## 4 Optimization Design
### Process of Optimization  
Using the parameterization design concept, the characteristic parameters of the model are abstracted based on its geometric structure. The following four design variables are adopted: \(M_{1}\) is the downstream slope coefficient of the second stage of the dam body, \(M_{2}\) is the upstream slope coefficient of second stage of the dam body, \(M_{3}\) is the downstream slope coefficient of the first stage of the dam body, and \(M_{4}\) is the upstream slope coefficient of the first stage of the dam body, as shown in Figure 16. Because the width of the dam body is uniform in the direction perpendicular to the river, the volume of the dam body with a width of one meter is selected as the objective function. When the objective function reaches its minimum, it means that the volume of the dam body is minimized, and this situation is considered as the minimum material consumption in this paper.  
According to relevant design specifications [35; 36; 37], it can be seen that the design requirements for this dam body are as follows: the range of upstream and downstream slope ratio of the first stage of the dam is 1:1.5\(\sim\)1:0.6; the range of upstream slope ratio of the second stage of the dam is 1:0.4\(-\)1:0.7; the range of downstream slope ratio of the second stage of the dam is 1:0.5\(-\)1:0.8; the vertical stress of the dam heel does not exhibit tensile stress; there is no tensile stress on the upstream surface of the dam body; the maximum compressive stress of the dam body does not exceed the maximum allowable compressive stress of the material; the stress ratio between the heel and toe of the dam does not exceed the allowable value of 1.5. In the early phase of engineering design, the material mechanics method was already used to conduct reliability calculations on the dam body, which meets the requirements of relevant specifications. This paper uses the finite element method for verification and obtains the following results: the vertical stress at the heel position of the dam body is \(-\)0.23 MPa, which is the compressive stress; the maximum stress in the upstream of the dam body is \(-\)0.01 MPa, which means it is all under compression stress conditions; the maximum value of compressive stress of the dam body is 2.41 MPa, which is less than the compressive strength of the material by 6 MPa; the compressive stress at the heel of the dam is 0.39 MPa; the compressive stress at the toe of the dam is 0.33 MPa; the stress ratio between the heel and toe of the dam is 1.18, meeting the requirements of the specification within the range of 1.5. All review results are safe. Therefore, when conducting optimization, only the geometric conditions of the dam body specified in the specifications are included in the constraint conditions. After finding the optimal solution, the parameters of the found optimal solution are inputted into ANSYS for numerical simulation to see whether the other conditions required by the relevant specifications are met. If not, the previous optimal solution is attempted, and so on. This can greatly improve optimization efficiency.  
Figure 16: Geometric dimension diagram of the dam body.