On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

• Published in: International journal for numerical methods in engineering 1998-06, Vol.42 (3), p.409-442
• Main Authors: Brank, Boštjan, Briseghella, Lamberto, Tonello, Nicola, Damjanić, Frano B.
• Format: Article
• Language: English
• Subjects: energy-momentum method; finite elements; non-linear shell dynamics
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/(SICI)1097-0207(19980615)42:3<409::AID-NME363>3.0.CO;2-B

Continuum and numerical formulations for non‐linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three‐dimensional continuum by employing standard assumptions of the first‐order shear‐deformation theories. Motion of the shell‐director is described by a singularity‐free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one‐step, second‐order accurate, time‐integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid‐point rule. Spatial discretization is based on the four‐noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long‐term dynamic computations of flexible shell‐like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four‐noded shell finite element in conjunction with the implicit energy and momentum conserving time‐integration algorithm. © 1998 John Wiley & Sons, Ltd.