Influence of parameterization method and optimization algorithm on airfoil optimization

In this study, the impact of parameterization methods and optimization algorithms on the optimization results of the RAE2822 airfoil is examined. Specifically, how variations in the number of design variables, spacing of control points, the scaling scale of the objective function, and line search step of different FFD parameterization methods affect the resistance convergence process are investigated. The results indicate that the number of design variables and the spacing of control points play a significant role in the speed and minimum value of resistance convergence. In contrast, the scaling scale of the objective function and the line search step has a relatively minor impact on resistance optimization.


Introduction
The essence of aerodynamic shape optimization is an optimization problem.Its basic elements are the optimization objective function, design variables, constraint function, and variable boundary.The selection or definition of design variables directly affects the optimization of the optimization problem, or it produces different optimization problems [1].According to whether the gradient method is used, the optimization problem is divided into gradient and non-gradient optimization methods [2][3][4][5].The definition of design variables in these two methods is the same.For aerodynamic shape optimization, design variables are formed by parameterization of the shape.Common parameterization methods include Hicks-Henne, B-spline, Class Function/ Shape Function Transformations (CST), Free-Form Deformation (FFD), etc. FFD method is applicable to 2 and 3D problems, has strong adaptability to optimized shape, and is widely used in aerodynamic optimization design.The FFD method takes the control points as the design variables of the optimization problem and realizes the control of the aerodynamic shape by establishing the mapping between the control points and the grid points.The position of the FFD control points affects the definition of the design variables.
Based on the SU2 suite [6], taking the two-dimensional RAE2822 airfoil as an example, this paper analyzes the influence of the FFD control point position on the optimization results.On the other hand, the optimization algorithm of the SU2 optimization framework uses the SLSQP algorithm in Python.When defining the optimization problem, involves setting the scale factor of the objective function and the step length of the line search, which also affect the optimization results, The objective of this paper is to study the performance of airfoil optimization under different FFD control point positions and optimization algorithm parameters.

Methodology
In this paper, the SU2 optimization framework is used to optimize the RAE2822 airfoil, which includes the flowfield solution module, the gradient solution module, the mesh deformation module, the geometric evaluation and gradient solution module and the optimization algorithm module based on Scipy, and the framework is shown as Fig. 1.The details of these modules are as follows:

CFD solver
The compressible RANS equations with the SA turbulence model control the flow.To handle the flow inviscid terms, the Jameson-Schmidt-Turkel scheme (2nd order in space) is employed, which offers an accurate yet efficient solution for the optimization problem addressed in this paper.The turbulence model is solved using a scalar upwind solver.The finite volume method is used to solve the governing equations.The Green-Gauss method is adopted to approximate the spatial gradients of the flow at the cell faces to determine the viscous fluxes.Time integration of the flow equations and turbulence model follows the Euler implicit scheme.The linear system is solved using FGMRES with ILU preconditioning.To compute the function sensitivity of flow variables, the discrete adjoint equations are solved using Algorithmic Differentiation.The original mesh used in this work is presented in Fig. 2 (a).

Mesh parameterization
For aerodynamics shape optimization problem, design variables are generally generated based on mesh coordinate parameterization method.FFD based on B-spline basis function is adopted as the parameterization method of optimization problem in this work and the relationship between global and parametric mesh coordinates is described as blew where B is the open uniform B-spline basis function and P denotes the control point of FFD (namely, design variable).When a parametric coordinate (u,v,w) is given, a global coordinate X(u,v,w) is determined and basic functions define the projection relationship of the two coordinates.Fig. 2(b) represents uniform and non-uniform FFD control points, respectively.

Mesh deformation
The operation of mesh deformation is performed on the design variables obtained from the meshcoordinated parameterization.In this process, linear elasticity equations are utilized to deform the volume mesh that surrounds the FFD control points, and the cell stiffness is adjusted based on the proximity to the nearest solid surface.

Optimizer
The SLSQP method employs Sequential Least SQuares Programming to minimize a function with multiple variables while satisfying a range of constraints, including bounds, equality, and inequality constraints.The SLSQP Optimization subroutine is the tool used to implement this method and was initially created by Dieter Kraft.[7].

Optimization problem description
The research presented in this paper is based on an optimization problem that aims to minimize the drag of the RAE2822 airfoil in viscous flow.The flow conditions include a free flow Mach number of 0.734 and a corresponding Reynolds number of 6.5E6.The optimization problem is formed by taking the drag coefficient of the airfoil as the objective function, with constraints including a fixed lift coefficient of 0.824, a minimum pitching moment coefficient of -0.092, and an initial value of the airfoil area not less than 0.07794.The design variables include the y-direction displacement of the FFD parameterization control points and the angle of attack.The optimization problem can be mathematically expressed as:   For the example of ndv=12, reducing the step size leads to smaller resistance.However, for the example of ndv=6, the resistance convergence result will be greater after reducing the step size.This indicates that the line search cannot affect the optimization results, and different line search steps will lead to the convergence of the optimization problem to different local optimal results.Figure 3(d) shows the optimization process of the scaling scale of the two objective functions in the case of FFD uniform control point with ndv=9.When the objective function is magnified by 10 times, the iterative steps resistance convergence are less for the FFD control point with ndv=9, but the optimal value of resistance is equivalent to the scale of the original objective function.The following describes the optimization results of ndv=9 parameterization.Translation: Fig. 4(a) and (b) show the Mach number distribution contour plots before and after optimization, respectively.It can be seen that there is a shock wave on the upper surface of the airfoil before optimization, while the optimized airfoil avoids the generation of the shock wave.Fig. 5 shows the pressure distribution and airfoil before and after optimization.From the pressure distribution point of view, the optimization result eliminates the shock wave at the 0.6c position on the upper surface of the airfoil.Compared with the pressure distribution of the initial shape, the optimized shape does not form a strong suction peak at the leading edge.The optimized result increases the forward loading and reduces the rear loading compared to the initial shape, indicating that the airfoil will produce a head-up trend to reduce drag in the optimization process, and therefore the optimal value appears at the feasible domain boundary of the optimization problem.From the optimized airfoil, it can be seen that the concave shape appears on the lower surface of the leading edge, and the curvature changes significantly, which is the reason for producing the forward loading.The aft part of the airfoil protrudes more than the original shape, resulting in a decrease in the effect of rear loading.The above shows that the optimization trend of the aerodynamic shape of the airfoil may not have large curvature changes along the entire chord, and the curvature changes may only occur locally (generally at the leading and trailing edges for twodimensional problems).Therefore, to achieve effective control of the local shape and improve the drag reduction effect, more control points are needed.Next, the moment constraint is examined.In previous iterations, the moment exhibits oscillations, and some points violate the constraint with a moment Cm less than -0.092.However, in subsequent iterations, no points violate the moment constraint, and the majority of points reach a moment C m of -0.092.At the final convergence point, the moment C m satisfies the equality constraint, demonstrating that in the optimization process of constant lift and drag reduction, without the moment constraint, the airfoil would experience a larger head-down moment.

Conclusion
In this paper, the aerodynamic shape optimization design of the RAE2822 airfoil with constant lift and drag reduction is carried out through the SU2 optimization framework.The influence of the number of design variables, the number of control points spacing design variables and the optimal design of control point spacing for different FFD parameterization methods are studied, which are mainly reflected in the descending speed and minimum value of drag convergence.In addition, the influence of the scaling scale of the objective function and the line search step on the resistance convergence process are also analyzed.The research results indicate that these two factors have a relatively minor impact on optimization results.

FMIA- 2023
Journal of Physics: Conference Series 2599 (2023) 012005 IOP Publishing doi:10.1088/1742-6596/2599/1/0120053 Minimize: C d Subject to: C l = 0.824 C m ≥ −0.092A ≥ 0.07794where C d , C l , and C m represent the drag, lift, and pitching moment coefficients, respectively.Meanwhile, A denotes airfoil area.To ensure that the lift constraint is met, the angle of attack is introduced as an additional design variable in the optimization process.

Figure 2 .
Figure 2. Mesh and FFD control points4.ResultsFigure3displays the convergence process of different methods for resistance.As shown in Figure3(a), the number of design variables significantly affects the optimization results.With an increase in the number of variables, the resistance of the convergent optimization results reduces, and the control ability of the airfoil shape becomes stronger.However, in multiple design variables, the resistance decreases slowly over dozens of iterations.The position of FFD control points also affects the convergence result of resistance when the number of design variables is the same.The use of non-uniform control points in the x-direction results in smaller spacing of control points at the leading and trailing edges of the wing, which optimizes the drag more than uniform FFD control points, as shown by the drag convergence curve of non-uniform FFD control points with ndv=9 and 12 in Figure3(b).The change of the two parameters of the optimization algorithm on the optimization results is displayed in Figure 3(c) and 3(d).Figure 3(c) shows the impact of the line search step size of two design variable examples on the

Figure 3 (
c) shows the impact of the line search step size of two design variable examples on the optimization results.

Figure 3 .
Figure 3. Drag convergence history: (a) Number of different design variables; (b) Position of different control points; (c) Different line search steps of the optimization algorithm; (d) Scaling factors of different objective functions

Figure 4 .
Figure 4. Mach number distribution before and after optimization.

Figure 5 .
Figure 5. Airfoil and pressure distribution before and after optimization.

Figure 6
Figure6presents the convergence history of the area and moment constraints.The airfoil's area rapidly converges to the constraint value of 0.07794 during the iteration process, indicating the activation of the constraint and the maintenance of the minimum objective function value at A=0.07794.Next, the moment constraint is examined.In previous iterations, the moment exhibits oscillations, and some points violate the constraint with a moment Cm less than -0.092.However, in subsequent iterations, no points violate the moment constraint, and the majority of points reach a moment C m of -0.092.At the final convergence point, the moment C m satisfies the equality constraint, demonstrating that in the

Figure 6 .
Figure 6.Change of constraints during iteration.