A Meshless Local Petrov-Galerkin (MLPG) Formulation for Static and Free Vibration Analyses of Thin Plates

A meshless method for the analysis of Kirchhoff plates based on the Meshless Local PetrovGalerkin (MLPG) concept is presented. A MLPG formulation is developed for static and free vibration analyses of thin plates. Local weak form is derived using the weighted residual method in local supported domains from the 4th order partial differential equation of Kirchhoff plates. The integration of the local weak form is performed in a regular-shaped local domain. The Moving Least Squares (MLS) approximation is used to constructed shape functions. The satisfaction of the high continuity requirements is easily met by MLS interpolant, which is based on a weight function with high continuity and a quadratic polynomial basis. The validity and efficiency of the present MLPG method are demonstrated through a number of examples of thin plates under various loads and boundary conditions. Some important parameters on the performance of the present method are investigated thoroughly in this paper. The present method is also compared with EFG method and Finite Element Method in terms of robustness and performance. keyword: Meshless Method; Meshless Local PetrovGalerkin (MLPG) method; Kirchhoff plates; Free Vibration; Numerical Analysis

[1]  S. Mukherjee,et al.  THE BOUNDARY NODE METHOD FOR POTENTIAL PROBLEMS , 1997 .

[2]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[3]  Satya N. Atluri,et al.  New concepts in meshless methods , 2000 .

[4]  N. Aluru,et al.  A Meshless Method for the Numerical Solution of the 2- and 3-D Semiconductor Poisson Equation , 2000 .

[5]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[6]  Ted Belytschko,et al.  Analysis of thin plates by the element-free Galerkin method , 1995 .

[7]  Hyun Gyu Kim,et al.  Analysis of thin beams, using the meshless local Petrov–Galerkin method, with generalized moving least squares interpolations , 1999 .

[8]  E. Oñate,et al.  A FINITE POINT METHOD IN COMPUTATIONAL MECHANICS. APPLICATIONS TO CONVECTIVE TRANSPORT AND FLUID FLOW , 1996 .

[9]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[10]  G. R. Liu,et al.  A local point interpolation method for stress analysis of two-dimensional solids , 2001 .

[11]  T. Belytschko,et al.  A new implementation of the element free Galerkin method , 1994 .

[12]  YuanTong Gu,et al.  Coupling of element free Galerkin and hybrid boundary element methods using modified variational formulation , 2000 .

[13]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[14]  Satya N. Atluri,et al.  Arbitrary Placement of Secondary Nodes, and Error Control, in the Meshless Local Petrov-Galerkin (MLPG) Method , 2000 .

[15]  Nathan Ida,et al.  Introduction to the Finite Element Method , 1997 .

[16]  Romesh C. Batra,et al.  DETERMINATION OF CRACK TIP FIELDS IN LINEAR ELASTOSTATICS BY THE MESHLESS LOCAL PETROV-GALERKIN (MLPG) METHOD , 2001 .

[17]  S. Atluri,et al.  A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach , 1998 .

[18]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[19]  S. BRODETSKY,et al.  Theory of Plates and Shells , 1941, Nature.

[20]  Hyun Gyu Kim,et al.  A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods , 1999 .

[21]  YuanTong Gu,et al.  A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids , 2001 .