Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Conforming polygonal finite elements

Abstract: SUMMARY In this paper, conforming finite elements on polygon meshes are developed. Polygonal finite elements provide greater flexibility in mesh generation and are better-suited for applications in solid mechanics which involve a significant change in the topology of the material domain. In this study, recent advances in meshfree approximations, computational geometry, and computer graphics are used to construct different trial and test approximations on polygonal elements. A particular and notable contribution is the use of meshfree (natural-neighbour, nn) basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons. This numerical formulation enables the construction of conforming approximation on n-gons (n 3), and hence extends the potential applications of finite elements to convex polygons of arbitrary order. Numerical experiments on second-order elliptic boundary-value problems are presented to demonstrate the accuracy and convergence of the proposed method. Copyright 2004 John Wiley & Sons, Ltd.

Numerical integration of the Galerkin weak form in meshfree methods

Abstract: The numerical integration of Galerkin weak forms for meshfree methods is investigated and some improvements are presented. The character of the shape functions in meshfree methods is reviewed and compared to those used in the Finite Element Method (FEM). Emphasis is placed on the relationship between the supports of the shape functions and the subdomains used to integrate the discrete equations. The construction of quadrature cells without regard to the local supports of the shape functions is shown to result in the possibility of considerable integration error. Numerical studies using the meshfree Element Free Galerkin (EFG) method illustrate the effect of these errors on solutions to elliptic problems. A construct for integration cells which reduces quadrature error is presented. The observations and conclusions apply to all Galerkin methods which use meshfree approximations.

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 field 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 ΩtrJ and ΩteI, respectively. These nodal trial and test functions are centered at the nodes J and I (which are the centers of the domains ΩtrJ and ΩteI), respectively, and, in general, vanish at the boundaries ∂ΩtrJ and ∂ΩteI of ΩtrJ and ΩteI, respectively. The local domains ΩtrJ and ΩteI can be of arbitrary shapes, such as spheres, rectangular parallelopipeds, and ellipsoids, in 3-Dimensional geometries. The sizes of ΩtrJ and ΩteI 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 specifically chosen so as to be the modified fundamental solutions to the differential equations in ΩteI, and to vanish at the boundary ∂ΩteI. The difficulty 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 fixed sub-domain ΩteI (which is the support for the test function centered at node I); or, alternatively the entry KIJ in the global stiffness matrix is generated by integrating over the intersections of the sub-domain ΩtrJ (which is the sub-domain, with node J as its center, and over which the trial function is non-zero), with ΩteI (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 finite element method (GFEM) as a general tool for numerical modeling, in the not too distant a future.