Abstract
The discrete elastic rod method (Bergou et al. 2008 ACM Trans. Graph . 27 , 63:1–63:12. ( doi:10.1145/1360612.1360662 )) is a numerical method for simulating slender elastic bodies. It works by representing the centreline as a polygonal chain, attaching two perpendicular directors to each segment and defining discrete stretching, bending and twisting deformation measures and a discrete strain energy. Here, we investigate an alternative formulation of this model based on a simpler definition of the discrete deformation measures. Both formulations are equally consistent with the continuous rod model. Simple formulae for the first and second gradients of the discrete deformation measures are derived, making it easy to calculate the Hessian of the discrete strain energy. A few numerical illustrations are given. The approach is also extended to inextensible ribbons described by the Wunderlich model, and both the developability constraint and the dependence of the energy on the strain gradients are handled naturally.
Highlights
The geometric non-linearity of thin elastic rods gives rise to a rich range of phenomena even when the strains are small, see e.g. [1,2] for recent examples
In the applications presented in the forthcoming sections, we find equilibrium configurations by minimizing Φ(X) in equation (4.3) using the sequential quadratic programming (SQP) method described by [29]; it is an extension of the Newton method for non-linear optimization problems, which can handle the non-linear constraints in equation (4.4)
We introduce the unknown ηi as an additional degree of freedom at each one of the interior nodes, and we use in equation (4.3) a strain energy density directly inspired by that of Wunderlich [15,16], Ei(κi, ηi−1, ηi, ηi+1)
Summary
The geometric non-linearity of thin elastic rods gives rise to a rich range of phenomena even when the strains are small, see e.g. [1,2] for recent examples. The unit quaternion qi introduced in equation (2.11) is the discrete analogue of the pullback (eI ⊗ dI (s)) · κ(s) of the rotation gradient κ(s) used in the continuous rod theory, whose components κJ (s) = eJ · [(eI ⊗ dI (s)) · κ(s)] = dJ (s) · κ(s) define the bending and twisting measures. Equations (2.17–2.18) show that the adjusted deformation measures ωi,J are the components of the rotation vector Ωi that maps one set of director frame (diI−1)I=1,2,3 to the other one (diI )I=1,2,3 across the vertex xi. These components can be calculated in any one of the adjacent director frame as they are identical.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
