Inverse Aerodynamic Design using Double Adjoint Formulation and Hicks-Henne Shape Functions

Inverse Aerodynamic Design using Double Adjoint Formulation and Hicks-Henne Shape Functions

Weiyang Lin, W. Kyle Anderson

The goal of this project is to implement the double adjoint method so the calculation will be mostly independent of design variables except vector-vector multiplications. The shape of the airfoil will be parameterized and controlled by the Hicks-Henne shape functions with control points on both upper and lower surfaces. The inverse design, which takes an arbitrary pressure distribution and change the shape of the airfoil to achieve it, can thus be constructed.

To gain the capability of controlling the position of the last point at the T.E., an additional shape function might be used, and they are together shown in Fig. 1 below.

Figure 1. The Hicks-Henne shape functions (black) and an additional shape function to control the trailing edge (red)

A mesh movement method referred to as "Spring Analogy" is implemented in this context, where the mesh is treated as s system of interconnected springs (edges). This system is constructed by representing each edge of triangle by a tension spring, where the stiffness of the spring is assumed inversely proportional to the length of its edge. Figures 2-4 show the example of mesh movement for inviscid mesh.

Figure 2. Example of mesh movement for inviscid mesh. I. Initial mesh

Figure 3. Example of mesh movement for inviscid mesh. I. Distorted mesh

Figure 4. Example of mesh movement for inviscid mesh. III. Close look of the distorted mesh at the leading edge

The optimization is based on the analysis with the 2D unstructured flow solver. A validation case has a given validated pressure distribution as the target, the initial guess is taken from the results of NACA-0012, and the corresponding results are shown in Figs. 5 and 6.

Figure 5. Converged shape and the corresponding flow-field in Mach

Figure 6. The target and converged pressure distributions and their shapes

Finally, with the assistant of Hicks-Henne shape functions, the inverse design process can be done to any arbitrary pressure distributions. 

For more information, refer to the attached PDF: inverse_design.pdf and/or contact the author. The report may be downloaded here [Link 1 Link 2].

Reference

Anderson, W. K., and Venkatakrishnan, V. "Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation," Computers & Fluids Vol. 28, No. 4, 1999, pp. 443-480. [Download PDF]