A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods

Abstract

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...

Solvability of partial differential equations by meshless kernel methods

“…But we cannot go into details here. Candidates for further analysis are the weak meshless local Petrov-Galerkin method (MLPG, [3,2,4]) or the generalized finite element method [7] based on techniques using partitions of unity [17,6]. Chances are good that such methods also work nicely for meshless kernel techniques, since they surely work for interpolation [25] and certain simple problems in weak form [24].…”

A MLPG(LBIE) numerical method for solving 2D incompressible and nearly incompressible elastostatic problems

“…On the other hand, excessive support domains increase the numerical errors as well as the computational cost because too many nodes are used for the MLS approximation scheme and the integration boundary becomes larger, thus, diminishing the locality of the proposed method. Furthermore, the more complex the domain of influence is the more rational the approximation function becomes [32].…”

“…After solving the linear system (37) and obtaining the nodal fictitious displacements and stresses, the approximation equations (16), (17) are employed and the true nodal values of the analysed model are retrieved. The evaluation of singular and hypersingular integrals is accomplished directly through advanced integration techniques explained in References [19,21,32]. For the evaluation of the hydrostatic pressure, the LBIE (34) is utilized after inserting the MLS approximation relations (16) and (17) for displacements and stresses, respectively, i.e.…”

Maximum-entropy meshfree method for compressible and near-incompressible elasticity

“…Atluri et al [31] proposed a methodology to integrate the weak form in the meshless local Petrov-Galerkin method without the need for background cells by using the support of the basis functions as the domain of integration. This approach was adopted and improved upon in the work of De and Bathe [32].…”