Section: New Results
Computational methods, numerical analysis and validation
Analysis and numerical approximation of macroscopic models of vehicular traffic
Participant : Paola Goatin.
Paola Goatin is parttime on secondment from University of Toulon.
Present research is focused on the mathematical analysis of traffic flow models on road networks or subject to unilateral constraints. In particular, [25] is devoted to a hyperbolic 2 phase model for traffic flow on a network. The model is rigorously described and the existence of solutions is proved, without any restriction on the network geometry.
In [49] , we deal with the mathematical description of toll gates along roads, or of the escape dynamics for crowds. This problem requires the introduction of unilateral constraints on the observable flow. We present a rigorous approach to these constraints, and numerical integrations of the resulting models are included to show their practical usability.
Paola Goatin defended her Habilitation thesis on the "Analysis and numerical approximation of some macroscopic models of vehicular traffic" [24] .
A new project entitled "Traffic management by macroscopic models" has been submitted to the ERC Starting Grant, in which the INRIA Sophia Antipolis Méditerranée Center has been proposed as the host institution.
Isogeometric analysis and design
Participants : JeanAntoine Désidéri, Régis Duvigneau, Frédéric Moresmau, Bernard Mourrain [ Galaad ProjectTeam ] , Mohammed Ziani.
Design optimization stands at the crossroad of different scientific fields (and related software): ComputerAided Design (CAD), Computational Fluid Dynamics (CFD) or Computational Structural Dynamics (CSM), parametric optimization. However, these different fields are usually not based on the same geometrical representations. CAD software relies on Splines or NURBS representations, CFD and CSM software uses gridbased geometric descriptions (structured or unstructured), optimization algorithms handle specific shape parameters. Therefore, in conventional approaches, several information transfers occur during the design phase, yielding approximations and nonlinear transformations that can significantly deteriorate the overall efficiency of the design optimization procedure.
The isogeometric approach proposes to definitely overcome this difficulty by using CAD standards as a unique representation for all disciplines. The isogeometric analysis consist in developing methods that use NURBS representations for all design tasks:

the geometry is defined by NURBS surfaces;

the computation domain is defined by NURBS volumes instead of meshes;

the solution fields are obtained by using a finiteelement approach that uses NURBS basis functions instead of classical Lagrange polynomials;

the optimizer controls directly NURBS control points.
Using such a unique data structure allows to compute the solution on the exact geometry (not a discretized geometry), obtain a more accurate solution (highorder approximation), reduce spurious numerical sources of noise that deteriorate convergence, avoid data transfers between the software. Moreover, NURBS representations are naturally hierarchical and allows to define multilevel algorithms for solvers as well as optimizers.
A prototype of isogeometric solver and design optimizer has been developed in collaboration with Galaad team. The heat conduction equation (elliptic) has been first considered to assess the convergence properties (highorder approximation schemes) and test the possible refinement preocedures (h and prefinement) [58] . The study of hierarchical optimization strategies for modelling and parameterization has been initiated. Moreover, collaborations with the Technical University of Munich for application to structural design and with National Technical University of Athens for application to hydrodynamic design are in progress. In the near future, the Euler equations for compressible aerodynamics (hyperbolic) will be studied to extend the method to more complex applications. (See Figure 1 for an illustration.)