Hostname: page-component-89b8bd64d-4ws75 Total loading time: 0 Render date: 2026-05-08T04:37:23.007Z Has data issue: false hasContentIssue false
  • English
  • Français

An Element Decomposition Method for the Helmholtz Equation

Part of: Probabilistic methods, simulation and stochastic differential equations Partial differential equations, boundary value problems

Published online by Cambridge University Press:  02 November 2016

Gang Wang ,
Xiangyang Cui  and
Guangyao Li
Gang Wang*
Affiliation:
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, P.R. China Collaborative Innovation Center of Intelligent New Energy Vehicle, Shanghai, 200092, P.R. China
Xiangyang Cui*
Affiliation:
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, P.R. China Collaborative Innovation Center of Intelligent New Energy Vehicle, Shanghai, 200092, P.R. China
Guangyao Li*
Affiliation:
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, P.R. China Collaborative Innovation Center of Intelligent New Energy Vehicle, Shanghai, 200092, P.R. China
*
*Corresponding author. Email addresses:wanggangny@163.com (G. Wang), cuixy@hnu.edu.cn (X. Y. Cui), gyli@hnu.edu.cn (G. Y. Li)
*Corresponding author. Email addresses:wanggangny@163.com (G. Wang), cuixy@hnu.edu.cn (X. Y. Cui), gyli@hnu.edu.cn (G. Y. Li)
*Corresponding author. Email addresses:wanggangny@163.com (G. Wang), cuixy@hnu.edu.cn (X. Y. Cui), gyli@hnu.edu.cn (G. Y. Li)
Get access

Abstract

It is well-known that the traditional full integral quadrilateral element fails to provide accurate results to the Helmholtz equation with large wave numbers due to the “pollution error” caused by the numerical dispersion. To overcome this deficiency, this paper proposed an element decomposition method (EDM) for analyzing 2D acoustic problems by using quadrilateral element. In the present EDM, the quadrilateral element is first subdivided into four sub-triangles, and the local acoustic gradient in each sub-triangle is obtained using linear interpolation function. The acoustic gradient field of the whole quadrilateral is then formulated through a weighted averaging operation, which means only one integration point is adopted to construct the system matrix. To cure the numerical instability of one-point integration, a variation gradient item is complemented by variance of the local gradients. The discretized system equations are derived using the generalized Galerkin weakform. Numerical examples demonstrate that the EDM can achieves better accuracy and higher computational efficiency. Besides, as no mapping or coordinate transformation is involved, restrictions on the shape elements can be easily removed, which makes the EDM works well even for severely distorted meshes.

Keywords

Helmholtz equationnumerical methodselement decomposition methodone-point integrationquadrilateral elementdispersion error

MSC classification

Secondary: 65C20: Models, numerical methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N22: Solution of discretized equations 68U20: Simulation

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1258 - 1282
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Thompson, L.L., A review of finite-element methods for time-harmonic acoustics, J. Acoust. Soc. Am., 119 (2006), 1315–1330.CrossRefGoogle Scholar
[2] Chen, J.T., Recent development of dual BEM in acoustic problems, Comput. Methods Appl. Mech. Engrg., 188 (2000), 833–845.CrossRefGoogle Scholar
[3] Ihlenburg, F., Babuska, I., Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation, Int. J. Numer. Meth. Engng., 38 (1995), 3745–3774.CrossRefGoogle Scholar
[4] Ihlenburg, F., Babuska, I., Finite element solution for the Helmholtz equation with high wave number. Part I: The h-version of the FEM, Computers Math. Applic., 30 (1995), 9–37.CrossRefGoogle Scholar
[5] Gaul, L., Wagner, M., Wenzel, W., Dumont, N., Numerical treatment of acoustic problems with the hybrid boundary element method, Int. J. Solids Struct., 38 (2001), 1871–1888.CrossRefGoogle Scholar
[6] Li, S., Huang, Q., A new fast multipole boundary element method for two dimensional acoustic problems, Comput. Methods Appl. Mech. Engrg., 200 (2011), 1333–1340.CrossRefGoogle Scholar
[7] Genechten, B.V., Atak, O., Bergen, B., Deckers, E., Jonckheere, S., Lee, J.S., Maressa, A., Vergote, K., Pluymers, B., Vandepitte, D., Desmet, W., An efficient wave based method for solving Helmholtz problems in three-dimensional bounded domains, Eng. Anal. Bound. Elem., 36 (2012), 63–75.CrossRefGoogle Scholar
[8] Zhang, G.M., Batra, R.C., Wave propagation in functionally graded materials by modified smoothed particle hydrodynamics (MSPH) method, J. Comput. Phys., 222 (2007), 374–390.CrossRefGoogle Scholar
[9] Sze, K.Y., Liu, G.H., Fan, H., Four- and eight-node hybrid-Trefftz quadrilateral finite element models for Helmholtz problem, Comput. Methods Appl. Mech. Engrg., 199 (2010), 598–614.CrossRefGoogle Scholar
[10] Li, W., Chai, Y.B., Lei, M., Liu, G.R., Analysis of coupled structural-acoustic problems based on the smoothed finite element method (S-FEM), Eng. Anal. Bound. Elem., 42 (2014), 84–91.CrossRefGoogle Scholar
[11] Li, E., He, Z.C., Xu, X., Liu, G.R., Hybrid smoothed finite element method for acoustic problems, Comput. Methods Appl. Mech. Engrg., 283 (2015), 664–688.CrossRefGoogle Scholar
[12] Mu, L., Wang, J.P., Ye, X., Zhao, S., A numerical study on the weak Galerkin method for the Helmholtz equation, Commun. Comput. Phys., 15 (2014), 1461–1479.CrossRefGoogle Scholar
[13] Harari, I., Magoules, F., Numerical investigation of stabilized finite element computations for acoustics, Wave Motion, 39 (2004), 339–349.CrossRefGoogle Scholar
[14] Yao, L.Y., Yu, D.J., Cui, X.Y., Zang, X.G., Numerical treatment of acoustic problems with the smoothed finite element method, Appl. Acoust., 71 (2010), 743–753.CrossRefGoogle Scholar
[15] Ihlenburg, F., Finite Element Analysis of Acoustic Scatting (Applied Mathematical Sciences), New York: Springer-Verlag, 1998.CrossRefGoogle Scholar
[16] Babuska, I. M, Sauter, S.A., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM Rev., 42 (2000), 451–484.Google Scholar
[17] Thompson, L.L., Pinsky, P. M., A Galerkin least squares finite element method for the two-dimensional Helmholtz equation, Int. J. Numer. Meth. Engng., 38 (1995), 371–397.CrossRefGoogle Scholar
[18] Franca, L.P., Dutra do Carmo, E.G., The Galerkin gradient least-squares method, Comput. Methods Appl. Mech. Engrg., 74 (1989), 41–54.CrossRefGoogle Scholar
[19] Franca, L.P., Farhat, C., Macedo, A.P., Lesoinne, M., Residual-free bubbles for the Helmholtz equation, Int. J. Numer. Meth. Engng., 40 (1997), 4003–4009.3.0.CO;2-Z>CrossRefGoogle Scholar
[20] Oberai, A.A., Pinsky, P.M., A multiscale finite element method for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 154 (1998), 281–297.CrossRefGoogle Scholar
[21] Farhat, C., Harari, I., Franca, L.P., The discontinuous enrichment method, Comput. Methods Appl. Mech. Engrg., 190 (2001), 6455–6479.CrossRefGoogle Scholar
[22] Tezaur, R., Kalashnikova, I., Farhat, C., The discontinuous enrichment method for medium-frequency Helmholtz problems with a spatially variable wavenumber, Comput. Methods Appl. Mech. Engrg., 268 (2014), 126–140.CrossRefGoogle Scholar
[23] Min, D.J., Yoo, H., Shin, C., Hyun, H.J., Suh, J.H., Weighted-averaging finite-element method for scalar wave equation in the frequency domain, J. Seism. Explor., 11 (2002), 197–222.Google Scholar
[24] Guddati, M.N., Yue, B., Modified integration rules for reducing dispersion error in finite element methods, Comput. Methods Appl. Mech. Engrg., 193 (2004), 275–287.CrossRefGoogle Scholar
[25] Gabard, G., Discontinuous Galerkin methods with plane waves for the displacement-based acoustic equation, Int. J. Numer. Meth. Engng., 66 (2006), 549–569.CrossRefGoogle Scholar
[26] Grigoroscuta-Strugaru, M., Amara, M., Calandra, H., Djellouli, R., A modified discontinuous Galerkin method for solving efficiently Helmholtz problems, Commun. Comput. Phys., 11 (2012), 335–350,CrossRefGoogle Scholar
[27] Babuska, I., Melenk, J.M., The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139 (1996), 289–314.Google Scholar
[28] Strouboulis, T., Hidajat, R., Partition of unity method for Helmholtz equation: q-convergence for plane-wave and wave-band local bases, Appl. Meth., 2 (2006), 181–204.CrossRefGoogle Scholar
[29] Yao, L.Y., Yu, D.J., Cui, X.Y., Zhou, J.W., A hybrid finite element-least square point interpolation method for solving acoustic problems, Noise Control Eng. J., 60 (2012), 97–112.CrossRefGoogle Scholar
[30] Yao, L.Y., Zhou, J.W., Zhou, Z., Li, L., Dispersion error reduction for acoustic problems using the finite element-least square point interpolation method, J. Vib. Acoust., 137 (2015), 021013.CrossRefGoogle Scholar
[31] Bouillard, Ph., Suleau, S., Element-free Galerkin solutions for Helmholtz problems: Formulation and numerical assessment of the pollution effect, Comput. Methods Appl. Mech. Engrg., 162 (1998), 317–335.CrossRefGoogle Scholar
[32] Wenterodt, C., Estorff, O.V., Dispersion analysis of the meshfree radial point interpolation method for the Helmholtz equation, Int. J. Numer. Meth. Engng., 77 (2009), 1670–1689.CrossRefGoogle Scholar
[33] Cui, X.Y., Liu, P.W., Li, G.Y., An element decomposition method with variance strain stabilization, Sci. China-Phys. Mech. Astron., 58 (2015), 074601.CrossRefGoogle Scholar
[34] Ihlenburg, F., Babuska, I., Reliability of finite element methods for the numerical computation of waves, Adv. Eng. Softw., 28 (1997), 417–424.CrossRefGoogle Scholar
[35] Harari, I., Nogueira, C.L., Reducing dispersion of linear triangular elements for the Helmholtz equation. J. Eng. Mech., 128 (2002), 351–358.CrossRefGoogle Scholar