Acta Mechanica Solida Sinica

, Volume 22, Issue 2, pp 116–124

# Elastic dynamic analysis of moderately thick plate using meshless LRPIM

• Ping Xia
• Shuyao Long
• Hongxue Cui
Article

## Abstract

A meshless local radial point interpolation method (LRPIM) for solving elastic dynamic problems of moderately thick plates is presented in this paper. The discretized system equation of the plate is obtained using a locally weighted residual method. It uses a radial basis function (RBF) coupled with a polynomial basis function as a trial function, and uses the quartic spline function as a test function of the weighted residual method. The shape function has the properties of the Kronecker delta function, and no additional treatment is done to impose essential boundary conditions. The Newmark method for solving the dynamic problem is adopted in computation. Effects of sizes of the quadrature sub-domain and influence domain on the dynamic properties are investigated. The numerical results show that the presented method can give quite accurate results for the elastic dynamic problem of the moderately thick plate.

## Key words

meshless method moderately thick plate local radial point interpolation method dynamic analysis Newmark method

## References

1. [1]
Lu, Y.Y., Belytschko, T. and Gu, L., A new implementation of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 1994, 113: 397–414.
2. [2]
Atluri, S.N. and Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics, 1998, 22: 117–127.
3. [3]
Gu, Y.T. and Liu, G.R., A meshless Local Petrov-Galerkin (MLPG) formulation for static and free vibration analyses of thin plates. Computer Modeling in Engineering & Sciences, 2001, 2(4): 463–476.
4. [4]
Long, S.Y., A local Petrov-Galerkin method for the elasticity problem. Acta Mechanical Sinica, 2001, 33(4): 508–518 (in Chinese).
5. [5]
Cai, Y.C., Zhu, H.H. and Wang, J.H., The meshless local Petrov-Galerkin method based on the Voronoi cells. Acta Mechanical Sinica, 2003, 35(2): 187–193 (in Chinese).Google Scholar
6. [6]
Long, S.Y., Liu, K.Y. and Hu, D.A., A new meshless method based on MLPG for elastic dynamic problems. Engineering Analysis with Boundary Elements, 2006, 30: 43–48.
7. [7]
Han, Z.D., Liu, H.T., Rajendran, A.M. and Atluri, S.N., The applications of meshless local Petrov-Galerkin (MLPG) approaches in high-speed impact, penetration and perforation problems. Computer Modeling in Engineering and Sciences, 2006, 14(2): 119–128.
8. [8]
Liu, H.T., Han, Z.D., Rajendran, A.M. and Atluri, S.N., Computational modeling of impact response with the RG damage model and the meshless Local Petrov-Galerkin (MLPG) approaches. Computers Materials and Continua, 2006, 4(1): 43–53.
9. [9]
Gao, L., Liu, K. and Liu, Y., Applications of MLPG method in dynamic fracture problems. Computer Modeling in Engineering and Sciences, 2006, 12(3): 181–195.
10. [10]
Liu, G.R. and Gu, Y.T., A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids. Journal of Sound and Vibration, 2001, 246(1): 29–46.
11. [11]
Liu, G.R., Yan, L., Wang, J.G. and Gu, Y.T., Point interpolation method based on local residual formulation using radial basis functions. Structural Engineering and Mechanics, 2002, 14(6): 713–732.
12. [12]
Wang, H. and Qin, Q.H., Some problems with the method of fundamental solution using radial basis functions. Acta Mechanica Solida Sinica, 2007, 20(1): 21–29.
13. [13]
Sun, H.T. and Wang, Y.H., The meshless virtual boundary method and its applications to 2D elasticity problems. Acta Mechanica Solida Sinica, 2007, 20(1): 30–40.
14. [14]
Zeng, X.Y., Zhu, A.J. and Deng, A.F., Natural element method for analysis of thick plates lying over winkler foundations. Chinese Journal of Solid Mechanics, 2008, 29(2): 163–169 (in Chinese).Google Scholar
15. [15]
Sun, J.D., Zhang, W.X. and Tong, L.W., Modal analysis of moderately thick plates by element-free method. China Civil Engineering Journal, 2006, 39(10): 29–33 (in Chinese).Google Scholar
16. [16]
Qian, L.F., Batra, R.C. and Chen, L.M., Free and forced vibrations of thick rectangular plates by using higher-order shear and normal deformable plate theory and meshless Petrov-Galerkin (MLPG) method. Computer Modeling in Engineering and Sciences, 2003, 4: 519–534.
17. [17]
Xiao, J.R., Batra, R.C., Gilhooley, D.F., Gillespie, Jr. J.W. and McCarthy, M.A., Analysis of thick plates by using a higher-order shear and normal deformable plate theory and MLPG method with radial basis functions. Computer Methods in Applied Mechanics and Engineering, 2007, 196: 979–987.
18. [18]
Wang, M.C. and Shao, M., Basic Theory and Numerical Method of the Finite Element Method. Beijing: Tsinghua University Press, 1997 (in Chinese).Google Scholar
19. [19]
Wang, J.G. and Liu, G.R., On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Computer Methods in Applied Mechanics and Engineering, 2002, 191: 2611–2630.