# Multi-Material Topology Optimization Taking into Account the Position of Material Interfaces in 3D
## 1 Introduction  
To achieve the goals of the Paris Climate Agreement, the CO\({}_{2}\) footprint of future products must be reduced. Lightweight construction is a key technology for this [1], as it enables the full potential of materials to be utilized and thus the weight of components and systems to be reduced. Particularly in the case of components that are installed in vehicles or aircraft, the weight reduction goes hand in hand with a reduction in CO\({}_{2}\). One approach to lightweight construction is multi-material design, which specifically combines several materials in order to utilize the different properties of the materials to achieve an objective. However, this advantage is linked to additional effort during product development. The structural optimization methods, such as topology optimization, support the product developer in creating weight optimized initial designs. The density-based approach to topology optimization was first introduced in the two publications by [PERSON] and [PERSON] [2] and [PERSON] [3]. [PERSON] and [PERSON] [4], [PERSON] et al. [5] and [PERSON] et al. [6] later developed the level set methods as an alternative approach for topology optimization. In this method, the geometry is represented by a level set function during optimization and the geometry is developed using Hamilton-Jacobi equation [6]. The Velocity Field Level Set Method (VFLSM) [7] is a level set topology optimization method that uses the Method of Moving Asymptotes (MMA) [8] for the straightforward integration of several constraints into level set-based topology optimization [7]. Further level set-based topology optimization methods are presented in [9]. For the optimization of multi-material systems, [PERSON] and [PERSON] [10] introduced the Color Level Set Method, which can represent \(2^{n}\) materials using \(n\) level set functions. An alternative method for multi-material topology optimization is the Reconciled Level Set Method (RCLSM) [11]. In the RCLSM, each material is described by one level set function. [PERSON] et al. [12] optimized smart energy harvesters employing the RCLSM, [PERSON] et al. [13] applied the RCLSM to optimize multi-material negative Poisson's ratio metamaterials and [PERSON] et al. [14] used the RCLSM to design ferromagnetic soft active structures. In [15], a new multi-material topology optimization method was developed based on RCLSM and VFLSM. Current publications on multi-material topology optimization focus on, e.g., additive manufacturing [16; 17] and the integration of fiber-reinforced composites into multi-material topology optimization [18]. In contrast to the articles on multi-material topology optimization listed above, the method developed in [15] remeshes the current design in each iteration during the optimization and thus enables the accurate evaluation of quantities on the material interfaces between two materials. Recently, various studies on the influence of interfaces on the optimization result in multi-material topology optimization have been carried out and published. [PERSON] et al. [19] were the first to investigate interface effects on multi-material optimization results in the context of level set topology optimization. [PERSON] et al. [20] used non-parametric shape optimization to optimize bimaterial interfaces in multi-material structures. For this purpose, they developed an objective function based on the quadratic strength criterion of [PERSON] [21]. [PERSON] et al. [22] integrated [PERSON]'s strength criterion into a 2D multi-material topology optimization method as a stress boundary condition and used it for the optimization of bimaterial interfaces. To do this, they remesh the 2D geometry in each iteration. [PERSON] et al. [23; 24] optimized bimaterial structures, taking into account the contact and cohesion between the two materials. They used an Extended Finite Element Method (XFEM) to determine the system response. In [25], they use XFEM and cohesive behavior in 2D multi-material topology optimization to model the separation behavior of the interface.