1.- Wed. Feb. 12, 11:00-11:30 : Olivier Pironneau (Paris VI-F) (pironneau@ann.jussieu.fr)

"Optimization in the presence of Discontinuities"

Abstract: There are many fluid flow problems with discontinuities in the data or in the flow. Among them three are quite important for applications: - flow through porous media with several geological layers, - transonic and supersonic flow with shocks, including shape design, - accoustics with sonic boom including shape design, Optimisation of these systems by standard gradient methods require the application of the techniques of the Calcul of Variations and an implicit asumption that a Taylor expansion exists with respect to the degrees of freedom of the problem. Take for example the flow in a transonic nozzle and the variation of the flow with respect to the inflow conditions; when these vary the shock moves and the derivative of the flow variables with respect to inflow conditions is a Dirac measure and so the Taylor expansion does not exists. By extending the calculus of variation via the theory of distribution it is possible to show however that the derivatives exists. But the result has serious numerical implications, in particular it favors the mixed finite element methods. We shall give numerical illustrations using the finite element method for an inverse problem for a Darcy flow, for the design of a transonic nozzle and for the design of a business supersonic airplane for sonic boom minimization.

2.- Wed. Feb. 12, 11:30-12:00 : Jean-Antoine Desideri (INRIA-Sophia-Antipolis-F) (Jean-Antoine.Desideri@inria.fr)

"Hierarchical Optimum-Shape Algorithms using Embedded Bezier Parameterizations"

Abstract: Several multi-level shape optimization algorithms are proposed for the minimization of a functional constrained by a PDE state equation subject to a shape-dependent boundary condition. The prototype test problem is the optimization of an airfoil shape immersed in a two-dimensionnal flow (of Eulerian type in the simplest case) to reduce drag; the shape is then subject to certain geometrical constraints (specified endpoints, vertical tangent at leading edge, given area, etc). In general applications, the field, or distributed state, is computed by some standard PDE approximation method (e.g. of finite-volume type) over a mesh, one boundary of which is optimized in shape, via design parameters in number much less than the boundary gridpoints. The shape is thus formally represented, prior to mesh-discretization, by a parameterization here specifically chosen to be of Bézier type, based on control points whose abscissas are prescribed, and ordinates optimized according to some physical criterion. Here we construct an a priori hierarchy of embedded Bézier parameterizations via the classical degree-elevation process as the support of a number of multi-level optimization algorithms, including one inspired from the Full Multi-Grid ( FMG) concept, and referred to as the Full and Adaptive Multi-Level Optimum-Shape Algorithm ( FAMOSA). Our construction yields rigorously-nested optimization search spaces. The framework allows us also to propose ways to employ reduced models, as well as to construct hybrid optimizers (combining gradient-based with evolutionary algorithms).

3.- Wed. Feb. 12, 12:00-12:30 : Francois Courty, Mariano Vazquez. Alain Dervieux (INRIA-Sophia-Antipolis-F) et Bruno Koobus (Montpellier II-F) (Alain.Dervieux@inria.fr, Francois.Courty@inria.fr, koobus@math.univ-montp2.fr, Mariano Vazquez@inria.fr)

"Analysis and solution of the optimality system in aerodynamical shape design"

Abstract: The optimality system involves the usual three equations, state, adjoint and functional stationarity. These can be used for advancing (toward final solution) respectively the state variable (i.e. flow variables), adjoint variables, and shape parameters. A formal analysis relying on the Hadamard formula shows that shape increment can then be less ergular than previous shape, which demonstrate the need of a functional preconditioner (as already noted by Jameson, Mohammadi, Ta'asan,...). Further, the loss of regularity is identified as 1. Our proposition for solving that point is an extension of the Bramble-Pasciak-Shu precontitioner. Turning to the discrete context, we observe that the complexity of the optimality system can be much reduced if the state and co-state equations are solved progressively/simultaneously. This idea is often refered as "one-shot" resolution. It is useful to control this process into a modern optimization loop. We adapt a Sequential Quadratic Programming algorithm to the progressive resolution of the three equation of optimality system. Applications to the sonic boom reduction of a supersonic aircraft will illustrate the above developements.

More about this work

4. Wed. Feb. 12, 12:30-13:00 : Jacques Periaux* , Mourad Sefrioui*, Eric Whitney ° and L. Gonzalez°

(Jacques.Periaux@dassault-aviation.fr)

* Dassault Aviation, Direction de la Prospective

° University of Sydney, Department of Aeronautical Engineering School of Aerospace, Mechanical and Mechatronic Engineering

" A new Hierarchical Asynchronous Parallel Evolutionary Algorithm for Multi-Objective Design Optimization problems in Aerodynamics"

Keywords: Multi-Objective Optimisation , Evolutionary Design, Parallel Computing, Pareto

Abstract: The goal of this lecture is to present a new Evolutionary Algorithm for multi criteria aerodynamic shape design optimization. This algorithm attempts to overcome some of the drawbacks associated with earlier evolutionary algorithms , the most important being that of large computational time. Based on recent results obtained with Hierarchical Parallel Genetic Algorithms (HPGA) we propose the use of a Hierarchical Asynchronous Parallel Evolutionary Algorithms (HAPEAs), where the search for the best solution takes place successively in separate hierarchical layers comprising different CFD solvers or resolutions. Each of the layers can be independently tuned to suit a given problem or CFD solver. Other strategies employed to reduce optimisation time are parallel computing and asynchronous evaluation. It is shown that when applied to Pareto multi-criteria optimisation problems the use of a combination of fast and simple model together with slower and more detailed models on an asynchronous parallel architecture results in the same design quality as obtained using a classical EA which would normally utilise only a complex model and involve much larger computational expense. Results indicate that overall the algorithm is easy to implement, robust and flexible to capture the solution or a set of solutions when applied to problems in aerodynamics such as supersonic nozzle inverse problems or multi-point airfoil optimization for the reduction of drag at different transonic regimes. Numerical experiments illustrate the possibilities of this approach for future applications in Multidisciplinary Design Optimization.

5. Wed. Feb. 12, 15:40-16:10 : Angelo Iollo (Politecnico di Torino), (angelo.iollo@polito.it) and Luca Zannetti (Politecnico di Torino), (zannetti@polito.it)

"Fluid dynamic adjoint optimization by solving constrained inverse problems "

Abstract: In fluid flow optimization  it is usually required to determine the shape of a given component to improve performance. In some cases, however, it is more natural to look for the flow solution which improves performance, obtaining the geometry by solving an inverse problem. The optimal flow solution can then be found by a descent method based on the inverse problem adjoint equations. Applications pertinent to turbomachine blading design will be presented.

6.- Wed. Feb. 12, 16:10-16:40 : Mohamed Masmoudi (Toulouse-F) (masmoudi@mip.ups-tlse.fr)

"Some applications of the topological asymptotic expansion"

Abstract: The aim of topological optimization is to find an optimal shape without any a priori information about the topology of the structure. Main contributions in this field are concerned with structural analysis and in particular the optimization of the compliance (external work) subject to a volume constraint. Most of the authors considered composite material optimization and the homogenization theory. The range of application of this approach is quite restricted. For this reason global optimization techniques like genetic algorithms and simulated annealing are used in order to solve more general problems. Unfortunately, these methods are very slow. The notion of topological asymptotic expansion gives an interesting alternative: - its range of application is very large, - using topological sensitivity information, we can build fast algorithms. The topological gradient (the leading term of the asymptotic expansion) is a function defined in the domain of computation. It gives the variation of a cost function if a small obstacle is created in the domain. The topological gradient is very easy to compute. Our minimization algorithm is based on a kind of level set approach, introduced by J. Céa in 1973 under the name of "fixed point method".

7. Wed. Feb. 12, 16:40-17:10 : Andreas Troxler, Rolf Jeltsch (ETH Zurich, Switzerland) (atroxler@sam.math.ethz.ch, rolf.jeltsch@sam.math.ethz.ch)

"Inverse Aerodynamic Shape Design of Gas Turbine Blades"

Abstract: A three-dimensional viscous/inviscid method for aerodynamic shape design of gas turbine blades is proposed. Prescribed data are the blade thickness and pressure loading distribution as well as the leading edge position. The corresponding blade shape and steady flow field are sought. The method is based on a finite volume discretization on a structured H-mesh. The resulting ordinary differential equations for the flow state and the (algebraic) pressure loading constraints form a DAE system. A half-explicit Runge-Kutta method is employed to solve this system: The flow state is advanced in time by a standard explicit Runge-Kutta scheme. At each stage the algebraic constraints are satisfied by updating the unknown circumferential positions of the blade surface nodes. The corresponding system of equations is small and sparse, as only the cells along the blade surfaces are involved. In comparison to the direct problem with known geometry the overhead per iteration is below 10%. In terms of number of iterations the method proves robust.

Back to AMAM Fluid Dynamics Minisymposium list