Mathematical Modelling of Structure Formation of Discrete Materials

Abstract

The mathematical modelling of different processes and events may be reduced, in most cases, to formulation of boundary-value problems for defined systems of differential equations. Series of statements and approximate methods for solving of such equations were developed by many authors. The most development have obtained variation methods, direct methods of mathematical physics and integral equation methods. These methods have specific capabilities and peculiarities, expanded class of observed problems, but were not completely eliminated most of principal contradictions. Nowadays, the most challenging method is finite element method (FEM). It has reached so high stage of development and popularity that can be no doubts of existence another approach competitive in capabilities and simplicity of realization (Segal et al., 1981; Wagoner & Chenot, 2001). The advantages of finite element method are free selection of nodal points, arbitrary shape of region and boundary conditions, simplicity of generalization for different models of bodies and problems of any dimensionality, natural accounting the non-uniformity of properties and other local effects, using of standard programs for a whole class of problems. A finite element method is well grounded, the equivalence of its different forms to differential and variation formulations and, also, to special cases of Ritz method, BubnovGalerkin method and least-squares method established (Zienkiewicz & Taylor, 2000). The first step of numerical solution is discretization of medium that allows reducing the problems with infinite number of degrees of freedom typical to continuous approach, to problems with finite number of unknown variables. Usually, discretization is including selection of certain number of nodal points with following implementation of two types of variables – nodal variables and special functions that are approximating the distributions of target parameters inside elements. In such case, the independent parameters are the nodal variables and distributions of target parameters that are determined by them (Zienkiewicz & Taylor, 2000). During finite element approximation the integration procedure is replaced by more simple algebraic operators expressed through nodal variables by summation on elements. Partial differential equations are replaced by system of algebraic equations written for sequence of nodes and special functions by functions for finite number of nodal variables. The subsequent calculation of target values and determination of parameters of state may be executed by standard methods of numerical analysis. The general requirements for selection of finite elements and approximating functions are determined by convergence criterions of FEM (Zienkiewicz & Taylor, 2000).