Inverse design of an isotropic suspended Kirchhoff rod: theoretical and numerical results on the uniqueness of the natural shape

Abstract : Solving the equations for Kirchhoff elastic rods has been widely explored for decades in mathematics, physics and computer science, with significant applications in the modeling of thin flexible structures such as DNA, hair, or climbing plants. As demonstrated in previous experimental and theoretical studies, the natural curvature plays an important role in the equilibrium shape of a Kirchhoff rod, even in the simple case where the rod is isotropic and suspended under gravity. In this paper, we investigate the reverse problem: can we characterize the natural curvature of a suspended isotropic rod, given an equilibrium curve? We prove that although there exists an infinite number of natural curvatures that are compatible with the prescribed equilibrium, they are all equivalent in the sense that they correspond to a unique natural shape for the rod. This natural shape can be computed efficiently by solving in sequence three linear initial value problems, starting from any framing of the input curve. We provide several numerical experiments to illustrate this uniqueness result, and finally discuss its potential impact on non-invasive parameter estimation and inverse design of thin elastic rods.
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Contributor : Florence Bertails-Descoubes <>
Submitted on : Monday, July 2, 2018 - 6:35:00 PM
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Florence Bertails-Descoubes, Alexandre Derouet-Jourdan, Victor Romero, Arnaud Lazarus. Inverse design of an isotropic suspended Kirchhoff rod: theoretical and numerical results on the uniqueness of the natural shape. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2018, 474 (2212), pp.1-26. ⟨10.1098/rspa.2017.0837⟩. ⟨hal-01827887⟩

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