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

Robert, Tristan; Peletier, MA Mark

doi:10.1002/1522-2616(200106)226:1<153::aid-mana153>3.0.co;2-2

Cited by 78 publications

(90 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The symmetry parameter m depends on the height L and is monotone decreasing in L, see in particular [Lachand-Robert and Oudet, 2005, Figure 6], [Lachand-Robert and Peletier, 2001, Table 1 and Figure 1]. Given m, we have to carry out the computations only on 1/(2 m) of the circle.…”

Section: Further Improvementsmentioning

confidence: 99%

The numerical solution of Newton’s problem of least resistance

Wachsmuth

2014

Math. Program.

View full text Add to dashboard Cite

In this paper we consider Newton's problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in R 2 . We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized problem could be solved by standard optimization software efficiently.Second, we conjecture that the optimal body has a certain structure. We exploit this structure and obtain a variational problem in R 1 . Deriving its Euler-Lagrange equation yields a program with two unknowns, which can be solved quickly.

show abstract

Section: Further Improvementsmentioning

confidence: 99%

The numerical solution of Newton’s problem of least resistance

Wachsmuth

2014

Math. Program.

View full text Add to dashboard Cite

show abstract

“…This kind of problem has a long history originating from the publication by I. Newton in his Principia of the famous problem on minimal resistance of convex axisymmetric bodies and continuing nowadays in a series of paper in 1990s and 2000s related to minimal resistance of convex (not necessarily symmetric) bodies [14,13,9,35,34]. If we consider nonconvex bodies, an explicit analytical expression for the resistance becomes impossible and one needs to use billiard techniques to minimize the resistance.…”

Section: Prefacementioning

confidence: 99%

“…It was found numerically in [34], however the properties of the solution are not well understood until now. In addition, the solution of the problem inf f ∈D(h) R(f ) in a narrower class D(h) was found analytically in [35]. We depict the solution of this problem for h = 2 in Fig.…”

Section: Newton's Aerodynamic Problemmentioning

confidence: 99%

Exterior Billiards

Plakhov¹

2013

View full text Add to dashboard Cite

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.

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 78 publications

References 7 publications

The numerical solution of Newton’s problem of least resistance

The numerical solution of Newton’s problem of least resistance

Exterior Billiards

The problem of the body of revolution of minimal resistance

Contact Info

Product

Resources

About