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

Abstract

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 ÿrst-order shear-deformation theories. Motion of the shell-director is described by a singularityfree 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 modiÿcation 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 ÿnite element formulation suitable for the long-term dynamic computations of exible 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 ÿnite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. ? 1998 John Wiley & Sons, Ltd.

An energy-momentum conserving algorithm for nonlinear transient analysis within the framework of hybrid elements

“…The sequence of deformed shapes at 2 second intervals is depicted in Figure 10. Figures 11, 12, and 13 show the time history of the linear and angular momentum vector components, and the total (kinetic and strain) energy; the results of the H27 element are in very good agreement with the results presented in [Brank et al 1998;Balah and Al-Ghamedy 2005]. In accordance with the design of the algorithm, these quantities are conserved after .…”

“…The dynamics of a short elastic cylinder, initially at rest, and subjected to an impulsive load have been studied by several researchers [Simo and Tarnow 1994;Brank et al 1998;Balah and Al-Ghamedy 2005]. The geometry, finite element mesh for the H27 element, material parameters, and loading conditions are shown in Figure 9.…”

“…Laursen and Meng [2001] and Gonzalez [2000] extended the method of Simo and Tarnow to nonlinear constitutive models, albeit by different methods. Brank et al [1998] and Sansour et al [2004] considered the application of these methods to the motion of shells. Betsch and Steinmann [2001] developed a time finite element method and introduced the assumed strain method in time which simplifies the design of energy-momentum conserving algorithms for nonlinear constitutive models.…”

A robust composite time integration scheme for snap-through problems

“…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]. Despite the achievement of unconditional stability, the energy conserving schemes still show difficulties for numerically stiff nonlinear problems [7, 1,2], and especially for snap-through buckling problems [31,32].…”

Degenerated shell element with composite implicit time integration scheme for geometric nonlinear analysis

“… introduced the energy‐momentum method (algorithmic conservation of energy) where the average of the strains was used to exactly enforce energy conservation. The algorithmic conservation approach had been widely developed and applied in the nonlinear dynamics of the shell structures . However, these methods are computationally costly that solve a scalar variable either at the integration point or over each element in an averaged sense and may result in non‐symmetric tangent stiffness matrices.…”