An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods

Abstract

A fully conserving algorithm is developed in this paper for the integration of the equations of motion in nonlinear rod dynamics. The starting point is a re-parameterization of the rotation field in terms of the so-called Rodrigues rotation vector, which results in an extremely simple update of the rotational variables. The weak form is constructed with a non-orthogonal projection corresponding to the application of the virtual power theorem. Together with an appropriate time-collocation, it ensures exact conservation of momentum and total energy in the absence of external forces. Appealing is the fact that nonlinear hyperelastic materials (and not only materials with quadratic potentials) are permitted without any prejudice on the conservation properties. Spatial discretization is performed via the finite element method and the performance of the scheme is assessed by means of several numerical simulations.

A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization

“…have appeared on the subject [4][5][6][7][8][9][10][11][12][13]. Given the significance of finite rotations within the geometrically exact beam theory, a number of investigations are also available on finite rotations and their different parametrizations, e.g.…”

“…These formulations are strain objective. Instead of total rotations, incremental rotations have been interpolated in a recent paper by Pimenta et al [13]. Hence, interpolation of incremental rotations remains invariant even after rigid rotations are superposed on the stored rotation at each node.…”

“…Thus (unlike [28]), rotation vectors corresponding to compound rotations are not extracted for interpolation. The authors in [13] have also stored the rotation variables and curvature strains at the Gauss points instead of nodes so as to avoid the interpolation of nodal rotations. Their strategy leads to a strain objective formulation.…”

Hybrid and multi-field variational principles for geometrically exact three-dimensional beams

“…the works presented by (Cardona and Geradin, 1988), (Simo and Vu-Quoc, 1991), (Pimenta and Yojo, 1993), (Simo et al, 1995), Ibrahimbegovic and Mikdad, 2000;Ibrahimbegovic and Knopf-Lenoir, 2003), (Saje et al, 1998), Jelenic and Crisfield, 1999), (Planinc and Saje, 1999), Atluri, 1988, 1989;Atluri, 1996, 1998;Atluri et al, 2001), (Betsch and Steinmann, 2002), (Zupan and Saje, 2003b), (Kapania and Li, 2003b,a), (Romero, 2004), (Mata et al, 2007), (Makinen, 2007), (Lens and Cardona, 2008), (Ghosh and Roy, 2008) and (Pimenta et al, 2008).…”

A robust composite time integration scheme for snap-through problems

“…Such observation has motivated the development of energy conserving schemes. Following the pioneering work by Simo and Tarnow [40], many studies extended this conserving algorithm to general hyperelastic materials [24,33,35,16] and arbitrary geometric nonlinearities [38,39]. Other work focused on applying these algorithms to specific finite element formulations (beam and shell elements) [41,42,15,48] and multi-body systems [7,10,28].…”