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SUMMARY:Optimization of the shape of regions supporting the boundary condi
tions of a physical problem
DTSTART:20240312T101500Z
DTEND:20240312T111500Z
DTSTAMP:20240422T005500Z
UID:indico-event-10130@indico.math.cnrs.fr
DESCRIPTION:Speakers: Charles Dapogny\n\nBroadly speaking\, shape optimiza
tion is the discipline of optimizing the design of a domain in two or thre
e space dimensions\, with respect to an objective and under certain constr
aints\, which are formulated as functions of the domain.In applications\,
these functions depend on the shape via to solution to a partial different
ial equation encoding the physical aspects of the problem under scrutiny\,
which is complemented with boundary conditions accounting for the effects
the exterior medium. Thus\, a mechanical structure is characterized by it
s displacement\, solution to the linear elasticity system\, equipped with
boundary conditions of homogeneous Dirichlet (modeling the fixation region
s of the structure)\, homogeneous Neumann (for the traction-free boundarie
s)\, or inhomogeneous Neumann (boundaries where loads are applied) types.M
ost often\, only one part of the boundary of the shape is optimized -- typ
ically\, the traction-free boundary in structural mechanics. The aim of th
is work is\, on the contrary\, to consider the optimization of these regio
ns bearing the boundary conditions of the physical problem at play. This q
uestion is considered from two complementary viewpoints.\n- On the one han
d\, we investigate the shape derivative of a function of the domain in the
sense of Hadamard\, when the involved deformations do not vanish where th
e boundary conditions change types: this allows to optimize how the region
s bearing boundary conditions may ``slide'' along the boundary of the shap
e.\n- On the other hand\, we investigate the sensitivity of the solution t
o a physical problem (and that of a related quantity of interest) when a s
mall region bearing a certain type of boundary conditions (typically\, of
homogeneous Dirichlet type) is nucleated within a region bearing other con
ditions (e.g. of Neumann type). This paves the way to a notion of ``topolo
gical derivative'' describing the change of boundary conditions on the bou
ndary of a given shape.Several numerical applications of these development
s will be discussed.These works have been realized in collaboration with E
ric Bonnetier\, Carlos Brito-Pacheco\, Nicolas Lebbe\, Edouard Oudet and M
ichael Vogelius.\n\nhttps://indico.math.cnrs.fr/event/10130/
LOCATION:Salle J. Cavailles (IMT)
URL:https://indico.math.cnrs.fr/event/10130/
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