In the first part of the talk, we present an original approach to reconstruct the vorticity distribution inside an axisymmetric vortex ring from some incomplete and possibly noisy measurements of the corresponding velocity field. The idea is to formulate  an inverse problem for identifying the structure of the vorticity distribution and solve this problem by a suitable numerical optimization algorithm. The novelty of the present study is that the vorticity function is reconstructed in a very general form with no assumptions other than smoothness and the behaviour at the endpoints of its domain of definition. The second part of the talk will be devoted to phase-change systems governed by the Navier-Stokes equations with Boussinesq approximation for the heat transfer. We present a new single domain numerical approach for phase-change systems with natural convection. The key ingredients of the method are the use of an adaptive finite element method with a new regularization of the functions representing the variation of thermodynamic properties at the solid-liquid interface, and a fully linearized Newton algorithm for the time integration of the system of equations. The advantage of our adaptivity method is to allow accurate and simultaneous tracking of different interfaces in the system.

Reference

[1]     I. Danaila, B. Protas, Optimal reconstruction of inviscid vortices, Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, 471, 20150323, 2015

[2]     I. Danaila, R. Moglan, F. Hecht, S. Le Masson, A Newton method with adaptive finite elements for solving phase-change problems with natural convection, J. of Computational Physics, 274, p. 826-840, 2014 

Professor Ionut Danaila obtained his PhD in fluid mechanics from Université Aix-Marseille II, France, in 1997. After one year postdoc in French Institute of Petroleum, he joined Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris 6), and worked as an associate professor from 1999-2011. In 2008,  professor Ionut Danaila completed his Habilitation in Applied Mathematics. Form 2011, he has been a full professor in applied mathematics,  Department of Mathematics Raphaël Salem, University of Rouen. He has a broad research interests in scientific computing and numerical analysis.