Publication de 6 numéros par an
ISSN Imprimer: 1543-1649
ISSN En ligne: 1940-4352
Indexed in
Multiscale Finite Elements for Acoustics: Continuous, Discontinuous, and Stabilized Methods
RÉSUMÉ
This work describes two perspectives for understanding the numerical difficulties that arise in the solution of wave problems, and various advances in the development of efficient discretization schemes for acoustics. Standard, low-order, continuous Galerkin finite element methods are unable to cope with wave phenomena at short wave lengths because the computational effort required to resolve the waves and control numerical dispersion errors becomes prohibitive. The failure to adequately represent subgrid scales misses not only the fine-scale part of the solution, but often causes severe pollution of the solution on the resolved scale as well. Since computation naturally separates the scales of a problem according to the mesh size, multi-scale considerations provide a useful framework for viewing these difficulties and developing methods to counter them. The Galerkin/least squares method arises in multiscale settings, and its stability parameter is defined by dispersion considerations. Bubble enriched methods employ auxiliary functions that are usually expressed in the form of infinite series. Dispersion analysis provides guidelines for the implementation of the series representation in practice. In the discontinuous enrichment method, the fine scales are spanned by free-space homogeneous solutions of the governing equations. These auxiliary functions may be discontinuous across element boundaries, and continuity is enforced weakly by Lagrange multipliers.
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Babuka, I., Ihlenburg, F., Paik, E. T., and Sauter, S. A., A Generalized Finite Element Method for Solving the Helmholtz Equation in Two Dimensions with Minimal Pollution. DOI: 10.1016/0045-7825(95)00890-X
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Burnett, D. S., A Three-Dimensional Acoustic Infinite Element Based on a Prolate Spheroidal Multipole Expansion. DOI: 10.1121/1.411286
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Harari, I. and Hughes, T. J. R., A Cost Comparison of Boundary Element and Finite Element Methods for Problems of Time-Harmonic Acoustics. DOI: 10.1016/0045-7825(92)90108-V
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Chen, J.-T. and Chen, K.-H., Applications of the Dual Integral Formulation in Conjunction with Fast Multipole Method in Large-Scale Problems for 2D Exterior Acoustics. DOI: 10.1016/S0955-7997(03)00122-X
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Harari, I. and Hughes, T. J. R., Galerkin/Least- Squares Finite Element Methods for the Reduced Wave Equation with Nonreflecting Boundary Conditions in Unbounded Domains. DOI: 10.1016/0045-7825(92)90006-6
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Franca, L. P. and Farhat, C., Bubble Functions Prompt Unusual Stabilized Finite Element Methods. DOI: 10.1016/0045-7825(94)00721-X
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Franca, L. P., Madureira, A. L., and Valentin, F, Towards Multiscale Functions: Enriching Finite Element Spaces with Local but Not Bubble-Like Functions. DOI: 10.1016/j.cma.2004.07.029
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Franca, L. P. and Valentin, F., On an Improved Unusual Stabilized Finite Element Method for the Advective-Reactive-Diffusive Equation. DOI: 10.1016/S0045-7825(00)00190-0
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Harari, I., Finite Element Dispersion of Cylindrical and Spherical Acoustic Waves. DOI: 10.1016/S0045-7825(00)00251-6
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Christon, M. A., The Influence of the Mass Matrix on the Dispersive Nature of the Semi- Discrete, Second-Order Wave Equation. DOI: 10.1016/S0045-7825(98)00266-7
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Christon, M. A. and Voth, T. E., Results of von Neumann Analyses for Reproducing Kernel Semi-Discretizations. DOI: 10.1002/(SICI)1097-0207(20000310)47:7<1285::AID-NME823>3.0.CO;2-3
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Harari, I., Reducing Spurious Dispersion, Anisotropy and Reflection in Finite Element Analysis of Time-Harmonic Acoustics. DOI: 10.1016/S0045-7825(96)01034-1
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Magoules, F., Meerbergen, K., and Coyette, J.-P., Application of a Domain Decomposition Method with Lagrange Multipliers to Acoustic Problems Arising from the Automotive Industry. DOI: 10.1142/S0218396X00000297
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Harari, I. and Gosteev, K., Bubble-Based Stabilization for the Helmholtz Equation. DOI: 10.1002/nme.1930
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Cessenat, O. and Despres, B., Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem. DOI: 10.1137/S0036142995285873
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Cessenat, O. and Despres, B., Using Plane Waves as Base Functions for Solving Time Harmonic Equations with the Ultra Weak Variational Formulation. DOI: 10.1142/S0218396X03001912
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Farhat, C., Harari, I., and Franca, L. P., The Discontinuous Enrichment Method. DOI: 10.1016/S0045-7825(01)00232-8
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Farhat, C., Harari, I., and Hetmaniuk, U., The Discontinuous Enrichment Method for Multiscale Analysis. DOI: 10.1016/S0045-7825(03)00344-X
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Zhang, L., Tezaur, R., and Farhat, C., The Discontinuous Enrichment Method for Elastic Wave Propagation in the Medium-Frequency Regime. DOI: 10.1002/nme.1619
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Massimi, P., Tezaur, R., and Farhat, C., A Discontinuous Enrichment Method for Three- Dimensional Multiscale Harmonic Wave Propagation Problems in Multi-Fluid and Fluid-Solid Media. DOI: 10.1002/nme.2334
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Tezaur, R., Zhang, L., and Farhat, C., A Discontinuous Enrichment Method for Capturing Evanescent Waves in Multiscale Fluid and Fluid/Solid Problems. DOI: 10.1016/j.cma.2007.08.023
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Grosu, E. and Harari, I., Studies of the discontinuous enrichment method for twodimensional acoustics. DOI: 10.1016/j.finel.2007.11.016
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Gabard, G., Exact Integration of Polynomial- Exponential Products with Application to Wave-Based Numerical Methods. DOI: 10.1002/cnm.1123
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Grosu, E. and Harari, I., Three-dimensional Element Configurations for the Discontinuous Enrichment Method for Acoustics. DOI: 10.1002/nme.2525
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Farhat, C., Harari, I., and Hetmaniuk, U., ADiscontinuous Galerkin Method with Lagrange Multipliers for the Solution of Helmholtz Problems in the Mid-Frequency Regime. DOI: 10.1016/S0045-7825(02)00646-1
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Farhat, C., Tezaur, R., and Weidemann- Goiran, P., Higher-Order Extensions of a Discontinuous Galerkin Method for Mid- Frequency Helmholtz Problems. DOI: 10.1002/nme.1139
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Harari, I., Tezaur, R., and Farhat, C., A Study of Higher-Order Discontinuous Galerkin and Quadratic Least-Squares Stabilized Finite Element Computations for Acoustics. DOI: 10.1142/S0218396X06002792
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Tezaur, R. and Farhat, C., Three-Dimensional Discontinuous Galerkin Elements with Plane Waves and Lagrange Multipliers for the Solution of Mid-Frequency Helmholtz Problems. DOI: 10.1002/nme.1575
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Huttunen, T., Seppala, E. T., Kirkeby, O., Karkkainen, A., and Karkkainen, L., Simulation of the Transfer Function for a Head-and-Torso Model over the Entire Audible Frequency Range. DOI: 10.1142/S0218396X07003469