Shape Optimization Design of Gravity Buttress of Arch Dam Based on Asynchronous Particle Swarm Optimization Method. Lei Xu

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1 Applied Mechanics and Materials Submitted: ISSN: , Vol. 662, pp Accepted: doi: / Online: Trans Tech Publications, Switzerland Shape Optimization Design of Gravity Buttress of Arch Dam Based on Asynchronous Particle Swarm Optimization Method Lei Xu College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing , China Keywords: Arch dam; Gravity buttress; Optimization design; Asynchronous particle swarm optimization method Abstract. The optimization design method was rarely used to design the gravity buttress of arch dam in the past. With this in mind, the parametric description of gravity buttress is given, and the auto-calculation of its exerting loads and the safety coefficient of anti-slide stability are realized subsequently. Then, the optimization design model of gravity buttress and the procedures of optimization design are presented using the asynchronous particle swarm optimization method. Finally, ODGB software, which is short for Optimization Design of Gravity Buttress software, is developed and verified. Introduction More and more designs of arch dam in China have to be faced with complicated and unfavorable topographic and geologic conditions, and setting gravity buttresses is a common way to solve this problem [1]. At present, the structure optimization design method has been widely used in the shape design of arch dam [2]. However, the optimization design method was rarely used to design the gravity buttress of arch dam in the past. With this in mind, the parametric description of gravity buttress is given firstly, and then, the optimization design model of gravity buttress and the procedures of optimization design are presented by using the asynchronous particle swarm optimization method. Finally, ODGB software, which is short for Optimization Design of Gravity Buttress software, is well developed and verified. The Optimization Design Model of Gravity Buttress Fig. 1 shows the 3D shape of a gravity buttress. Generally, the shape of gravity buttress can be defined by a set of parameters, which is called as shape parameters. For the shape of gravity buttress, the width and length of the top surface of gravity buttress are T and L respectively;β 1, β 2, and β 3 represent the slope angles of upstream surface, downstream surface and the surface connected to rock; α is used to describe the angle of between the center line of the top surface of gravity buttress and the center line of the top surface of arch dam; h 1 and h 2 are the altitudes of the top surface and the bottom surface; and the number of control attitudes is N. As to h 1, it is usually equal to the altitude of arch dam, while h 2 depends on the topographic and geological condition of dam site and should be determined before optimization design. The value of h 1, h 2 and N is considered as fixed value, and the shape of gravity buttress can be obtained through spatial geometry analysis if the value of shape parameters (which are T, L, β 1, β 2, β 3 andα) is determined. Therefore, these shape parameters are considered as the design variables. On the other hand, the mechanical constraint condition of optimization design mainly takes the anti-slide stability safety of gravity buttress into account as the shape and size of gravity buttress are generally controlled by anti-slide stability, and the rigid block equilibrium method is used to calculate the anti-slide stability safety coefficient. In addition, the loads exerted on gravity buttress and the safety coefficient of anti-slide stability are automatically calculated through programming. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (ID: , Pennsylvania State University, University Park, USA-06/03/16,15:02:41)

2 Applied Mechanics and Materials Vol Top Surface Surface Connected to Rock Downstream Surface Surface Connected to Dam Upstream Surface Bottom Surface Fig. 1 3-D sharp of gravity buttress The objective function and constraint conditions of the optimization model can be given as follows: m min ( X ), X R hi ( X ) 0, i 1,2,, l1 (1) g j ( X ) 0, j 1,2,, l2 where is the objective function (namely, the volume of gravity buttress) with m sharp parameters represented by X, h i (X)=0 is the i th equality constraint, while g j (X) 0 represents the j th inequality constraint. Asynchronous Particle Swarm Optimization Method Particle swarm optimization method, hereafter refers to as PSO method, is widely used for its high efficiency and convenience for application [3]. As to PSO method, each potential solution of optimization problem is a particle in the search space, and a certain number of particles constitute a particle swarm. In a m dimensional search space, n particles constitute a particle swarm X={x 1,, x a,, x n }. The position of the ath particle is described by x a ={x a1, x a2,, x am } T, and its speed is described by v a ={v a1, v a2,, v am } T. p a ={p a1, p a2,, p am } T represents the best position of the ath particle, while the position of the global best particle is described by p g ={p g1, p g2,, p gm } T. Then the particle updates its position and speed according to the following Eqs. (2) and (3): v w v c rand ( ) p x k 1 k k k k ad ad 1 ad ad k c rand ( ) p x 2 k 1 k k ad ad ad gd k ad x x v (3) where a=1, 2,, n, d=1, 2,, m, k represents the particle generation, w k is inertia weight, which controls the search capability of PSO method. Rand ( ) represents a random number which uniformly distributed between 0 and 1, and c 1 and c 2 are acceleration constants and are usually set as 2.0. From Eq. (2) and Eq. (3), it can be found that the information of the global best position gbest cannot be shared in time. In order to avoid this shortcoming of the PSO method, the behavior of each particle is considered as an independent process, and the global best position gbest is updated immediately when the fitness of any particle is less than the fitness of the current global best position. The revised method described above is called asynchronous particle swarm optimization method, which is short for APSO method. Fig. 3 shows the sharing mechanism of the global best optimization (2)

3 162 Mechanical Engineering, Materials and Information Technology II information of APSO method. Compared with PSO method, APSO method has higher optimization efficiency. The global best optimization information Particle 1 Particle m Particle 2 Particle Swarm Fig. 2 The sharing mechanism of APSO method The convergence criterions adopted in the optimization design can be expressed as follows: (4) where is the global best position of the k th generation of the particle swarm, is the global best position of the k-1 th generation of the particle swarm, is the anti-slide stability safety coefficient corresponding to is the minimum anti-slide stability safety coefficient, represent the convergence tolerances. The Procedures of Optimization Design and Software Development The first step of the optimization design is auto-calculation of the whole thrust force and its angle by calling the module TL, and initial particle swarm is randomly generated subsequently. Then, the third step of the optimization design is the calculations of coordinates of controlling points of shape, exerting loads, volume, and anti-slide stability safety coefficients for each particle. The next step of the optimization design is the update of particle according to Eq. (2) and Eq. (3). The above steps cycle until the convergence criterions are satisfied. ODGB software (short for Optimization Design of Gravity Buttress software), which is developed on the platform of Visual Studio 2008which is short for, is composed of a main module and nine functional modules. In order to verify the feasibility of the proposed method and developed software, the following application is conducted by using ODGB software. A concrete double curvature arch dam is going to be built in southwest China, and gravity buttress is adopted to improve the unfavorable geologic condition. Fig. 3 shows the FEM mesh of dam. The number of particles is set to 20, and the convergence tolerances are set to The friction coefficient f and cohesion strength c are set to 0.92 and 0.85 MPa respectively, and is set to 3.0. The bottom surface altitudes of both the two gravity buttresses are 480 m, and the angles of the surfaces connected to rock of both the two gravity buttresses are set to 90 for the purpose of prevent the interaction between the gravity buttresses and the rock with poor quality. Owing to the application requirement and the topographic condition, the width of the top surface of gravity buttresses should be no less than 6 m, and the value of α is allowed to vary between 40 and 90 in the optimization (5)

4 Applied Mechanics and Materials Vol design. The length of the top surface of gravity buttress is set to 15 m due to the practical topographic condition. In addition, the value ofβ 1 and β 2 is taken as the same in the optimization design in order to obtain a more regular shape and be convenient to construct, and the value is allowed to vary between 0 and 90 in the optimization design considering the practical situation. The above mentioned limitation of the shape parameters are actually the constraint conditions of this optimization design and can be easily expressed in simple mathematical form. Fig. 3 FEM mesh of arch dam Table 1 gives the main results of the optimization design. From Table 1, it is shown that the safety coefficients of both the left and right gravity buttresses are quite close to the minimum value given by the design specification, and all the constraint conditions described above are satisfied. Table 1 The optimal shape parameters, volume (V), bottom areas (A) and safety coefficients (F) β 1,β 2 ( ) α( ) T (m) V (m 3 ) A (m 2 ) F Left gravity buttress Right gravity buttress Conclusions The parametric description of gravity buttress is given firstly in this paper, and the auto-calculation of its exerting loads and the safety coefficient of anti-slide stability is realized subsequently. Then, the optimization design model of gravity buttress and the procedures of optimization design are presented by using APSO method. Finally, ODGB software is well developed, and its feasibility is verified through an application to a practical arch dam located in southwest China. Acknowledgment This work was supported by projects of the National Natural Science Foundation of China (Grant No , ) References [1] H. Y. Yang, Y. D. Zhou, F. Jin, et al, Integral Non-linear Stability Analyis and Safety Evaluation for High Gravity Pier-arch Dam at Sizhai Reservoir, Chinese Journal of Advances in Science and Technology of Water Resources, Vol (2007) [2] L. S. Sun, W. H. Zhang and N. G. Xie, Multi-objective Optimization for Shape Design of Parabolic Double-curvature Arch Dams, Dam Engineering, Vol.17 1 (2006) [3] L. Xu, T. J. Zhang, Development of Automatic System for Inversion of Mechanical Parameters and Advanced Prediction of Mechanical Response of Surrounding Rock of Underground Cavern in the Period of Construction, Chinese Journal of Sichuan University (Engineering Science Edition),Vol ( 2013)

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