非均质材料动力响应及大位移小应变问题的多尺度有限元分析

New multiscale finite element analysis methods with high precision and efficiency will be developed for the dynamic response and large displacement - small strain problems of heterogeneous materials, to overcome the limitations and difficulties effectively in the aspects of computational cost and efficiency. For the dynamic response problem, the dynamic equation of the unit cell will be transformed into a series of equivalent static equations by using the time integration algorithms. Based on the equivalent stiffness of the unit cell, the displacement base function of coarse element can be constructed to take the inertial effect into consideration completely. Thus the contribution of the mass of the unit cell to the structural response can be reflected effectively, and the computational accuracy can be improved obviously. To simulate and predict the dynamic behaviors of heterogeneous materials with complex boundaries such as crack, a new coarse element is constructed on the premise of completely considering the inertial effect of the unit cell by using the static condensation method and the penalty function method. For large displacement - small strain problem, a transformation between the initial and current configurations of coarse element will be built by employing co-rotational approach. The tangent stiffness matrix and nodal load vector of the coarse element can be derived and calculated. The nonlinear equations will be solved only on the macroscopic scale. The calculation cost will be reduced greatly and the computational efficiency will be improved obviously. In addition, the microscopic results of the unit cell can be obtained by the downscaling computation by virtue of the established transformation and the constructed displacement base function of the coarse element on any load step.

针对非均质材料的动力响应和大位移小应变问题，发展新的高精度高效率的多尺度有限元分析方法，有效克服目前所面临的计算量和计算效率方面的局限和困难。对动力响应问题，采用时间积分算法将单胞动力方程转化为一系列的等效静力方程，在单胞等效刚度上构造宏观单元的位移基函数，使其严格考虑单胞的惯性效应，有效反映单胞质量对结构响应的贡献，提高计算精度。借鉴静态凝聚法和罚函数法，在严格考虑单胞惯性效应的前提下，构造新的宏观单元以模拟和预测含有裂纹等复杂边界的非均质材料的动力学行为。对大位移小应变问题，借鉴共旋坐标法，建立宏观单元初始构型与当前构型之间的转换关系，推导并计算宏观单元的切线刚度矩阵和节点载荷向量，在宏观尺度上进行平衡迭代，极大地降低计算量，提高计算效率。此外，在任一载荷步，利用宏观单元的所建立的转换关系和构造的位移基函数进行降尺度计算，得到单胞微观尺度上的解。

自然界中绝大多数材料都是非均质的，很多材料的特征尺寸与结构特征尺寸相差甚大，对这类问题的求解，传统计算方法面临着计算量大、计算时间长等困境，故发展高效高精度的多尺度计算方法非常重要。.本项目针对非均质材料与结构的动力响应、几何非线性等问题，发展了新的高精度高效率的多尺度有限元分析方法，有效地克服了目前所面临的计算量和计算效率方面的局限和困难。对具有周期非均质单胞组成的结构，基于阶谱函数的逼近思想，成功发展了新的阶谱多尺度有限元方法，该方法对静力和动力均有效，且具有较高的计算精度。对动力响应问题，采用静态凝聚法和罚函数法，构造了新的多节点宏观单元以模拟和预测含有裂纹等复杂边界的非均质材料的动力学行为，发展了一种新的广义多尺度有限元方法。对几何非线性问题，借鉴共旋坐标法，建立了宏观单元初始构型与当前构型之间的转换关系，推导并计算了宏观单元的切线刚度矩阵和节点载荷向量，在宏观尺度上进行平衡迭代，极大地降低了计算量，提高了计算效率。此外，本项目还进一步将所提出的多尺度计算方法拓展到非均质压电复合材料以模拟其压电耦合几何非线性力学行为。再者，本项目还基于所提出的多节点粗单元，发展了一种新的自适应节点技术，使得粗单元的节点分布更加合理，可在保证计算精度的同时节约计算量和计算时间。