# Aerodynamic shape optimization

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Aerodynamic shape optimization, or aerodynamic design optimization consists in maximizing the performance of a given body (such as an airfoil or wing) by changing its shape. The aerodynamic performance is usually evaluated using computer fluid dynamics (CFD) and the optimization can be done using a number of algorithms. The process is iterative: It starts with a given shape and then changes that shape to improve the performance while satisfying the specified constraints. In this page we summarize the work our lab has done in aerodynamic shape optimization. This is not intended to be a comprehensive literature review; for such reviews, you can see the introductions of our papers and the works they cite.

## Example: From a circle to a supercritical airfoil

In the video below, we start from a circle and minimize the drag with respect to the shape while enforcing constraints on the area and chord (He et al., 2019). The process is fully automatic and the final result is the re-invention of a modern supercritical airfoil.

### The adjoint approach for computing derivatives

The key enabler in aerodynamic shape optimization is the combination of gradient-based optimization, which is necessary to handle the hundreds of shape variables involved, with an adjoint method that computes the required gradients efficiently. Note that the adjoint method itself is not an optimization strategy, it is just a way to compute the gradients and it is independent of the specific gradient-based optimization algorithm that is used. Gradient-based optimization requires the derivatives of the objective function (e.g., drag) and constraint functions (e.g., lift, moment) with respect to all the design variables (e.g., angle of attack, shape variables).

The adjoint method and our implementation is explained by (Mader et al., 2008). The overall approach is to selectively use automatic differentiation on parts of the CFD code to compute the partial derivative terms in the discrete adjoint equations. This paper described an initial adjoint method that worked well but was not very efficient. (Martins et al., 2016) briefly describe other versions of the adjoint implementation that are more efficiently and are currently used in our aerodynamic shape optimization studies. (Martins & Hwang, 2013) present a broader overview of methods for computing derivatives, putting the adjoint method into context.

The AIAA Aerodynamic Design Optimization Discussion Group developed a series of benchmark cases. (Lyu et al., 2015) solve the RANS-based wing optimization problem, try to find multiple local minima, and solve a number of related wing design optimization problems, including a multipoint optimization. The initial and optimized geometries and meshes are provided on this page. We have also created multipoint cases (Kenway & Martins, 2016) for the ADODG, and a full configuration case (Chen et al., 2016) where we look at the effect that trim has on the wing design. The video below show the single point aerodynamic shape optimization.

## Constraining buffet onset

Designs obtained from aerodynamic shape optimization at transonic conditions are likely to not satisfy buffet requirements. In this paper (Kenway & Martins, 2017), we develop a way to enforce buffet margin constraints and show the difference it makes in the optimal designs.

## Stability constraints and flying wing design

In this paper (Mader & Martins, 2013), we develop methods for making sure that optimized wings are longitudinally stable and analyse trade-offs between reverse camber and sweep in flying wing design8. We also do a more detailed study of the blended wing body in this paper (Lyu & Martins, 2014), where we perform RANS-based optimization considering a constraint on static margin.

## Morphing wings

In this paper (Lyu & Martins, 2015), we evaluate the aerodynamic performance gains for a hypothetical morphing technology using aerodynamic shape optimization, where the wing shape is allowed to change during flight.

## Aerostructural wing design

We have also developed the capability to optimize wings taking into account the flexibility by coupling the aerodynamic forces and structural displacements. A coupled adjoint approach (Kenway et al., 2014) enables us to do perform the simultaneous optimization of the wing shape and wing box structural sizing and perform wing aerostructural optimization trade studies (Kenway & Martins, 2014).

## Applications beyond aircraft design

Our approach to aerodynamic shape optimization has also been used in other applications, such as wind turbine blades (Dhert et al., 2017), and hydrofoils (Garg et al., 2015).

## Bibliography

1. Robust aerodynamic shape optimization—from a circle to an airfoil

X. He, J. Li, C. A. Mader, A. Yildirim, and J. R. R. A. Martins

Aerospace Science and Technology, 8748–61, 2019

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2. Aerodynamic Shape Optimization of Wind Turbine Blades Using a Reynolds-Averaged Navier–Stokes Model and an Adjoint Method

T. Dhert, T. Ashuri, and J. R. R. A. Martins

Wind Energy, 20(5):909–926, 2017

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3. Buffet-Onset Constraint Formulation for Aerodynamic Shape Optimization

G. K. W. Kenway, and J. R. R. A. Martins

AIAA Journal, 55(6):1930–1947, 2017

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4. Aerodynamic Shape Optimization of the Common Research Model Wing-Body-Tail Configuration

S. Chen, Z. Lyu, G. K. W. Kenway, and J. R. R. A. Martins

Journal of Aircraft, 53(1):276–293, 2016

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5. Multipoint Aerodynamic Shape Optimization Investigations of the Common Research Model Wing

G. K. W. Kenway, and J. R. R. A. Martins

AIAA Journal, 54(1):113–128, 2016

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6. Details
7. High-fidelity Hydrodynamic Shape Optimization of a 3-D Hydrofoil

N. Garg, G. K. W. Kenway, Z. Lyu, J. R. R. A. Martins, and Y. L. Young

Journal of Ship Research, 59(4):209–226, 2015

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8. Aerodynamic Shape Optimization of an Adaptive Morphing Trailing Edge Wing

Z. Lyu, and J. R. R. A. Martins

Journal of Aircraft, 52(6):1951–1970, 2015

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9. Aerodynamic Shape Optimization Investigations of the Common Research Model Wing Benchmark

Z. Lyu, G. K. W. Kenway, and J. R. R. A. Martins

AIAA Journal, 53(4):968–985, 2015

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10. Scalable Parallel Approach for High-Fidelity Steady-State Aeroelastic Analysis and Adjoint Derivative Computations

G. K. W. Kenway, G. J. Kennedy, and J. R. R. A. Martins

AIAA Journal, 52(5):935–951, 2014

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11. Multipoint High-Fidelity Aerostructural Optimization of a Transport Aircraft Configuration

G. K. W. Kenway, and J. R. R. A. Martins

Journal of Aircraft, 51(1):144–160, 2014

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12. Aerodynamic Design Optimization Studies of a Blended-Wing-Body Aircraft

Z. Lyu, and J. R. R. A. Martins

Journal of Aircraft, 51(5):1604–1617, 2014

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13. Stability-Constrained Aerodynamic Shape Optimization of Flying Wings

C. A. Mader, and J. R. R. A. Martins

Journal of Aircraft, 50(5):1431–1449, 2013

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14. Review and Unification of Methods for Computing Derivatives of Multidisciplinary Computational Models

J. R. R. A. Martins, and J. T. Hwang

AIAA Journal, 51(11):2582–2599, 2013

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15. ADjoint: An Approach for the Rapid Development of Discrete Adjoint Solvers

C. A. Mader, J. R. R. A. Martins, J. J. Alonso, and d. W. E. van

AIAA Journal, 46(4):863–873, 2008

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