Existence of minimizers for Newton's problem of the body of minimal resistance under a single impact assumption

Comte, Myriam; Lachand-Robert, Thomas

doi:10.1007/bf02790266

Cited by 43 publications

(41 citation statements)

References 3 publications

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“…Because of this fact, the choice of the class C of admissible functions is a delicate issue. A number of different choices have been explored in the literature [4,5,6,7,2], but the most interesting one, from a mathematical viewpoint, seems to be…”

Section: Introductionmentioning

confidence: 99%

Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions

Robert

Peletier

2001

Math. Nachr.

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Section: Introductionmentioning

confidence: 99%

Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions

Robert

Peletier

2001

Math. Nachr.

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“…This observation was first made by Buttazzo and Kawohl in [14]. The techniques of making dimples and grooves were further developed by Comte and Lachand-Robert in [18,19,20] when studying generalizations of Newton's problem in classes of nonconvex bodies.…”

Section: Bodies Modified In a Neighborhood Of Their Boundarymentioning

confidence: 96%

Exterior Billiards

Plakhov¹

2013

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The book contains an account of results obtained by the author and his collaborators on billiards in the complement of bounded domains and their applications in aerodynamics and geometrical optics.We consider several problems related to aerodynamics of bodies in highly rarefied media. It is assumed that the medium particles do not interact with each other and are elastically reflected when colliding with the body boundary; these assumptions drastically simplify the aerodynamics and allow to reduce it to a number of purely mathematical problems.First we examine problems of minimal resistance in the case of translational motion of bodies. These problems generalize the Newton problem of least resistance; the difference is that the bodies are generally nonconvex in our case and therefore the particles can make multiple reflections from the body surface. It is proved that typically the infimum of resistance equals zero; thus, there exist 'almost perfectly streamlined' bodies.Next we consider the generalization of Newton's problem on minimal resistance of convex axisymmetric bodies to the case of media with thermal motion of particles. Two kinds of solutions are found: first, Newton-like bodies and second, shapes obtained by gluing together two Newton-like bodies along their rear ends.Further, we state results on characterization of billiard scattering by nonconvex and rough bodies; next we solve some special problems of optimal mass transportation. These two groups of results are applied to problems of minimal and maximal resistance for bodies that move forward and at the same time slowly rotate. It is found, in particular, that the resistance of a three-dimensional convex body can be increased at most twice and decreased at most by 3.05% by roughening its surface.Next, we consider a rapidly rotating rough disc moving in a rarefied medium on the plane. It is shown that the force acting on the disc is not generally parallel to the direction of the disc motion, that is, has a nonzero transversal component. This phenomenon is called Magnus effect (proper or inverse, depending on the direction of the transversal component). We show that the kind of Magnus effect depends on the kind of disc roughness, and study this dependence. The problem of finding all admissible values of the force acting on the disc is formulated in terms of a vector-valued problem of optimal mass transportation.Finally, we describe bodies that have zero resistance when translating through a medium, and state results on existence or non-existence of bodies with mirror surface invisible in one or several directions. We also consider the problem of constructing retroreflectors: bodies with specular surface that reverse the direction of any incident beam of light.

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“…Since the early 1990s, there have been obtained new interesting results related to the problem of minimal resistance in various classes of admissible bodies [1][2][3][4][5][6][7][8][9]11,12]. In particular, there has been considered the wider class of convex (generally non-symmetric) bodies inscribed in a given cylinder [1,3,4,7,9].…”

Section: Introductionmentioning

confidence: 99%

The problem of the body of revolution of minimal resistance

Plakhov

Aleksenko

2008

ESAIM: COCV

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Abstract. Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of convex axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but generally nonconvex bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the value of minimal resistance are compared with the corresponding results for Newton's problem and for the problem in the intermediate class of axisymmetric bodies satisfying the single impact assumption [Comte and Lachand-Robert, J. Anal. Math. 83 (2001) 313-335]. In particular, the minimal resistance in our class is smaller than in Newton's problem; the ratio goes to 1/2 as (length)/(width of the body) → 0, and to 1/4 as (length)/(width) → +∞.Mathematics Subject Classification. 49K30, 49Q10.

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Existence of minimizers for Newton's problem of the body of minimal resistance under a single impact assumption

Cited by 43 publications

References 3 publications

Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions

Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions

Exterior Billiards

The problem of the body of revolution of minimal resistance

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