论文标题
弹性张力紧张框架中的灾难
Catastrophe in Elastic Tensegrity Frameworks
论文作者
论文摘要
我们讨论由刚性条和弹性电缆制成的弹性张力框架，具体取决于许多参数。对于任何固定参数值，框架的稳定平衡位置是通过最小化受代数约束的能量函数来确定的。随着参数平稳变化，稳定的平衡可能会消失。这种平衡的丧失称为“灾难”，因为尽管参数的变化很小，但该框架将经历大规模的变化。使用非线性代数，我们表征了参数空间的一个半gebraic子集，即灾难集，该集合从这个受约束优化问题的参数化家族中检测到局部极值的合并，因此检测到可能的灾难。数值非线性代数的工具可以可靠，有效地计算所有稳定的平衡位置以及灾难集本身。
We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called `catastrophe' since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra we characterize a semialgebraic subset of the parameter space, the catastrophe set, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself.
