A NURBS-based Multi-Material Interpolation (N-MMI) for isogeometric topology optimization of structures

In this paper, the main intention is to propose a new Multi-material Isogeometric Topology Optimization (M-ITO) method for the optimization of multiple materials distribution, where an improved Multi-Material Interpolation model is developed using Non-Uniform Rational B-splines (NURBS), namely the “NURBS-based Multi-Material Interpolation (N-MMI)”. In the N-MMI model, three key components are involved: (1) multiple Fields of Design Variables (DVFs): NURBS basis functions with control design variables are applied to construct DVFs with the sufficient smoothness and continuity; (2) multiple Fields of Topology Variables (TVFs): each TVF is expressed by a combination of all DVFs to present the layout of a distinct material in the design domain; (3) Multi-material interpolation: the material property at each point is equal to the summation of all TVFs interpolated with constitutive elastic properties. DVFs and TVFs are in the decoupled expression and optimized in a serial evolving mechanism. This feature can ensure the constraint functions are separate and linear with respect to TVFs, which can be beneficial to lower the complexity of numerical computations and eliminate numerical troubles in the multi-material optimization. Two kinds of multi-material topology optimization problems are discussed, i.e., one with multiple volume constraints and the other with the total mass constraint. Finally, several numerical examples in 2D and 3D are provided to demonstrate the effectiveness of the M-ITO method.