Bullet Train Aerodynamic Optimization
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This project aimed to determine the optimal nose geometry of the leading and trailing cars of a bullet train. The metric used to determine this optimal geometry is the coefficients of drag for each parameterized design.
Modeling Details
In order to parameterize the design, four design variables were chosen. Angle 1 (α) and angle 2 (β) determine the head shape of the train, while angle 3 (γ) determines the bluntness of the nose. Displayed in Figures 1 and 2. The last design variable was the number of cars per train (N). Parameters included the power of the train, as well as general car dimensions, modeled after the N700 Series Shinkansen.
Figure 1: Angle 1 (α) and Angle 2 (β)
Figure 2: Angle 3 (γ)
To perform the optimization, a CAD model of the train was first generated in Solidworks. The model consisted of an identical leading and trailing car, uniform center cars, and Jacob’s Bogies connecting the cars. The model is shown in Figures 3 - 5.
Figure 3: Car Lengths
Figure 4: Cross Section Dimensions
Figure 5: Jacob’s Bogie Modeled
Metamodeling
In order to create a metamodel, a set of boundaries was set for each variable. The upper bound and lower bound of each variable is shown in Table 1.
With the information above, Latin Hypercube Sampling was used to create 12 entries for each variable. A 12 x 4 array was then created. The original CAD design was then adjusted according to the combinations of variables generated by the Latin Hypercube. For each model, a computational fluid dynamics model was performed using ANSYS CFX. In this simulation the mesh settings were chosen as tetrahedral, with a target element quantity of 5*10^-2, a standard resolution of 2, and a growth rate of 1.2. These mesh settings were chosen with simulation run-time in mind, keeping it low due to the large quantity of simulations run.
Figure 6: Boundary Conditions for Simulation
For the simulation setup, an inlet speed of 75 m/s was set with an outlet pressure of 0 Pa. The fluid model was set to isothermal Air at 25 C and 1 atm. The turbulence was set to a shear stress transport model. The boundary details of the train are modeled as a smooth no slip wall. The figures below are pressure contours with velocity streamlines of trains with different variables
Figure 7: α =15°, β=15°, γ=20°, Number of Cars = 3
Figure 8: α =50°, β=50°, γ=10°, Number of Cars = 3
Figure 9: α =50°, β=30°, γ=20°, Number of Cars = 7
The Coefficient of Drag (Cd ) is calculated with the following expression:
Which is the applied form of the following equation:
Where (Fd) is the drag force, rho is the mass density of the fluid, u is the flow speed of the object relative to the fluid, and A is the reference area.
The maximum velocity of the train can be calculated using the following equation due to the following assumptions: The train is a maglev train, meaning no energy is lost to friction with the tracks. At maximum velocity, the acceleration is zero so the force of the train is equal to the force of the drag acting on the train.
Derived from the following equations:
Model Constraints and Conditions
In order for the optimization results of this model to interact with the other optimizations, a constraint of maximum velocity has been set, and is currently at 83 m/s for this model.
Optimization
Simulation findings:
Due to the limited number of data points of this model, I chose to fit a linear regression model to the data to predict a coefficient of drag given a set of variables. The linear regression surrogate model had the following statistics:
Number of Observations: 12
Error degrees of Freedom: 7
Root Mean Squared Error: 0.0998
R-Squared: 0.916
Adjusted R-Squared: 0.867
F-statistic vs. constant model: 19
P-value: 0.000735
The linear regression model demonstrates a strong fit to the data, with an R² of 0.916 indicating that approximately 91.6% of the variance in the coefficient of drag is explained by the predictors. The model is statistically significant, as evidenced by an F-statistic of 19 and a p-value of 0.000735, confirming that it outperforms a constant (intercept-only) model. The adjusted R² of 0.867 accounts for the small sample size (12 observations) and suggests that the model remains robust despite this limitation. The root mean squared error (RMSE) of 0.0998 indicates that the average prediction error is approximately 0.1 units in the drag coefficient.
Given the function output by the linear regression surrogate model, a genetic algorithm was used to optimize the train's aerodynamic design by minimizing an objective function while satisfying constraints on velocity and design variables. It works by generating a population of candidate solutions, evaluating their fitness based on the objective function, and iteratively improving the solutions through processes like selection, crossover, and mutation. The algorithm handles nonlinear constraints, such as ensuring the train's velocity does not exceed 83 m/s, and incorporates integer constraints for one design variable. Using a Genetic algorithm was a good choice because it is robust in solving complex optimization problems with non-differentiable objectives and constraints, avoids getting stuck in local optima, and efficiently explores the global search space for feasible solutions.
Optimized Train Geometry
References
"N700 Series Shinkansen." Wikipedia, Wikimedia Foundation, 4 May 2024, https://en.wikipedia.org/wiki/N700_Series_Shinkansen. Schito, Paolo, et al.
"Numerical Analysis of the Effect of Different Nose Shapes on Train Aerodynamic Performance." Preprints, 2 May 2025, http://www.preprints.org/manuscript/202407.2258/v1. Preprint.
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