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Engineer positions
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Internships
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PhD positions
Isogeometric analysis based on triangular supports
The construction of a computational grid from CAD (Computer-Aided Design) data is currently a bottleneck that limits the use of simulation and optimization methods in industry. A new approach has been proposed in 2005 [1], named isogeometric analysis, which consists in solving PDEs (Partial Differential Equations) on a parametric computational domain originating from CAD directly. This approach bypasses the grid generation phase and targets to fusion CAD and FEM (Finite-Element Method).
A first experiment has been carried out recently by Opale and Galaad project-teams [2], on the basis of B-Spline surfaces or volumes as computational domains. However, the construction of the computational domain is still difficult and the local refinement is tedious because the B-Spline domain is based on a tensorial representation.
Therefore, we propose to study the use of a parametric domain based on triangular supports. The doctoral student will first examine the possible bases associated to triangulations and establish their main properties, in terms of dimension, regularity, complexity, power of approximation and local refinement. Numerical algorithms to construct complex computational domains will be developed. Then, the definition of numerical schemes to solve PDEs on such a computational domain will be considered. Elliptic PDEs (heat conduction, linear elasticity) as well as hyperbolic problems (compressible Euler equations) will be examined and local refinement strategies based on error estimators will be tested on model problems.
This ambitious program requires an expertise in both geometric modeling and numerical schemes for PDEs. Therefore, a collaborative work between Opale and Galaad Project-Teams is proposed.
[1] T. Hughes, J. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, nurbs, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering, (194):4135–4195, 2005.
[2] G. Xu, B. Mourrain, R. Duvigneau, and A. Galligo. Parametrization of computational domain in isogeometric analysis: methods and comparison. Computer Methods in Applied Mechanics and Engineering, 200(23-24), 2011.Contact : Regis Duvigneau
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Post-Doc positions
Crowd motion modeling by conservation laws
The existing literature on mathematical modeling of human crowds is especially concerned with models at the microscopic scale, in which pedestrians are described individually in their motion by a set of ordinary differential equations. A different approach to the problem, not yet as much developed in the specialized literature, consists in using partial differential equations at the macroscopic scale, that is in describing the evolution in time and space of some density of pedestrians rather than following each of them individually. The starting point is, like for the previously discussed vehicular traffic models, the analogy with classical fluid dynamics: An Eulerian point of view is adopted, appealing to the conservation of mass of pedestrians supplemented by either suitable closure relations linking the mean pedestrains’ velocity to their density, or an analogous balance law for the momentum.
Typical guidelines in devising this kind of models are the concepts of preferred direction of motion and discomfort at high densities. The above mentioned models have been treated by means of numerical simulations. On the contrary, the analytical properties of the solutions such as their existence, uniqueness and long-time behavior are not known in general.
The PhD fellow will study macroscopic pedestrian traffic flow models consisting in hyperbolic PDEs from a analytical and numerical point of view.
In particular, the PhD fellow will investigate the analytical properties of the solutions such as their existence, uniqueness and long-time behavior, as well as the construction of numerical schemes for efficiently computing solutions. The considered equations do not fit the hypotheses of the results available in the literature, and need the construction of new “ad hoc” methods. Numerical simulations will provide a useful support for determining which models can capture the targeted features.
In a second time, the PhD fellow will address the problem of combining microscopic and macroscopic methods in hybrid models in which the individuality of each “particle” is maintained. He/she will analyze the well-posedness such models. This would allow to model individual trajectories starting from the global dynamics, and to treat related control problems.
Contact : Paola Goatin
For more details: pdf