Numerical Convergence Study of Heterogeneous Multi-scale Method Piezoelectric Composites

In this paper, we focus on multi -- scale modeling and simulation of piezo electric composite materials, Piezo electricity is the ability of some materials to generate an electric field or electric potential in response to applied mechanical stress.And it isreversible in piezoelectric materials.This property can be reflected in the constitutive relations as coupling of mechanical field and electrical field. First, for those materials with periodic microstructure, we deduced the macroscopic model from the microscopic model of piezoelectric composite material and carried out the corresponding convergence analysis under the flame of homogenization. Second, for general cases, we also designed the multi-scale model under the frame of HMM and gave out error estimate between the HMM solutions and the homogenization solutions.Finally we give out numerical examples.The macroscopic models of type 1-3 piezoelectric coinpositematerials are got by the two methods mentioned above. Also, we gi ve out the comparison and discussion about the numerical results we get in the numerical examples.