几何形变与椭圆曲线
Auxetic deformations and elliptic curves
摘要：
检测拉胀行为的问题起源于材料科学和数学晶体学，指的是柔性周期性杆和关节框架在某些方向拉伸时变宽而不是收缩的特性。唯一已知的算法解决方案，用于检测无穷小的增长是基于相当沉重的机械固定维半定规划。在本文中，我们提出了一种新的，更简单的算法方法，适用于一个自然的家庭的三维周期杆和关节框架与3自由度。本类包括大多数沸石结构，这是重要的应用程序在计算材料科学。我们发现拉胀变形的存在与相关的椭圆曲线的性质有关。通过定义曲线的三次型的经典Aronhold不变量，得到了一种快速识别拉胀能力的算法。
英文摘要：
The problem of detecting auxetic behavior, originating in materials science and mathematical crystallography, refers to the property of a flexible periodic bar-and-joint framework to widen, rather than shrink, when stretched in some direction. The only known algorithmic solution for detecting infinitesimal auxeticity is based on the rather heavy machinery of fixed-dimension semi-definite programming. In this paper we present a new, simpler algorithmic approach which is applicable to a natural family of 3D periodic bar-and-joint frameworks with 3 degrees-of-freedom. This class includes most zeolite structures, which are important for applications in computational materials science. We show that the existence of auxetic deformations is related to properties of an associated elliptic curve. A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants of the cubic form defining the curve.
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