# Multi-Material Optimization for Lattice Materials Based on Nash Equilibrium
## 1 Introduction
Additionally, it is necessary to add appropriate constraints to prevent overlap between different materials, ensuring that the final optimization result aligns with subsequent manufacturing requirements. Currently, in the field of multi-material optimization research, the most widely studied method is the topology optimization method for multi-material composite structures based on SIMP (Solid Isotropic Material with Penalization). [PERSON] and [PERSON] [15] combined SIMP with the Voigt bound, H-S bound, and R-V material interpolation methods to address multi-material topology optimization problems. Meanwhile, [PERSON] [25] also proposed a mixed material interpolation method for the topology design of multi-physics compliant mechanisms. This method is based on the SIMP interpolation method to interpolate solid and empty materials, and the H-S upper and lower boundary weight method to interpolate solid materials. However, due to the large number of design variables involved in this optimization method, the optimization solution process is slow. When studying the optimization problem of composite materials, [PERSON] and [PERSON] [26] proposed a Discrete Material Optimization (DMO) method, which utilizes gradient information and mathematical programming to solve discrete optimization problems. Here, the equivalent material properties of the unit are obtained as the linear weighted average values of each material property. The advantage of this multi-material interpolation method lies in the gradual decrease in weight coefficients corresponding to other design variables as the value of a particular design variable increases. Meanwhile, [PERSON] and [PERSON] [27] described an optimization method based on the phase field approach, using a generalized Cahn-Hilliard model to address the topological optimization of multi-material structures. Due to its straightforward concept and ease of differentiation, this multi-material interpolation method is also widely applied for material interpolation in topology optimization problems of multi-material composite structures. [PERSON] and [PERSON] [28] extended the SIMP and Rational Approximation of Material Properties (RAMP) interpolation models for two-phase materials, thereby providing two material interpolation models that can be applied to interpolate any number of materials. [PERSON] and [PERSON] [29] introduced an efficient heuristic topology generator for the topology optimization of graded multi-material structures. This method does not necessitate gradient calculation and is straightforward to implement numerically. [PERSON] et al. [30] extended a Matlab code for multi-material topology optimization on the classic 88-line code [31] and made the code openly available for scholars to facilitate learning and the exchange of ideas. While several methods have been discussed above, game theory can also be applied to address the allocation of different materials within the design domain in multi-materialoptimization. Game theory, a branch of applied mathematics, serves as an analytical tool for making optimal choices in decision-making problems. In structural optimization, the goal is to find the best solution within a limited set of options. From a certain standpoint, it can be argued that game theory essentially functions as an optimization method. [PERSON] and [PERSON] [32] summarized research that combines game theory and optimization algorithms, highlighting the close correlation between game theory and optimization concepts. Many studies utilize their combinations to address a wide array of problems. [PERSON] et al. [33] listed the applications of evolutionary algorithms based on game theory in structural optimization and skeletal structures. Additionally, the Nash evolutionary algorithm demonstrates superior performance compared to other algorithms in terms of convergence speed and optimization effectiveness when addressing structural problems such as the reconstruction inverse problem (RIP) [34], fully stressed design (FSD) [35], and minimum constrained weight (MCW) [36].