Research and Modelling Seminar: Phase field methods for the numerical approximation of evolving interfaces with geometric evolution laws

Research and Modelling Seminar: Phase field methods for the numerical approximation of evolving interfaces with geometric evolution laws

From October 26th to 30th, Prof. Elie Bretin, from the University of Lion will give some lectures about "Phase field methods for the numerical approximation of evolving interfaces with geometric evolution laws". Here a short abstract:

The motivation of this course is to give an overview of the phase field methods for the numerical approximation of  evolving interfaces  
with geometric evolution laws. In the last decades, a lot of attention has been devoted to this method for its efficiency and applications concern for instance image processing (denoising, segmentation, interface reconstruction from
discretized data), material science (motion of grain boundaries in alloys, crystal growth), biology  (modeling of vesicles and blood cells) or Shape optimization. In this course, we first introduce the phase phase field method  in the
simplest context of the  mean curvature flow (which can be viewed as the L2 gradient flow of the perimeter) and we present an analysis of the Cahn Hilliard energy and of the Allen Cahn equation from a theoretical and a numerical point
of view.  Then, having in mind some applications in image processing, we explain how to adapt the phase field method in the case of additional volume constraints, for multiphase models or to approximate the Willmore energy.

Detailed description of the course :

Lesson 1 : Introduction to the context of geometric interface energies with applications in images processing. Definition of the curvature of an interface and shape derivatives formulas for classical surface energies (Volume,
perimeter, Willmore energy and anisotropic perimeter)    

Lesson 2 : Definition and elementary properties of the mean curvature flow. Classical numerical  methods to approximate mean curvature flow
(parametric approach, level set method, phase field). Gamma convergence of the Cahn Hilliard energy to the perimeter (Mordica-Mortola theorem).
        
Lesson 3 : Definition and properties of the Allen Cahn equation (existence, Comparison principle). Formal asymptotic expansion of its solutions.
Comparison of  two numerical schemes (Euler implicit && Lie splitting method).   

Lesson 4 : Phase field models to treat the case of additional volume constraint and to approximate multiphase perimeters.
Applications to optimal partitions and to image segmentation.

Lesson 5 : Comparison of two phase fields models to minimize the Willmore energy. Applications to the optimal shape of red cells and for the regularization  of discrete contours.  

Lesson 6 : Numerical illustrations with Matlab of the double bubble conjecture in the isotropic and anisotropic case.

For the timetable, please look at the Calendar.

Marco Caliari
Data pubblicazione
venerdì 30 ottobre 2015 - 10.42.31
ultima modifica
mercoledì 4 novembre 2015- 10.33.51
Oggetto
Research and Modelling Seminar: Phase field methods for the numerical approximation of evolving interfaces with geometric evolution laws
Pubblicato da
Marco Caliari
Methods for applied mathematics (seminar course) (2015/2016)
Research and modelling seminar (seminar course) (2015/2016)
Laurea magistrale in Mathematics
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