Closed Euler elasticae

Sachkov, Yu. L.

doi:10.1134/s0081543812060211

Cited by 24 publications

(20 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here, we denote the line element of u by ds(u) to emphasize the dependence on u. Now, we discretize the chain rule (17). We first change the time derivatives to time differences by expressing d dt E[u] and u t as E[u] − E[v] and u − v, respectively.…”

Section: Derivation Of An Energy Dissipative Numerical Schemementioning

confidence: 99%

Energy dissipative numerical schemes for gradient flows of planar curves

Kemmochi

2017

Bit Numer Math

View full text Add to dashboard Cite

In this paper, we develop an energy dissipative numerical scheme for gradient flows of planar curves, such as the curvature flow and the elastic flow. Our study presents a general framework for solving such equations. To discretize time, we use a similar approach to the discrete partial derivative method, which is a structure-preserving method for the gradient flows of graphs. For the approximation of curves, we use B-spline curves. Owing to the smoothness of B-spline functions, we can directly address higher order derivatives. In the last part of the paper, we consider some numerical examples of the elastic flow, which exhibit topology-changing solutions and more complicated evolution. Videos illustrating our method are available on YouTube.2010 Mathematics Subject Classification. 65M60, 35A35, 53C44.

show abstract

Section: Derivation Of An Energy Dissipative Numerical Schemementioning

confidence: 99%

Energy dissipative numerical schemes for gradient flows of planar curves

Kemmochi

2017

Bit Numer Math

View full text Add to dashboard Cite

show abstract

“…Similar to the isoperimetric inequality from the Introduction, one can prescribe the length of a closed curve and ask for the shape that minimizes elastic energy. This is known to be a disk for a recent proof see [17] or inspect (1) below in the case n = 2 and notice that it suffices to minimize among convex curves. Alternatively, one can prescribe the enclosed area of a closed curve without intersection points and ask for the shape with minimal elastic energy.…”

Section: Open Problemmentioning

confidence: 99%

Two dimensions are easier

Kawohl

2016

Arch. Math.

View full text Add to dashboard Cite

show abstract

“…The convergence of the area A(Ω ε ) to A(Ω) follows from the expression of the area in terms of the gauge function: 11) and the uniform convergence. Let us consider the perimeter of Ω ε , given by…”

Section: Proofmentioning

confidence: 99%

“…Let us also mention some related works. In [11], Yu. L. Sachkov studies the "closed elasticae", that is the closed curves which are stationary points of the elastic energy.…”

Section: Introductionmentioning

confidence: 99%

Elastic energy of a convex body

Bianchini

Henrot

Takahashi

2015

Mathematische Nachrichten

View full text Add to dashboard Cite

In this paper a Blaschke-Santaló diagram involving the area, the perimeter and the elastic energy of planar convex bodies is considered. More precisely we give a description of setwhere A is the area, P is the perimeter and E is the elastic energy, that is a Willmore type energy in the plane. In order to do this, we investigate the following shape optimization problem:where C is the class of convex bodies with fixed perimeter and µ 0 is a parameter. Existence, regularity and geometric properties of solutions to this minimum problem are shown.

show abstract

Closed Euler elasticae

Cited by 24 publications

References 13 publications

Energy dissipative numerical schemes for gradient flows of planar curves

Energy dissipative numerical schemes for gradient flows of planar curves

Two dimensions are easier

Elastic energy of a convex body

Contact Info

Product

Resources

About