Aerodynamic Optimal Shape Design Based on Body-Fitted Grid Generation

Farzad Mohebbi,Mathieu Sellier

doi:10.1155/2014/505372

Farzad Mohebbi, Mathieu Sellier

Open Access

https://doi.org/10.1155/2014/505372

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Abstract

This paper is concerned with an optimal shape design problem in aerodynamics. The inverse problem in question consists in finding the optimal shape an airfoil placed in a potential flow at a given angle of attack should have such that the pressure distribution on its surface matches a desired one. The numerical method to achieve this aim is based on a body-fitted grid generation technique (elliptic, O-type) to generate a mesh over the airfoil surface and solve for the flow equation. The O-type scheme is used due to its ability to generate a high quality (fine and orthogonal) grid around the airfoil surface. This paper describes a novel and very efficient sensitivity analysis scheme to compute the sensitivity of the pressure distribution to variation of grid node positions and both the conjugate gradient method (CGM) and a version of the quasi-Newton method (i.e., BFGS) are used as optimization algorithms to minimize the difference between the computed pressure distribution on the airfoil surface and desired one. The elliptic grid generation technique allows us to map the physical domain (body) onto a fixed computational domain and to discretize the flow equation using the finite difference method (FDM).

Highlights

Thanks to the advent of modern high speed computers over the last few decades, computational fluid dynamics (CFD) has been extensively employed as an analysis and as a design optimization tool
This paper describes a novel and very efficient sensitivity analysis scheme to compute the sensitivity of the pressure distribution to variation of grid node positions and both the conjugate gradient method (CGM) and a version of the quasi-Newton method (i.e., BFGS) are used as optimization algorithms to minimize the difference between the computed pressure distribution on the airfoil surface and desired one
The numerical algorithm consists of three steps, namely, grid generation and flow equation solver to find the pressure on the airfoil surface, sensitivity analysis to compute the gradient of the objective function with respect to the design variables, and an optimization method to minimize the functional and reach optimum solution

Summary

Introduction

Thanks to the advent of modern high speed computers over the last few decades, computational fluid dynamics (CFD) has been extensively employed as an analysis and as a design optimization tool. Among the methodologies often employed in shape optimization are gradient-based techniques. These techniques may be applied to minimize a specified objective function. The objective function can be, for example, a measure of difference between the pressure distribution on the airfoil surface and a desired one, and it would be desirable to minimize this objective function. The procedure employed is based on the elliptic grid generation, a novel sensitivity analysis (based on finite difference method), and an optimization method. The airfoil surface is parameterized using the grid points and the Bezier curve. Three different types of design variables were considered: the grid points, the Bezier curve control points, and the maximum thickness of NACA00xx airfoils. It will be shown that the proposed sensitivity analysis method reduces the computation cost significantly even for large number of the design variables

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