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

• Published in: Computational mechanics Vol. 42; no. 5; pp. 715 - 732
• Main Authors: Pimenta, P. M., Campello, E. M. B., Wriggers, P.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.10.2008; Springer Nature B.V
• Subjects: Algorithms; Classical and Continuum Physics; Computational Science and Engineering; Computer simulation; Energy conservation; Engineering; Equations of motion; Finite element method; Nonlinear dynamics; Nonlinear equations; Original Paper; Parameterization; Rotation; Theoretical and Applied Mechanics; Nonlinear dynamics; Rods; Energy conservation; Time integration; Momentum conservation
• ISSN: 0178-7675; 1432-0924
• DOI: 10.1007/s00466-008-0271-5

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.