Moving Finite Element Method

Abstract

A novel discretization method is proposed and developed for numerical solution of boundary value problems governed by partial differential equations. The spatial variation of field variables is approximated by using Lagrange finite elements for interpolation without discretization of the analysed domain into the mesh of finite elements. Only the net of nodal points is used for discrete degrees of freedom on the analysed domain and its boundary. The governing equations are considered at interior nodal points while the boundary conditions at nodal points on the boundary. The finite elements are created for each nodal point properly instead of using fixed finite elements like in standard Finite Element Method. In this way, we can eliminate interfaces between elements as well as the difficulties with continuity of derivatives of field variables on such interfaces. Both the strong and weak formulations are implemented for governing equations. The reliability (accuracy and efficiency) of the new method has been verified in numerical simulations for 2D problems of heat conduction in solids with possible continuous gradation of the heat conduction coefficient. Keywords: Lagrange finite element, bi-quadratic and bi-cubic approximation, strong and weak formulations.( 1 ) and the boundary conditions

Functional gradation of material coefficients and size –effects in heat conduction: Numerical simulations

“…Without going into details, we note that the moving finite elements are created automatically as shown in [5,6] for 2D problems with using 9-node biquadratic Lagrange element. Recall that higher-order continuity is achieved, since there are no element interfaces.…”

Mesh-Free Analysis of Plate Bending Problems by Moving Finite Element Approximation

“…In contrast to the classical FEM, the analysed domain is not discretized into the mesh of finite elements, but only a mesh of nodes is utilized [4]. A finite element is associated with each node and this element is created automatically according to the position of the reference node on the bounded domain    [4]. In this paper, we shall consider the bi-quadratic Lagrange elements (with 9 nodes).…”

“…In this paper, we shall consider the bi-quadratic Lagrange elements (with 9 nodes). Construction of the moving finite elements (FE) associated with particular nodes a x follows the rules [4]…”

“…The numerical implementation of the strong formulation can be obtained very easily by using the expressions for the derivatives at nodal points in both the governing equations and boundary conditions. In the case of weak formulation, some additional implementation is needed for the geometry of local subdomains and boundary contours, and it can be done as shown in [4]. There is no restriction on the shape of the local subdomain around a nodal point, but the subdomain should lie within the moving FE associated with the considered nodal point.…”

“…The standard finite elements (FE) give only C 0 continuity, while the recently developed mesh-free approximations utilize the higher order continuous shape functions, but these are not expressed in terms of elementary functions and hence, such mesh-free approximations result into time consuming evaluation of shape functions. Utilization of advantages of the polynomial interpolation (like in FEM) and the element-free discretization of the analysed domain (like in mesh-free methods) has been proposed originally in works [1]- [3] and recently elaborated in [4]- [6]. In this paper, the moving finite element (MFE) approximation [4] is developed and implemented for the solution of plate bending boundary value problems, with studying numerical aspects of the proposed method.…”