Numerical solution of a bending-torsion model for elastic rods
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Numerische Mathematik https://doi.org/10.1007/s00211-020-01156-6
Numerical solution of a bending-torsion model for elastic rods Sören Bartels1 · Philipp Reiter2 Received: 16 November 2019 / Revised: 14 October 2020 / Accepted: 15 October 2020 © The Author(s) 2020
Abstract Aiming at simulating elastic rods, we discretize a rod model based on a general theory of hyperelasticity for inextensible and unshearable rods. After reviewing this model and discussing topological effects of periodic rods, we prove convergence of the discretized functionals and stability of a corresponding discrete flow. Our experiments numerically confirm thresholds, e.g., for Michell’s instability, and indicate a complex energy landscape, in particular in the presence of impermeability. Mathematics Subject Classification 65N12 · 57M25 · 65N15 · 65N30 · 74K10
Contents 1 Introduction . . . . . 2 Elastic rods . . . . . . 3 Density . . . . . . . . 4 Discretization . . . . 5 Iterative minimization 6 Experiments . . . . . References . . . . . . . .
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1 Introduction Long slender objects—such as springy wires made of plastic or metal—can be approximated by curves. In many cases, equilibrium shapes are characterized in terms of the
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Sören Bartels [email protected] Philipp Reiter [email protected]
1
Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg im Breisgau, Germany
2
Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
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S. Bartels, Ph. Reiter
bending energy, i.e., (half of) the total squared curvature. The latter has a long history, dating back to Bernoulli, and can be seen as the starting point of elasticity theory. The bending energy depends just on the centerline of an object and does not incorporate other physical effects such as twisting, friction, or shear. For instance, only relying on the bending energy one cannot explain why a telephone cable tends to curl. It also does not preclude self-penetration. In this paper we extend the study of inextensible elastic curves by the first author [4] to inextensible and unshearable elastic rods. To this end we discretize the minimization problem ⎧ 2 cb L 2 ct L ⎪ ⎪ ⎨ Minimize Irod [y, b] = b · (y × b) dx |y | dx + 2 0 2 0 in the set A = {(y, b) ∈ H 2 × H 1 : L rod [y, b] = rod ⎪ ⎪ bc bc , ⎩ |y | = |b| = 1, y
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