Barycentric Finite Element Method And Its Application In Numerical Simulation For The Effective Moduli Of Heterogeneity Materials

The real mechanical properties of materials could be obtained by analysis of thematerial mesostructure. Dividing the heterogeneous materials into polygonal meshesbased on the mesostructure, it is convenient and efficient to simulate the properties ofheterogeneous materials. The conventional finite element method (FEM) is based onthe displacement interpolation. However, applying FEM to an element with n nodes isa difficult problem. The interpolation could not ensure the compatibility ofdisplacements with nterm polynomial between elements. Even the quadrilateralelements have to introduce the isoparametric techniques to ensure the compatibility.In this dissertation, the mean value coordinates were constructed directly on thearbitrary polygon by geometric method. Furthermore, the Barycentric Finite ElementMethod (BFEM) based on polygonal element was introduced using mean valuecoordinates as shape functions.In this dissertation the following subjects were investigated.(1). The mean value coordinates interpolation are constructed directly onarbitrary polygon element by geometric method. Some properties of mean valuecoordinates are presented. The algebraic expressions and the computing process arepresented. Using these expressions, the computing program could be compiledconveniently. Compared with the Laplace interpolation, the mean value coordinatesneed not the isoparametric transformation. Distinguish from the Wachspress typeinterpolation, the unknown parameters are not be included in mean value coordinatesinterpolation. So, it is convenient to code the computing program. (2). The error estimation of Wachspress type interpolation is investigated. Thecompact formulations of Wachspress'interpolations are given. Using the properties ofWachspress'interpolations and the bivariate Taylor expression, the error estimationinequality of Wachspress'interpolations is presented. The error of Wachspress'interpolation decreases with the reduction of the polygonal element's size. If theWachspress's interpolation is regarded as shape function FEM, the numericalsolutions will converge the exact solution.(3). Using the mean value coordinates interpolation as the trial function and testfunction, the BFEM for elastic problems is presented by Galerkin method. By theresults of numerical examples, BFEM is a numerical method with high precision inelasticity problems. Compared with the conventional FEM, the number of nodes andelements are decreased and the computing efficiency is boosted evidently.(4). The effective moduli of heterogeneous materials are numerical simulatedusing BFEM. In the simulation, the representative unit cells are regarded as thecomputational models. The influences of reinforced phase's geometric mesostructureare discussed by polygonal elements. The simulations show that the size of reinforcedphases is the essential factor to the effective moduli. Except the influence of size, theorientations and shapes of reinforced phases assume the other important role. But theeffect of orientation is more obviously than the effect of shape.The BFEM is a polygonal element method. BFEM overcomes the difficulty ofconventional FEM on polygonal element. The compatibility displacementinterpolation is constructed on polygon directly. Using the polygonal elements, themesh of computed zone can be partitioned flexibly. Based on the real mesostructure ofmaterials, the results of numerical simulations are approached the real propertiesmuch more. The BFEM take advantage in the numerical simulation of heterogeneousmaterials.