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Abstract This paper shows a new design technique for high-speed trains using a multi-objective optimization method to balance multiple aerodynamic properties. The technique is composed of an evolutionary algorithm, a shape parameterization technique using B-spline curves and Coon’s patches, and a computational simulation using a message passing interface. In order to demonstrate the capability of the method, the train nose shape is designed to optimize the aerodynamic drag and aerodynamic forces affecting the other trains. After the evolutionary calculation of the tenth generation with 512 individuals, physically reasonable Pareto solutions are successfully obtained. Keywords: high-speed train, shape optimization, multi-objective optimization, evolutionary algorithm, computational fluid dynamics, aerodynamic property. 1 Introduction Designs of train shapes are determined with consideration of many factors such as mechanical structures, intensities, aerodynamic properties, driver visibility, manufacture’s cost, ease of maintenance. Amid them, the aerodynamic properties are critical for high-speed trains. What one should take consideration for the aerodynamic properties are as follows: tunnel micro-pressure waves, an aerodynamic drag, flow-induced car vibrations in tunnels, response to crosswind, aerodynamic forces affecting the other trains and trackside structures, etc. Among them, the tunnel micro-pressure wave has been considered to be the most important factor in designing nose shapes of the high-speed trains in Japan. The tunnel micro-pressure wave is an impulsive pressure wave, which are radiated from the tunnel exit with an explosive sound when the train nose enters the tunnel at a high speed. Since Iida et al. [1] obtained the optimum cross-sectional area variation (but not three-dimensional shape) of the train nose for reducing the tunnel micro-pressure wave

Paper 135 Multi-Objective Design Optimization of the Nose of a High-Speed Train M. Suzuki1 and K. Nakade2 1 Department of Vehicle and Mechanical Engineering Faculty of Science and Technology, Meijo University, Nagoya, Japan 2 Railway Technical Research Institute, Tokyo, Japan

©Civil-Comp Press, 2012 Proceedings of the Eleventh International Conference on Computational Structures Technology, B.H.V. Topping, (Editor), Civil-Comp Press, Stirlingshire, Scotland

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using a flow simulation and a nonlinear optimization method, several studies have been conducted [2, 3]. The latest Shinkansen trains have been designed based on these studies. As for the aerodynamic drag, wind tunnel experiments and on-track tests have been conducted and a guide principle for reducing the drag was proposed [4]. It was reported that the pressure fluctuation on the train side causes the flow-induced car vibration in the tunnel and several train shapes have been examined to reduce the pressure fluctuation by wind tunnel and on-track tests [5]. Since the aerodynamic forces acting on the train under the crosswind depend not only on the vehicle shape but also wayside structures, wind tunnel experiments with systematic classification of the trains and the wayside structures have been executed [6]. The optimum nose configuration of train for reducing the pressure variation at the train passage, which causes the aerodynamic forces affecting the other train and the trackside structures, was obtained by using an axisymmetric flow simulation combined with a nonlinear programming and confirmed by a model experiment [7]. As stated above, studies for improving each single aerodynamic property of the high-speed train have been conducted extensively. However, it is not easy to satisfy plural aerodynamic properties simultaneously. In the field of aerospace and aeronautical engineering, a multi-objective optimization has been extensively studied and already applied for a case of designing a real aircraft [8]. On the other hand, few studies of the multi-objective optimization have been conducted in the field of railway. Though Tyll et al. [9] proposed a concept of a multi-objective optimization for a magnetic elevated vehicle, they examined only the optimum two-dimensional nose shape, but not the three-dimensional one. Therefore, in this study, we develop a multi-objective optimization method for the three-dimensional train shape with the prospect of supporting the actual train design process.

2 Method 2.1 Outline of the method A flow chart showing the multi-objective optimization process is provided in Figure 1. First, initial values, those of designing variables defining initial train shapes, are given randomly. Then the flow field around the train is calculated and the aerodynamic properties are estimated. Next, new possible designing variables are set by the evolutionary algorithm, before returning to the flow simulation stage. The process is repeated until the objective functions converge. 2.2 Definition of the train shape The shape parameterization techniques are very important aspect of the optimization. Higher degree of flexibility in representing shapes by fewer parameters is required. In this study, first, cross-sectional shapes are defined by B-spline curves, which is one of the most popular approaches for airfoil designs [10]. Next, surfaces between the cross-sectional shapes are determined by bilinear Coons patches [11] (Figure 2).

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Figure 1: Flow chart of the proposed method

Figure 2: Representation of the train nose shape

Here, we define the cross-sectional shape using the third-order B-Spline curve with four control points as an example (Figure 3). To reduce the computational cost, the distance d from the corner to the control points V0 and V1 is set to be constant. This means curvature of the corner is constant. In addition to the constraint condition of

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the maximum height and width of the train, the cross-sectional area is set to be in accordance with the optimum cross-sectional area variation for reducing the tunnel micro-pressure wave. The optimum cross-sectional area is defined by the following equation [1]:

1 1 (1)

where :the distance from the front end, :the length of the train nose, πb : the area of the maximum part of the nose and set to be 11 m2 in this study. The design variables and are reported to be 4.18 and 0.75 respectively when ⁄ =5. Adding this as the constraint condition reduces the number of design variables of each cross-section to one, which is the aspect ratio. After defining a set of cross-sections along the nose by the B-spline curves, the surface between each cross-section is interpolated by the bilinear Coons patch. In this study, we use five cross-sections for representing the nose shape. Therefore, the number of design variables for creating the train shape comes to five.

Figure 3: Definition of the cross-section by the third-order B-spline curve

2.3 Multi-objective evolutionary algorithm An optimization problem consists in maximizing or minimizing an objective function under a constraint condition. If we have multiple objective functions, it is called a multi-objective optimization problem. The multi-objective optimization present not a unique solution, but a set of compromised solutions called Pareto optimal solutions, which show the trade-off information of the competing objectives [12] (Figure 4). The optimization methods are generally categorized into two groups, deterministic methods and stochastic ones [12]. If one want to search the maximum values of the objective functions and these functions have a single peak in their

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solution space, the deterministic method is suitable. On the while, in cases that the objective functions have plural peaks, the deterministic method has the risk of getting stuck in a local optima. The stochastic method is suitable in these cases. In general, aerodynamic problems have the plural peaks. Amidst the stochastic approaches, there is an evolutionary algorithm, which mimics biological evolution. In the evolutionary algorithm, an initial population of design candidates called individuals is randomly generated at first. A fitness function of each individual, which is related with the objective function, is evaluated. Matting pairs of the individuals with higher fitness values are selected to produce offspring for the next generation by exchanging and mutating their design parameters. Then the fitness functions of the new generation are evaluated and the matting pairs are selected to reproduce the next generation again. In the process, one can expect to have the individuals with better objective vales. The evolutionary algorithm can sample as many solutions as the number of individuals during the alternation of generations and is suitable to find Pareto optimal solutions. Moreover, the algorithm is easily programed for parallel computation. Each process of the evolutionary algorithm is explained as follows.

Figure 4: Pareto Solution

Selection: One may say that the individuals with parents of the higher fitness values possibly have the higher fitness values comparing with ones with the parents of the lower fitness values. Then the individuals to be parents are selected in a stochastic process corresponding to their fitness values. Several selection methods are proposed such as roulette selection, ranking selection and tournament selection [10]. We employ Stochastic Universal Sampling (SUS) [13] to prevent the loss of population diversity. SUS uses a roulette, which is divided into parts in proportion as their fitness values, with plural indicators to select a certain number of parents each wheel spin (Figure 5). We have two indicators in this study.

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Figure 5: Stochastic universal sampling

Figure 6: Pareto ranking method

The fitness values are evaluated by Pareto ranking method [14]. In Pareto ranking method, ranks of Pareto optimal solutions are assigned one. The ranks of the other will be assigned in response to their locations in the solution space (Figure 6). In a population of a generation, if an individual is dominated by (inferior to) p piece of individuals, the rank of the individuals is assigned 1+p. For instance, since the individual D in Figure 6 is inferior to the individuals A, B and C, the rank of the individual D is 4. The fitness value is set to be an inverse number of the rank. It is preferable that Pareto optimal solutions are uniformly distributed in the solution space to keep the population diversity. Then the fitness values of the

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individuals in the dense populated area of the solutions are lowered and ones in the depopulated area are raised. The modified fitness value is as follows [15]:

∑ ,⁄ (2) where is the size of the population. is a function of the population density as follows:

,1 , ,

0 ,

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where , is Euclidean distance between the individuals and in the space of the objective functions [12]. is set to be 0.25. is a parameter that controls to increase or decrease the fitness values in response to the population density and is determined by solving the following equation [14].

∏ ∏0 (4)

where , and are maximum and minimum values of each objective function and is the number of objective functions. In a case of two objective functions, Equation (4) becomes:

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2 0 (5)

Crossover: Genes are exchanged between the selected individuals. Here, the genes mean the design parameters. Tough there are two ways of representing the design parameters, binary representation and floating-point representation, we adopt the floating representation which is conceptually close to the real design space. BLX- Method [16], which has a wide exploring area, is employed for combining the design parameters of two parents.

γ · 1 · (6)

1 γ · · (7) where and are the design parameters of the children, and

are those of the parents. And 1 2 where u is a uniform random number between 0 and 1. According to , the design parameters of the children are assigned to be outside between the parents’ design parameters in a stochastic manner. is set to be 0.5.

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Mutation: To explore the wider area where the crossover of the present population cannot reach, parts of the genes are forcedly mutated in a stochastic manner. There are two ways of mutation in general. One is a uniform mutation which adds a uniform random number to each design parameter at a probability called a mutation rate. Another one is Gaussian mutation which adds a number with the unit normal distribution. Since the former is generally employed in the evolutionary algorithm with floating-point representation [12], we adopt the uniform mutation method. The mutation rate is set to be 0.1. Alternation of generations: A simple way of alternating generations is that the parent population is always replaced by the offspring population. The lifetime of each individual is one generation in this system. The offspring, however, do not always have better fitness values than the parents. Then the parents and the children compete with each other. The individuals with higher fitness values among them within the population size survive to the next generation [17]. 2.4 Objective functions and a numerical flow simulation In this study, we adopt two aerodynamic properties as the objective functions: the aerodynamic drag on the front nose of the train, and the pressure variation around the car, which causes the aerodynamic forces affecting the other trains and the trackside structures. To estimate these properties, we need to analyse the flow around the train nose accurately. Although the high-speed trains have streamlined-nose and there are no large separations around it, we should consider the boundary layer. Thus, a steady three-dimensional viscous flow simulation is conducted. Baldwin-Lomax model [18] is employed as a turbulence model. The numerical scheme is based on the MAC method [19]. The Reynolds number based on the train height is set to be 105. As the computational flow simulation places a heavy burden on computational resources and time, the reduction of these costs is the key to make the optimization feasible. This study reduces the computational cost by parallel computation with Message Passing Interface (MPI) [20]. By allocating each processor to each individual in the process of the evolutionary algorithm, the aerodynamic estimations of all individuals are conducted simultaneously. The grid system used in the flow simulation is generated in the following procedure. First, the surface grid is created on the surface, which is represented by the Coons patches with the B-spline curves as described before, by using a surface grid generation method based on unstructured grid [21]. Then the grid in the whole area is created by a parabolic-hyperbolic hybrid scheme [22]. The train has a length of 2.5 cars. However, no gaps between the cars are considered. Numbers of the surface grid on the train are 141 in the flow direction and 56 in the circumferential direction, respectively. Numbers of the whole grid in each direction are (192,55,46) and the total amounts to about 490,000 points. Figure 7 shows an example of the grid system. A previous computation using the same numerical scheme and a grid system with almost same grid spacing around the train nose as this one showed good agreement with experiments in the pressure distribution [23].

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The aerodynamic drag of the nose is evaluated by pressure drag acting from the top edge to the end of the nose. The pressure variation around the car is estimated by the difference between the maximum and the minimum pressures along a line where the side of the on-coming train (but in the absence of it in the computation) is located (Figure 8).

Figure 7: An example of grid system

Figure 8: Sampling location of pressure variation

(Note no on-coming train in the computation.)

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3 An example of optimization To show the feasibility of this method, the train shape was optimized with the above described two objective functions. The evolutionary calculation was implemented until 10th generation with 512 individuals. The computation was conducted on Cray XT-4 and it took elapsed computational time of about 3 hours per generation.

Figure 9 indicates the objective value distribution at each generation. The horizontal axis is the aerodynamic coefficient and the vertical one is the pressure coefficient which indicates the magnitude of the pressure variation. At the first generation, the individuals distribute widely since the train shapes are randomly given. As the generation goes on, the individuals congregate in the neighbours of Pareto front. Figure 10 demonstrates Pareto optimal solutions and examples of their train shape. We found 109 Pareto optimal solutions. The train of the minimum drag has a two-dimensional wedge shape. This agrees with a result of comparing 16 classified train shapes qualitatively [24]. On the while, the train of the minimum pressure variation along the car has a three-dimensional shape with gradually expanding in both upward and sideward directions.

Figure 9: Objective value distribution at each generation

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Figure 10: Pareto solutions and examples of nose shapes 4 Concluding remarks We developed the optimization method of the three-dimensional train shape to satisfy the plural aerodynamic properties simultaneously using the evolutionary algorithm and the numerical flow simulation. The example optimization with the two objective functions, i.e. the aerodynamic drag and the pressure variation along the car demonstrates its feasibility. In this study, we optimized the relatively simple train shapes using the two objective functions. More practical shapes with more objective functions can be employed. In the future study, this method will be applied to the actual train design process.

References [1] M. Iida, T. Matsumura, K. Nakatani, T. Fukuda, T. Maeda, “Effective nose

shape for reducing tunnel sonic boom”, QR of RTRI, 38(4), 206-211, 1997. [2] T. Ogawa, K. Fujii, “Theoretical algorithm to design a train shape for

alleviating the booming noise at a tunnel exit”, Transactions of the Japan Society of Mechanical Engineers, 62(599), 2679-2686, 1996. (in Japanese)

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0.165

0.170

0.175

0.180

0.185

0.190

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Cp

Aerodynamic drag Cd

Shape of minimumpressure variation

Shape of minimumaerodynamic drag

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[3] K. Kikuchi, M. Iida, T. Fukuda, “Optimization of train nose shape for reducing micro-pressure wave radiated from tunnel exit”, Journal of Low Frequency Noise, Vibration and Active Control, 30(1), 1-19, 2011.

[4] A. Ido, “Method for estimating the reduced quantity of aerodynamic drag of trains”, Transaction of the Japan Society of Mechanical Engineers, 69(685), 2037-2043, 2003. (in Japanese)

[5] M. Suzuki, K. Nakade, A. Ido, “Countermeasures for reducing unsteady aerodynamic force acting on high-speed train in tunnel by use of modifications of train shapes”, Journal of Mechanical Systems for Transportation and Logistics, 2(1), 1-12, 2009.

[6] K. Tanemoto, M. Suzuki, H. Saito, T. Imai, “Wind tunnel tests on aerodynamic characteristics of train/vehicles in cross winds and protection fences”, RTRI Report, 18(9), 17-22, 2004. (in Japanese)

[7] K. Kikuchi, N. Yamauchi, M. Iida, M. Yanagizawa, “Aerodynamic optimization of nose configuration for reducing pressure variation at train passage”, Transaction of the Japan Society of Mechanical Engineers, 65(632), 1355-1361, 1998. (in Japanese)

[8] T. Kumano, S. Jeong, S. Obayashi, Y. Ito, K. Hatanaka, H. Morino, “Multidisciplinary design optimization of wing shapes for a small jet aircraft using Kriging model, AIAA paper, No.2006-0932, 2006.

[9] J.S. Tyll, J.A. Schetz, “Concurrent aerodynamic shape/cost design of magnetic levitation vehicles using MDO techniques”, AIAA paper, No.98-4935, 1998.

[10] A. Oyama, “Wing design using evolutionary algorithms”, Ph.D. dissertation, Tohoku University, 2000.

[11] G. Farin, “Curves and surfaces for computer aided geometric design”, Academic press, 1990.

[12] S. Obayashi, “New stage of CFD application –Numerical optimization”, Journal of the Japan Society of Mechanical Engineers, 105(999), 64-69, 2002.

[13] J.E. Baker, “Reducing bias and inefficiency in the selection algorithm”, Proceedings of the Second International Conference on Genetic Algorithms and Their Application, 14-21, 1987.

[14] C.M. Fonseca, P.J. Fleming, “Genetic algorithms for multiobjective optimization: Formulation, discussion and generalization”, Proceedings of the 5th International Conference on Genetic Algorithms, 416-423, 1993.

[15] D.E. Goldberg, “Genetic algorithms in search, optimization and machine learning”, Addison Wesley, 1989.

[16] L.J. Eshelman, J.D. Schaffer, “Real-coded genetic algorithms and interval schemata”, Proceedings of Foundations of Genetic Algorithms Workshop, 187-202, 1992.

[17] L.J. Eshelman, “The CHC adaptive search algorithm: How to have safe search when engaging in non-traditional genetic recombination”, in Foundations of genetic algorithms, Morgan Kaufmann Publishers, 265-283, 1991.

[18] B.S. Baldwin, H. Lomax, “Thin layer approximation and algebraic model for separated turbulent flows”, AIAA paper, No.78-257, 1978.

[19] M. Suzuki, K. Kuwahara, “Stratified flow past a bell-shaped shape”, Fluid Dynamics Research, 9(1-3), 1-18, 1992.

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[20] www.open-mpi.org [21] M. Suzuki, “Surface grid generation with a linkage to geometric generation”,

International Journal of Numerical Methods for Fluids, 17, 163-176, 1993. [22] S. Nakamura, M. Suzuki, “Noniterative three dimensional grid generation

using a parabolic-hyperbolic hybrid scheme”, AIAA paper, No.87-0277, 1987. [23] M. Suzuki, “Aerodynamic studies of the vibration of a train in a tunnel”, RIRI

Report, Special No.8, 1996. (in Japanese) [24] A. Ido, M. Iida, T. Maeda, “Wind tunnel tests for nose and tail of train”, RTRI

report, 7(7), 59-66, 1993. (in Japanese)