小变形初始形状识别的无条件收敛迭代格式
An unconditionally convergent iterative scheme for initial shape identification in small deformations
摘要：
在本研究中感兴趣的问题是：“给定一个受到机械载荷的物体，如何定义初始几何形状，使变形后的几何形状精确地匹配预定的形状？”这个问题在成形工艺中特别相关，其中变形和期望几何形状之间的容许失配可能低于微米。这里提出的方法使用所需的初始几何形状作为第一“猜测”，然后迭代地更新一组边界点的位置，使得它们在变形构型中的位置越来越接近所需的位置。该计划被证明是无条件收敛的小变形的意义上，任意小的变形和所需的形状之间的失配可以实现。此外，由于它完全基于几何考虑，因此收敛不应受到材料性质的影响，即它独立于本构律。通过一个例子说明了该方法的成功。
英文摘要：
The question of interest in the present study is: ``Given a body subject to mechanical loads, how to define the initial geometry so that the deformed one matches precisely a prescribed shape?'' This question is particularly relevant in forming processes where the tolerated mismatch between the deformed and desired geometries may be lower than a Micron. The method proposed here uses as a first ``guess'' to the required initial geometry the desired one, then it updates iteratively the locations of a set of boundary points so that their locations in the deformed configuration come closer and closer to the desired ones. The scheme is shown to converge unconditionally for small deformations in the sense that arbitrarily small mismatch between the deformed and desired shape can be achieved. Moreover, since it is based entirely on geometric considerations, the convergence should not be affected by the nature of material, i.e. it is independent of the constitutive law. The success of the method is illustrated by considering an example.
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