Element-Free Discretization Method with Moving Finite Element Approximation

Abstract

The popularity of standard Finite Element Method (FEM) results from its simplicity and wide applicability in engineering computations. The simplicity consists in polynomial interpolation of field variables within finite elements, while the wide applicability in rather close similarity of engineering problems based mostly on variational formulations. On the other hand, the standard FEM suffers from limited continuity on element intersections, global meshing difficulty in analysis of complex geometry, and global equations may not be parallelized efficiently. All these shortcomings can be removed in the mesh free formulations, where the arbitrary net of nodal points is used for discretization instead of subdivision of the analyzed domain into finite elements. The spatial approximations based on cluster of nodes are not expressed in terms of elementary functions and evaluation of the shape functions prolongs the computation. To get rid of disadvantages of the standard FEM and mesh free formulations, the moving finite element (MFE) method has been proposed. In this paper, we pay attention to explanation of the basis of the MFE approximation and to its combination with the strong as well as local weak formulations of boundary value problems for stationary heat conduction in solids with functionally graded heat conduction coefficient. Finally, in numerical test examples the reliability (accuracy and numerical stability) and computational efficiency are compared for three variants of the MFE method and the standard FEM.