Integrated Metaheuristic Differential Evolution Optimization Algorithm and Pseudo Static Analysis of Concrete Gravity Dam

A differential evolution-algorithm-based optimum design method is presented for concrete gravity dams under stability, principal stress, and sliding constraints. A gravity dam is a large scale hydraulic structure providing its stability based on weight of concrete volume. Hence, optimization of dam cross-section leads to an economic and low cost structure. For this aim, a general dam section is reconstructed with seven proper horizontal and vertical geometric parameters which take into account all possible cross section shapes. Weight of dam is considered as goal function and the optimization problem of geometric parameters is solved using DE algorithm. The DE algorithm written as a MATLAB code are applied to Four benchmark gravity dams including Middle Fork, Richard, Pine Flat, and Friant. The comparison of DE optimum solutions with real dimension of dams and another optimization method in literature shows the performance of the DE algorithm. In mentioned benchmark dams, there are 26.82%, 30.11%, 25.31%, and 20.93% of weight reduction Compared to real values, respectively. Also, optimization results of DE algorithm are compared with literatures. The comparison shows 3.55%, 5.1%, 19.13% and 12.14% reduction of weight compared to GA and PSOD algorithms, respectively.


Introduction
Optimum design can be defined as the best acceptable design, based on a predetermined criterion of qualitative merit.The importance of designing structures with minimum weight was first considered by the aerospace industries in which the design of the plane is controlled majorly by weight than cost.In other industries related to civil engineering, cost may have the first importance even though the weight of system impresses its cost and function.So, optimization issue plays a great role in the ideal world.Existing optimization methods can be generally classified as analytical and numerical approaches.In Analytical approaches, the optimal point of continuous and differentiable goal function is determined using differential calculus.Due to nonlinear, discontinuous and non-differentiable goal function of real problems with various design constraints, analytical optimization may not be effective in civil engineering.Whilst the numerical approaches, especially metaheuristic algorithms, show high acceptability for solving complex and nonlinear design problems in civil engineering.
In concrete gravity Dams, weight and strength of the concrete maintain stability and equilibrium against external loads.Due to high volume of consumer concrete, a shape optimization design leads to a remarkably reduction of construction cost.Hence, researchers have shown an Impressive interest in employing metaheuristic approaches to obtain optimal shape of gravity dams.Genetic algorithm optimum design of gravity dam is presented in [22].In this research, optimal top width of gravity dam is optimized with genetic algorithm under stability and stress constrains.Also, [23] were developed genetic algorithm based model for optimal design of a gravity dam under seismic excitation with reservoirdam foundation interactions.The particle swarm algorithm coupled with ANSYS software is used to carry out optimum design of gravity dam [24].Three recently developed metaheuristic algorithms including charged system search (CSS), colliding bodies optimization (CBO) and its enhanced edition (ECBO) are utilized for optimizing the gravity dam by [25].The shape optimization problem of gravity dam is considered under stability and principal stress constraints.GA and PSOD optimization algorithm are utilized to minimize the cross section of gravity dam by [26].Dam-waterfoundation interaction and nonlinear effects due to earthquake of gravity dams are considered in optimization problem [27].This paper presents differential evolution based optimization model of each possible cross section of gravity dams.The DE algorithm, a smart metaheuristic algorithms based on probability law with local search ability, is introduced by [28].A general cross section of gravity dam is constructed using various unknown geometric parameters.The horizontal and vertical forces actin on gravity dam and corresponding moments about toe which may affect the pseudo static analysis of gravity dam are written based on unknown geometric parameters.Then, total weight per unit length of dam is chosen as goal function and the optimization problem is stablished using DE algorithm under stability, principal stress, and sliding constraints.The DE algorithm is written in MATLAB software and applied to four real gravity dams and the optimization result are compared with real size.Furthermore, optimization results of DE algorithm are compared with GA and PSOD algorithms to show the ability of DE algorithm in determining the optimal global point.

Gravity Dam and Differential Evolution Based Shape Optimization
A gravity dam was designed against proper failures.The main failure modes may be occurred due to (i) overturning/rotation about the toe; (ii) compressing or crushing; and (iii) shear sliding.The failure modes should be checked for two full and empty reservoir cases.A gravity dam design is accepted when the following stability, principal stress, and sliding constraints are satisfied.
(1) Stability Constrain Where,   and   are resisting and overturning moments acting about toe.  and   are total vertical and horizontal forces acting on dam. is coefficient of friction and  , ,  , ,  , ,  , ,  , and  , are principal direct and shear stresses at heel, principal direct and shear stresses at toe and allowable compressive and shear stress in concrete, respectively.Figure 1.shows a general parametric cross section of gravity dam in which the cross section are defined based on seven geometry parameters i.e., ( 1 ,  2 ,  3 ,  4 ,  1 ,  2 ,  3 ).Indeed, these seven unknown parameter form the variables of optimization problem.By dividing Figure 1. to three main parts, all horizontal and vertical forces on gravity dam and corresponding moment about toe are presented in Table 1.  . 2 Weight per unit length part 3 Static earthquake load due to W3 Coefficient of dynamic effect of earthquake force (4)


Where, m is coefficient of crest position.The angle between vertical axis and downstream and upstream are indicated by  and .Also   , ,   , and   ′ imply on uplift pressure, minimum stress at heel, hydrodynamic pressure of water in absence and presence of earthquake are shown by, respectively.The optimization problem is explained at follow.Total weight per unit length of dam (sum of three parts of cross section) is considered as objective function.Gravity dam weight per unit length optimization subjected to constrain can be expressed as: (7)   x is the vector of design variables containing the optimization variables as ( 1 ,  2 ,  3 ,  4 ,  1 ,  2 ,  3 ).The indices i and j are number of design variables and design constrains, respectively.Finally, six design constraints are considered as below: The differential evolution (DE) is a more advanced version of genetic algorithm (GA) developed by Storn and Price.The DE is an exceptional easy, very fast and powerful without encoding and decoding stages designed for optimization problems in continuous search space.It is a stochastic search algorithm and derivative-free method that search the global optimal point.The main parameters of DE approaches are NP (population size), PCR (crossover probability), and F (weight factor).In first step of DE approach, number of variables and their Lower and upper limits are determined.After considering initial values for NP, PCR, and F, a random value are attributed for j th variable of i th member as below: (9)    10 10 , 1 , 0 In second step, the goal function of each member are calculated and the lowest value in the second row are stored in fmin.The third step is mutation where for each member, three various and random members of population are selected and the following calculation are executed: (10)  

  
In the fourth Crossover step, a   population series are produced that inherits their properties from   and   as below: (11 The last step is Selection.The main different between DE and GA approaches is selection step.In the GA, the selection of an answer as a parent is dependent on its merit value.Whilst in DE algorithm, all the answers have an equal chance to be selected.In selection step, the fittest answer is selected and passed onto the next generation.The basic structure of differential evolution (DE) algorithm is illustrated in Figure 2.

Results and Discussion
As mentioned, the purpose of this study is to optimize the shape of concrete gravity dam that will reduce the volume of concrete used.Hence, Differential Evolution algorithm is utilized for a pseudo static analysis based optimization of gravity dams.In this study total weight per unit length of dam is chosen as goal function and the optimization problem is established using DE algorithm under stability, principal stress, and sliding constraints.To illustrate the effectiveness of the proposed method, four concrete gravity dams including Middle Fork, Richard, Pine Flat, and Friant are selected as the real word case studies in this paper.The details of the considered dams are explained in the sequence.
The Middle Fork dam has the structural height of 32 m and has been constructed to control floodwater on the Middle Fork River, Colorado area in USA.The upstream floodwater and normal height level are 30 m and 28.96 m respectively.Also, the corresponding height level of downstream floodwater are 4.57 m and 3.5 m.According to USRB code, drainage galleries taken into account with the distance .05mfrom upstream water.Second gravity dam is Richard that has the structural height of 49.68 m and has been constructed to control floodwater on the Savana River, Georgia USA.The upstream floodwater and normal height level are 46.63 m and 45.26 m, respectively.Also, the corresponding height level of downstream floodwater are 8.83 m and 4.88 m.In this dam, drainage galleries is as same as Middle Fork dam.Pine Flat is the third specimen for optimization task.This dam has the structural height of 121.92 m and has been constructed on the Kings River in California.The dam is constructed to control of floodwater, agriculture water supplement of downstream and production of hydropower energy.The upstream floodwater and normal height level are 116.28m and 114 m, respectively.Also, the corresponding height level of downstream floodwater are 15 m and 10 m.According to USRB code, drainage galleries taken into account with the distance 0.05 m from upstream water.
The last selected dam is Friant.This dam has the structural height of 85.95 m and has been constructed on the San Joan River in California, USA.The Friant dam is also constructed to control of floodwater, agriculture water supplement of downstream and production of hydropower energy.The upstream floodwater and normal height level are 85.1 m and 84.73 m and the corresponding height level of downstream floodwater are 2 m and 1 m.
The weights per unit length of four considered benchmark dams are 952.33,2379, 13631, and 7170 ton/m, respectively.Number of initial population and maximum iteration for DE algorithm are selected as 50 and 500 for all optimization process.Convergence curve of performance index for the optimization are shown in Figure 3.According to Figure 3, the Objective function has been converged after 250, 150, 310, and 350 Iterations for four mentioned dams, respectively.As seen, the DE algorithm has a high convergence accuracy and speed in addition to its ability to find the optimal global point.In maximum state, the DE algorithm has been converged in less than 350 iterations for Pine Flat Gravity Dam.

Middle Fork Gravity Dam
Richard Gravity Dam Pine Flat Gravity Dam Friant Gravity Dam

Figure 3. Convergence curve of performance index in each generation for gravity dams
The optimum value of variables and corresponding objective function of benchmark dams attained by the DE algorithm and another optimization method in literature for comparison are presented in Table 2.The results show the ability of Differentiate Evolution algorithms in optimizing the dimensions of concrete gravity dams, and thus reducing the amount of concrete used.Furthermore it can be concluded that the dam dimensions decreased rather than real dimensions.Applying the DE algorithm, we obtained 26.82%, 30.11%, 25.31%, and 20.93% of reduction in weight per unit length for considered dams relative to real value, respectively.The Real and optimal cross section of four dams are presented in Figure 4 to 7. in which the reduction of the dimensions of dams is evident clearly.Finally, optimization results of DE algorithm are compared with GA and PSOD in addition to real value to study the effectiveness of the DE algorithm.The comparison shows an acceptable reduction of dimensions and weight of dams relative to GA and PSOD algorithms i.e. 3.55%, 5.1%, 19.13% and 12.14% of reduction in weight per unit length of considered dams, respectively.This considerable reduction is achieved due to the ability of DE algorithm in determining the optimal global point.

Conclusion
Safety and economy are two main primary goals in civil engineering problems.The economy will has special importance when a huge amount of material is used.A gravity dam is a large scale hydraulic structure providing its stability based on weight of concrete volume.Hence, optimization of dam cross-section leads to an economic and low cost structure.In this paper, a general parametric cross section of gravity dam is defined based on seven unknown horizontal and vertical geometric parameters.Weight of dam is considered as objective function and the optimization problem is solved using DE algorithm.The DE algorithm, written as a MATLAB code, are applied to four benchmark gravity dams including Middle Fork, Richard, Pine Flat, and Friant.The comparison of DE optimum solutions with real dimension of dams shows the performance of the DE algorithm.In mentioned benchmark dams, there are 26.82%,30.11%, 25.31%, and 20.93% of reduction in weight per unit length, respectively.Furthermore, DE algorithm optimal cross section shows a remarkable reduction of objective function in comparison with GA and PSOD algorithms.

Figure 1 .
Figure 1.Parametric cross section of gravity dam Therefore, stability, principal stress, and sliding constraints Equation 1 to 3. can be rewritten as below:

Figure 2 .
Figure 2. Basic flowchart of Differential Evolution (DE) algorithm

Figure 3 .Figure 4 .Figure 5 .Figure 6 .
Figure 3. Real and DE based optimal cross section of Middle Fork gravity dam

Table 2 . Comparison of DE optimal cross section of Benchmark dams with real size and references [25]
(m)   (m)   (m)   (m)   (m)   (m)   (m) (ton) Middle Fork Gravity Dam