# 1 Airfoil parametrization

The numerical search for the optimum shape of an airfoil is of great interest for aircraft and turbine designers. A typical aerodynamic optimization procedure often consists four stages:^{[1]}

- Shape parametrization/control
- Mesh generation/deformation
- Flow solution
- Optimization

However, the whole process is computationally intense and requires a large amount of individual flow filed simulations resulting in very long CPU runtime. In order to optimize an airfoil shape using several of algorithms, the first step is to define the airfoil geometry, or in another words, to describe the airfoil with some suitable parameters which control its shape.

Shape parametrization method can be categorized as either **constructive** or **deformative**. Constructive methods represent airfoil based on series of parameters while deformative methods take an existing airfoil then deform it to create the new shape. Masters^{[2]} presented a comprehensive review of constructive and deformative methods with a detailed comparison of seven airfoil parametrization methods.

Many parametrization techniques are in use nowadays , some may have better shape control while others may show faster convergence rate. Anyway, there are several objectives that any parametrization technique should satisfy:^{[3]}

- Minimize the number of degrees of freedom.
- Be able to represent a wide range of existing airfoils.
- Parameters should be simple to formulate and impose.
- The parametrization should result in effective and efficient optimization.

Earlier, Hick^{[4]} represented the airfoil with a set of **shape functions**, and then a class of parametrization techniques has been presented based on the linear combination of shape functions. Kuruvila^{[5]} proposed an polynomial function to parametrize upper and lower curves of an airfoil using orthogonal shape functions. Onur^{[6]} developed a Fourier series expansion of the Wagner functions which can represent a large class of airfoils. Another approach using combination of shape functions is proposed by **Sobieczky**^{[7]}; many known classical as well as more sophisticated airfoils can be duplicated by this parametric section(**PARSEC**) method with good accuracy and the key concept is that design parameters should be based on the aerodynamic performance. Recently, some other modified PARSEC methods have also been proposed which provide more flexibility in defining the airfoil geometry compared with that of the conventional PARSEC method^{[8]}.**B-Spline** method is also an effective and accurate representation for shape optimization of airfoil. However, the drawback of the regular B-Spline representation is its inability to represent a conic sections accurately. Therefore, a special form of B-Spline, nonuniform rational B-Spline(**NURBS**) is adapted. Lepine^{[9]} investigates the performance of NURBS for the aerodynamic design of a wing, and this method has been shown to be capable of accurately representing a large family of airfoils with 13 or fewer control points.**Rogalsky**^{[10]} proposes a new **Bezier-PARSEC** (BP) parametrization method which accelerates the convergence of aerodynamic design using Differential Evolution. The results show the number of function evaluations required by Differential Evolution has been reduced significantly.

Airfoil consists of two curves namely camber line and thickness distribution.The upper and lower curves of the airfoil can be obtained by adding and subtracting the **thickness** distribution to and from the **camber** line. So the airfoil can be constructed by camber and thickness curves parametrization or directly built by upper and lower curves. Next, three common parametrization method will be introduced: PARSEC method completes upper and lower curves directly while B-Spline and Bezier-PARSEC method use camber-thickness definition, on which the aerodynamic properties of an airfoil are more directly dependent.

# 2 Bezier Parametrization

## 2.1 Bezier curve

A Bezier curve(B-Spline) is controlled with the help of its control points in a plane. It passes through initial and final control points but not each intermediate control point which defines the shape of the airfoil.

A $n$ degree Bezier curve is defined by $n+1$ control points:

$$

P(t)=\sum_{i=0}^{n} P_{i} B_{i}^{n}(t)

\tag{2-1}$$

where $P_{i}$ are control points, $t\in[0,1]$ and $B_{i}^{n}(t)$ are $n$ basis functions:

$$

B_{i}^{n}=\frac{n !}{i !(n-i) !}(1-t)^{n-i} t^{i}

\tag{2-2}$$

For example, a three-degree B-Spline having $n=3$ :

$$

\begin{aligned} B_{0}^{3} &=(1-t)^{3} \\ B_{1}^{3} &=3 t(1-t)^{2} \\ B_{2}^{3} &=3 t^{2}(1-t) \\ B_{i}^{n} &=t^{3} \end{aligned}

\tag{2-3}$$

There are four control points $(x_i,y_i) , i=0,1,2,3$.

So this curve can be described by the equations below:

$$

\left\{\begin{array}{l}{x(t)=x_{0}(1-t)^{3}+3 x_{1} t(1-t)^{2}+3 x_{2} t^{2}(1-t)+x_{3} t^{3}} \\ {y(t)=y_{0}(1-t)^{3}+3 y_{1} t(1-t)^{2}+3 y_{2} t^{2}(1-t)+y_{3} t^{3}}\end{array}\right.

\tag{2-4}$$

## 2.2 Advantages

This parametrization has many properties that are

attractive for aerodynamic design.

- The end points are automatically fixed at the two end vertices.
- At an endpoint, the curve is tangent to

the vector between that endpoint and the closest control point, making it simple to join curves with first order continuity. - The curve always lies within the convex figure defined by the extreme points of the polygon.
- The curve is nth order continuous throughout and never oscillates wildly away from its defining control points.

## 2.3 Disadvantages

Higher computational cost comparing to the shape functions method

^{[11]}Discontinuous second derivative at the camber and thickness crests which can result in inaccurate flow field simulations.

# 3 PARSEC

Parsec method is a effective method of airfoil parametrization. 11 basic Parameters are used to completely define the airfoil shape. An airfoil shape is defined by basic geometric parameters instead of the coefficients of shape functions themselves.

These parameters are :

- Leading edge radius $r_{LE}$
- upper crest location $X_{UP},Z_{UP}$
- lower crest location $X_{LO},Z_{LO}$
- upper and lower curvature $Z_{XXUP},Z_{XXLO}$
- trailing edge coordinate $Z_{TE}$
- trailing edge direction $\alpha_{TE}$
- trailing edge wedge angle $\beta_{TE}$
- trailing edge thickness $\Delta_{TE}$

This method describes the airfoil shape by a linear combination of shape functions :

$$

Z_{k}=\sum_{n=1}^{6} a_{n,k}X_{k}^{n-\frac{1}{2}}

\tag{3-1}$$

The subscript $k$ will take values 1 and 2 for upper and lower surface respectively, and coefficients $a_{n,k}$ can be solved with the help of defined geometric parameters.

Parsec method has become one of the most robust method for airfoil parametrization due to its ability to avoid a more arbitrary and less physically based airfoil geometry. Della^{[12]}presented an optimization technique that deals with PARSEC parametrization method and Nash equilibrium genetic algorithm, Grey^{[13]}compared PARSEC with another airfoil shape parametrization and studied the active subspaces present in transonic lift and drag coefficients.## 3.1 Advantages

- Oyama
^{[14]}compared the performance of B-Spline method, orthogonal shape functions and Sobieczky method. The results show EA(Evolutionary Algorithms) using Sobieczky method has reached to the airfoil shape of best performance while EA using B-Spline curves is slightly less than Sobieczky. - This parametrization gives the design space wide enough, which helps finding a global optimum.
- The ability to minimize an optimization phenomena known as epistasis( the nonlinear manner in which the objective function is dependent on the design parameters)

## 3.2 Disadvantages

- Doesn't provide sufficient control over the trailing edge shape where important flow phenomenon can take place.(Change difficulty at the trailing edge)
^{[15]}

# 4 Bezier-Parsec Parametrization

This method combines Bezier and PARSEC parametrization techniques, which have advantages of both technique. This parametrization uses four Bezier curves to define the leading edge and trailing edge of the camber line and thickness distribution, and then solves the coefficients of Bezier curves using PARSEC parameters.

For example, **BP 3434** parametrization($ijkl$ represents the order of leading and tailing edge of thickness curve and camber curve) uses third degree Bezier curves for two leading edges curves and fourth degree Bezier curves for trailing edges to define the airfoil^{[16]}^{[17]}.

## 4.1 Advantages

- Increased robustness and convergence speed for aerodynamic optimization using genetic algorithms
- Fit the known shape of a wide range of existing airfoil profiles

## 4.2 Disadvantages

- The Bezier control parameters in Bezier-PARSEC parametrization are hard to control

# 5 References

- 1.MASTERS D A, TAYLOR N J, RENDALL T. Review of Aerofoil Parametrisation Methods for Aerodynamic Shape Optimisation[C]//53rd AIAA Aerospace Sciences Meeting. Kissimmee, Florida: American Institute of Aeronautics and Astronautics, 2015. ↩
- 2.MASTERS D A, TAYLOR N J, RENDALL T C S. Geometric Comparison of Aerofoil Shape Parametrization Methods[J]. AIAA Journal, 2017, 55(5): 1575–1589 ↩
- 3.P. SALUNKE N, AHAMAD R. A. J, CHANNIWALA S A. Airfoil Parametrization Techniques: A Review[J]. American Journal of Mechanical Engineering, 2014, 2(4): 99–102. ↩
- 4.HICKS R, HENNE P. Wing design by numerical optimization[C]//Aircraft Systems and Technology Meeting. Seattle,WA,U.S.A.: American Institute of Aeronautics and Astronautics, 1977. ↩
- 5.KURUVILA G, TA’ASAN S, SALAS M. Airfoil design and optimization by the one-shot method[C]//33rd Aerospace Sciences Meeting and Exhibit. Reno,NV,U.S.A.: American Institute of Aeronautics and Astronautics, 1995. ↩
- 6.ONUR N. Application of Wagner Functions in Symmetrical Airfoil Design[J]. Journal of Aircraft, 1997, 34(2): 259–261. ↩
- 7.LI P, SEEBASS R, SOBIECZKY H. Manual aerodynamic optimization of an oblique flying wing[C]//36th AIAA Aerospace Sciences Meeting and Exhibit. Reno,NV,U.S.A.: American Institute of Aeronautics and Astronautics, 1998. ↩
- 8.SHAHROKHI A, JAHANGIRIAN A. Airfoil shape parametrization for optimum Navier–Stokes design with genetic algorithm[J]. Aerospace Science and Technology, 2007, 11(6): 443–450. ↩
- 9.LEPINE J, GUIBAULT F, TREPANIER J-Y. Optimized Nonuniform Rational B-Spline Geometrical Representation for Aerodynamic Design of Wings[J]. AIAA Journal, 2001, 39(11): 2033–2041. ↩
- 10.ROGALSKY T P. Acceleration of differential evolution for aerodynamic design[M], 2004. DELLA VECCHIA P, DANIELE E, DʼAMATO E. An airfoil shape optimization technique coupling PARSEC parametrization and evolutionary algorithm[J]. Aerospace Science and Technology, 2014, 32(1): 103–110. ↩
- 11.SONG W, KEANE A. A Study of Shape Parametrisation Methods for Airfoil Optimisation[C]//10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Albany, New York: American Institute of Aeronautics and Astronautics, 2004. ↩
- 12.DELLA VECCHIA P, DANIELE E, DʼAMATO E. An airfoil shape optimization technique coupling PARSEC parametrization and evolutionary algorithm[J]. Aerospace Science and Technology, 2014, 32(1): 103–110. ↩
- 13.GREY Z J, CONSTANTINE P G. Active subspaces of airfoil shape parametrizations[J]. arXiv:1702.02909 [math], 2017. ↩
- 14.OYAMA A, OBAYASHI S, NAKAHASHI K, 等. Fractional Factorial Design of Genetic Coding for Aerodynamic Optimization[J]. 1999. ↩
- 15.SHAHROKHI A, JAHANGIRIAN A, FOULADI N. Navier- Stokes Optimization Using Genetic Algorithm and a Flexible Parametric Airfoil Method[J]. 2006: 4. ↩
- 16.DERKSEN R W, ROGALSKY T. Optimum aerofoil parametrization for aerodynamic design[C]//Algarve, Portugal: 2009: 197–206. ↩
- 17.DERKSEN R W, ROGALSKY T. Bezier-PARSEC: An optimized aerofoil parameterization for design[J]. Advances in Engineering Software, 2010, 41(7–8): 923–930. ↩