Multi-objective Optimization in Sports Equipment Design Using Expected Hypervolume Improvement: A Discus and a Race Car Rear Wing

Technical Summary: Multi-objective Optimization in Sports Equipment Design Using Expected Hypervolume Improvement

Problem Statement
Sports equipment design frequently involves balancing multiple, often conflicting, objectives such as aerodynamic performance, structural integrity, and cost. These trade-offs result in a set of Pareto-optimal solutions rather than a single global optimum. A significant challenge in this field is that performance evaluations often rely on computationally expensive simulations, particularly for aerodynamic components in throwing events (e.g., discus) and motorsports (e.g., race car rear wings). Consequently, there is a critical need for optimization frameworks that can identify promising designs while minimizing the number of required evaluations. While previous studies have utilized Expected Improvement (EI) for single-objective problems, this study addresses the gap in applying efficient criteria to multi-objective sports equipment optimization.

Methodology
The authors propose a framework utilizing Expected Hypervolume Improvement (EHVI) as an acquisition function to guide the selection of design variables. The workflow integrates the following components:

1. Initial Sampling: Design variable sets are generated using Latin Hypercube Sampling (LHS).
2. Simulation & Analysis:
   • CAD Modeling: Parametric models are created for the specific equipment.
   • CFD/EFD: Computational Fluid Dynamics (ANSYS Fluent for discus; OpenFOAM with Large Eddy Simulation for the rear wing) or Experimental Fluid Dynamics are used to determine aerodynamic coefficients ($C_D$, $C_L$, $C_m$).
   • Trajectory Simulation: For the discus, flight trajectories are simulated (1,000 Monte Carlo runs) to evaluate flight distance and robustness.
3. Surrogate Modeling: Gaussian Process Regression (GPR) is constructed to predict the mean and variance of objective functions based on the design variables.
4. Optimization Loop:
   • The EHVI metric is calculated to quantify the expected improvement in the hypervolume dominated by the current Pareto front.
   • A Genetic Algorithm (GA) maximizes the EHVI to identify the most promising new design set ($x^*$).
   • The process iterates (CAD $\to$ CFD $\to$ GPR $\to$ EHVI maximization) until stopping criteria are met.
5. Case Studies:
   • Men's Discus: Four design variables (diameter, flat center diameter, thickness, metal rim radius) optimized for flight distance and robustness.
   • Race Car Rear Wing: Seven design variables (airfoil shapes at centerline and mid-span, angle of attack) optimized for drag coefficient ($C_D$), lift coefficient ($C_L$), and pitching moment coefficient ($C_m$).
6. Benchmarking: The method was first validated against a standard welded-beam design problem.

Key Results

• Benchmark Performance: In the welded-beam test problem, EHVI successfully identified 81% of the reference Pareto-optimal solutions (13 out of 16). However, the selected solutions were observed to cluster near the center of the objective space rather than covering the extremes.
• Discus Optimization: EHVI identified 5 out of 6 Pareto-optimal solutions (83%). Self-Organizing Map (SOM) analysis revealed that the radius of the metal rim ($r_m$) is the dominant design parameter for maximizing flight distance, likely due to its role in delaying flow separation and stall. Robustness dependence on design variables was found to be more complex.
• Rear Wing Optimization: EHVI identified 5 out of 12 Pareto-optimal solutions (42%). The angle of attack was identified as the primary driver for the trade-off between drag and downforce.
  • Minimum Drag Case: The wing inclination aligned with the rear-deck angle, utilizing a larger effective wing area.
  • Maximum Downforce Case: The wing was oriented at approximately 40 degrees relative to the rear deck, with a narrower geometry near the centerline.
• Limitations Observed: In both sports cases, the solutions selected by EHVI tended to cluster near the center of the objective space. The method showed reduced efficiency in identifying extreme Pareto-optimal solutions and ensuring sufficient diversity, particularly in the higher-dimensional rear-wing case (3 objectives, 7 variables) compared to the discus case (2 objectives, 4 variables).

Significance and Claims
The paper claims that EHVI is a viable criterion for efficiently selecting design variables in multi-objective sports equipment optimization, capable of identifying Pareto-optimal solutions and key physical parameters (such as rim radius and angle of attack). The study demonstrates the framework's applicability across diverse sporting contexts, from throwing events to motorsports.

However, the authors maintain a modest stance regarding the method's limitations. They explicitly state that while EHVI is effective, it currently exhibits a limited ability to identify extreme Pareto-optimal solutions and to ensure sufficient diversity among the selected designs, as the algorithm tends to favor solutions near the center of the objective space. The paper does not propose future applications or modifications but presents these findings as a demonstration of the current capabilities and constraints of the EHVI approach in this specific domain.