论文标题
磁性弹性的形状编程
Shape programming of a magnetic elastica
论文作者
论文摘要
我们考虑具有可能不均匀的永久性磁化的悬臂梁，其形状由施加的磁场控制。我们将光束建模为平面弹性曲线，并假设磁场通过一对分布式夫妇的方式作用于光束，该夫妇将磁化强度朝向其方向。给定目标形状列表，我们寻找磁化配置文件的设计和一个控件列表，以便在平均意义上，由控件作用时梁时假定的形状与目标尽可能近。为此，我们制定并解决了一个最佳设计和控制问题，从而最大程度地限制了我们通过直接和间接方法研究的功能。特别是，我们证明存在最小化器，解决了相关的Lagrange-Multiplier公式（除非非传统情况），并且至少对于控制磁场的足够低强度而言是独一无二的。为了达到后一个结果，我们使用两个嵌套的定点参数，这些参数依靠问题的拉格朗日 - 塑料公式，这也暗示了数值方案。还讨论了各种相关的公开问题。
We consider a cantilever beam which possesses a possibly non-uniform permanent magnetization, and whose shape is controlled by an applied magnetic field. We model the beam as a plane elastic curve and we suppose that the magnetic field acts upon the beam by means of a distributed couple that pulls the magnetization towards its direction. Given a list of target shapes, we look for a design of the magnetization profile and for a list of controls such that the shapes assumed by the beam when acted upon by the controls are as close as possible to the targets, in an averaged sense. To this effect, we formulate and solve an optimal design and control problem leading to the minimization of a functional which we study by both direct and indirect methods. In particular, we prove that minimizers exist, solve the associated Lagrange-multiplier formulation (besides non-generic cases), and are unique at least for sufficiently low intensities of the controlling magnetic fields. To achieve the latter result, we use two nested fixed-point arguments relying on the Lagrange-multiplier formulation of the problem, a method which also suggests a numerical scheme. Various relevant open question are also discussed.
