# Multi-Material Optimization for Lattice Materials Based on Nash Equilibrium
## 1 Introduction  
Lattice materials [1], characterized by high porosity and periodicity, have attracted great interest in the field of structural light weight [2; 3]. In addition to their excellent mechanical properties, they also have functional properties such as energy absorption [4] and thermal applications [5; 6]. So they are widely used in fields such as the automotive industry [7], aerospace industry [8; 9], biomedical industry [10], etc. Recently, with the advancement in additive manufacturing (AM) technology [11], effective manufacturing methods for lattice materials with complex geometric shapes have emerged. Consequently, research on the configuration of lattice materials at the mesoscale and the design of topological configurations at the macroscale have garnered significant attention from scholars. Unlike conventional materials, the performance of lattice materials at the macroscale depends on the topologies and size of the unit cell inside the lattice at the mesoscale [12]. This implies that different geometry units have different mechanical properties. Therefore, in order to obtain better stiffness of the lattice structure, scholars have deepened the research on the mechanical properties of lattice materials. Due to the large amount of mesoscopic topologies [13] inside the lattice materials, direct structural modeling and analysis would result in a huge workload, and universal finite element methods are no longer applicable. Equivalent computation approaches, such as the representative volume element (RVE) method [14] and homogenization method [15], have been proposed to reduce computational costs. [PERSON] [16] compared the prediction results of the equivalent properties of lattice materials with periodic mesostructures using these two methods. [PERSON] et al. [17] proposed the extended multiscale finite element method (EMsFEM). This method not only takes into account the absolute size of the mesostructure of lattice materials but also enables easy implementation. It can optimize the design of lattice materials for complex engineering structures with practical shapes, loads, and boundaries. Furthermore, there have been efforts by scholars to employ particle-based methods, such as the Elastic Network Model [18], Molecular Dynamics [19] and Monte Carlo [20], in order to predict the mechanical characteristics of lattice materials. This approach involves treating the nodes of the lattice as individual particles and analyzing the material's properties by simulating the interactions between these particles. Such a method is particularly well suited for investigating the mesostructure and localized properties of materials. Assisted by methods for calculating the equivalent properties of lattice materials, we can delve deeper into the design of lattice materials to explore their performance, enabling the feasibility of multi-material optimization for lattice materials. Topology optimization is a technique that enables the rational distribution of materials within a specific design domain, based on specific boundary conditions and constraints [15]. For over thirty years, different topology optimization methods [21; 22; 23; 24] have been proposed. However, most studies focused on single-material optimization. In some specialized fields where there are higher requirements for structural materials, single-material solutions cannot meet the usage demands. As a result, multi-material optimization has garnered widespread attention. Therefore, there is considerable interest in developing topology optimization methods for multi-material structures. Compared to optimizing single-material structures, optimizing multi-material structures presents greater challenges. Firstly, a reasonable optimization formulation is needed to effectively represent different materials within the design domain while also facilitating sensitivity analysis.