非均质材料动力分析的广义多尺度有限元法

A GENERALIZED MULTISCALE FINITE ELEMENT METHOD FOR DYNAMIC ANALYSIS OF HETEROGENEOUS MATERIAL

自然界和工程中的大部分材料都具有多尺度特征,当考察尺度小到一定程度后,都将表现出非均质性.针对非均质材料的动力问题,提出了一种广义多尺度有限元方法,其基本思想是利用静态凝聚法以及罚函数法构造能够反映单元内部材料非均质特性的多尺度位移基函数.与传统扩展多尺度有限元法中的基函数构造方式不同,广义多尺度有限元法的基函数无需通过在子网格域上多次求解椭圆问题得到,而可直接通过矩阵运算获得.其主要步骤如下:利用数值基函数将一个非均质单胞等效为一个宏观单元,进而形成整个结构的等效刚度矩阵,并得到宏观网格的节点位移,最后再次利用数值基函数得到微观尺度上的位移结果.该广义多尺度有限元法是扩展多尺度有限元法的一种新的拓展,可模拟具有更加复杂几何的非均质单胞的力学行为.通过数值算例,模拟了非均质材料的静力问题、广义特征值问题以及瞬态响应问题,计算结果表明:在边界条件一样的情况下,广义多尺度有限元法的计算结果与传统有限元的计算结果保持高度一致.与传统有限元相比,该方法在保证计算精度的同时极大地提高了计算效率.研究结果表明,广义多尺度有限元法能够很好地模拟非均质单胞的力学行为,具有良好的工程应用潜力.

Almost all natural materials, as well as industrial and engineering materials, have multiscale features. This paper presents a generalized multiscale finite element method for dynamic analysis of heterogeneous materials. In this method, multiscale base functions, which can reflect the internal heterogeneity of materials within the coarse-scale element, are constructed by using the static condensation method and penalty function method. And these functions, which are different from those in the traditional extended multiscale finite element method（EMs FEM）, are obtained by matrix operations analytically rather than solving elliptic problem many times on the sub-grid domain numerically. The main steps of this proposed method are described as follows. Firstly, a single heterogeneous unit cell will be equivalent into a macroscopic element by virtue of the constructed multiscale base functions. Then, the stiffness matrix of heterogeneous structure can be calculated on the macroscopic scale. Thus, the macroscopic nodal displacements of coarse-scale mesh can be obtained. Finally, the microscopic nodal displacements of sub grids within the unit cell can be calculated by usingthe numerical base functions once again. This generalized multiscale finite element method is a new expansion of the EMs FEM, which can simulate the mechanical behavior of heterogeneous unit cell with more complex geometric configurations. The static problem, generalized eigenvalue problem and transient response problem of the heterogeneous material are then simulated. It can be found that the calculation results of the generalized multiscale finite element method maintain highly consistent with those from the traditional FEM. Compared with the traditional FEM, the present multiscale method has significantly improved the computational efficiency while ensuring the computational accuracy, which has a good application potential.