DOI QR코드

DOI QR Code

Multi-scale finite element analysis of acoustic waves using global residual-free meshfree enrichments

  • Wu, C.T. (Livermore Software Technology Corporation) ;
  • Hu, Wei (Livermore Software Technology Corporation)
  • 투고 : 2013.03.01
  • 심사 : 2013.05.05
  • 발행 : 2013.09.01

초록

In this paper, a multi-scale meshfree-enriched finite element formulation is presented for the analysis of acoustic wave propagation problem. The scale splitting in this formulation is based on the Variational Multi-scale (VMS) method. While the standard finite element polynomials are used to represent the coarse scales, the approximation of fine-scale solution is defined globally using the meshfree enrichments generated from the Generalized Meshfree (GMF) approximation. The resultant fine-scale approximations satisfy the homogenous Dirichlet boundary conditions and behave as the "global residual-free" bubbles for the enrichments in the oscillatory type of Helmholtz solutions. Numerical examples in one dimension and two dimensional cases are analyzed to demonstrate the accuracy of the present formulation and comparison is made to the analytical and two finite element solutions.

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참고문헌

  1. Babuska, I., Ihlenburg, F., Paik, E. and Sauter, S. (1995), "A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution", Comput. Meth. Appl. M., 128(3-4), 325-359. https://doi.org/10.1016/0045-7825(95)00890-X
  2. Baiges, J. and Codina, R. (2013), "A variational multiscale method with subscales on the element boundaries for the Helmholtz equation", Int. J. Numer. Meth. Eng., 93(6), 664-684. https://doi.org/10.1002/nme.4406
  3. Belytschko, T. and Mullen, R. (1978), "On dispersive properties of finite element solutions", Modern Problems in Elastic Wave Propagation, John Wiley & Sons, Ltd.
  4. Belytschko, T., Lu, Y.Y. and Gu, L. (1994), "Element-free Galerkin methods", Int. J. Numer. Meth. Eng., 37(2), 229-256. https://doi.org/10.1002/nme.1620370205
  5. Bouillard, P. and Suleau, S. (1998), "Element-free Galerkin solutions for Helmholtz problems: formulation and numerical assessment of pollution effect", Comput. Meth. Appl. Mech. Eng., 161, 317-335.
  6. Franca, L.P., Madureira, A.L. and Valentin, F. (2005), "Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions", Comput. Meth. Appl. Mech. Eng., 194(27-29), 3006-3021. https://doi.org/10.1016/j.cma.2004.07.029
  7. Farhat, C., Harari, I. and Franca, L.P. (2001), "The discontinuous enrichment method", Comput. Meth. Appl. Mech. Eng., 190, 6455-6479. https://doi.org/10.1016/S0045-7825(01)00232-8
  8. Harari, I. and Hughes, T.J.R. (1991), "Finite element method for the Helmholtz equation in an exterior domain: Model problems", Comput. Meth. Appl. Mech. Eng., 87(1), 59-96. https://doi.org/10.1016/0045-7825(91)90146-W
  9. Harari, I. and Hughes, T.J.R. (1992), "Galerkin/least squares finite element method for the reduced wave equation with non-reflecting boundary conditions", Comput. Meth. Appl. Mech. Eng., 98(3), 441-454.
  10. Harari, I. and Gosteev, K. (2007), "Bubble-based stabilization for the Helmholtz equation", Int. J. Numer. Meth. Eng., 70(10), 1241-1260. https://doi.org/10.1002/nme.1930
  11. Harari, I. (2008), "Multiscale finite elements for acoustics: continuous, discontinuous, and stabilized methods", Int. J. Numer. Meth. Eng., 6, 511-531.
  12. Hu, W., Wu, C.T. and Koishi, M. (2012), "A displacement-based nonlinear finite element formulation using meshfree-enriched triangular elements for the two-dimensional large deformation analysis of elastomers", Finite Elem. Anal. Des., 50, 161-172. https://doi.org/10.1016/j.finel.2011.09.007
  13. Hughes, T.J.R., Scovazzi, G. and Franca, L.P. (2004), "Multiscale and stabilized methods", Encyclopedia of Computational Mechanics, John Wiley & Sons, Ltd, 3.
  14. Hughes, T.J.R. and Sangalli, G. (2007), "Variational multiscale analysis: the fine-scale Green's function, projection, optimization, localization, and the stabilized methods", SIAM J. Numer. Anal., 45(2), 539-557. https://doi.org/10.1137/050645646
  15. Ihlenburg, F. and Babuska, I. (1995), "Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of FEM", Comput. Math. Appl., 30(9), 9-37.
  16. Ihlenburg, F. and Babuska, I. (1997), "Finite element solution of the Helmholtz equation with high wave number Part II: The h-p version of FEM", SIAM J. Numer. Anal., 34(1), 315-358. https://doi.org/10.1137/S0036142994272337
  17. Lai, S.J., Wang, B.Z. and Duan, Y. (2010), "Solving Helmholtz equation by meshless radial basis functions method", Prog. Electromagnetics Res. B, 24, 351-367. https://doi.org/10.2528/PIERB10062303
  18. Liu, W.K., Jun, S. and Zhang, Y.F. (1995), "Reproducing kernel particle methods", Int. J. Numer. Meth. Fl., 20(8-9), 1081-1106. https://doi.org/10.1002/fld.1650200824
  19. Liu, W.K., Hao, W., Chen, Y., Jun, S. and Gosz, J. (1997), "Multiresolution reproducing kernel particle methods", Comput. Mech., 20, 295-309. https://doi.org/10.1007/s004660050252
  20. Oberai, A.A. and Pinsky, P.M. (1998), "A multiscale finite element method for the Helmholtz equation", Comput. Meth. Appl. Mech. Eng., 154(3-4), 281-297. https://doi.org/10.1016/S0045-7825(97)00130-8
  21. Oberai, A.A. and Pinsky, P.M. (2000), "A residual-based finite element method for the Helmholtz equation", Int. J. Numer. Meth. Eng., 49(3), 399-419. https://doi.org/10.1002/1097-0207(20000930)49:3<399::AID-NME844>3.0.CO;2-5
  22. Park, C.K., Wu, C.T. and Kan, C.D. (2011), "On the analysis of dispersion property and stable time step in meshfree method using generalized meshfree approximation", Finite Elem. Anal. Des., 47(7), 683-697. https://doi.org/10.1016/j.finel.2011.02.001
  23. Suleau, S. and Bouillard, P. (2000), "One-dimensional dispersion analysis for the element-free Galerkin method for the Helmholtz equation", Int. J. Numer. Meth. Eng., 47(6), 1169-1188. https://doi.org/10.1002/(SICI)1097-0207(20000228)47:6<1169::AID-NME824>3.0.CO;2-9
  24. Uras, R.A., Chang, C.T., Chen, Y. and Liu, W.K. (1997), "Multi-resolution reproducing kernel particle methods in Acoustics", J. Comput. Acoust., 5, 71-94. https://doi.org/10.1142/S0218396X9700006X
  25. Voth, T.E. and Christon, M.A. (2001), "Discretization errors associated with reproducing kernel methods: one-dimensional domains", Comput. Meth. Appl. Mech. Eng., 190(18-19), 2429-2446. https://doi.org/10.1016/S0045-7825(00)00245-0
  26. Wenterodt, C. and von Estorff, O. (2009), "Dispersion analysis of the meshfree radial point interpolation method for the Helmholtz equation", Int. J. Numer. Meth. Eng., 77(12), 1670-1689. https://doi.org/10.1002/nme.2463
  27. Wu, C.T. and Koishi, M. (2009), "A meshfree procedure for the microscopic analysis of particle-reinforced rubber compounds", Interact. Multiscale Mech., 2(2), 129-151. https://doi.org/10.12989/imm.2009.2.2.129
  28. Wu, C.T., Park, C.K. and Chen, J.S. (2011), "A generalized approximation for the meshfree analysis of solids", Int. J. Numer. Meth. Eng., 85(6), 693-722. https://doi.org/10.1002/nme.2991
  29. Wu, C.T. and Hu, W. (2011), "Meshfree-enriched simplex elements with strain smoothing for the finite element analysis of compressible and nearly incompressible solids", Comput. Meth. Appl. Mech. Eng., 200(45-46), 2991-3010. https://doi.org/10.1016/j.cma.2011.06.013
  30. Wu, C.T., Hu, W. and Chen, J.S. (2012), "A meshfree-enriched finite element method for compressible and near-incompressible elasticity", Int. J. Numer. Meth. Eng., 90(7), 882-914. https://doi.org/10.1002/nme.3349
  31. Wu, C.T. and Koishi, M. (2012), "Three-dimensional meshfree-enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites", Int. J. Numer. Meth. Eng., 91(11), 1137-1157. https://doi.org/10.1002/nme.4306
  32. Wu, C.T., Guo, Y. and Askari, E. (2013), "Numerical modeling of composite solids using an immersed meshfree Galerkin method", Composit. B, 45(1), 1397-1413. https://doi.org/10.1016/j.compositesb.2012.09.061
  33. Yao, L.Y., Yu, D.J., Cui, X.Y. and Zang, X.G. (2010), "Numerical treatment of acoustic problems with the smoothed finite element method", Appl. Acoust., 71(8), 743-753. https://doi.org/10.1016/j.apacoust.2010.03.006
  34. You, Y., Chen, J.S. and Voth, T.E. (2002), "Characteristics of semi- and full discretization of stabilized Galerkin meshfree method", Finite Elem. Anal. Des., 38(10), 999-1012. https://doi.org/10.1016/S0168-874X(02)00090-2

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