Weak form quadrature element analysis of geometrically exact shells

Highlights

• A quadrature element formulation for nonlinear analysis of shells is presented.

• A geometrically exact shell model is employed.

• The total Lagrangian updating scheme of rotational quaternion is adopted.

• The locking problems can be avoided without additional efforts in the formulation.

Abstract

This paper presents a weak form quadrature element formulation of the stress resultant geometrically exact shell model proposed by Simo. A total Lagrangian updating scheme of rotation is adopted by the use of rotational quaternion. In addition to its conciseness for the coincidence of discrete nodes and integration points, the weak form quadrature element formulation exhibits computational feasibility as well as avoidance of shear and membrane locking problems for its high-order approximation property in nonlinear shell analysis. Several numerical examples are presented to illustrate the effectiveness of the proposed formulation.

Introduction

Nonlinear analysis of shells is often of concern to designers and a great deal of research work has been conducted over the last few decades. As an analog and extension of the geometrically exact beam model [1], [2], the geometrically exact shell model proposed by Simo [3] has shown its superiority in analyzing shells undergoing large displacements and rotations. The Cosserat hypothesis [4] is utilized in this model to describe the shell with a two-dimensional surface and an inextensible director attached at each point, and the classical Mindlin–Reissner shell hypothesis that excludes drilling rotations is adopted in the model. The term ‘geometrically exact’ is well justified in light of no translation or rotation update approximation in the formulation, thus facilitating large rotations and minimizing loading steps.

A large amount of research work was conducted since the inception of the geometrically exact shell model. Simo and his co-workers presented a finite element formulation of the model [5] and some extensions of the model to shells with variable thickness [6] and shells with plastic deformation [7]. Wriggers and Gruttmann derived a finite rotation shell formulation on the basis of engineering strains and work-conjugated Biot stress resultants [8]. Vu-Quoc et al. proposed a geometrically exact sandwich shell model [9]. Ibrahmbegović et al. used rotation vector parameters and updated Lagrangian scheme to describe finite rotations of shells [10]. Pimenta and Campello developed a shell model based on an idea of initial deformations [11]. Betsch and Sänger proposed a conserving framework for dynamic analysis of geometrically exact shells and applied it to flexible multibody dynamics [12].

Numerical implementation of nonlinear analysis of shells is a demanding task in view of possible volatile load–displacement curves. Adding to the considerable computational effort, finer finite element meshes are usually needed due to complex configuration of shells undergoing large displacements and rotations. Moreover, finite element analysis of geometrically exact shells is inflicted by shear and membrane locking phenomena, a major drawback in low-order shell elements. Various skills have been proposed by researchers to eliminate locking problems, but they usually generate other unwanted effects or complicate the numerical implementation process of the formulations.

The weak form quadrature element method (QEM) [13], [14], [15], [16] has been applied to analysis of various structures including geometrically exact beams [17], [18] since its inception. It starts with approximation of integrands in the variational formulation of a problem instead of selecting shape functions. The derivatives at integration points are approximated by the use of differential quadrature analogs and algebraic equations are eventually established. The initial conception of the QEM is to directly exploit the derivative approximation feature of the differential quadrature analog in the weak form solution of a problem. When Lobatto or Clenshaw–Curtis quadrature is used for numerical integration, the method intersects with the spectral element method [19]. The QEM has shown its advantages in nonlinear analysis of frame structures as well as linear analysis of many structural problems. Contrasting with finite element meshes, a structure is often divided into rather few elements and accuracy of results is usually improved by increasing element integration order according to convergence requirements. The locking problems are naturally overcome due to the use of high-order approximation in the method. In addition, the coincidence of integration points and discrete nodes of the QEM enables a more concise formulation as well as renders it free from objectivity problems when dealing with structures with large rotations.

In this paper, a weak form quadrature element formulation of geometrically exact shell model is presented. Six benchmark problems for geometrically nonlinear analysis of shells are studied and the results are compared with those obtained using finite element methods. The excellent agreement with available results at relatively low computational cost demonstrates the effectiveness of the proposed formulation. It should be noted that only quasi-static analysis of shells is discussed here, but the extension of the present formulation to dynamic analysis is straightforward, which is analogous to that of the previous finite element formulations [12], [20]. For dynamic analysis, the non-diagonal consistent inertia matrix that makes the calculation cumbersome may be disadvantageous to high-order approximations. However, some inexact integration schemes applied in the spectral method, which may also be usable in the QEM, can effectively diagonalize the mass matrix [21]. The extension of the present work to dynamic analysis will be reported in the near future.