Parameterisation Techniques

Parameterisation is essentially numerical coding of the design variables to obtain a general form of representing a particular shape. It is a key part of the optimisation process. Samareh109 states that a good parameterisation technique for multidisciplinary shape optimisation should be:

• automated

• provide consistent geometry changes across all disciplines • provide sensitivity derivatives

• fit into product development time

• have a direct connection to the CAD system for design • produce a compact and effective set of design variables

However, the initial main focus in this case will be the last point made. This is because the optimisation method desired is one that will not require grid regeneration until the end of the process. The coupled method has been used by a number of authors such as Le Pape36where for the elsA solver, an in-house analytic grid-generator was used to create a grid for each evaluation. However, in this project, the optimisation technique is to be decoupled from the high-fidelity CFD model and linked to it through a lower fidelity model.

H`ajek describes a number of techniques used to parameterise aerofoils in the light of aerody- namic optimisation110. This paper states that the choice of technique strongly influences the optimisation. Similarly, Castonguay and Nadarajah111 also performed a study on four different parameterisation method for the optimisation of aerofoils using an inverse method. Below are some of the methods described in both these papers:

Mesh point approach: Here eac mesh point can be independently moved. This is one of the easiest to implement and allows a lot of flexibility in the design but can lead to discontinuous geometries and possibly too many control variables111.

Joukowski transformation: Consists of transforming a circle in the complex plane via the transformation

z=ϵ+ 1

ϵ (1.5)

The circle should pass through the pointϵ = 1. The aerofoil shape is controlled by varying the centre of the circle. This method was especially advantageous in the past because it enabled the plane potential flow to be analytically solved.

Splines: These use piece-wise polynomial approximations of curves. B-splines use the same idea but are slightly more complex in that they are built on linear combinations of base functions. Non-Uniform Rational B-Splines (NURBS) are a further development. Ghaly and Mengistu112 show that B´ezier curves were used because of the simplicity of their implementation. However, they are global and hence a change in a single control point changes the shape of the entire blade and so the designer has less control over local regions. To accommodate this, B-splines were used. They work just like B´ezier curves but have more complex interpolation functions. These allow more local control, but are still not accurate in describing conic sections such as rounded leading edges etc. Therefore, non-ration B-splines (NURBS) were used since these allowed local control

and could also reproduce conic sections as accurately as required by industrial standards.

Hick-Henne ‘bump’ functions49: Modelling small or moderate perturbations of ‘baseline’ airfoil shapes used especially in inverse design. The perturbation is expressed as a linear super- position of ‘bump’ functions of the form

y=sint(xβ) (1.6)

where β is used to control the maximum of the bump function, located at x = π/21/β _{and t}

controls the width of the function (typicallyt= 3). It allows specific regions to be refined thereby reducing the number of variables and their nature ensures continuous gradients of the shapes. However, they are not orthogonal and hence are incapable of representing the full set of continous functions111.

PARSEC5, 111: used mainly for subsonic and transonic aerofoils. The aerofoil shape is expressed as an unknown linear combination of suitable base functions and selecting 12 important geometric characteristics as the control variables in such a way that the shape can be determined by solving a linear system with these variables. The variables are shown in Figure 1.7. The advantage of

Figure 1.7: PARSEC variables of an aerofoil for parameterisation5_.

PARSEC is that no baseline shape is needed, a wide range of aerofoil shapes can be generated, constraints are easy to impose and the impact of individual PARSEC design parameters on aero- dynamic properties can be easily predicted. The disadvantage though, is that it cannot cover as wide a range as spline curves.

B-splines, NURBS: For complex geometries, the use of B´ezier curves, splines and NURBS is quite popular, especially in the optimisation of compressor blades39, 66 mainly because they allow a greater degree of freedom with the use of fewer variables and can produce smooth, discon- tinuity free curves. However, the implementation of constraints is not as straightforward as using numerical representations of physical measurements. Philip Schneider gives a detailed explanation of how NURB curves work and his article.113 Joh114 used NURBS with 36 points to fully describe the M6 wing. This shows the versatility of the technique but also the high number of parameters used to describe a simple body. In Mengistu and Ghaly’s paper112, they describe a method for finding the minimum number of control points to represent turbomachinery blades using NURBS within a specified tolerance. An objective function to be minimised is a representation of the error and the weights and control point coordinates are the parameters being optimised for. In this way, coordinate point data curves can be converted to NURB curves and vice versa. In the case of Mengistu and Ghaly112_{, simulated annealing (SA) was used to optimise for the minimum}

number of points.

In Samad39_{, cubic spline interpolation was used to define the blade (i.e. a separate cubic spline} exists for each interval - these are popular because they produce smooth curves and are easy to implement). In total 20 variables were brought down to 6 variables.

Le Pape36 _{used Bezi´er curves to parameterise all the distributions of the variables in rotor opti-} misation except for the aerofoils.

The use of splines. B´ezier curves and NURBS allow localised control and reduced number of variables and the ability to represent continuous curves as well as discontinuities in the geometry. However, the link between the values and the design variables are not intuitive to the user. Kulfan and Bussoletti115 also carried out some work on parameterisation. They divided the parameterisation into two classes, one for the shape function such as aerofoils etc. and the other for body cross-sections such as fuselages, nacelles etc. For an aerofoil, the leading edge and trailing edge are specified and the curve in between these extremes is what is modified. Berstein polyno- mials were used to do this. This technique was adapted to parameterise aerofoils and wings116. Vanderplaat also describes a parameterisation method in the appendix of his paper3 where the shape of an aerofoil is defined with an equation:

t=δ(A√x+a1x+a2x2+...anxn )

(1.7) where t is the thickness of the aerofoil andδ is the thickness to chord ratio for the initial aerofoil. The square root term yields a parabolic LE term. The coefficients A,a1,a2,...,anare the variables

perturbed by the optimisation function.

Other parameterisation methods include Fourier series, piece-wise polynomial, and orthogonal polynomial. NACA has in effect a parameterisation system of 4, 5 and 6-digit sections. In gen- eral, most helicopter optimisation problems rarely exceed 10 - 30 design variables17. The aim in terms of parameterisation, would be to maintain a low number of design parameters to optimise, while giving the designer as much flexibility in the specific design variables required as possible.