The implementation of B-splines to Hashin and Shtrikman variational principle based FFT method for the homogenization of composite

Abstract

Fast Fourier transform (FFT) has been successfully used to estimate the effective properties of composites with meso periodic structure for more than two decades. Numerous improvements have been done to make it adequate for infinite contrast, bound elastic energy and composite voxels. The Hashin and Shtrikman variational principle based FFT method handles these three problems simultaneously and is very attractive. In this paper, B-splines were adopted to discretize the polarized field. The approximate polarized field was substituted into the Hashin and Shtrikman variational principle to obtain the discretized system. The constitutive matrix of a composite voxel was calculated by both the energetically consistent way and the laminate mixing rule. Numerical example of a square matrix with a circular inclusion shows that the results are more accurate than the remaining cases when the first and second order B-splines, as well as the laminate mixing constitutive matrix, are used. For higher order B-spline, the summation to compute the discrete Green operator for periodic boundary conditions converges faster while the discretized polarized field needs more iterations to converge. Only when the polarized field is sufficiently smooth, high order B-spline improves the numerical accuracy and efficiency.