## 5 Example Analysis and Discussion
### Reliability-Based Optimization of a Vehicle Disc Brake System  
A vehicle disc brake system that contains a brake disc and a pair of brake pads is depicted in Figure 6. The brake pads are made of back plates and friction materials. According to [PERSON] et al. (2019), a continuum 3D finite element model of the vehicle disc brake system includes 26,125 elements and 37,043 nodes. Since the lightweight structure is a critical requirement in the preliminary design stage of a vehicle, the mass of the back plate needs to be minimized. Therefore, back plate thickness \(h_{3}\) serves as the objective function, and the general URBDO model of vehicle disc brake can be formulated as follows:  
\[ \begin{array}{ll}\text{find}&\mathbf{d}=h_{3}\\ \text{min}&f(\mathbf{d})=h_{3}\\ \text{s.t.}&\mathcal{M}\left\{g(\mathbf{d};\mathbf{\xi})+0.01\geq 0\right\}\geq R_{ \text{U}}\\ &g(\mathbf{d};\mathbf{\xi})=0.046287+0.20458\mu-0.059821p-0.00036549h_{1}\\ &-0.010037h_{2}+0.013836h_{3}+0.24308\mu p-0.0037884\mu h_{1}\\ &+0.0023358\mu h_{2}-0.016918\mu h_{3}+0.029287 ph_{1}-0.015872 ph_{2}\\ &-0.0028339\mu h_{3}+0.0007175h_{1}h_{2}-0.00046158h_{1}h_{3}\\ &-0.00036486h_{2}h_{3}-0.39076\mu^{2}-0.015968p^{2}-0.0011936h_{1}^{2}\\ &+0.000269h_{2}^{2}+0.00062638h_{3}^{2}\\ 13\text{mm}\leq h_{3}\leq 16\text{mm},\ \mathbf{\xi}=\left[h_{1},h_{2},\mu,p \right]^{\text{T}},\ 0.5\leq R_{\text{U}}\leq 0.999\end{array}  \tag{18}\]  
where \(h_{1}\) is the friction material thickness, \(h_{2}\) the disc thickness, \(h_{1}\) and \(h_{2}\) are assumed as linear uncertain variables with \(h_{1}\sim\mathcal{L}(14.5\text{ mm},\ 15.5\text{ mm})\) and \(h_{2}\sim\mathcal{L}(19.5\text{ mm},\ 20.5\text{ mm})\), respectively. \(\mu\) is the friction coefficient, \(p\) indicates the brake pressure, \(\mu\) and \(p\) follow the normal uncertainty distribution with \(\mu\sim\mathcal{N}(0.35,\ 0.01)\) and \(p\sim\mathcal{N}(0.5\text{ MPa},\ 0.02\text{ MPa})\) respectively, and \(R_{\text{U}}\) is the prescribed reliability level according to the engineering requirement. In this problem, the URBDO model of the brake system can be implemented by the presented CEP method for different reliability levels. To confirm the optimal results of the proposed approach, the classical FORM-based RBDO (reliability index approach, RIA) method is also applied to the brake system problem when all input factors are considered to follow normal or uniform probability distributions. In other words, \(h_{1}\sim U(14.5\text{ mm},\ 15.5\text{ mm})\), \(h_{2}\sim U(19.5\text{ mm},\ 20.5\text{ mm})\), \(\mu\sim N\big{(}0.35,\ 0.01^{2}\big{)}\), and \(p\sim N\big{(}0.5\text{ MPa},\ 0.02^{2}\text{ MPa})\). The optimal results of the RBDO and URBDO model are plotted in Figure 7.  
According to Figure 7, with an increase in predetermined reliability requirements, more material is needed to fabricate the brake system. According to the optimal results, variation trends of back plate thickness computed by URBDO and RBDO models are similar. When the prescribed reliability \(R_{\text{pre}}\) is in \(R_{\text{pre}}\in[0.50,\ 0.99335]\), the back plate thickness estimated by URBDO model is lower than the one calculated by the RBDO model. However, the back plate thickness calculated by the URBDO model is higher than the one estimated by the RBDO model when the prescribed reliability is in \(R_{\text{pre}}\in[0.99335,\ 0.999]\). This is because the impact of epistemic uncertainty is omitted in the RBDO model. Consequently, the RBDO model is not suitable when human subjective judgment appears in structural  
Figure 6: A vehicle disc brake system. (**a**) CAD model; (**b**) Finite element model. design problems. The URBDO model based on the uncertain measure can evaluate the design of engineering structure more accurately and credibility under epistemic uncertainty. The CEP method and the USGA algorithm are simultaneously applied to the brake disc system for the desired reliability level \(R_{\text{U}}=0.9987\) to discuss the performance of two proposed approaches for solving the URBDO model.