An energy‐conserving co‐rotational procedure for the dynamics of shell structures

Abstract

The simplest facet‐shell formulation involves the combination of the constant‐strain membrane triangle with a constant‐curvature bending triangle. The paper first describes an alternative co‐rotational procedure to the one initially proposed by Peng and Crisfield in 1992. This new formulation introduces a spin matrix which allows a simpler formulation for the consistent tangent stiffness matrix. The paper then moves to the dynamics of the element. To obtain stable solutions, an energy‐conserving mid‐point time‐integration scheme is developed. This scheme exactly conserves the total energy when external forces are constant and when the physical system does not present any damping. The performance of this scheme is compared with other more conventional implicit schemes through a set of numerical examples involving large‐scale rotations.

Discrepancies of energy values in dynamics of three intersecting plates

“…Figure 6 of [6]). References [7,8] do not throw additional light on this matter. Analysis of finite element method (FEM) mesh used in [6,7] reveals that along edges C and D (cf.…”

“…The problem has been studied, among others, in [6][7][8]; yet in the latter reference the load was defined differently than in the two former papers. Consequently, the results from [8] cannot be discussed directly here. The comparison of results reported in [6,7] indicates significant discrepancies between computed total energy of the structure.…”

“…After the driving loads die out the structure experiences free motion where, by definition, the total energy, linear momentum vector and angular momentum vector of the structure must remain constant. The problem has been studied, among others, in [6][7][8]; yet in the latter reference the load was defined differently than in the two former papers. Consequently, the results from [8] cannot be discussed directly here.…”

An energy–momentum conserving algorithm for non‐linear hypoelastic constitutive models

“…This adequate evaluation was given for a Saint Venant-Kirchho hyperelastic material. This scheme was further extended to shells [14][15][16][17], to composite laminates [18] and to multi-body dynamics [10,19]. A generalization to other hyperelastic models was given by Laursen [20], who iteratively solves a new equation for each Gauss point to determine the adequate second Piola-Kirchho stress tensor.…”

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].…”