous enrichment method that has been shown to offer superior performance to the classical Galerkin finite element method for a number of constant wavenumber Helmholtz problems and has also outperformed two representative methods that use plane waves – the partition of unity and the ultra-weak variation formulation methods. In this paper, the discontinuous enrichment method is extended to the variable wavenumber Helmholtz equation. To this effect, the concept of enrichment functions based on free-space solutions of the homogeneous form of the governing differential equation is enlarged to include free-space solutions of approximations of this equation obtained in this case by successive Taylor series expansions of the wavenumber around a reference point. This leads to plane wave enrichment functions based on the piece-wise constant approximation of the wavenumber, and to Airy wave enrichment functions. Several elements based on these enrichment functions are constructed and evaluated on benchmark problems modeling sound-hard scattering by a disk submerged in an acoustic fluid where the speed of sound varies in space. All these elements are shown to outperform by a substantial margin their continuous polynomial counterparts.

MSC:
76Q05 Hydro- and aero-acoustics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Babuška, I.; Melenk, J. M., The partition of unity method, Int. J. Numer. Methods Engrg., 40, 727-758, (1997) · Zbl 0949.65117
[2] Cessenat, O.; Despres, B., Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem, SIAM J. Numer. Anal., 35, 255-299, (1998) · Zbl 0955.65081
[3] Djellouli, R.; Farhat, C.; Macedo, A.; Tezaur, R., Finite element solution of two-dimensional acoustic scattering problems using arbitrarily shaped convex artificial boundaries, J. Comput. Acoust., 8, 81-100, (2000) · Zbl 1360.76131
[4] Farhat, C.; Harari, I.; Franca, L. P., The discontinuous enrichment method, Comput. Methods Appl. Mech. Engrg., 190, 6455-6479, (2001) · Zbl 1002.76065
[5] Farhat, C.; Harari, I.; Hetmaniuk, U., The discontinuous enrichment method for multiscale analysis, Comput. Methods Appl. Mech. Engrg., 192, 3195-3210, (2003) · Zbl 1054.76048
[6] Farhat, C.; Harari, I.; Hetmaniuk, U., A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime, Comput. Methods Appl. Mech. Engrg., 192, 1389-1419, (2003) · Zbl 1027.76028
[7] Farhat, C.; Tezaur, R.; Wiedemann-Goiran, P., Higher-order extensions of a discontinuous Galerkin method for mid-frequency Helmholtz problems, Int. J. Numer. Methods Engrg., 61, 1938-1956, (2004) · Zbl 1075.76572
[8] Farhat, C.; Wiedemann-Goiran, P.; Tezaur, R., A discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of short wave exterior Helmholtz problems on unstructured meshes, J. Wave Motion, 39, 307-317, (2004) · Zbl 1163.74344
[9] Farhat, C.; Kalashnikova, I.; Tezaur, R., A higher-order discontinuous enrichment method for the solution of high Péclet advection-diffusion problems on unstructured meshes, Int. J. Numer. Methods Engrg., 81, 604-636, (2010) · Zbl 1183.76805
[10] Gabard, G.; Gamallo, P.; Huttunen, T., A comparison of wave-based discontinuous Galerkin, ultra-weak and least-square methods for wave problems, Int. J. Numer. Methods Engrg., 85, 380-402, (2011) · Zbl 1217.76047
[11] Grosu, E.; Harari, I., Studies of the discontinuous enrichment method for two-dimensional acoustics, Finite Elem. Anal. Des., 44, 272-287, (2008)
[12] Grosu, E.; Harari, I., Three-dimensional element configurations for the discontinuous enrichment method for acoustics, Int. J. Numer. Methods Engrg., 78, 1261-1291, (2009) · Zbl 1183.76806
[13] Huttunen, T.; Monk, P.; Kaipio, J. P., Computational aspects of the ultra-weak variational formulation, J. Comput. Phys., 182, 27-46, (2002) · Zbl 1015.65064
[14] Ihlenburg, F., Finite element analysis of acoustic scattering, (1998), Springer-Verlag New-York · Zbl 0908.65091
[15] L.-M. Imbert-Gerard, B. Després. Generalized plane wave numerical methods for magnetic plasma. in: Proceedings of the 10th International Conference on the Mathematical and Numerical Aspects of Waves Vancouver, Canada, 233-236, 2011.
[16] I. Kalashnikova, The Discontinuous Enrichment Method for Multi-Scale Transport Problems, Ph.D. Thesis, Stanford University, Stanford, CA, 2011.
[17] Kalashnikova, I.; Tezaur, R.; A discontinuous enrichment method for variable coefficient advection-diffusion at high Péclet number, C. Farhat., Int. J. Numer. Methods Engrg., 87, 309-335, (2013)
[18] Kovalevsky, Louis; Ladevèze, Pierre; Riou, Hervé, The Fourier version of the variational theory of complex rays for medium-frequency acoustics, Comput. Methods Appl. Mech. Engrg., 225-228, 142-153, (2012) · Zbl 1253.76118
[19] Laghrouche, O.; Bettess, P.; Perrery-Debain, E.; Trevelyan, J., Plane wave basis finite-elements for wave scattering in three dimensions, Commun. Numer. Methods Engrg., 19, 715-723, (2003) · Zbl 1031.65132
[20] Laghrouche, O.; Bettess, P.; Perrey-Debain, E.; Trevelyan, J., Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed, Comput. Methods Appl. Mech. Engrg., 194, 367-381, (2005) · Zbl 1143.65395
[21] Laghrouche, O.; El-Kacimi, A.; Trevelyan, J., Extension of the PUFEM to elastic wave propagation in layered media, J. Comput. Acoust., 20, 2, (2012) · Zbl 1360.74076
[22] Luostari, T.; Huttunen, T.; Monk, P., Improvements for the ultra weak variational formulation, Int. J. Numer. Methods Engrg., 94, 598-624, (2013) · Zbl 1352.65528
[23] Massimi, P.; Tezaur, R.; Farhat, C., A discontinuous enrichment method for three-dimensional multiscale harmonic wave propagation problems in multi-fluid and fluid-solid media, Int. J. Numer. Methods Engrg., 76, 400-425, (2008) · Zbl 1195.74292
[24] Massimi, P.; Tezaur, R.; Farhat, C., A discontinuous enrichment method for the efficient solution of plate vibration problems in the medium-frequency regime, Int. J. Numer. Methods Engrg., 84, 127-148, (2010) · Zbl 1202.74201
[25] Melenk, J. M.; Babuška, I., The partition of unity method finite element method: basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139, 289-314, (1996) · Zbl 0881.65099
[26] Monk, P.; Wang, D. Q., A least-squares method for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 175, 121-136, (1999) · Zbl 0943.65127
[27] Ortiz, P.; Sanchez, E., An improved partition of unity finite element model for diffraction problems, Int. J. Numer. Methods Engrg., 50, 2727-2740, (2001) · Zbl 1098.76576
[28] Petersen, S.; Farhat, C.; Tezaur, R., A space-time discontinuous Galerkin method for the solution of the wave equation in the time domain, Int. J. Numer. Methods Engrg., 78, 275-295, (2008) · Zbl 1183.76813
[29] Pluymers, B.; van Hal, B.; Vandepitte, D.; Desmet, W., Trefftz-based methods for time-harmonic acoustics, Arch. Comput. Methods Eng., 14, 343-381, (2007) · Zbl 1170.76332
[30] Polyanin, A. D.; Zaitsev, V. F., Handbook of exact solutions for ordinary differential equations, (2003), Chapman & Hall Boca Raton, FL · Zbl 1015.34001
[31] Riou, H.; Ladevèze, P.; Sourcis, B., The multiscale VTCR approach applied to acoustics problems, J. Comput. Acoust., 16, 4, 487-505, (2008) · Zbl 1257.74067
[32] Rouch, P.; Ladevèze, P., The variational theory of complex rays: a predictive tool for medium - frequency vibrations, Comput. Methods Appl. Mech. Engrg., 192, 3301-3315, (2003) · Zbl 1054.74602
[33] Strouboulis, T.; Babuška, I.; Hidajat, R., The generalized finite element method for Helmholtz equation: theory, computation, and open problems, Comput. Methods Appl. Mech. Engrg., 195, 4711-4731, (2006) · Zbl 1120.76044
[34] Tezaur, R.; Macedo, A.; Farhat, C.; Djellouli, R., Three-dimensional finite element calculations in acoustic scattering using arbitrarily shaped convex artificial boundaries, Int. J. Numer. Methods Engrg., 53, 1461-1476, (2002) · Zbl 0996.76058
[35] Tezaur, R.; Farhat, C., Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems, Int. J. Numer. Methods Engrg., 66, 796-815, (2006) · Zbl 1110.76319
[36] Tezaur, R.; Zhang, L.; Farhat, C., A discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid/solid problems, Comput. Methods Appl. Mech. Engrg., 197, 1680-1698, (2008) · Zbl 1194.74476
[37] Wang, D.; Tezaur, R.; Toivanen, J.; Farhat, C., Overview of the discontinuous enrichment method, the ultra-weak variational formulation, and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons, Int. J. Numer. Methods Engrg., 89, 4, 403-417, (2012) · Zbl 1242.76143
[38] Zhang, L.; Tezaur, R.; Farhat, C., The discontinuous enrichment method for elastic wave propagation in the medium-frequency regime, Int. J. Numer. Methods Engrg., 66, 2086-2114, (2006) · Zbl 1110.74860
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.