Abstract

This paper is concerned with the growth-driven shape-programming problem, which involves determining a growth tensor that can produce a deformation on a hyperelastic body reaching a given target shape. We consider the two cases of globally compatible growth, where the growth tensor is a deformation gradient over the undeformed domain, and the incompatible one, which discards such hypothesis. We formulate the problem within the framework of optimal control theory in hyperelasticity. The Hausdorff distance is used to quantify dissimilarities between shapes; the complexity of the actuation is incorporated in the cost functional as well. Boundary conditions and external loads are allowed in the state law, thus extending previous works where the stress-free hypothesis turns out to be essential. A rigorous mathematical analysis is then carried out to prove the well-posedness of the problem. The numerical approximation is performed using gradient-based optimisation algorithms. Our main goal in this part is to show the possibility to apply inverse techniques for the numerical approximation of this problem, which allows us to address more generic situations than those covered by analytical approaches. Several numerical experiments for beam-like and shell-type geometries illustrate the performance of the proposed numerical scheme.

中文翻译：

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通过差异增长进行超弹性形状编程

摘要

本文关注生长驱动的形状编程问题，其中涉及确定可以在超弹性体上产生变形以达到给定目标形状的生长张量。我们考虑全局兼容增长的两种情况，其中增长张量是未变形域上的变形梯度，以及不兼容的情况，它放弃了这种假设。我们在超弹性最优控制理论的框架内制定了该问题。豪斯多夫距离用于量化形状之间的差异；驱动的复杂性也包含在成本函数中。州法律允许边界条件和外部载荷，从而扩展了之前无应力假设至关重要的工作。然后进行严格的数学分析以证明问题的适定性。使用基于梯度的优化算法执行数值近似。我们这一部分的主要目标是展示应用逆向技术来对该问题进行数值逼近的可能性，这使我们能够解决比分析方法所涵盖的更通用的情况。梁状和壳型几何形状的几个数值实验说明了所提出的数值方案的性能。