## Development of a software for multipoint laminar airfoil optimization

A multipoint single objective optimization procedure for laminar airfoil design has been implemented. The analysis block integrates the geometric module with a fluid dynamic solution based on a coupled panel and integral boundary layer method. The decision making criterion is the Nelder-Mead algorithm based on the simplex method.

### Parametric geometry generation

The geometry is generated using a NURBS curve. A NURBS (Non–Uniform Rational B–Splines) is a generalization of the B-spline formulation. It is the combination of a set of piecewise rational functions with control points and associated weights . The defining equation is:

where is the curve parameter and are the B-spline basis functions defined as

for and

for .

See reference [1] for further details on NURBS formulation.

The control points of the curve are the variables of the optimization problem. The geometric constraints are implemented through opportune limitations of the variables space and by penalties in the objective function.

### Fluid dynamic solution

The aerodynamic analysis is performed using the well–known XFoil code developed by prof. Drela at MIT [2]. It is a coupled panel and integral boundary layer code in which a simplified transition criterion, based on a formulation of the semiempirical method, is also available. The code is driven to automatically generate, every iteration, an aerodynamic solution, in terms of coefficients and pressure distribution, for each design point.

### The optimization criterion

The optimization criterion is the Nelder-Mead algorithm [3]. It is a direct search algorithm (no cost function derivative is used) based on the update of a polyhedron (simplex) containing point in a dimensional space where is the number of design variables. The simplex is moved toward the optimum applying the *Reflection*, *Contraction* and *Expansion* operators (figure 1).

**Fig. 1: The Simplex operators**

In order to perform a single objective optimization using multipoint design conditions, the objective function is defined as a weighted combination of the design points targets according to:

### Example of application

A laminar airfoil with a wide range of operative lift conditions has been optimized at a Reynolds number equal to 4 million. The geometry was generated by a control polygon composed of 13 points, each defined by its X and Y coordinates. Not all points coordinates were adopted as design variables since some of them had to be constrained. The positions of leading and trailing edge was imposed as well as the leading edge tangent and the angle between the trailing edge tangents. The target was to obtain significant laminar flow in a lift coefficient range between 0.6 and 1.4. In order to guarantee acceptable performance also in case of laminar flow loss, a design target with fully turbulent flow was also added. Three design points were then adopted:

- Design point 1: D
_{t}= C_{d}at C_{l}= 1.4 (free transition) - Design point 2: D
_{t}= C_{d}at C_{l}= 0.6 (free transition) - Design point 3: D
_{t}= C_{d}at C_{l}= 1.2 (fully turbulent)

The objective function was defined as follows:

The number of optimization variables was 17. The minimum thickness constraints was 14% of the chord and has been imposed as a penalty on the objective function. The optimization was started from the solution of the NACA 4418. The following video shows the convergence history of the optimization cycle.

**Fig. 2: Video of the optimization convergence history**

### Conclusions

A single objective multipoint laminar airfoil optimization procedure has been developed by integrating a NURBS geometry generation module and a coupled panel/integral boundary layer CFD solver. The design method had been tested on the optimization of an airfoil with operative lift coefficient ranging from 0.6 to 1.4. The procedure showed to be very robust and efficient. A reduction of more than 20% on the objective function value was obtained despite the introduction of acceptable performance requirement with fully turbulent flow.

### Bibliography

[1] Demetri Terzopoulos and Hong Qin, “*Dynamic NURBS with Geometric Constraints for Interactive Sculpting*”, Department of Computer Science, University of Toronto, Published in ACM Transactions on Graphics, 13(2), April 1994, 103–136.

[2] Mark Drela, “

*Xfoil: An Analysis and Design System for Low Reynolds Number Airfoils*”, MIT Dept. of Aeronautics and Astronautics, Cambridge, Massachusetts.

[3] J.A. Nelder and R. Mead, “

*A simplex method for function minimization*” Computer Journal, Volume 7, Issue 4, 1965, pp. 308-313.