A local symmetric weak form (LSWF) for linear potential problems is developed, and a truly meshless method, based on the LSWF and the moving least squares approximation, is presented for solving potential problems with high accuracy. The essential boundary conditions in the present formulation are imposed by a penalty method. The present method does not need a``®nite element mesh'', either for purposes of interpolation of the solution variables, or for the integration of the``energy''. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No post-smoothing technique is required for computing the derivatives of the unknown variable, since the original solution, using the moving least squares approximation, is already smooth enough. Several numerical examples are presented in the paper. In the example problems dealing with Laplace & Poisson's equations, high rates of convergence with mesh re®nement for the Sobolev norms jj Á jj 0 and jj Á jj 1 have been found, and the values of the unknown variable and its derivatives are quite accurate. In essence, the present meshless method based on the LSWF is found to be a simple, ef®-cient, and attractive method with a great potential in engineering applications.

The Galerkin ®nite element method (GFEM) owes its popularity to the local nature of nodal basis functions, i.e., the nodal basis function, when viewed globally, is non-zero only over a patch of elements connecting the node in question to its immediately neighboring nodes. The boundary element method (BEM), on the other hand, reduces the dimensionality of the problem by one, through involving the trial functions and their derivatives, only in the integrals over the global boundary of the domain; whereas, the GFEM involves the integration of the``energy'' corresponding to the trial function over a patch of elements immediately surrounding the node. The GFEM leads to banded, sparse and symmetric matrices; the BEM based on the global boundary integral equation (GBIE) leads to full and unsymmetrical matrices. Because of the seemingly insurmountable dif®culties associated with the automatic generation of element-meshes in GFEM, especially for 3-D problems, there has been a considerable interest in element free Galerkin methods (EFGM) in recent literature. However, the EFGMs still involve domain integrals over shadow elements and lead to dif®culties in enforcing essential boundary conditions and in treating nonlinear problems.The object of the present paper is to present a new method that combines the advantageous features of all the three methods: GFEM, BEM and EFGM. It is a meshless method. It involves only boundary integration, however, over a local boundary centered at the node in question; it poses no dif®culties in satisfying essential boundary conditions; it leads to banded and sparse system matrices; it uses the moving least squares (MLS) approximations. The method is based on a Local Boundary Integral Equation (LBIE) approach, which is quite general and easily applicable to nonlinear problems, and non-homogeneous domains.The concept of a``companion solution'' is introduced so that the LBIE for the value of trial solution at the source point, inside the domain X of the given problem, involves only the trial function in the integral over the local boundary oX s of a sub-domain X s centered at the node in question. This is in contrast to the traditional GBIE which involves the trial function as well as its gradient over the global boundary C of X. For source points that lie on C, the integrals over oX s involve, on the other hand, both the trial function and its gradient. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple and algorithmically very ef®cient in the present LBIE approach.In the example problems dealing with Laplace and Poisson's equations, high rates of convergence for the Sobolev norms jj Á jj 0 and jj Á jj 1 have been found.In essence, the present EF-LBIE (Element Free-Local Boundary Integral Equation) approach is found to be a simple, ef®cient, and attractive alternative to the EFG methods that have been extensively popularized in recent literature.

The essential features of the Meshless Local Petrov-Galerkin (MLPG) method, and of the Local Boundary Integral Equation (LBIE) method, are critically examined from the points of view of a non-element interpolation of the ®eld variables, and of the meshless numerical integration of the weak form to generate the stiffness matrix. As truly meshless methods, the MLPG and the LBIE methods hold a great promise in computational mechanics, because these methods do not require a mesh, either to construct the shape functions, or to integrate the Petrov-Galerkin weak form. The characteristics of various meshless interpolations, such as the moving least square, Shepard function, and partition of unity, as candidates for trial and test functions are investigated, and the advantages and disadvantages are pointed out. Emphasis is placed on the characteristics of the global forms of the nodal trial and test functions, which are non-zero only over local sub-domains X J tr and X I te , respectively. These nodal trial and test functions are centered at the nodes J and I (which are the centers of the domains X J tr and X I te ), respectively, and, in general, vanish at the boundaries oX J tr and oX I te of X J tr and X I te , respectively. The local domains X J tr and X I te can be of arbitrary shapes, such as spheres, rectangular parallelopipeds, and ellipsoids, in 3-Dimensional geometries. The sizes of X J tr and X I te can be arbitrary, different from each other, and different for each J, and I, in general. It is shown that the LBIE is but a special form of the MLPG, if the nodal test functions are speci®-cally chosen so as to be the modi®ed fundamental solutions to the differential equations in X I te , and to vanish at the boundary oX I te . The dif®culty in the numerical integration of the weak form, to generate the stiffness matrix, is discussed, and a new integration method is proposed. In this new method, the Ith row in the stiffness matrix is generated by integrating over the ®xed sub-domain X I te (which is the support for the test function centered at node I); or, alternatively the entry K IJ in the global stiffness matrix is generated by integrating over the intersections of the sub-domain X J tr (which is the sub-domain, with node J as its center, and over which the trial function is nonzero), with X I te (which is the sub-domain centered at node I over which the test function is non-zero). The generality of the MLPG method is emphasized, and it is pointed that the MLPG can also be the basis of a Galerkin method that leads to a symmetric stiffness matrix. This paper also points out a new but elementary method, to satisfy the essential boundary conditions exactly, in the MLPG method, while using meshless interpolations of the MLS type. This paper presents a critical appraisal of the basic frameworks of the truly meshless MLPG/LBIE methods, and the numerical examples show that the MLPG approach gives good results. It now apears that the MLPG method may replace the well-known Galerkin ®nite element method (GFEM) as a gene...

The Element free Galerkin method, which is based on the Moving Least Squares approximation, requires only nodal data and no element connectivity, and therefore is more¯exible than the conventional ®nite element method. Direct imposition of essential boundary conditions for the element free Galerkin (EFG) method is always dif®cult because the shape functions from the Moving Least Squares approximation do not have the delta function property. In the prior literature, a direct collocation of the ®ctitious nodal values u used as undetermined coef®cients in the MLS approximation, u h x u h x U Á u, was used to enforce the essential boundary conditions. A modi®ed collocation method using the actual nodal values of the trial function u h x is presented here, to enforce the essential boundary conditions. This modi®ed collocation method is more consistent with the variational basis of the EFG method. Alternatively, a penalty formulation for easily imposing the essential boundary conditions in the EFG method with the MLS approximation is also presented. The present penalty formulation yields a symmetric positive de®nite system stiffness matrix. Numerical examples show that the present penalty method does not exhibit any volumetric locking and retains high rates of convergence for both displacements and strain energy. The penalty method is easy to implement as compared to the Lagrange multiplier method, which increases the number of degrees of freedom and yields a non-positive de®nite system matrix.

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The meshless local Petrov-Galerkin (MLPG) approach is an effective method for solving boundary value problems, using a local symmetric weak form and shape functions from the moving least squares approximation. In the present paper, the MLPG method for solving problems in elasto-statics is developed and numerically implemented. The present method is a truly meshless method, as it does not need a``®nite element mesh'', either for purposes of interpolation of the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation are imposed by a penalty method, as the essential boundary conditions can not be enforced directly when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present MLPG method. The numerical examples show that the present MLPG approach does not exhibit any volumetric locking for nearly incompressible materials, and that high rates of convergence with mesh re®nement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.

In this paper, the conventional moving least squares interpolation scheme is generalized, to incorporate the information concerning the derivative of the ®eld variable into the interpolation scheme. By using this generalized moving least squares interpolation, along with the MLPG (Meshless Local Petrov±Galerkin) paradigm, a new numerical approach is proposed to deal with 4th order problems of thin beams. Through numerical examples, convergence tests are performed; and problems of thin beams under various loading and boundary conditions are analyzed by the proposed method, and the numerical results are compared with analytical solutions.

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