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Séminaire EDP, Modélisation et Calcul Scientifique (commun ICJ & UMPA)

Numerical study of 1D optimal structures

by Edouard Oudet (LJK, Université Grenoble Alpes)

mardi 28 février 2017 de au (Europe/Paris)
at ICJ, UCBL - La Doua, Bât. Braconnier ( salle Fokko du Cloux )
We focus our attention on shape optimization problems in which one
dimensional connected objects are involved. Very old and classical
problems in calculus of variation are of this kind: euclidean Steiner's
tree problem, optimal irrigation networks, cracks propagation, etc.

In a first part we quickly recall some previous work in collaboration
with F. Santambrogio related to the functional relaxation of the
irrigation cost. We establish a $\Gamma$-convergence of Modica and
Mortola's type and illustrate its efficiency from a numerical point of
view by computing optimal networks associated to simple sources/sinks
configurations. We also present more evolved situations with non Dirac
sinks in which a fractal behavior of the optimal network is expected.

In the second part of the talk we restrict our study to the euclidean
Steiner's tree problem. We recall recent numerical approach which have
been developed the last five years to approximate optimal trees:
partitioning formulation, relaxation with geodesic distance terms and
energetic constraints. We describe the first results obtained in
collaboration with A. Massaccesi and B. Velichkov to certify the
optimality of a given tree. With our discrete parametrization of
generalized calibration, we are able to recover the theoretical optimal
matrix fields which certify the optimality of simple trees associated to
the vertices of regular polygons.

Finally, we focus on the delicate problem of the identification of the
optimal structure. Based on  a recent approach obtained in collaboration
with G. Orlandi and M. Bonafini, we describe the first convexification
framework  associated to the Euclidean Steiner tree problem which
provide relevant tools from a numerical point of view.