ABSTRACT
The element-free Galerkin (EFG) method requires a mesh of background cells for integration in computing the systemmatrices. The reason behind the need for background cells for integration is the use of the Galerkin weak form for generating the discrete system equations. Is it possible not to use the weak form? The answer is yes: meshfree methods that operate on strong forms, such as the irregular finite difference method [1,2], finite point method [3], and local point collocation methods [4-8] have been developed. However, these kinds of methods are generally not very stable against node irregularities, and the results obtained can be less accurate. Efforts are still being made to stabilize these methods, especially in the direction of using local radial functions with properly devised regularization techniques [7,8]. In using the weighted residual method, if we try to satisfy the equation point-by-point
using information in a local domain of the point as we do in the point collocation methods, the integration form can then be implemented locally by carrying out numerical integration over the local domain. The meshless local Petrov-Galerkin (MLPG) method originated by Atluri and Zhu [9] uses the so-called local weak form of the Petrov-Galerkin residual formulation. MLPG has been fine-tuned, improved, and extended over the years [10-17]. This chapter details theMLPGmethod for two-dimensional (2D) solid mechanics problems. In the MLPG implementation, moving least squares (MLS) approximation is employed
for constructing shape functions. Therefore, similar to the EFG method, there is an issue of imposition of essential boundary conditions. The original MLPG proposed in [10,11] uses the penalty method. In the formulation in [13], a method called direct interpolation is used. This chapter formulates both methods, in addition to the orthogonal transformation method for free-vibration problems. A number of benchmark examples are presented to illustrate the procedure and effect-
iveness of the MLPG method. The effects of different parameters including the dimensions of different domains of MLPG on the accuracy of the results are also investigated via these examples. Although the node-by-node procedure in MLPG is quite similar to that of the collocation
method, the MLPG is more stable against nodal irregularity due to the use of a local weak form of locally integrated weighted residuals. Due to the use of the MLS shape functions in the local Petrov-Galerkin formulation, the MLPG can reproduce the polynomials that are included in the basis of MLS shape functions. This fact will be evidenced in the examples of patch tests.
