Analytical and Numerical Developments in Optimal Shape
Design for aerospace : An overview.
Name
Olivier Pironneau
Organization Université Paris VI and IUF (www.ann.jussieu.fr/pironneau)
Country France
Dassault Aviation Contact Mourad Sefrioui, DGT/DPR
Keywords
Optimization, Optimal shape Design, Gradient Methods, Finite Element methods
Objectives
Recall the mathematical conditions for well posedness of an OSD problem. Recall the connection between continuous and discrete algorithms
Present recent developments in topological optimization Present some state of the art solutions.
Applications
Optimal shape design (OSD) is now a necessity in airplane design because even a few percent of drag reduction means a lot . Optimization of 3D wings are routinely done in aerodynamic computations at Boeing and Airbus; wing body configuration and aero elasticity are more difficult, moreover the challenge is still in multi objective design. Applications to the car industry is well underway. Optimization of pipes, heart valves and even MEMS and fluidic devices is also done.
Results
1. Problens can be studied as infinite dimensional controls with state variable in partial differential equations and constraints. Existence of solution is guaranteed undeer mild hypothesis in 2D and under the flat cone property in 3D. Tikhonov regularization is easily done with penalization of the surface of the shape.
2. In Variational form results translate without modifications to the discrete cases if discretized by the finite element method or finite volume methods. Gradient methods are efficient a convergent even though it is always preferable to use second order methods when possible. Geometric
constraints can be handle at no cost but more complex constraints involving the state variables are a real challenge.
3. Multicriteria optimization and Pareto optimality have not been solved in a satisfactory way by differentiable optimization, either because the problems are too stiff and/or there are too many local minima. Evolutionary algorithms offer an expensive alternative, when the number of optimization parameter is small. The black box aspect of this solution is a real asset in industrial context.
4. Topological optimization is a very powerful tool for optimizing the coefficients of PDEs. Hence in microfluidic it has been used to optimize the thickness and even the shape of the device. It is ideal for structure optimization where the answer can be a composite material . It does not look to be a promising technique for high Reynolds number flow.
Fig 1 : Drag optimization on k-epsilon high Reynolds flow by a gradient method. (courtesy of B. Mohammadi )
Fig 2 : Minimization of the sonic boom of a supersonic business jet (courtesy of B. Mohammadi)
