The project has several objectives : to analyze mathematically coupled PDE systems involving one or more disciplines in the perspective of geometrical optimization or control; to construct, analyze and experiment numerical algorithms for the efficient solution of PDEs (coupling algorithms, model reduction), or multi-criterion optimization of discretized PDEs (gradient-based methods, evolutionary algorithms, hybrid methods, artificial neural networks, game strategies); to develop software platforms for code-coupling and for parallel and distributed computing. Major applications include the multi-disciplinary optimization of aerodynamic configurations (wings in particular) in partnership with Dassault Aviation and Piaggio Aero France, and the geometrical optimization of antennas in partnership with France Télécom and Thalès Air Défense (see Opratel Virtual Lab.).

Optimization problems involving systems governed by PDEs, such as optimum shape design in aerodynamics or electromagnetics, are more and more complex in the industrial setting.

In certain situations, the major difficulty resides in the costly evaluation of a functional by means of a simulation, and the numerical method to be used must exploit at best the problem characteristics (regularity, local convexity).

In many other cases, several criteria are to be optimized and some are
non differentiable and/or non convex. A large set of parameters,
sometimes of different types (boolean, integer, real or functional),
are to be taken into account, as well as constraints of various types
(physical and geometrical, in particular).
Additionally, today's most interesting optimization pre-industrial
projects are multi-disciplinary, and this complicates the
mathematical, physical and numerical settings.
Developing robust optimizers is therefore an essential
objective to make progress in this area of scientific computing.

Genetic Algorithms (GAs), and more generally Evolutionary Strategies
(ES), are methods based on natural selection. They rely on the analogy
with one of the best known Darwinian principles :
survival of the fittest. GAs operate on a population of
individuals that evolve in the course of generations,
according to pseudo-stochastic operators, towards
an optimal individiual, solution to an optimization problem.
These individuals are referred to as chromosomes and can be coded
as binary strings. They evolve selectively according to their fitness function value, that is the value of the functional to be
optimized.

GAs differ from the more classical deterministic methods
(steepest descent, conjugate gradient, one-shot methods)
in three principal ways : (i) they do not necessitate the
explicit calculation of the gradient, or even higher-order
derivatives; (ii) they operate on a population of individuals
instead of a single representative; (iii) they involve
semi-stochastic operators.
As a consequence, they are notably very robust, often successful in
optimizing multimodal, non-convex and non-differentiable functions,
and better equiped to avoid stagnation in local minima

In this area, the project aims at developing numerical approaches by ES or hybrid methods, for the treatment of more and more general optimization problems, but also, at enhancing the computational cost-efficiency of these methods by various techniques of numerical analysis (model reduction, e.g. by POD; convergence acceleration, in particular by multi-level methods; best approximation and shape-parameterization; hybridization of optimizers; parallel and grid computing; etc.).

In view of enhancing the robustness of algorithms in shape optimization or shape evolution, modeling the moving geometry is a challenging issue. The main obstacle between the geometrical view point and the numerical implementation lies in the basic fact that the shape gradients are distributions and measures lying in the dual spaces of the shape and geometrical parameters. These dual spaces are usually very large since they contain very irregular elements. While obviously in any finite dimensional approach we are back in the Hilbert framework, and dual spaces are identified implicitly to the shape parameter spaces. Ignoring this question leads to well-known instabilities which necessitate some smoothing procedure in order to stabilize the shape large evolution. This point is sharp in the ``narrow band'' techniques where the lack of stability implies to reinitialize the underlying level equation at each step.

The mathematical understanding of these questions is sought via the full analysis of the continuous modeling of the evolution. How can we ``displace'' a smooth geometry in the direction opposite to a non smooth field, that is going to destroy the boundary itself, or its smoothness, curvature, and at least generate oscillations.

The notion of Shape Differential Equation is an answer to
this basic question and it rises from the functional analysis
framework to be developed in order to manage the lack of duality in a
quantitative form. These theoretical complications are simplified when
we are back to a Hilbert framework, which in some sense, is possible,
but to the undue expense of a large order of the
differential operator implied as duality operator. This operator can
always be chosen as a ``had hoc'' power of an elliptic system. In this
direction, the key point is the optimal regularity of the solution to
the considered system (aerodynamical flow, electromagnetic field, etc.)
up to the moving boundary whose regularity is itself governed by
the evolution process.

We are driven to analyse the fine properties concerning the minimal
regularity of the solution. We make intensive use of the ``extractor
method'' that we developed in order to extend the I. Lasiecka and
R. Triggiani ``hidden regularity theory''.
For example, it was well known (before this theory)
that when a domain without regularity assumption on the solution

In view of algorithms for shape optimization, we consider the
continuous evolution virtual domain deformation.
The main issue is the validity of such large evolution when

We denote oriented distance function

If the domains are Sobolev domains, that is if

Several different results have been derived for this equation under
boundedness assumptions of the following kind:

The existence of such bound has been proved first for the problem of
best location of actuators and sensors,
and have since been extended to a large class of boundary value problems.
The asymptotic analysis (in time minimal regularity for the geometries.

The intrinsic geometry is the main ingredient to treat convection by a
vector fields transverse field concept

Grids for complex problem solving is one of the priorities of the IST chapter in the 6th Framework Program of the European Community. One of the challenges of the 21st century in the computer science area lies in the integration of various expertise in complex application areas such as simulation and optimisation in aeronautics, automotive and nuclear simulation. Indeed, the design of the reentry vehicle of a space shuttle calls for aerothermal, aerostructure and aerodynamics disciplines which all interact in hypersonic regime, together with electromagnetics. Further, efficient, reliable, and safe design of aircraft involve thermal flows analysis, consumption optimisation, noise reduction for environmental safety, using for example aeroacoustics expertise.

The integration of such various disciplines require powerful computing
infrastructures and particular software coupling techniques.
Simultaneousl, advances in computer technology advocates the use of
massively parallel PC-clusters including thousands of processors
connected by high-speed gigabits/sec wide-area networks.
This conjunction makes it possible for an unprecedented
cross-fertilisation of numeric technology and computer science. New
approaches including evolutionary algorithms, parametrisation,
multi-hierarchical decomposition lend themselves seamlessly to
parallel implementations in such computing infrastructures. This
opportunity is being dealt by project Opale
since its very beginning.
A software integration platform has been designed by project
Opale for
the definition, configuration and deployment of multidisciplinary
applications on distributed heterogeneous infrastructure

The main drawback still remains however in the deployment and control
of complex distributed applications on grids by the end-users. Indeed,
the deployement of the computing grid infrastructures and of the
applications in such environments still requires specific expertise by
computer science specialists. However, the users, which are experts in
their particular application fields, e.g. aerodynamics, are not
necessarily experts in distributed and grid computing.
Being accustomed to Internet browsers, they want similar interfaces to
interact with grid computing and problem-solving environments. A first
approach to solve this problem is to define component-based
infrastructures, e.g. the Corba Component Model, where the
applications are considered as connection networks including various
application codes. The advantage is here to implement a uniform
approach for both the underlying infrastructure and the application
modules. Still however, it requires specific expertise not directly
related to the application domains of each particular user.
A second approach is to make use of grid services, defined as
application and support procedures to standardise access and
invocation to remote support and application codes. This is usually
considered as an extension of Web services to grid infrastructures.
A new approach, which is currently being explored by the Opale
project, is the design of a virtual computing environment able to hide
the underlying grid-computing infrastructures to the users. An
international collaborative project has been set up in 2003 on this
subject involving the Opale project at INRIA.