Section: New Results
Numerical algorithms for optimization and optimum-shape design
Our research themes are related to optimization and control of complex multi-disciplinary systems governed by PDEs. They include algorithmic aspects (shape parameterization, game strategies, evolutionary algorithms, gradient/evolutionary hybridization, model reduction and hierarchical schemes), theoretical aspects (control and domain decomposition), as well as algorithmic and software aspects (parallel and grid computing).
These general themes for Opale are given some emphasis this year through the involvement of our project in the ANR/RNTL National Network on Multi-Disciplinary Optimization "OMD”.
Hierarchical (multilevel) and adaptive shape parameterization
Participants : Jean-Antoine Désidéri, Régis Duvigneau, Abderrahmane Benzaoui.
Multilevel shape optimization algorithms and application to 3D aerodynamic Problems
We have proposed to exploit the classical degree-elevation process to construct a hierarchy of nested Bézier parameterizations. The construction yields in effect a number of rigorously-embedded search spaces, used as the support of multilevel shape-optimization algorithms mimicking multigrid strategies. In particular, the most general, FAMOSA, Full Adaptive Multilevel Optimum Shape Algorithm , is inspired by the classical Full Multigrid Method .
The FAMOSA method has been applied to the context of three-dimensional flow for the purpose of shape optimization of a transonic aircraft wing (pressure-drag minimization problem) [51] using the Free-Form Deformation (FFD) approach to handle nested levels of parameterization (see Figure 2 ).
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Multi-level algorithms based on an algebraic approach
The previous hierarchical approach, based on the degree-elevation property of Bézier curves, has been extended to other parameterization types in order to be able to solve general parametric optimization problems. The proposed approach rely on the construction of a hierarchical basis of the design space, originating from the eigenmodes of the Hessian matrix of the cost functional.
We have experimented the method on simple analytic functions and then on shape reconstruction problems, using various approximations of the Hessian matrix (exact, finite-difference, local metamodel, global least-squares) [53] .
Finally, this approach has been applied to the multidisciplinary design of a supersonic business jet (aerodynamics, structure, propulsion, flight mechanics), proposed by Dassault-Aviation as multi-disciplinary optimization benchmark for "OMD" project [29] , [40] .
Multidisciplinary optimization
Participants : Jean-Antoine Désidéri, Régis Duvigneau, Aurélien Goudjo [ Univ. of Abomey-Calavi, Bénin, from October to December 2009 ] , Abderrahmane Habbal, Malik Haris [ Erasmus Mundus International Master in Mathematical Engineering 'MathMods' ] .
In the most competitive engineering fields, such as aeronautics, multicriterion and multidisciplinary design has gained importance in order to cope with new and acute needs of society. In the literature, contributions to single discipline and/or single-point design optimization abound. In recent years, for purely-aerodynamic design, we had proposed to introduce a new approach combining the adjoint method with a formulation derived from game theory for multipoint design problems [22] . Transonic flows around lifting airfoils were analyzed by Euler computations. Airfoil shapes were optimized according to various aerodynamic criteria. The notion of player was introduced. In a competitive Nash game, each player attempts to optimize its own criterion through a symmetric exchange of information with others. A Nash equilibrium is reached when each player constrained by the strategy of the others, cannot improve further its own criterion. Specific real and virtual symmetric Nash games were implemented to set up an optimization strategy for design under conflict.
When devising a numerical shape-optimization method in the context of a practical engineering situation, the practitioner is faced with an additional difficulty related to the participation of several relevant physical criteria in a realistic formulation. For some problems, a solution may be found by treating all but one criteria as additional constraints. In some other problems, mainly when the computational cost is not an issue, Pareto fronts can be identified at the expense of a very large number of functional evaluations. However the difficulty is very acute when optimum-shape design is sought w.r.t. an aerodynamic criterion as well as other criteria for two main reasons. The first is that aerodynamics alone is costly to analyze in terms of functional evaluation. The second is that generally only a small degradation of the performance of the absolute optimum of the aerodynamic criterion alone is acceptable (suboptimality) when introducing the other criteria.
We have proposed a numerical methodology for the treatment of such
problems of concurrent engineering [4] .
After completion of the parametric, possibly-constrained minimization of a
single,
primary functional JA , approximations of the gradient and
the Hessian matrix are available or calculated using data
extracted from the optimization loop itself. Then,
the entire parametric space (a subset of 
) is split into two
supplementary subspaces on the basis of a criterion
related to the second variation. The construction is such that from the
initial convergence point of the primary functional, normalized perturbations
of the parameters lying in one of the two subspaces, of specified dimension p
n , cause the least possible degradation to the primary functional. The
latter subspace is elected to support the parameterization of a secondary
functional, JB , in a concurrent optimization realized by an algorithm
simulating a Nash game between players associated with the two functionals.
We prove a second result indicating that the original global optimum point
of the full-dimension primary problem is Pareto-optimal for a
trivial concurrent problem. This latter result permits us to define a
continuum of Nash equilibrium points originating from the initial
single-criterion optimum, in which the designer could potentially make a
rational election of operating point.
Following the thesis of B. Abou El Majd, a wing-shape aero-structural optimization was successfully realized despite the strong antagonism of the criteria in conflict in the concurrent reduction of the wing drag in Eulerian flow and a stress integral of the structural element treated as a shell subject to linear elasticity [50] and [39] (see Figure 3 ).
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The technique of territory splitting is currently being extended to encompass cases where all the criteria are of comparable importance (“equitable splits”). In a more global optimization process under developement, the optimization is carried out in two phases. In the first, said to be “cooperative”, all the criteria under consideration are iteratively improved. This phase relies on a general result of convex analysis yielding to the definition of the so-called Multiple-Gradient Descent Algorithm [57] . In the second phase, said to be “competitive”, viable trade-offs are identified as particular Nash equilibrium points in the smooth continuation of the termination point of the MGDA phase [31] .
Metamodel-based optimization
Participants : Praveen Chandrashekarappa, Régis Duvigneau.
Design optimization in Computational Fluid Dynamics or Computational Structural Mechanics is particularly time consuming, since several hundreds of expensive simulations are required in practice. Therefore, we are currently developing approaches that rely on metamodels , i.e. models of models, in order to accelerate the optimization procedure by using different modelling levels. Metamodels are inexpensive functional value predictions that use data computed previously and stored in a database. Different techniques of metamodelling (polynomial fitting, Radial Basis Functions, Kriging) have been developed and validated on various engineering problems[54] . Our developments have been particularly focused on the construction of algorithms that use both metamodels and models based on PDE's solving to drive a semi-stochastic optimization, with various couplings :
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A strong coupling approach consists in using metamodels to pre-evaluate candidate designs and select those which are exactly evaluated by simulation at each iteration. Then, the optimization algorithm relies only on exact evaluations. For the Particle-Swarm Optimization (PSO) algorithm, an adaptive method has been proposed, that allows the algorithm to automatically adjust the number of exact evaluations required at each iteration[26] .
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A weak coupling approach consists in using metamodels only to solve a set of optimization subproblems iteratively. In that case, kriging is employed to predict both function value and modelling error. The subproblems considered (lower bound minimization, probability of improvement maximization, expected improvement maximization) indicate which simulations should be performed to improve the model as well as determine the best design[62] .
The efficiency of these approaches has been studied on two benchmark test-cases proposed in the framework of the "Design Database Workshop" organized in December by University of Jyvaskyla (Finland): an inverse problem (pressure reconstruction) using a three-body airfoil[44] , and a flow control problem for a transonic airfoil[46] .
Uncertainty estimation and robust design
Participants : Régis Duvigneau, Massimiliano Martinelli.
A major issue in design optimization is the capability to take uncertainties into account during the design phase. Indeed, most phenomena are subject to uncertainties, arising from random variations of physical parameters, that can yield off-design performance losses.
To overcome this difficulty, a methodology for robust design is currently developed and tested, that includes uncertainty effects in the design procedure, by maximizing the expectation of the performance while minimizing its variance.
Two strategies to propagate the uncertainty are currently under study :
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the use of metamodels to predict the uncertainties of the objective function from the uncertainties of the input parameters of the simulation tool. During the optimization procedure, a few simulations are performed for each design variables set, for different values of the uncertain parameters in order to build a database used for metamodels training. Then, metamodels are used to estimate some statistical quantities (expectation and variance) of the objective function and constraints, using a Monte-Carlo method.
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the use of the automatic differentiation tool Tapenade (developped by Tropics Project-Team) to compute first and second order derivatives of the performance with respect to uncertain parameters. The first order derivatives are computed by solving the adjoint system, that is built by using Tapenade in reverse mode. For the computation of the second derivatives, two strategies can be employed: the use of two successive tangent mode differentiations or the use of the tangent mode on the result of the reverse mode differentiation. The efficiency of these strategies depends on the number of the parameters considered. Once these derivatives have been computed, one can easily derive statistic estimations by integrating the Taylor series expansion of the performance multiplied by the probability density function. This work is carried out in collaboration with Tropics Project-Team.
Uncertainty estimation has been carried out in the particular framework of flow control, for an oscillatory rotating cylinder, in order to measure the sensitivity of optimal control parameters (frequency, amplitude) to variable flow conditions[42] .
Various robust optimization approaches (statistical, min-max, multi-point) have been compared for the optimization of the wing shape of a Falcon business aircraft subject to four uncertain parameters: free-stream velocity, angles of attack, yaw and pitch[41] .
Application of shape optimization algorithms to naval hydrodynamics
Participants : Jean-Antoine Désidéri, Régis Duvigneau, Antoine Maurice, Yann Roux [ K-Epsilon ] .
The shape optimization algorithms developed by Opale have been applied to challenging problems in naval hydrodynamics.
In the framework of the collaboration with K-Epsilon company, the optimization of a mast section for a sailing race boat has been performed, on the basis of unsteady Navier-Stokes simulations. We have also initiated the study of bow design optimization for fishing boats, in order to minimize the flow resistance induced by free-surface elevation [61] (see Figure 4 ).
A collaboration with the fluid mechanics laboratory of Ecole Centrale de Nantes (CNRS UMR 6598), is currently setting-up in order to develop optimization strategies adapted to this particular context.
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Multiobjective Shape optimization applied to Nonlinear Structural Dynamics
Participants : Jean-Antoine Désidéri, Régis Duvigneau, Abderrahmane Habbal, Gaël Mathis [ Arcelor Mittal Automotive Research Division ] , Zahra Shirzadi.
The reduction of the carbon dioxide CO2 emitted by cars is directly related to the reduction of their overall weight. When designing vehicles which comply to "green" environment standards, the automotive industry has however to fulfil security requirements, particularly those involving crash and fatigue. The so-called high performance steel -HPS- is therefore a promising material, since it allows to design very thin structures which demonstrate good mechanical properties. The metal forming process of such a steel pieces is however complex and requires the development of new and efficient numerical tools permitting a good understanding as well as for an optimal control of the mechanical behavior. In collaboration with Arcelor Mittal Automotive Research Division, the Opale team is conducting research activity in the framework of multidisciplinary optimisation of HPS structures.
In a first step, via Z. Shirzadi's internship, we led a study related to the design of the shape of a beverage can, using an axisymmetric elastoplastic model. The aim was to design a can with respect to dome reversal -DR- and reversal pressure -RP- criteria. We have used a metamodel approach, based on Neural Networks, to capture the Pareto Front of the problem. The database was built by means of the direct simulation code LS-DYNA (to compute exact nonlinear responses). The first results have proved that the considered costs are antagonistic, and allowed us to capture a part of the front. Perspectives are to push further the can design case, taking profit from the axisymmetric 2D computations (much more economical than the 3D counterpart in the nonlinear dynamic case), to implement the normal boundary intersection method -NBI- in order to efficiently capture the Pareto Front (see Figure 5 ). In a longer term perspective, it is envisaged to develop efficient algorithms to compute (in particular) Nash equilibria for games involving criteria representative of crash, fatigue, blank holder force, forming defects, and so on, and to apply the methodology to the design of complex 3D steel elements. Such work will be carried out through a doctoral program within the Opale team.
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Numerical shape optimization of axisymmetric radiating structures
Participants : Benoît Chaigne, Claude Dedeban [ France Télécom R & D ] , Jean-Antoine Désidéri.
This activity aims at constructing efficient numerical methods for shape optimization of three-dimensional axisymmetric radiating structures incorporating and adapting various general numerical advances [69] (multi-level parameterization, multi-model methods, etc) within the framework of the time-harmonic Maxwell equations.
The optimization problem consists in finding the shape of the structure that minimizes a criterion related to the radiated energy. In a first formulation, one aims at finding the structure whose far field radiation fits a target radiation pattern. The target pattern can be expressed in terms of radiated power (norm of the field) or directivity (normalized power). In a second formulation, we assume that the structure is fed by a special device named the waveguide. In such a configuration, one wishes to reduce the so-called reflexion coefficient in the waveguide. Both formulations make sense when the feeding is monochrome (single frequency feeding). For multiple frequency optimization, several classical criteria have been used and various multipoint formulations considered (min-max, aggregated criterion, etc.).
Concerning the numerical simulation, two models have been considered: a simplified approximation model known as “Physical Optics” (PO) for which the far field is known explicitly for a given geometry; a rigorous model based on the Maxwell equations. For the latter, the governing equations are solved by SRSR, a 3D solver of the Maxwell equations for axisymmetric structures provided by France Télécom R&D.
A parametric representation of the shape based on Free-Form deformation (FFD) has been considered. For the PO model, the analytical gradient w.r.t. the FFD parameters has been derived. An exact Hessian has been obtained by Automatic Differentiation (AD) using Tapenade (developed by Tropics Project-Team). Both gradient and Hessian have been validated by finite differences. For the Maxwell equations model, the gradient is computed by finite differences.
Both global and local points of view have been considered for solving the optimization problem. An original multilevel semi-stochastic algorithm [64] showed great robustness for global optimization. In the case of an optimization with multiple frequency w.r.t. the radiation diagram, numerical experiments showed that an algorithm treating the frequency points hierarchically demonstrated improved robustness. For local optimization, a quasi-Newton method with BFGS update of the Hessian and linear equality constraints, has been developed. A numerical spectral analysis of the projected Hessian or quasi-Hessian for some shapes has exhibited the geometrical modes that are slow to converge. Based on this observation, several multi-level strategies to facilitate the convergence of these modes precisely, have been proposed and tested. Successful results have been obtained for both PO and Maxwell model.
In order to provide a theoretical basis to this multilevel method, a shape reconstruction problem has been considered. The convergence of an ideal two-level algorithm has been studied. In a first step, the matrix of the linear iteration equivalent to the two-grid cycle is computed. Then, by means of similar transformations and with the help of Maple, the eigenvalues problem has been solved. Hence, the spectral radius of the ideal cycle is deduced. Provided that an adequate prolongation operator is used, we were able to show that the convergence rate is independent of the dimension of the search space. The detailed proof is to be find in [56] .
Finally, a two-criterion optimization problem has been considered: both radiation pattern and reflexion coefficient need to be optimized. As an alternative to the classical but costly Multi Objective Evolutionnary Algorithms (MOEA), a two-player Nash game strategy has been adopted. The split of the territory is guided by a preliminary local sensitivity analysis: after a primary criterion has been chosen and optimized, one seeks an appropriate subspace, in which a sensitivity is minimal, using the diagonalization of an approximation of the Hessian matrix (which is positive definite). This subspace is thus assigned to the second player. Consequently, the numerical Nash game allows to reduce the secondary criterion while almost preserving the primary criterion.
All of these multiobjective techniques have been applied and validated on a realistic test-case provided by France Télécom. This required a prototype software to be developed. Details on methods and representative examples are reported in the thesis [23] .
