## 2 Materials and Methods  
The shock absorber is a part mounted in every vehicle's suspension system. Its role is not only to provide the damping response, but also frequently to prevent the wheel from colliding with the (car) body. Therefore, it must withstand severe axial loads, especially when the vehicle goes through an obstacle, such as a pothole. As the energy generated by this impact is usually very high, the loads transferred by the shock absorber may cause it to bend, crack, or buckle, the last being addressed in this paper. In addition, the loads coming from the assembly process or the modular spring (Figure 1a) may induce significant stress in the structure, affecting the overall strength performance. The engineer's work is to reduce the mass of this component, utilizing one of many topology or shape optimization techniques. During the design phase, it is important to predict the proper mechanical response and to ensure that the damper will survive the given impact. It can be achieved by constraining a minimum force at which stability is lost, as shown in Figure 1b. As a result, the following optimization problem may be formulated:  
Objective function \(f\):  
\[f=\underset{\mathbf{K}\mathbf{V}}{\text{min}}\rho\int_{V}\text{d}V, \tag{1}\]  
subjected to constraint \(q\):  
\[q(\mathbf{K}\mathbf{V})>0,\ q(\mathbf{K}\mathbf{V})=F_{REQUIRED}-F_{LIMIT}( \mathbf{K}\mathbf{V}), \tag{2}\]  
where  
\(\mathbf{K}\mathbf{V}=[KV_{1},KV_{2},\dots,KV_{N}]\) is a vector of design variables controlling the shape of the structure;  
\(\rho\)--material density;  
V--the volume of the part being optimized;  
\(F_{LIMIT}\)--maximum force achieved by the shock absorber before losing stability;  
\(F_{REQUIRED}\)--the minimum force at which the shock absorber is allowed to lose stability. The method for obtaining the value of the limit load \(F_{LIMIT}\) is usually a simulation using the finite element method. In each iteration of the design process, this constraint must  
Figure 1: Strength calculation: (**a**) Boundary conditions, (**b**) considered stability problem (presented as load vs. deflection curves). be fulfilled. As the mass of the part is minimized, the fulfillment of the stability constraint is ensured by the optimization algorithm. On the other hand, the mass optimization (minimization) task may be resolved in many ways; however, the authors focus on the application of the derivative-free level-set method applied for nonlinear 3D structures. The shape of the part is controlled by changing the design variable vector **KV**, which affects the distribution of the auxiliary scalar field \(\Phi\). For each of the finite elements, the value of the calculated field \(\Phi\) is compared with the assumed threshold value in the following way:  
\[\left\{ \begin{aligned} \Phi\geq\textit{threshold}\rightarrow\textit{ material}\\ \Phi<\textit{threshold}\rightarrow\textit{void}\end{aligned} \right., \tag{3}\]  
Using this attitude, the material can be removed from the structure by adjusting the knot values at each of the radial basis function centers. A more detailed explanation of this method can be found in [21], since the authors developed a methodology suitable for the optimization of unstable structures. The exemplary distribution of the knots is shown in Figure 2.  
The main disadvantage of this approach is the time-consuming nonlinear finite element analysis needed to obtain information about the stability response of the system, for each design vector. Therefore, the authors decided to use surrogate models to represent the actual nonlinear system. This allowed for the verification of hundreds of thousands of design combinations in a small time fraction compared to a single FEM simulation, for the purpose of obtaining the best parameters of the optimization algorithm (tuning optimization algorithm parameters). The finite element method was used to generate the samples that were then used to feed the metamodels (surrogate models).