The improved element-free Galerkin method for three-dimensional wave equation

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• 作者：Zan Zhang (12)
  Dong-Ming Li (3)
  Yu-Min Cheng (12) ymcheng@shu.edu.cn
  Kim Moew Liew (3)
• 关键词：Weighted orthogonal function &#8211 ; Improved moving least squares (IMLS) approximation &#8211 ; Improved element ; free Galerkin (IEFG) method &#8211 ; Penalty method &#8211 ; Temporal discretization &#8211 ; Wave equation
• 刊名：Acta Mechanica Sinica
• 出版年：2012
• 出版时间：June 2012
• 年：2012
• 卷：28
• 期：3
• 页码：808-818
• 全文大小：787.5 KB
• 参考文献：1. Selvadurai, A.P.S.: Partial Differential Equations in Mechanics. Springer, Berlin (2000)
  2. Nettel, S.:Wave Physics:Oscillations-Solitons-Chaos.Springer, Berlin (2003)
  3. Hirose, A.: Introduction to Wave Phenomena. Wiley, New York (1985)
  4. Wazwaz, A.M.: Partial differential equations: methods and applications. Swets & Zeitlinger B.V., Lisse, Netherlands (2002)
  5. Lapidus L., Pinder, G. F.: Numerical Solution of Partial Differential Equations In Science And Engineering. Wiley, New York (1982)
  6. Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Mathematics and Computers in Simulation 71, 16–30 (2006)
  7. Dehghan, M.: On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numerical Methods for Partial Differential Equations 21, 24–40 (2005)
  8. Dehghan, M.: A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications. Numerical Methods for Partial Differential Equations 22, 220–257 (2006)
  9. Dehghan, M.: The one-dimensional heat equation subject to a boundary integral specification. Chaos, Solitons and Fractals 32, 661–675 (2007)
  10. Dehghan, M.: Time-splitting procedures for the solution of the two-dimensional transport equation. Kybernetes 36, 791–805 (2007)
  11. Kamruzzaman, M.: A new meshless collocation method for partial differential equations. Communications in Numerical Methods in Engineering 24, 1617–1639 (2008)
  12. Xiong, Z.G.: Convergence of FEM with interpolated coefficients for semilinear hyperbolic equation. Journal of Computational and Applied mathematics 24, 313–317 (2008)
  13. Avilez-Valente, P., Seabra-Santos, F.J.: A Petrov-Galerkin finite element scheme for the regularized long wave equation. Computational Mechanics 34, 256–270 (2004)
  14. Feng, Y.D., Wang, Y.S., Zhang. Z.M.: Analysis of transient wave scattering from an inclusion with a unilateral smoothly contact interface by BEM. Archive of Applied Mechanics 73, 337–347 (2003)
  15. Gamallo, P., Astley. R.J.: The partition of unity finite element method for short wave acoustic propagation on non-uniform potential flows. International Journal for Numerical Methods in Engineering 65, 425–444 (2006)
  16. Carlos J.S., Alves, S., Valtchev, S.: Numerical comparison of two meshfree methods for acoustic wave scattering. Engineering Analysis with Boundary Elements 29, 371–382 (2005)
  17. Lu, Y.Y., Belytschko, T., Tabbara, M.,: Element-free Galerkin method for wave propagation and dynamic fracture. Computer Methods in Applied Mechanics and Engineering 26, 131–153 (1995)
  18. Bouillard, P., Lacroix, V., De Bel, E.: A wave-oriented meshless formulation for acoustical and vibro-acoustical applications. Wave Motion 39, 295–305 (2004)
  19. Liu, G.R., Mesh Free Methods: Moving Beyond the Finite Element Method. CRC Press, Boca Raton (2003)
  20. Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: Difference approximation and diffuse elements. Computational Mechanics 10, 583–594 (1992)
  21. Lancaster, P., Salkauskas, K.: Surfaces generated by moving least-squares methods. Mathematics Computation 37, 141–158 (1981)
  22. Liew, K.M., Cheng, Y., Kitipornchai, S.: Boundary elementfree method (BEFM) for two-dimensional elastodynamic analysis using Laplace transform. International Journal of Numerical Methods in Engineering 64, 1610–1627 (2005)
  23. Liew, K.M., Cheng, Y., Kitipornchai, S.: Boundary elementfree method (BEFM) and its application to two-dimensional elasticity problems. International Journal of Numerical Methods in Engineering 65, 1310–1332 (2006)
  24. Liew, K.M., Zou, G.P., Rajendran, S.: A spline strip kernel particle method and its application to two-dimensional elasticity problems. International Journal for Numerical Methods in Engineering 57, 599–616 (2003)
  25. Zhang, Z., Zhao, P., Liew, K.M.: Improved element-free Galerkin method for two-dimensional potential problems. Engineering Analysis with Boundary Elements 33, 547–554 (2009)
  26. Zhang, Z., Zhao, P., Liew, K.M.: Analyzing three-dimensional potential problems with the improved element-free Galerkin method. Computational Mechanics 44, 273–284 (2009)
  27. Frank, C.G., Liu, W.K.: Implementation of boundary conditions for meshless methods. Computer Methods in Applied Mechanics and Engineering 163, 205–230 (1998)
  28. Mukherjee Y.X., Mukherjee S.: On boundary conditions in the element-free Galerkin method. Computational Mechanics 19, 264–270 (1997)
  29. Ventura G.: An augmented Lagranging approach to essential boundary conditions in meshless methods. International Journal for Numerical Methods in Engineering 53, 825–843 (2002)
  30. Zhang X., Liu X.H., Song K.Z., et al.: Least-square collocation meshless method. International Journal for Numerical Methods in Engineering 51, 1089–1100 (2001)
  31. Krysl P., Belytschko T.: Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions. Computer Methods in Applied Mechanics and Engineering 148, 257–277 (1997)
  32. Chung H.J., Belytschko T.: An error estimate in the EFG method. Computational mechanics 21, 91–100 (1998)
  33. Gavete L., Cuesta J.L., Ruiz A.: A procedure for approximation of the error in the EFG method. International Journal for Numerical Methods in Engineering 53, 677–690 (2002)
  34. Gavete L., Gavete M.L.: A posteriori error approximation in EFG method. International Journal for Numerical Methods in Engineering 58, 2239–2263 (2003)
  35. Gavete, L., Faclon, S.: An error indicator for the element free Galerkin method. European Journal of Mechanics-A/Solids 20, 327–341 (2001)
• 作者单位：1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, 200072 Shanghai, China2. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, 200072 Shanghai, China3. Department of Civil and Architectural Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR, China
• ISSN：1614-3116

文摘

The paper presents the improved element-free Galerkin (IEFG) method for three-dimensional wave propagation. The improved moving least-squares (IMLS) approximation is employed to construct the shape function, which uses an orthogonal function system with a weight function as the basis function. Compared with the conventional moving least-squares (MLS) approximation, the algebraic equation system in the IMLS approximation is not ill-conditioned, and can be solved directly without deriving the inverse matrix. Because there are fewer coefficients in the IMLS than in the MLS approximation, fewer nodes are selected in the IEFG method than in the element-free Galerkin method. Thus, the IEFG method has a higher computing speed. In the IEFG method, the Galerkin weak form is employed to obtain a discretized system equation, and the penalty method is applied to impose the essential boundary condition. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the wave equations and the boundary-initial conditions depend on time, the scaling parameter, number of nodes and the time step length are considered for the convergence study.