July 9, 2019

Bi-level airfoil optimization

Industry: Aerospace | Product: pSeven | Company: TsAGI (Central Aerohydrodynamic Institute)


The helicopter horizontal tail stabilizer provides a back-reaction in pitch. In normal mode, its lifting force Cγ increases with the angle of attack. However, at critical angles, the airfoil can stall, and lifting force will decrease. This effect limits the working range of fixed stabilizer. In this study, the goal is to find the airfoil shape to provide a maximum angle of attack range with a positive derivative of Cγ over \(α\).


Bi-level (nested) optimization approach is required in order to define the maximum angle of attack range at each iteration of airfoil geometry optimization. The logic of the study imposes special requirements for automation platform as well as for optimization capabilities.


We perform bi-level global optimization of airfoil geometry to determine the one with a maximal range of angles of attack \(Δα\) with a positive derivative of the lift coefficient Cγ over \(α\).

For demo purposes, the airfoil is described by a well-known simple NACA 4digit parametrization, 3 parameters:

  • maximum camber \(m\)
  • maximum camber position \(p\)
  • maximum thickness \(t\)

To determine the lift coefficient, a 2D steady flow CFD simulation was conducted in FloEFD. Total mesh size was about 190000 cells. A special structure of the mesh in FloEFD (created with mesh areas setup technique) allows using the same model for a wide range of angles of attack, including nearly-critical. Flow velocity is fixed to 0.25 Mach (Re ~ 106).


As expected, the main inaccuracy appears near the critical angles of attack, where the flow is unstable and the uncontrolled turbulence arises. This effect results in additional noise in responses (the values of critical angles, which are to be found).

The optimization procedure consists of two levels. The bi-level optimization is an hierarchical problem, where one optimization loop is embedded (nested) within another. It imposes the specific requirements on the workflow engine:

  • Triggering of the optimization process by outer signal
  • Changing the lower-level optimization setup during the runtime (at each iteration of upped level optimization)
  • Updating the lower-level model if needed
  • Combining several lower-level models
  • Caching the previously obtained data


General optimization scheme

In pSeven, the Optimizer block is configurable even during runtime, and the workflow engine allows to create hierarchical loops, supporting nested optimization. It provides a special direct integration blocks to modify CAD geometry and CFD simulation settings in FloEFD, as well as in other CFD simulation tools, and to run the solver to create automated workflows.

airfoil 3-1

airfoil 3-2

On the lower level, we searched for maximum and minimum of the lift coefficient (2 separate optimization problems!) as a function of angles of attack for a given airfoil, and determined the range of angles of attack with positive dCγ/dα. Two approaches were applied for comparison: “direct”, or optimization based directly on CFD simulation, and the optimization based on predictive (approximation) model, built in advance, to speed up the process.

A predictive  model of Cγ as a function of α was built using pSeven for a given airfoil using a fixed set of α values. Such models were used as a replacement for the computational model in lower-level cycles. This approach allows to save up to 20% of simulation run, since we use the predictive model to locate both maximum and minimum of Cγ.


Optimization history for AoA range 

On the upper level, the geometry of the airfoil was varied to find one with the widest range of angles of attack. Surrogate-based optimization (SBO) algorithm of pSeven was used, selected automatically by SmartSelection.


Workflow in pSeven


As a result, the optimal profile was discovered with only 40 iterations at the upper optimization level, i.e. 40 airfoil geometries tested, and with 19 lower-level simulations for each airfoil (760 FloEFD runs in total).

The range of angles of attack Δα with a positive derivative of the lift coefficient over α for this airfoil was increased by 6 degrees compared to initial NACA 2412 airfoil.

This workflow can be easily extended to a more sophisticated parameterization of the airfoil, as well as for any other problem where nested optimization is required.


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