【作者】 张景涛；

【导师】 王兆清；

【作者基本信息】 山东建筑大学 ， 工程力学， 2008， 硕士

【摘要】 依据非均质材料的细观几何结构对材料进行数值模拟，能够获得材料真实的力学性能。采用多边形单元对模拟材料进行网格划分，可以方便有效的实现非均质材料力学性能的数值模拟。经典有限元法难以在多边形单元上构造出满足位移协调性要求的多项式形式的插值函数，即便是在四边形单元上，也需要借助于等参变换技术的应用。本文基于几何方法直接在多边形上构造出多边形单元的平均值坐标形式插值函数，提出用于非均质材料力学性能模拟的重心有限元法。本文以重心有限元方法的建立为主线，开展了以下工作的研究：一、采用几何方法构造出多边形单元上的平均值坐标插值形函数。证明了多边形单元上平均值坐标插值的有关性质。给出了平均值坐标插值的代数表达式和计算流程，利用该表达式可以方便地编写计算程序。构造的平均值坐标插值与Laplace型插值相比，不需要进行等参变换；与Wachspress型插值相比，不含有待定参数，方便程序的编写。二、对Wachspress型插值进行了误差估计。给出了Wachspress插值形函数的简化公式，利用Wachspress插值形函数的性质和二元函数的Taylor展开式，给出了Wachspress插值的误差估计不等式。Wachspress插值的误差随单元尺寸的缩小而缩小，说明如果利用Wachspress插值作为多边形有限元的试函数，随着单元尺寸的缩小，多边形有限元的解将收敛于精确解。三、以多边形单元上的平均值坐标插值作为试函数，采用Galerkin法建立了求解弹性力学问题的重心有限元法。数值算例表明，重心有限元法在求解弹性力学问题时得到的数值解有较高的精度。与传统有限元相比，减少了单元节点和单元的数量，减少了前处理的工作量，提高了计算效率。四、利用重心有限元法对非均质材料的有效模量进行了数值模拟。以代表单胞为计算模型，用多边形单元计算讨论了增强相的细观几何结构对复合材料有效模量的影响。计算结果表明，增强相的尺寸是影响材料有效模量的最基本的因素。除去尺寸因素，增强相的方位和形状也是对材料有效模量有较大影响的因素。方位因素所产生的效应较之形状因素更加明显。基于多边形单元平均值坐标插值的重心有限元法，克服了传统有限元法对多边形单元难以构造满足协调性要求的多项式形式位移插值的难题。由于采用多边形单元，使得区域网格划分更加灵活，实现基于材料真实结构的数值模拟，使计算结果更加接近材料的真实性能。重心有限元法在非均质材料的数值模拟中具有较大的优势。更多还原

【Abstract】 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.更多还原