## 1 Introduction  
Topology optimization is a method that can support the product developer in the lightweight design of structures and is therefore a central tool in product development to reduce CO\({}_{2}\) emissions from products [1]. It is also employed, for example, in the design of heat exchangers [2], to identify the optimal flow path [3] or in the context of medicine for the design of implants [4]. The most widespread method for topology optimization is density-based topology optimization, the development of which began with the publications by [PERSON] and [PERSON] [5] and [PERSON] [6]. In addition to the density-based topology optimization methods, there are many other topology optimization methods. These include, for example, the optimization criteria-based methods [7], the phase field methods [8] or the nature-mimicking evolutionary structural optimization [9, 10]. Another topology optimization method is based on the level set method and was primarily developed by [PERSON] and [PERSON] [11], [PERSON] et al. [12] and [PERSON] et al. [13]. In this method, the geometry is described during optimization using a discretized level set function. The geometry is evolved by applying the classical shape derivative to determine the front velocity and then solving a Hamilton-Jacobi equation [13]. In addition to the level set-based topology optimization approach just described, there are other variants of using the level set method for topology optimization. An overview is given in [14], and more recently developed methods can be found, for example, in [15, 16]. One such method is the Velocity Field Level Set Method (VFLSM) [17]. It allows the use of mathematical programming methods, such as the Method of Moving Asymptotes (MMA) [18] in combination with the level set method, thereby enabling the straightforward integration of several constraints into level set-based topology optimization [17]. [PERSON] and [PERSON] [19] developed the so-called Color Level Set method for the optimization of multi-material systems. For this purpose, they refined a description of multi-material systems known from image processing for level set topology optimization. It allows the description of \(2^{n}\) materials using \(n\) level set functions. An alternative approach to optimizing multi-material systems is the Reconciled Level Set Method (RCLSM) [20], in which each material is described by its own level set function. The individual level set functions are evolved independently of each other, and any overlaps that arise are subsequently resolved using a correction process [20]. [PERSON] et al. [21] applied the RCLSM in the context of multi-material topology optimization of smart energy harvesters, and [PERSON] et al. [22] used it for topology optimization of multi-material negative Poisson's ratio metamaterials. [PERSON] et al. [23] combined the RCLSM with the extended level set method to design ferromagnetic soft active structures using topology optimization. There are three different approaches to represent the current design for the evaluation of the system response during level set-based topology optimization. In the first approach, a density field of a constant background Finite Element (FE) mesh is adjusted based on the values of the level set function. A smoothed Heaviside function is usually used for the adjustment. This variant is similar to density-based topology optimization methods [13]. The second approach applies immersed finite element methods, such as XFEM [24] or CutFEM [25] methods. These use a constant background mesh that is cut by the level set functions. The third approach is to adapt a mesh or remesh the level set function in each iteration [26]. With this approach, the standard Finite Element Method (FEM) can be used. [PERSON] et al. [27] developed a level set topology optimization with iterative mesh adaptation. The unstructured mesh is iteratively adapted using classical mesh modification techniques such as edge collapse, vertex smoothing and edge swap. [PERSON] et al.