[Paper Review] Geodesic Convolutional Shape Optimization

Pierre Baqué, Edoardo Remelli+2 more|arXiv (Cornell University)|Feb 12, 2018

Advanced Numerical Analysis Techniques26 citations

TL;DR

This paper proposes a geodesic convolutional neural network (GCNN)-based surrogate model for fast and accurate aerodynamic shape optimization. By remeshing 3D surfaces using a poly-cube map, the method enables efficient GPU-accelerated inference and differentiable optimization, outperforming state-of-the-art Gaussian process methods by 5–20% on 2D airfoils and enabling optimization of complex 3D car-like shapes previously intractable with traditional parameterization-dependent approaches.

ABSTRACT

Aerodynamic shape optimization has many industrial applications. Existing methods, however, are so computationally demanding that typical engineering practices are to either simply try a limited number of hand-designed shapes or restrict oneself to shapes that can be parameterized using only few degrees of freedom. In this work, we introduce a new way to optimize complex shapes fast and accurately. To this end, we train Geodesic Convolutional Neural Networks to emulate a fluidynamics simulator. The key to making this approach practical is remeshing the original shape using a polycube map, which makes it possible to perform the computations on GPUs instead of CPUs. The neural net is then used to formulate an objective function that is differentiable with respect to the shape parameters, which can then be optimized using a gradient-based technique. This outperforms state- of-the-art methods by 5 to 20% for standard problems and, even more importantly, our approach applies to cases that previous methods cannot handle.

Motivation & Objective

• To overcome the computational bottleneck of traditional CFD-based aerodynamic shape optimization, which requires expensive re-simulation for each shape change.
• To enable gradient-based optimization of complex 3D shapes without relying on low-dimensional, hand-crafted parameterizations.
• To develop a surrogate model that generalizes across diverse shape geometries, including those not seen during training.
• To make deep learning-based shape optimization practical by enabling efficient GPU computation on arbitrary meshes via poly-cube remeshing.

Proposed method

• Train a Geodesic Convolutional Neural Network (GCNN) to predict aerodynamic forces (e.g., drag, pressure) from 3D surface meshes, using CFD simulations (e.g., Ansys Fluent) as ground truth.
• Remesh the original shape using a poly-cube map to transform irregular meshes into structured grids suitable for efficient GCNN operations on GPUs.
• Formulate the objective function (e.g., drag minimization) as a differentiable function of vertex positions by backpropagating through the trained GCNN.
• Perform gradient-based optimization (e.g., L-BFGS) on the GCNN-approximated objective to explore the shape space efficiently.
• Leverage transfer learning by fine-tuning the GCNN on a small set of real-world shapes (e.g., car CAD models) to improve generalization.
• Use a deformation function fC with 21 parameters to generate diverse training shapes, enabling generalization to unseen geometries.

Experimental results

Research questions

• RQ1 Can a GCNN surrogate model trained on synthetic, non-car-like shapes generalize to predict aerodynamic performance on real 3D car geometries?
• RQ2 To what extent does GCNN-based optimization outperform traditional Gaussian Process (Kriging) methods in terms of accuracy and convergence speed?
• RQ3 Can the proposed method optimize complex 3D shapes with high degrees of freedom that are infeasible for parameterization-based methods?
• RQ4 How effective is poly-cube remeshing in enabling GPU-accelerated GCNN inference on arbitrary 3D meshes for shape optimization?
• RQ5 Does fine-tuning the GCNN on a few real-world examples significantly improve performance on target geometries without retraining from scratch?

Key findings

• The GCNN-based method achieved 77.2% regression accuracy on synthetic test shapes (SYNT-TEST), outperforming standard CNNs and demonstrating robustness to distribution shift.
• On real car shapes (CARS-TEST), the GCNN achieved 51.5% accuracy after pre-training on synthetic data, improving to 70.3% after fine-tuning on real car examples.
• For 2D NACA airfoil optimization, the GCNN method reduced drag by 5–20% more than state-of-the-art GP-based approaches, demonstrating superior optimization performance.
• The GCNN-based optimization found a better drag-minimizing shape than GP methods with only 50 simulator calls, even though the network had never seen a car before.
• The method successfully optimized a 3D car-like shape under geometric constraints (e.g., passenger and engine compartments), achieving a 4.71 drag coefficient after 100 simulations—significantly better than the GP online method (12.56 drag at 100 calls).
• The use of poly-cube remeshing enabled efficient GCNN inference on GPUs, making the approach practical where standard mesh convolutions would be prohibitively slow.