47 research outputs found
Partitions of Minimal Length on Manifolds
We study partitions on three dimensional manifolds which minimize the total
geodesic perimeter. We propose a relaxed framework based on a
-convergence result and we show some numerical results. We compare our
results to those already present in the literature in the case of the sphere.
For general surfaces we provide an optimization algorithm on meshes which can
give a good approximation of the optimal cost, starting from the results
obtained using the relaxed formulation
On the Blaschke-Lebesgue theorem for the Cheeger constant via areas and perimeters of inner parallel sets
The first main result presented in the paper shows that the perimeters of
inner parallel sets of planar shapes having a given constant width are minimal
for the Reuleaux triangles. This implies that the areas of inner parallel sets
and, consequently, the inverse of the Cheeger constant are also minimal for the
Reuleaux triangles. Proofs use elementary geometry arguments and are based on
direct comparisons between general constant width shapes and the Reuleaux
triangle
A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results
We consider the multiphase shape optimization problem
where
is a given constant and is a bounded open set
with Lipschitz boundary. We give some new results concerning the qualitative
properties of the optimal sets and the regularity of the corresponding
eigenfunctions. We also provide numerical results for the optimal partitions
Volume computation for Meissner polyhedra and applications
The volume of a Meissner polyhedron is computed in terms of the lengths of
its dual edges. This allows to reformulate the Meissner conjecture regarding
constant width bodies with minimal volume as a series of explicit finite
dimensional problems. A direct consequence is the minimality of the volume of
Meissner tetrahedras among Meissner pyramids
Semaine d'Etude Mathématiques et Entreprises 6 : Représentation des fonctions de réponse radiométrique
Ce rapport rassemble les différentes pistes de réponses apportées au problème posé par l'entreprise Kolor lors de la 6ème Semaine d'Étude Maths-Entreprises de juin 2013. Le problème porte sur la représentation de fonctions de réponse radiométrique utilisées dans de nombreux logiciels de manipulation de photographies. Une grande partie du travail effectué a consisté à comprendre le problème et ses enjeux afin de le formaliser de façon mathématique. Après une description formelle des outils envisagés, nous les évaluons par rapport aux contraintes du problème afin de déterminer leurs avantages et inconvénients. D'un point de vue pratique, les outils sont présentés dans l'objectif d'être éventuellement développés et intégrés à un logiciel existant. Nous avons donc tenté, dans la mesure du possible, de prendre en compte la simplicité de ces outils que ce soit du côté développement ou du côté utilisation. Ce rapport s'articule en six parties. Après une description pratique du problème, nous en détaillons les principales caractéristiques. Dans une troisième partie, nous décrivons les trois outils envisagés. Les deux parties suivantes constituent l'étude des outils par rapport aux contraintes du problème. Finalement nous donnons une conclusion de cette étude
Phase field approach to optimal packing problems and related Cheeger clusters
In a fixed domain of we study the asymptotic behaviour of optimal
clusters associated to -Cheeger constants and natural energies like the
sum or maximum: we prove that, as the parameter converges to the
"critical" value , optimal Cheeger clusters
converge to solutions of different packing problems for balls, depending on the
energy under consideration. As well, we propose an efficient phase field
approach based on a multiphase Gamma convergence result of Modica-Mortola type,
in order to compute -Cheeger constants, optimal clusters and, as a
consequence of the asymptotic result, optimal packings. Numerical experiments
are carried over in two and three space dimensions
Numerical shape optimization among convex sets
This article proposes a new discrete framework for approximating solutions to shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate discrete convex shapes and is easily implementable using standard optimization software. The framework can handle various objective functions ranging from geometric quantities to functionals depending on partial differential equations. Width or diameter constraints are handled using the support function. Functionals depending on a convex body and its polar body can be handled using a unified framework
Mixed volumes and the Blaschke-Lebesgue theorem
The mixed area of a Reuleaux polygon and its symmetric with respect to the origin is expressed in terms of the mixed area of two explicit polygons. This gives a geometric explanation of a classical proof due to Chakerian. Mixed areas and volumes are also used to reformulate the minimization of the volume under constant width constraint as isoperimetric problems. In the two dimensional case, the equivalent formulation is solved, providing another proof of the Blaschke-Lebesgue theorem. In the three dimensional case the proposed relaxed formulation involves the mean width, the area and inclusion constraints