# Shape Optimization of Single-Curvature Arch Dam Based on Sequential Kriging-Genetic Algorithm
## Methodology
### Construction of Mathematical Optimization Model
#### 2.2.2 Constraints  
Geometric constraints, stress constraints, and stability constraints should be considered in the design optimization of a single-curvature arch dam to meet engineering safety requirements.  
**Geometrical constraints.** To facilitate construction, the degrees of overhang of the upstream and downstream surfaces of a single-curvature arch dam should be limited. The degree of overhang can be expressed as follows:  
\[K_{u}=y^{\prime}_{cu}(H)=0 \tag{6}\]  
\[K_{d}=y^{\prime}_{cd}(0)=y^{\prime}_{cu}(0)+B^{\prime}_{c}(0)=b_{1} \tag{7}\]  
where \(K_{u}\) and \(K_{d}\) are the real maximal degrees of overhang of the upstream and downstream surfaces of the arch dam, respectively. These degrees must satisfy the following conditions:  
\[K_{u}\leq[K_{u}],K_{d}\leq[K_{d}] \tag{8}\]  
where \([K_{u}]\) and \([K_{d}]\) are the allowable maximal degrees of overhang of the upstream and downstream surfaces, respectively. Following the Code for Design of Concrete Arch Dams, \([K_{u}]=[K_{d}]=0.3\) in this study.  
**Stress constraints.** Under the actions of gravitational load, hydrostatic pressure, uplift pressure, silt pressure, and temperature load, the principal stresses at each control point of an arch dam must satisfy the following conditions:  
\[\sigma_{1}/[\sigma_{1}]\leq 1,\sigma_{3}/[\sigma_{3}]\leq 1 \tag{9}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) represent the first principal stress and the third principal stress, respectively. Moreover, the maximum value of \(\sigma_{1}\) and the minimum value of \(\sigma_{3}\) were used in this study. \([\sigma_{1}]\) and \([\sigma_{3}]\) denote the allowable values of the principal tensile stress and the principal compressive stress, respectively. This study used the FEM to analyze the structure of an arch dam. According to the Code for Design of Concrete Arch Dams, the equivalent stress should be calculated as follows:  
\[\sigma_{1 eqp}/[\sigma_{1 eqp}]\leq 1,\sigma_{3 eqp}/[\sigma_{3 eqp}]\leq 1 \tag{10}\]  
where \(\sigma_{1 eqp}\) is the maximum equivalent stress and \(\sigma_{3 eqp}\) is the minimum equivalent stress. \([\sigma_{1 eqp}]\) and \([\sigma_{3 eqp}]\) are the allowable values of the maximum equivalent stress and the minimum equivalent stress, respectively. According to practical engineering design requirements, this study set the values of \([\sigma_{1 eqp}]\) and \([\sigma_{3 eqp}]\) to be equal to those of \([\sigma_{1}]\) and \([\sigma_{3}]\).  
**Stability constraints.** Stability constraints can be divided into three types: safety factor constraint of anti-sliding stability, thrust angle constraint of the arch abutment, and center angle constraint of the arch ring. In this study, the center angle of the arch ring was limited,  
\[\varphi/[\varphi]\leq 1 \tag{11}\]  
where \(\varphi\) is the real central angle of the arch; \(\varphi=\theta_{L}+\theta_{R}\), where the definitions of \(\theta_{R}\) and \(\theta_{L}\) are the same as those mentioned previously. \([\varphi]\) is the largest allowable central angle of the arch.  
In this study, the values of \(\sigma_{1}\), \(\sigma_{3}\), \(\sigma_{1 eqp}\), \(\sigma_{3 eqp}\), and \(V\) were calculated using ANSYS.