65 research outputs found
Two-dimensional body of maximum mean resistance
A two-dimensional body, exhibiting a slight rotational movement, moves in a
rarefied medium of particles which collide with it in a perfectly elastic way.
In previously realized investigations by the first two authors, Plakhov &
Gouveia (2007, Nonlinearity, 20), shapes of nonconvex bodies were sought which
would maximize the braking force of the medium on their movement. Giving
continuity to this study, new investigations have been undertaken which
culminate in an outcome which represents a large qualitative advance relative
to that which was achieved earlier. This result, now presented, consists of a
two-dimensional shape which confers on the body a resistance which is very
close to its theoretical supremum value. But its interest does not lie solely
in the maximization of Newtonian resistance; on regarding its characteristics,
other areas of application are seen to begin to appear which are thought to be
capable of having great utility. The optimal shape which has been encountered
resulted from numerical studies, thus it is the object of additional study of
an analytical nature, where it proves some important properties which explain
in great part its effectiveness.Comment: Accepted (April 16, 2009) for publication in the journal "Applied
Mathematics and Computation
Minimizing within convex bodies using a convex hull method
International audienceWe present numerical methods to solve optimization problems on the space of convex functions or among convex bodies. Hence convexity is a constraint on the admissible objects, whereas the functionals are not required to be convex.To deal with, our method mix geometrical and numerical algorithms. We give several applications arising from classical problems in geometry and analysis: Alexandrov's problem of finding a convex body of prescribed surface function; Cheeger's problem of a subdomain minimizing the ratio surface area on volume; Newton's problem of the body of minimal resistance. In particular for the latter application, the minimizers are still unknown, except in some particular classes. We give approximate solutions better than the theoretical known ones, hence demonstrating that the minimizers do not belong to these classes
Bodies of constant width in arbitrary dimension
International audienceWe give a number of characterizations of bodies of constant width in arbitrary dimension. As an application, we describe a way to construct a body of constant width in dimension n, one of its (n-1)-dimensional projection being given. We give a number of examples, like a four-dimensional body of constant width whose 3D-projection is the classical Meissner's body
Optimal roughening of convex bodies
A body moves in a rarefied medium composed of point particles at rest. The
particles make elastic reflections when colliding with the body surface, and do not
interact with each other. We consider a generalization of Newton’s minimal resistance problem: given two bounded convex bodies C1 and C2 such that C1 ⊂ C2 ⊂ R3 and
∂C1 ∩ ∂C2 = ∅, minimize the resistance in the class of connected bodies B such
that C1 ⊂ B ⊂ C2. We prove that the infimum of resistance is zero; that is, there
exist ”almost perfectly streamlined” bodies
On the two-dimensional rotational body of maximal Newtonian resistance
We investigate, by means of computer simulations, shapes of nonconvex bodies
that maximize resistance to their motion through a rarefied medium, considering
that bodies are moving forward and at the same time slowly rotating. A
two-dimensional geometric shape that confers to the body a resistance very
close to the theoretical supremum value is obtained, improving previous
results.Comment: This is a preprint version of the paper published in J. Math. Sci.
(N. Y.), Vol. 161, no. 6, 2009, 811--819. DOI:10.1007/s10958-009-9602-
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
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