A new one‐point quadrature, general non‐linear quadrilateral shell element with physical stabilization

Abstract

A new four-node shell element with a single-point quadrature to be used for explicit time integration is presented in this paper. The physical stabilization is applied, which enables explicitly evaluating the stabilizing forces on basis of the general degenerated shell formulation and which does not require any input parameters. An optimized choice of the moduli is performed in order to compute the stabilized forces for non-linear material so that the element's behaviour is improved with respect to similar physical stabilization elements.The cost e ciency of the element is demonstrated by the numerical examples in comparison with a fully integrated 4-node element. The new element is then implemented in and validated with the RADIOSS crash program. ? 1998 John Wiley & Sons, Ltd.

A new discontinuous Galerkin method for Kirchhoff–Love shells

“…In such structures the locking results in excessive stiffness when the membrane and bending modes are mixed. This is generally solved by considering reduced integration [12,13], or by using a mixed formulation sometimes combined with enhanced assumed strains methods [14][15][16][17]). When considering mixed methods, for which the displacement, rotation and stress fields can be both unknown and discontinuous, it has been shown that the discontinuous Galerkin method can reduce the locking effect for Reissner-Mindlin plates [18], for Timoshenko beams [19] and for shells [20,21].…”

“…The inter-element boundary terms arising from the discontinuous Galerkin formulation guaranteeing its consistency, symmetry and stability are integrated by recourse to interface elements [3,22]. The potential membrane locking behavior arising from the coupling of membrane and bending modes is addressed either by recourse to reduced integration [12,13], or by adopting an Enhanced Assumed Strains (EAS) approach [14][15][16][17]. However, this is not necessary if the polynomial approximation is higher than two.…”

A new assumed strain solid-shell formulation “SHB6” for the six-node prismatic finite element

“…Numerous efficient plate and shell elements have been developed based on mixed formulations or shear projection techniques in order to avoid locking problems. For this purpose, two approaches have been mainly used to formulate shell theories, which range from the degenerated three-dimensional concept originated by Ahmad [1], and subsequently adopted by Hughes and Liu [2], to the more classical shell descriptions (see, e.g., Bathe and Dvorkin [3,4], Simo et al [5], Cheung and Chen [6], Onate and Castro [7], Wriggers and Gruttmann [8], Ayad et al [9], Chapelle and Bathe [10], Zeng and Combescure [11]). However, the planestress assumptions, which are the most commonly adopted in these formulations, require special treatment for the integration of the constitutive equations and represent one of the major drawbacks of shell derivations.…”

The ?constant? bending moment three-noded triangle

“…The importance of explicit non-linear codes is manifest in safety applications where they have been used, for example, in the crashworthiness assessment [4] of automotive vehicles or to investigate the e ect of explosions [5] in aircraft-a matter of grave concern today. A common device in non-linear ÿnite element analysis is to reduce the computational cost of quadrilateral elements by evaluating the element sti ness matrix with one-point numerical quadrature only, as described by Zhu and Zacharia [6] and Zeng and Combescure [7]. This is tantamount to employing constant approximations to the ÿeld variables.…”

“…In References [6; 7], for example, the spurious mechanisms are controlled by intricate 'physical stabilization methods' where extra sti ness is added into the element sti ness matrix. Apparently this approach can also avoid some so-called 'locking' problems [7] caused by the injudicious choice of ÿnite element trial functions. As another example, the non-linear explicit dynamic code DYTRAN [8], a modern development of DYNA3D and PISCES [9] which includes uid-structure interaction, has the option to introduce either an artiÿcial viscosity or an artiÿcial sti ness as a remedy for inherent numerical di culties.…”