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CATEGORIES:Applied Topology Colloquium
SUMMARY:A consistent relaxation of optimal design problems
for coupling shape and topological derivatives -
Professor Samuel Amstutz\, Applied Mathematics at
Ecole Polytechnique
DTSTART:20220204T140000Z
DTEND:20220204T130000Z
UID:TALK4779AT
URL:/talk/index/4779
DESCRIPTION:I will present a procedure for approximating a ‘bl
ack and white’ shape and topology optimization pro
blem by a density optimization problem\, allowing
for the presence of ‘grayscale’ regions. The const
ruction relies on a regularizing operator for smea
ring the characteristic functions involved in the
exact optimization problem\, and on an interpolati
on profile\, which endows the intermediate density
regions with fictitious material properties. In p
articular\, this framework\nincludes the classical
SIMP model.\n\nUnder mild hypotheses on the smoot
hing operator and on the interpolation profile\, w
e prove that the features of the approximate densi
ty optimization problem (material properties\, obj
ective function\, etc.) converge to their exact co
unterparts as the smoothing parameter vanishes. No
tably\, the Fréchet derivative of the approximate
objective functional with respect to the density f
unction converges to either the shape or the topol
ogical derivative of the exact objective\, dependi
ng on whether it is evaluated at the boundary of t
he domain or in its interior. These results shed n
ew light on the connections between these two diff
erent notions of sensitivities for domain function
als and on the construction of consistent interpol
ation schemes. This also\napplies to the bi-materi
al case.\n\nThe concepts of shape and topological
derivatives will be first recalled. Then the appro
ximation procedure will be explained. Related algo
rithms\, including level-set formulations and the
incorporation of perimeter penalization\, will be
finally discussed and illustrated by numerical\nex
amples in the linear elasticity context.
LOCATION:Nuffield Building\, G17
CONTACT:Jack Sykes
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