Abstract
The meshless local Petrov-Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving leastsquares approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method was a truly meshless one as it did not need a finite element or boundary element mesh, either for purpose of interpolation of the solution, or for the integration of the energy. All integrals could be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions were enforced by the penalty method. Several numerical examples were presented to illustrate the implementation and performance of the present method. The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply-supported edge conditions. No post processing procedure is required to computer the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.
Similar content being viewed by others
References
Atluri S N, Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics [J].Comput Mech, 1998,22(1):117–127.
Atluri S N, Cho J Y, Kim H G. Analysis of thin beams, using the meshless local Petrov-Galerkin method, with generalized moving least squares interpolation[J].Comput Mech, 1999,24(4): 334–347.
Atluri S N, Kim H G, Cho J Y. A critical assessment of the truly meshless local Pettrov-Galerkin (MLPG), and local boundary integral equation (LBIE) methods[J].Comput Mech, 1999,24(2): 348–372.
Atluri S N, Zhu T. New concepts in meshless methods[J].International Journal for Numerical Methods in Engineering, 2000,47(3):537–556.
Belytschko T, Lu Y Y, Gu L. Element-free Galerkin methods[J].International Journal for Numerical Methods in Engineering, 1994,37(2):229–256.
LONG Shu-yao. A local Petrov-Galerkin method for the elasticity problem[J].Acta Mechanica Sinica 2001,33(4):508–518. (in Chinese)
Costa J A. The boundary element method applied to plate problems [D]. Southampton, UK: Southampton University, 1986.
Timoshenko S, Woinowsky-Krieger S.Theory of Plates and Shells[M]. 2nd ed. New York: McGraw-Hill, 1959.
Author information
Authors and Affiliations
Additional information
(Communicated by CHENG Chang-jun)
Foundation items: the National Natural Science Foundation of China (10372030); the Natural Science Foundation of Hunan, China (02JJY4071)
Biographies: XIONG Yuan-bo (1959≈)
Rights and permissions
About this article
Cite this article
Yuan-bo, X., Shu-yao, L. Local Petrov-Galerkin method for a thin plate. Appl Math Mech 25, 210–218 (2004). https://doi.org/10.1007/BF02437322
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02437322
Key words
- thin plate
- meshless local Petrov-Galerkin method
- moving least square approximation
- symmetric weak form of equivalent integration for differential equation