A geometric nonlinear multi-material topology optimization method based on univariate combination interpolation scheme

The multi-material topology optimization design is a significant area of research, especially when considering geometric nonlinearity. Traditional topology optimization methods are primarily developed based on linear problems and often face the issue where the number of design variables increases proportionally with the number of candidate materials. Additionally, the interphases obtained using stair-step interpolation formulations are often enclosed within adjacent materials, leading to impractical designs and suboptimal results. To address these challenges, a univariate combination interpolation-based multi-material topology optimization method is proposed and applied to multi-material topology optimization considering geometric nonlinearity. Firstly, the univariate characteristic function is utilized to map the single design variable field into multiple topology density fields, each represented by a distinct topology density function. These topology density fields are then processed using a smoothing algorithm based on the convolution-based density filtering method. Subsequently, a physical density field is established through a regularized Heaviside function. By integrating the univariate characteristic function with the convolution density filtering technique, a series of topology density functions with adequate smoothness and continuity is embedded within the Discrete Material Optimization (DMO) interpolation formulation, forming the composite interpolation model. Due to the non-convexity of the topology optimization problem, a continuation strategy for penalty parameter and smoothness parameter adaptive adjustment is introduced to enhance the robustness and optimization efficiency of the algorithm. The Method of Moving Asymptotes (MMA) gradient optimization algorithm is employed to update the design variables iteratively. Finally, a series of two-dimensional and three-dimensional numerical examples considering geometric nonlinearity is presented, with the objective of minimizing compliance under volume constraints. The results indicate that the proposed method effectively combines the advantages of the DMO method with the univariate characteristic function in multi-material topology optimization considering geometric nonlinearity, which successfully addresses the challenges posed by interphase materials between stiff and compliant materials. Moreover, the number of design variables is independent of the number of candidate materials, demonstrating the successful extension of the proposed method to problems involving geometric nonlinearity.