# Design Optimization of a Hydrodynamic Brake with an Electrorheological Fluid
## 4 Optimization Method for the Hydrodynamic Brake Design
### The Algorithm for Design Optimization of the Hydrodynamic Brake  
On the basis of the criteria given in Table 4, the following objective functions are assumed:  
\[\rm{F_{G}=G-0.5\cdot M} \tag{38}\] \[\rm{F_{G1}=G/M}\] \[\rm{F_{P}=0.1\cdot P_{e}-M}\] \[\rm{F_{P1}=P_{e}/M}\] \[\rm{F_{T}=T-0.3\cdot M}\] \[\rm{F_{T1}=T/M} \tag{40}\]  
The objective functions F\({}_{\rm G}\), F\({}_{\rm P}\), F\({}_{\rm T}\), F\({}_{\rm G1}\), F\({}_{\rm F1}\), F\({}_{\rm F1}\) described by Formulas (38)-(40) cannot be written using simple formulas, in which there are the design variables, due to the complicated form of the applied mathematical model. The values of the criteria G, M, P\({}_{\rm e}\), T were calculated on the basis of the formulas given in Table 4. The numerical coefficients appearing in the Formulas (38)-(40) are arbitrarily selected during initial calculations, so that minimal values of the objective functions F\({}_{\rm Gmin}\), F\({}_{\rm Pmin}\), F\({}_{\rm min}\) are as close to zero as possible. The objective function, which is the difference between G and M, is also assumed in the articles [49,50]. The minimalization of the objective functions (38)-(40) leads to finding the design variables, for which the unit of the torque M will be assigned with,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Designation** & **Requirements** & **Main Calculation Formula Numbers** \\ \hline Torque & M & Maximal & (16), (17), (25), (26) \\ Weight & G & Minimal & (4)–(7) \\ Electric power & P\({}_{\rm e}\) & Minimal & (30), (31), (33), (34) \\ Time & T & Minimal & (35)–(37) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Requirements concerning the HB optimization criteria.  
respectively, the lowest weight G, the lowest electrical power P\({}_{\rm e}\), and the shortest time T. The design variables intervals r\({}_{\rm w}\), R, n and constraints concerning d\({}_{1}\) and U\({}_{\rm h}\) are the ones given in Table 3. The optimization calculations for the HB are aimed at minimizing the selected objective function F described with one of the Formulas (38)-(40).The calculations are performed as follows:  
* the decision variables r\({}_{\rm w}\), R, and n are randomly selected from the assumed intervals with the use of a random number generator;
* the value of \(\omega_{1}\) is determined and g\({}_{\rm b}\) = 0.02R is calculated;
* calculations are performed according to the formulas of the mathematical model;
* the constraints of d\({}_{1}\) and U\({}_{\rm h}\) are checked;
* when the randomly selected values of the design variables meet the conditions of the constraints, the objective function F is calculated, its result is saved, and the calculations are repeated;
* the smaller value of the objective function from the previous F\({}_{\rm p}\) and present step of the calculations F\({}_{\rm a}\) is selected;
* the values of the design variables r\({}_{\rm w}\), R, and n are saved, and on the basis of these values, an objective function of a smaller value is obtained.  
The scheme of calculations carried out in one calculation step of a computer program is shown in Figure 19.