一种简单方案在弹性结和自避不可伸长曲线近似中的稳定性
Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves
AI总结：
本文提出一种半隐式数值方案，用于在特定同调类中最小化曲线的弯曲能，通过加权弯曲能与切点功能的和，证明了稳定性结果，保证了能量衰减和弧长参数化保持，通过数值实验探讨了如何获得弹性结的全局极小值。
摘要：
我们讨论了一种半隐式数值方案，该方案允许在特定同调类中最小化曲线的弯曲能。为此，我们考虑了弯曲能和切点功能的加权和。基于后者二阶导数的估计和均匀双利普希茨半径，我们证明了稳定性结果，表明在演化过程中能量衰减以及弧长参数化的保持。最后，我们展示了一些数值实验，探索能量景观，旨在探讨如何在结类中获得弯曲能的全局极小值，即所谓的弹性结。
英文摘要：
We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of the bending energy and the tangent-point functional. Based on estimates for the second derivative of the latter and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization. Finally we present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in knot classes, so-called elastic knots.
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