Multi-scale Modeling for Piezoelectric Composite Materials
  In this paper, we focus on multi-scale modeling and simulation of piezoelectric composite materials. A multi-scale model for piezoelectric composite materials under the framework of Heterogeneous Multi-scale Method(HMM) is proposed. For materials with periodic microstructure, macroscopic model is derived from microscopic model of piezoelectric composite material by asymptotic expansion. Convergence analysis under the framework of homogenization theory is carried out. Moreover, error estimate between HMM solutions and homogenization solutions is derived. A 3-D numerical example of 1-3 type piezoelectric composite materials is employed to verify the error estimate. 
Qian Zhang, Xingye Yue
math-ph math.MP