Section: New Results

Other topics

Numerical analysis of POD-based Galerkin approximations

Participants : Dominique Chapelle, Asven Gariah, Philippe Moireau, Jacques Sainte-Marie.

In [2] , we proposed a numerical analysis of Proper Orthogonal Decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and — whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

This work is now being continued in order to handle parameter-dependent models, and thence estimation problems.

Sail modeling

Participants : Dominique Chapelle, Daniele Trimarchi, Marina Vidrascu.

A dynamic Finite Element method – based on non-linear MITC shell finite elements implemented in the MITCNL software – has been proposed and assessed for the analysis of downwind sail-type structures, see [12] . The main purpose was here to investigate the development of wrinkling, a phenomenon commonly observed in practice for such structures. Considering the wrinkling in this type of analysis is of great interest, since wrinkling affects the stress distribution in the fabric. Further developments primarily regard various refinements of the model, in order to represent some even more realistic sail configurations such as with non-isotropic material models, corner reinforced zones, and cable boundary conditions. Of course, another very important perspective – and work in progress, indeed – concerns the use of such sail models coupled with the wind flow in a fluid-structure interaction framework.