## 2 Methods
### Single-Material
#### 2.2.2 Velocity Field Level Set Method  
The description of the part surface during optimization is completed using the level set function \(\varphi:\mathbb{R}^{3}\rightarrow\mathbb{R}\). An example of a level set function is depicted in Figure 1. The level set function is positive inside the part, zero on the part surface and negative outside the part (in the rest of the design domain). In summary, the following applies:  
\[\begin{cases}\varphi(\mathbf{x})>0,&\forall\mathbf{x}\in\Omega\quad(\text{part})\\ \varphi(\mathbf{x})=0,&\forall\mathbf{x}\in\partial\Omega\quad(\text{part surface})\\ \varphi(\mathbf{x})<0,&\forall\mathbf{x}\in D\backslash\Omega\end{cases} \tag{6}\]In this article, a special form of the level set function, the so-called Signed Distance Function (SDF), is used to numerically stabilize the evolution of the surface described below [13]. The function values of the SDF are the signed distances to the surface inside and outside the part. For example, in the part, \(\varphi(\mathbf{x})=d(\mathbf{x},\partial\Omega)\) applies, whereby the function \(d\) returns the distance between \(\mathbf{x}\) and \(\partial\Omega\). SDFs also have the property that \(\|\
abla\varphi(\mathbf{x})\|=1\)[30].  
The evolution of the level set function and thus the surface (\(\varphi(\mathbf{x})=0\)) in a previously given or calculated normal velocity field \(v^{n}\) is described by the Hamilton-Jacobi equation [30]:  
\[\frac{\partial\varphi}{\partial t}+v^{n}\|\
abla\varphi\|=0. \tag{7}\]  
The normal velocity field \(v^{n}\) is linked to the design variables \(\mathbf{\eta}\) by linear interpolation in the topology optimization using VFLSM [17]:  
\[v^{n}(\mathbf{x})=\sum_{i=1}^{n_{d}}b_{i}(\mathbf{x})\eta_{i}. \tag{8}\]  
Here, \(b_{i}(\mathbf{x})\) is the global basis function of the linear FEM associated with node \(i\) and \(\eta_{i}\) is the respective design variable. When discretizing the normal velocity field, other basis functions can be used in addition to the basis functions from the linear FEM. An overview of this is given in [17].  
The design variables are varied by the optimization algorithm based on previously calculated sensitivities. In the VFLSM, the derivation of the sensitivities is based on the shape derivative [17]. The shape derivative of the compliance component of the objective function is based on the simplification described above (no volume forces and no variation in the Neumann and Dirichlet boundary) [13; 17]:  
\[DJ_{c}(\Omega)(\mathbf{\theta})=\int_{\partial\Omega}\mathbf{\epsilon(\mathbf{u})}\cdot \mathbb{C}_{1}[\varepsilon(\mathbf{u})]\mathbf{\theta}\cdot\mathbf{nd}\xi, \tag{9}\]  
and the shape derivative of the regularization component [13]:  
\[DJ_{r}(\Omega)(\mathbf{\theta})=\int_{\partial\Omega}\kappa\mathbf{\theta}\cdot\mathbf{nd }\xi. \tag{10}\]  
The mean curvature \(\kappa\) of the surface \(\partial\Omega\) is defined as \(\kappa=\
abla\cdot\mathbf{n}\)[13]. Using \(v^{n}=\mathbf{\theta}\cdot\mathbf{n}\), the derivative of the objective function with respect to the design variables \(\eta_{i}\) is given by [17]:  
\[\frac{\partial J}{\partial\eta_{i}}=w_{c}\int_{\partial\Omega}\mathbf{\epsilon( \mathbf{u})}\cdot\mathbb{C}_{1}[\varepsilon(\mathbf{u})]b_{i}d\xi+w_{r}\int_{\partial \Omega}\kappa b_{i}d\xi,\quad i=1,...,n_{d}. \tag{11}\]  
Figure 1: (**a**) A 2D level set function (blue), which represents a rectangle with two holes. The intersection between the level set function and the plane (green) results in the surface of the rectangle (orange). (**b**) Representation of the level set function and the corresponding part colored in blue.  
Similarly, the following derivative for the volume constraint can be derived [17]:  
\[\frac{\partial f_{V1}}{\partial\eta_{i}}=-\frac{1}{V_{D}}\int_{\partial\Omega}b_{ i}d\xi,\quad i=1,...,n_{d}. \tag{12}\]