A single variable-based method for concurrent multiscale topology optimization with multiple materials

A concurrent multiscale topology optimization method which exploits a single type design variable to characterize the material properties for any number of microstructures or materials is proposed by using the Smolyak method merged with the Non-Uniform Rational B-Spline(NURBS) method. Two kinds of design variables are assigned to describe the macro and micro scale topology density fields, respectively A multiscale nested form interpolation scheme is constructed from a sequence of discrete terms coupled with the penalty principle which contributes to the selection of candidate microstructures or base materials. The proposed method eliminates the dependency of the dimension of design variables on the number of material types, and significantly reduces the number of design variables. The characteristic function which indicates the existence of each microstructure or base material is introduced to enforce an arbitrary number of volume constraints at the macro and micro scales. Sensitivity analysis of the proposed optimization formulation is also conducted. Finally, the proposed method has been verified and illustrated against multiple examples found in the literature.