January 19, 201515:00-16:00

Carlos MORA CORRAL, Universidad Autónoma de Madrid, Madrid, Spain

ELLIPTIC APPROXIMATION OF FREE DISCONTINUITY PROBLEMS:CAVITATION AND FRACTURE

I will present some classic variational problems of the type of free discontinuity, such as the Mumford-Shah functional for image  segmentation and the perimeter functional for phase transitions in  liquids. The presence of the free discontinuity makes the problem numerically unfeasible. I will explain an approximation procedure of those problems, based on gamma-convergence, by elliptic functionals that are tractable numerically. That approximation regularizes the problem and relies on a singular perturbation and the introduction of a new phase-field function that tracks the free discontinuity set. Then, I will present a variational model in nonlinear elasticity that allows for cavitation and fracture. It is again a free discontinuity problem where the total energy to minimize is the sum of the elastic energy plus the energy produced by cavitation and fracture. The free discontinuity relies in the fact that the cavity set and the crack set are unknowns of the problem. We explain the corresponding approximation procedure and present some numerical experiments.

This is a joint work with D. Henao and X. Xu.

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Scientific Seminar