# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Shape Optimization of a Concrete Gravity Dam Under Multiple Earthquake Loads  
The test case (iii) solved in this study was the shape optimization of a concrete gravity dam under multiple earthquake loads. The real structure is located in the Shafaroud region of Northern Iran: the dam height (\(H_{\text{dam}}\)) is 150 m, water elevation in the normal state in the dam upstream (\(I_{\text{water}}\)) is 145 m, and the sediment height (\(I_{\text{ise}}\)) is 7 m. The 3D schematic of the dam is shown in Figure 8a. Here, the concrete volume per unit width of the dam was minimized by optimizing the dam cross-section shown in Figure 8b with the nine shape variables (x\({}_{1}\),x\({}_{2}\),x\({}_{3}\),x\({}_{4}\),x\({}_{5}\),x\({}_{6}\),x\({}_{7}\),x\({}_{8}\),x\({}_{9}\)) indicated in the figure. Figure 8: (a) 3D schematic of the gravity dam; (b) cross-section of the concrete gravity dam indicating shape variables and main geometric dimensions. The optimization problem was stated as follows:  
\[\textit{Minimize}\ W(\overrightarrow{X})=H_{dam}x_{1}+\frac{1}{2}x_{2}x_{3}+x_{3 }(x_{4}+x_{6})+\frac{1}{2}x_{4}x_{5}+x_{5}x_{6}+\frac{1}{2}x_{6}x_{7}+\frac{1}{ 2}x_{8}x_{9} \tag{20}\]  
\[\textit{Subject to}\ \left\{ \begin{array}{l}G_{1}(\overrightarrow{x})=4-SFS \leq 0\\ G_{2}(\overrightarrow{X})=1.5-SFO\leq 0\\ G_{3-4}(\overrightarrow{X})=0\leq\sigma_{U}\leq\sigma_{MAX}\\ G_{5-6}(\overrightarrow{X})=0\leq\sigma_{D}\leq\sigma_{MAX}\\ G_{7}(\overrightarrow{X})=x_{4}+x_{6}-x_{8}\leq 0\\ H_{8}(\overrightarrow{X})=(x_{2}+x_{4}+x_{6}-H_{dam})^{2}=0\end{array} \right. \tag{21}\]  
where SFS is the safety factor against dam sliding; SFO is the safety factor against dam overturning; \(\sigma_{\rm U}\), \(\sigma_{\rm D}\), and \(\sigma_{\rm max}\) are the vertical fatigue specific loads in upstream and downstream and the corresponding fatigue limit, respectively. Side constraints of shape variables are as follows: \(1\leq x_{1}x_{3}\leq 20\) m; \(1\leq x_{2}\leq 50\) m; \(5\leq x_{4}x_{5}x_{6}x_{7}\leq 100\) m; \(50\leq x_{8}x_{9}\leq 150\) m.  
In Equation (21), the SFS factor (involved in the inequality constraint \(G_{1}\)) is defined as (f\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\))/\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\)\(\cdot\), where f = 0.7 is the static friction coefficient, \(\sigma\) = 3.5 MPa is the allowable shear tension in the cutting surface, and b = (x\({}_{1}\)+x\({}_{3}\)+x\({}_{5}\)+x\({}_{7}\)+x\({}_{9}\)) is the dam base length. The dam is subject to the vertical forces denoted as F\({}_{\rm v}\): concrete weight W, vertical components of water pressure Fh\({}_{\rm v}\), and sediment pressure F\({}_{\rm SV}\) directed downward; uplift force generated by the dam basement F\({}_{\rm uplift}\), and earthquake force F\({}_{\rm E}\) directed upward. The dam is also subject to horizontal forces F\({}_{\rm H}\) in the positive X-direction: an additional earthquake force of magnitude 2F\({}_{\rm E}\), the horizontal components of water pressure Fh\({}_{\rm h}\), and sediment pressure F\({}_{\rm SO}\). The SFO factor (involved in the inequality constraint \(G_{2}\)) is defined as \(\Sigma\)M\({}_{\rm R}\)/\(\Sigma\)M\({}_{\rm o}\) where M\({}_{\rm R}\) and M\({}_{\rm o}\), respectively, are the torque of resisting forces (W, Fh\({}_{\rm v}\), P\({}_{\rm SV}\)) and the torque of driving forces (F\({}_{\rm uplift}\), 2F\({}_{\rm E}\), Fh\({}_{\rm H}\), P\({}_{\rm SO}\)) on the dam. Vertical fatigue loads \(\sigma_{\rm U}\) and \(\sigma_{\rm D}\) (involved in inequality constraints \(G_{5}\) and \(G_{6}\)) are defined as (\(\Sigma\)F\({}_{\rm v}\)/b \(-\)\(\delta\)\(\Sigma\)M\({}_{\rm o}\)/b\({}^{2}\)) and (\(\Sigma\)F\({}_{\rm v}\)/b + 6\(\Sigma\)M\({}_{\rm o}\)/b\({}^{2}\)), respectively.