功能梯度材料动力学问题的POD模型降阶分析  

ANALYSIS FOR DYNAMIC RESPONSE OF FUNCTIONALLY GRADED MATERIALS USING POD BASED REDUCED ORDER MODEL

作　　者：郑保敬[1] 梁钰 高效伟[2] 朱强华[2] 吴泽艳[1] Zheng Baojing;Liang Yu;Gao Xiaowei;Zhu Qianghua;Wu Zeyan(College of Hydraulic and Environmental Engineering,China Three Gorges University,Yichang 443002,Hubei,Chin;State Key Laboratory of Structural Analysis for Industrial Equipment,School of Aeronautics andAstronautics,Dalian University of Technology,Dalian 116024,China)

机构地区：[1]三峡大学水利与环境学院,湖北宜昌443002 [2]大连理工大学航空航天学院工业装备结构分析国家重点实验室,大连116024

出　　处：《力学学报》2018年第4期787-797,共11页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金资助项目(11602129;11672061)

摘　　要：为了快速分析非均质材料结构在复杂载荷作用下的动态响应,提出一种模型降阶方法,只需计算结构在简单均质材料情况下的动力学问题,进而用其计算结果对非均质材料结构进行分析.首先,采用结构内部任意一点处的材料参数值作为整个结构的材料参数,利用有限元分析软件计算该均质材料结构在动态载荷作用下的位移场建立数据库,该数据库包含计算模型各个节点(自由度为N)在某时间段内L个时刻的位移;其次,对数据库中的信息按照时间离散的特定方式组集成瞬像矩阵,并利用特征正交分解方法对其进行分解,得到该模型的L个特征正交基底,选取其中能反应模型主要特征的H个(其中H<L?N)作为一组最优基底,通过这组基底建立模型的低阶离散控制方程;最后,求解低阶离散微分方程组,得到功能梯度材料结构在复杂载荷作用下的位移场.文中分别给出二维和三维算例,比较了降阶模型和全阶模型计算结果,验证了该方法的有效性,并且计算效率能提高1～2个数量级.In order to quickly analysis the response of heterogeneous materials under dynamic loads, a reduced order method was presented in this paper which only needed to compute dynamic characteristics of homogeneous material under sudden load and got the results for analysis complex non-homogeneous material. Firstly, we used the finite element method to compute the displacement field of homogeneous materials under sudden load, and then discretized data samples was obtained to establish a database which including every moment displacement information of all degrees of freedom（order of N） during a period of time. Secondly, dealing with database by specific way of time discretization, a snapshot matrix was formed. The matrix was decomposed into L orthogonal basis by proper orthogonal decomposition method and we picked up the major H basis from that. Till now we achieved the goal that reducing the model（H 〈L 〈〈 N）. Finally, the H basis were used to obtain order-reduced governing dynamic equation. Different dynamic loads of time dependent were applied to the model, and the dynamic response of non-homogeneous material would be achieved by solving order-reduced governing dynamic equations. The displacement fields of traditional FEM and proposed ROM were compared. 2D and 3D examples showed that the computing scales reduced one or two orders of magnitude.

关 键 词：特征正交分解 模型降阶 动力响应 功能梯度材料 

分 类 号：O321[理学—一般力学与力学基础]