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

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 $L$) 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 $H$ orthogonal basis by proper orthogonal decomposition method and we picked up the major $H<L?N$ basis from that. Till now we achieved the goal that reducing the model ($H$). Finally, the $4-5$ 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.