Multi-point airfoil shape optimization

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


Aerodynamic design is one of the main challenges of aircraft wing design. Optimization has long been considered as a means to solve this problem in a formal and general manner. The goal of this case study is to determine optimal geometry of 2D wing section (airfoil), which satisfies various geometrical and aerodynamic constraints and has the lowest possible drag.

In this study, a three-point optimization problem is considered, i.e. the objective function to be minimized is a weighted sum of drag coefficients for three different operating points. In particular, three Mach numbers are considered (0.70, 0.75 and 0.78) with appropriate fixed values of lift coefficients. Two geometrical constraints and two aerodynamic constraints are imposed. Geometry constraints are as follows:

  • Maximum profile thickness 
  • Profile tail thickness 

Aerodynamic constraints for 3 operating points are as follows:

  • Maximum lift coefficient for all the angles of attack, for low-speed operating point (Mach number equals to 0.2) should be larger or equal to 1.5;
  • Moment coefficient at zero lift for Mach number equals to 0.75 should be larger or equal to 0.125.


The main challenges of this problem are as follows:

  • High-dimensional geometry parameterization,
  • Big computational time of aerodynamic solver and its non-robustness with respect to geometry variations.

All these challenges are efficiently solved using pSeven!


In this study, VISTRAN & MULTIVIS CFD solver[1] has been used. The goal is to find the airfoil that would provide minimum drag coefficient, satisfying two geometrical and two aerodynamic constraints at the same time. The objective's structure took into account three different operating points, i.e. different Mach numbers. So, processing of a single airfoil actually means a series of calculations in VISTRAN in order to evaluate the objective and the four constraints. 

Low-dimensional and accurate airfoil shape description is one of the key ingredients for an efficient solution of the optimization problem. First, a general description of an airfoil by 58 points (x, y) along its contour, at prescribed x (chord-wise) positions, is constructed. This description cannot be directly used as a reasonable parameterization, since it does not take into account smoothness and other subtle geometric properties of an airfoil shape, and hence has much more degrees of freedom than needed. Therefore, Dimension Reduction tool is applied to reduce the dimensionality of this description. To this end, a database of approximately 200 real airfoils is collected, and GT DR is applied to construct a smooth mapping of a few parameters to the 58-dimensional space of airfoil descriptions, so as to closely approximate the 200 database shapes. Finally, just ten parameters are determined to represent the complex geometry of transonic airfoil shape.

pSeven has been used to wrap VISTRAN CFD solver and to automate the whole simulation process – from airfoil geometry generation up to post-processing of VISTRAN results. A bird's-eye view of the constructed workflow is given below.

It is important to note that calculating the maximal lift coefficient for all the angles of attack implies search of a maximum of a function. This task was solved with pSeven by creating an additional optimization block inside of the main optimization workflow. This is a unique solution currently available in pSeven only.

In order to minimize the computation time and to search for global optimum pSeven’s surrogate-based optimization method has been used with the maximum allowed number of CFD solver calls equal to 600.


At the end, pSeven made only 390 evaluations to converge to an optimal solution. This would not be possible using conventional gradient-based or stochastic optimization methods. Optimal airfoil geometry for the considered optimization problem setup is presented below:


Some characteristics of optimal solution are given in the table below:

[1] VISTRAN CFD solver has been developed S.V.Lyapunov, A.V.Volkov in TsAGI.