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Airfoil Optimization using Design-by-Morphing

Haris Moazam Sheikh, Sangjoon Lee, Jinge Wang, Philip S. Marcus

TL;DR

This paper presents Design-by-Morphing (DbM), a universal parameterization that morphs homeomorphic baseline shapes to create an unconstrained design space for 2D airfoil optimization. By linearly combining baseline airfoil collocation vectors and allowing extrapolation with negative weights, DbM enables radical, high-fidelity shape variation without fixed geometric constraints. The authors demonstrate strong reconstruction of the UIUC airfoil database (≈99.87% with MAE $<1\%$) and show that extrapolation expands the design space beyond conventional parameterizations. Using NSGA-II with two objectives, $f_1=CLD_{max}$ and $f_2=\Delta\alpha$, they obtain a Pareto-front of 80 optimal airfoils, including diverse clusters, illustrating the method’s capacity to yield novel, high-performance shapes. The work suggests DbM’s broad applicability to design-space creation for fluid machinery and potentially 3D problems via extensions coupled with Bayesian optimization or other evaluators.

Abstract

We present Design-by-Morphing (DbM), a novel design methodology applicable to creating a search space for topology optimization of 2D airfoils. Most design techniques impose geometric constraints and sometimes designers' bias on the design space itself, thus restricting the novelty of the designs created, and only allowing for small local changes. We show that DbM methodology does not impose any such restrictions on the design space and allows for extrapolation from the search space, thus granting truly radical and large search space with a few design parameters. In comparison to other shape design methodologies, we apply DbM to create a search space for 2D airfoils. We optimize this airfoil shape design space for maximizing the lift-over-drag ratio, $CLD_{max}$, and stall angle tolerance, $Δα$. Using a bi-objective genetic algorithm to optimize the DbM space, it is found that we create a Pareto-front of radical airfoils exhibiting remarkable properties for both objectives.

Airfoil Optimization using Design-by-Morphing

TL;DR

This paper presents Design-by-Morphing (DbM), a universal parameterization that morphs homeomorphic baseline shapes to create an unconstrained design space for 2D airfoil optimization. By linearly combining baseline airfoil collocation vectors and allowing extrapolation with negative weights, DbM enables radical, high-fidelity shape variation without fixed geometric constraints. The authors demonstrate strong reconstruction of the UIUC airfoil database (≈99.87% with MAE ) and show that extrapolation expands the design space beyond conventional parameterizations. Using NSGA-II with two objectives, and , they obtain a Pareto-front of 80 optimal airfoils, including diverse clusters, illustrating the method’s capacity to yield novel, high-performance shapes. The work suggests DbM’s broad applicability to design-space creation for fluid machinery and potentially 3D problems via extensions coupled with Bayesian optimization or other evaluators.

Abstract

We present Design-by-Morphing (DbM), a novel design methodology applicable to creating a search space for topology optimization of 2D airfoils. Most design techniques impose geometric constraints and sometimes designers' bias on the design space itself, thus restricting the novelty of the designs created, and only allowing for small local changes. We show that DbM methodology does not impose any such restrictions on the design space and allows for extrapolation from the search space, thus granting truly radical and large search space with a few design parameters. In comparison to other shape design methodologies, we apply DbM to create a search space for 2D airfoils. We optimize this airfoil shape design space for maximizing the lift-over-drag ratio, , and stall angle tolerance, . Using a bi-objective genetic algorithm to optimize the DbM space, it is found that we create a Pareto-front of radical airfoils exhibiting remarkable properties for both objectives.
Paper Structure (16 sections, 8 equations, 17 figures, 3 tables)

This paper contains 16 sections, 8 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Design-by-Morphing
  3. Baseline Shapes and Morphing
  4. Intersection Control
  5. Selection of Baseline Shapes
  6. Airfoil Design Capacity Test
  7. Optimization Methodology
  8. Airfoil Evaluation
  9. Optimization
  10. Results
  11. Discussion
  12. Conclusion
  13. Aerodynamic Optimization Objectives
  14. Baseline Airfoil Shapes and Evaluation
  15. Optimization Test Functions and Validation
  16. ...and 1 more sections

Figures (17)

  • Figure 1: An example of DbM. The coordinates of the baseline shapes are weighted, summed, and normalized to form the coordinates of a morphed shape.
  • Figure 2: Application of DbM to 2D airfoils. Column 1 shows the baseline shapes. Column 2 depicts the elements of the collocation vectors of the baseline shapes plotted as a function of the index $i$ of the collocation vector. Column 3 shows the weighted elements of the collocation vector plotted as a function of the index $i$ of the collocation vector. Column 4 shows the resultant collocation vector of the morphed shape and the morphed shape itself.
  • Figure 3: Conditioning for intersection removal. (\ref{['fig:DbM-intersect_a']}) Intersections are detected; (\ref{['fig:DbM-intersect_b']}) A blown-up image of one intersection, with the shape coordinates direction depicted by arrows; (\ref{['fig:DbM-intersect_c']}) Intersection removed by flipping the vector between the intersection; (\ref{['fig:DbM-intersect_d']}) The 'zero-area' removed by linear interpolation and then smoothed over, as shown by hatted y-coordinates.
  • Figure 4: Twenty-five baseline airfoil shapes chosen from the UIUC database uiuc_database. See Appendix \ref{['app:baseline']} for further details.
  • Figure 5: Geometric demonstration of mean absolute error (MAE) between target and reconstructed airfoil surfaces.
  • ...and 12 more figures