Mapping method for sensitivity analysis of composite material property

Composite properties are dependent on the microstructure of materials, which is depicted with a base cell. The parameters for representing the microstructure should include the shape parameters of the base cell and those used to describe the distribution of materials in the base cell. The goal of material design optimization is to find appropriate values of these parameters to make the materials have specific properties. Design optimization needs the sensitivity information of the material properties with respect to the shape parameter of the base cell and the material distribution parameters. Moreover, sensitivity calculation is often expensive. Thus, it is very important to develop an efficient sensitivity analysis method. In this paper, a mapping method is proposed for predicting the material properties and computing their sensitivities with respect to the shape parameters of the base cell. Through mapping transformation, solutions to the micro-scale homogenization problem defined on the domain of a base cell can be obtained by solving a homogenization problem defined on an initial given domain. The composite properties and their sensitivities with respect to the shape parameters of the base cell are explicitly expressed in terms of the properties and their sensitivities of a virtual material with respect to the distribution parameters. This virtual material has an initially given base cell domain. Thus re-meshing for discretizing the problem is avoided and computing cost savings are realized. Numerical examples show that the proposed method is accurate and efficient in both the prediction of material properties and sensitivity calculation.