Epi-convergence of discrete elastica
Epi-convergence of discrete elastica
复制标题
离散弹性方程的落收敛
DOI：
10.1080/00036810108840955
复制
发表时间：
2001
影响因子：
1.1
通讯作者：
T. Richardson
中科院分区：
文献类型：
--
作者：
A. Bruckstein;A. Netravali;T. Richardson
以特定方向通过特定位置并使能量泛函最小化的曲线称为弹性曲线。虽然物理样条函数很容易假设能量最小的配置，但要找到涉及曲率的非线性函数积分的变分问题的数值解仍然是一个相当艰巨的挑战，这类问题的近似解会产生令人满意的结果，计算机辅助设计领域严重依赖多项式或有理曲线设计。在本文中，我们讨论了一种方法离散化的非线性样条设计的问题，一种替代更传统的方法离散化的微分方程，解决所涉及的变分问题。我们表明，离散化的能量泛函（即，考虑多边形近似的曲线，并找到那些最小化的“能量”直接定义在转弯角和段长度）是一种方法，更简单，导致的解决方案，在非常小的段长度的限制，收敛到最佳的连续解决方案。
Curves that pass through specified locations with specified orientations and minimize an energy functional are called elastica. While physical splines readily assume minimal energy configurations, finding the numerical solutions of variational problems involving integrals of nonlinear functions of the curvature remains quite a formidable challenge.Approximate solutions of such problems yield satisfactory results and the computeraided design field relies heavily on polynomial or rational curve designs. In this paper we discuss a method for discretizing the problem of nonlinear spline design, an alternative to the more traditional approach of discretizing the differential equations that solve the variational problems involved. We show that discretizing the energy functionals (i.e, considering polygonal approximations of the curves and finding the ones that minimize their “energy” defined directly in terms of turn angles and segment length) is an approach that is simpler and leads to solutions that, in the limit of very small segment lengths, converge to the optimal continuous solutions.