This presentation gives an overview of the capacities of optimal shape design methods as actual engineering tools utilized for industrial applications, mostly for aerodynamic shape design. Numerical formulation and implementation are recalled and illustrations of applications are discussed.

The most significant outcomes of the workshop on “Optimum Design in Aerodynamics” concern the progress accomplished in the following methodologies:

During the last years with a constant increasing activity, an increasing number of scientific meetings gathering expert researchers and engineers have taken place in the field of Computational Fluid Dynamics.

This section contains the synthesis of the different contributions presented for the resolution of the workshop test cases. This synthesis contains, for each test case, the reasons that make the test case interesting for the workshop, the special difficulties for the resolution, its possible interest for a future workshop in the context of the ECARP Brite/Euram project and a comparison of the results presented by different contributors.

ABSTRACT This paper is devoted to shape optimization for Partial Differential Equations (PDE) systems related to Computational Fluid Dynamics (CFD). Numerical approximation of the PDE's relies on schemes satisfying discrete maximum principles and using unstructured meshes generated from the shape parameters. The theory of control is applied to the discrete design problem with the resulting constrained optimization problem solved by gradient based algorithms. An automatic differentiation procedure for Fortran codes is extensively used to carry out the CFD sensitivity analysis.

A methodologv for the integrafed design of hypersonic vehicles has been presented in [lJ and 121; it relies on nose to tail CFD simulafions obtained using a domain decomposition method which infegrates different flow solvers. chosen lo adapt locally the modelization to the flow physics, with fhe objective of attaining the yust needed accuracy. We present here a progress report on an eflort to validate this methodology, through an analysis of the sensitivig of fhe predictions of overall peformances to fhe level of modelization, and through fhe rebuilding of the aerothermodynamic caracteristics of an Wegra fed Validation Object, for which wind funnel data must be mailable. The objective is to obtain a quanti/ication of fhe uncertainlies on the predicfions of a vehicle's glohal peformances, by transposing to flight conditions the results of the ground analysis. J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HOW to solve shape optimization with un1 Approximate-gradient structured meshes ?We consider exact gradients, descent or one-shot algorithms and hierarchical parametrization. approach A~~l i ca t ionsa re transonic flows governed by equaLet 7 be a set of control parameters for the shape of an tions around an airfoil or in a 3D nozzle. airfoil, and W(y) the corresponding flow variables, implicitly defined from the discretized steady Euler equation written as follows Introduction Q(r , W(7)) = 0 , (1) The optimization of complex systems as those arising from aerodynamics is a challenging field from both intuition standpoint and implementation efficiency. The complexity of the multi-point turbulent 3D aerodynamics is difficult to master intuitively and optimization is no more a convex problem. The analysis of such systems stands at (or yet beyond) the limits of the possibilities of existing methods and computers. In order to answer the first point, powerful non-convex optimization algorithms have to be devised and a part of this paper will concentrate on non-numerical Genetic Algorithms. For addressing the efficiency problems, sensitivity analyses are attempted. The variable-domain question will be considered from two standpoints: in 2D, a variablemesh method is applied and an approximate sensitivity analysis is proposed and assessed; in 3D, a transpiration approach is chosen and combined with a hierarchical gradient method. The plan is the following: Section 1 : Approximate-gradient approach [2]), Section 2 : Towards 3D unstructured multilevel for shapes (see [9]). Section 3 : Genetic algorithms for solving aerodynamic optimum shape design problems. if the mesh is of fixed topology, with a deformation parametrized by 7, then the discrete flow variables W are a smooth function of y. Introducing a discrete cost function : the gradient is given by: where II is an adjoint state, solution of the linear system: and the cost functional to be minimized is given as follows : j(7) = function of q ( y ) (5) where Pl(y) is the pressure variable from W(y), and where I belongs to a set of points on the airfoil wall. In these conditions, the whole chain can be exactly differentiated, the gradient of j is expressed with an adjoint-state and a conjugate gradient algorithm can be applied to the minimization of j. In the study of this section, the Euler system (I) , is discretized on the triangulations by means of unstructured, MUSCL or centraldifferenced, finite-volume methods ([4]). The applica'INRIA, B.P. 93,06902 Sophia Antipolis Cedex, France tion of the ODYSSEE automated differentiator in order ~DASSAULT Aviation, 78, Quai Marcel Dassault, 92214 Saintto derive the exact adjoint is studied in ([lll); Cloud, France in the present study, the linearized adjoint Euler system ~ N A I , PR of China and Dassault (1) is based on a first-order Van Leer Flux Vector split§Laforia, Tour 46/00, 2e elage, University of Paris V1, BP 169, ting (i41) to allow an easier by-hand differentiation. 75252 Paris Cedex, France Om1995 bv Dervieux. Published bv the American Institute of The airfoil shape is defined from 20 to 80 spline control Aeronautics and Astronautics, Inc, with permission ordinates. The mesh is deformed by an elasticity sytem RAE Zep MdJ.73 d=2 #kg (woigb 0 1 I) Cp initid Cpopt ----Figure 1: Successive airfoil shapes during one optimizaFigure 2: Shock drag reduction: initial and final shapes.

Abstract In the past few years much effort has been devoted to the design of numerical methods applying on unstructured meshes, in order to obtain more complicated methods that can be more easily used, due to mesh generality. This work has been concentrated on mono-element meshes (only one type of element) and particularly on triangles with P1 or Po degrees of freedom. Now, there are strong motivations for considering different elements and multi-element meshes in two possible directions. Firstly, combining triangles and quadrangles can be advantageous in particular for the capture of boundary layers. Secondly, considering higher-order elements is a door to higher-order accuracy in general geometries. For this purpose we start from a Lagrange- Galerkin formulation (Lagrange interpolation, Galerkin variational formulation) and consider upwind extensions. Many upwind methods applying on unstructured meshes are related to Lagrange-Galerkin finite element discretisations. One class of methods, that conserves the finite element philosophy, is the so-called SUPG or Galerkin Least-Square formulation, which has been successfully applied to solve 3-D problems [l]. Recently, distributive schemes ensuring a maximum principle and preserving a linear solution of the advective equation have been studied by several teams (among others [12]). The extension to systems is performed through a multiwave decomposition. Another kind of method, initiated by INRIA and Dassault Aviation, relies on an interpretation of a Galerkin formulation as a Finite Volume centered approximation applied on a dual mesh [2], [3], [4], where the central differences integration is replaced by a Riemann solver [5]. This methodology, which is nowadays widely used, has proved to be reliable as an industrial tool to solve aerodynamic flow problems [6].

In the framework of Euler solvers on unstructured meshes, a class of methods is based on an interpretation of a Galerkin formulation as a finite volume approximation on a dual mesh made of cells. In this paper, we propose a new definition for the dual cells, that may improve the accuracy and grid-insensitiveness for this methodology.

A Navier Stokes solver based on a Galerkin - Least Square formulation is used; entropy variables are introduced to ensure dimensional consistency and satisfy the stability inequality as the second law of Thermodynamics. Convergence to the steady state solution is obtained with an implicit technique using a preconditioned GMRES linear solver. This method has been developped in close cooperation with F. Chalot, T.J.R Hughes, Z. Johan and F. Shakib at Stanford University.

The goal of this paper is to discuss the application on critical problems encountered during reentry of space vehicles of an Euler flow solver developped in the past using unstructured meshes to handle equilibrium and/or nonequilibrium reactive flow simulations. Results in two dimensions for three test cases selected among those proposed around a double ellipse geometry at high Mach number and high angle of attack are presented.

Finite volume TVD schemes derived for the Euler equations are extended to the Navier-Stokes system. The numerical diffusion introduced in the approximation of the convective part is chosen through a total variation analysis taking in account the physical diffusion. Two dimensional numerical simulations are presented, using an algorithm to solve cheaply the steady equations.

We present an overview of a new finite element method for the compressible Euler equations. Our discretization is based on entropy variables. The method is developed within the framework of a Petrov-Galerkin formulation. Two perturbations are added to the weighting function; one is a generalization of the SUPG operator and the other is designed to enhance shock capturing capability. We present results of tests selected among the GAMM Workshop problems. The calculations were performed collaboratively by Stanford University and the Avions Marcel Dassault-Breguet Aviation company.