The Hellan-Herrmann-Johnson and TDNNS method for linear and nonlinear shells

In this paper we extend the recently introduced mixed Hellan-Herrmann-Johnson (HHJ) method for nonlinear Koiter shells to nonlinear Naghdi shells by means of a hierarchical approach. The additional shearing degrees of freedom are discretized by H(curl)-conforming N\'ed\'elec finite elements entailing a shear locking free method. By linearizing the models we obtain in the small strain regime linear Kirchhoff-Love and Reissner-Mindlin shell formulations, which reduce for plates to the originally proposed HHJ and TDNNS method for Kirchhoff-Love and Reissner-Mindlin plates, respectively. By using the Regge interpolation operator we obtain locking-free arbitrary order shell methods. Additionally, the methods can be directly applied to structures with kinks and branched shells. Several numerical examples and experiments are performed validating the excellence performance of the proposed shell elements

Formulation and implementation of conforming finite element approximations to static and eigenvalue problems for thin elastic shells

Bibliography: pages 132-135.In deriving asymptotic error estimates for a conforming finite element analyses of static thin elastic shell problems, the French mathematician Ciarlet (1976) proposed an approach to the formulation of such problems. The formulation he uses is based on classical shell theory making use of Kirchhoff-Koiter assumptions. The shell problem is posed in two-dimensional space to which the real problem, in three-dimensional space, is related by a mapping of the domain of the problem to the shell mid-surface. The finite element approximation is formulated in terms of the covariant components of the shell mid-surface displacement field. In this study, Ciarlet's formulation is extended to include the eigenvalue problem for the shell. In addition to this, the aim of the study is to obtain some indication of how well this approach might be expected to work in practice. The conforming finite element approximation of both the static and eigenvalue problems are implemented. Particular attention is paid to allowing generality of the shell surface geometry through the use of an approximate mapping. The use of different integration rules, in-plane displacement component interpolation schemes and approximate geometry schemes are investigated. Results are presented for shells of different geometries for both static and eigenvalue analyses; these are compared with independently obtained results

Subdivision Shell Elements with Anisotropic Growth

A thin shell finite element approach based on Loop's subdivision surfaces is proposed, capable of dealing with large deformations and anisotropic growth. To this end, the Kirchhoff-Love theory of thin shells is derived and extended to allow for arbitrary in-plane growth. The simplicity and computational efficiency of the subdivision thin shell elements is outstanding, which is demonstrated on a few standard loading benchmarks. With this powerful tool at hand, we demonstrate the broad range of possible applications by numerical solution of several growth scenarios, ranging from the uniform growth of a sphere, to boundary instabilities induced by large anisotropic growth. Finally, it is shown that the problem of a slowly and uniformly growing sheet confined in a fixed hollow sphere is equivalent to the inverse process where a sheet of fixed size is slowly crumpled in a shrinking hollow sphere in the frictionless, quasi-static, elastic limit.Comment: 20 pages, 12 figures, 1 tabl

Computational model for elasto-plastic and damage analysis of plates and shells

Shells and plates are very important for various engineering applications. Analysis and design of these structures is therefore continuously of interest to the scientific and engineering community. Accurate and conservative assessments of the maximum load carried by the structure, as well as the equilibrium path in both elastic and inelastic range are of paramount importance. Elastic behaviour of shells has been very closely investigated, mostly by means of the finite element method. Inelastic analysis on the other hand, especially accounting for damage effects, has received much less attention from the researchers. A computational model for finite element, elasto-plastic and damage analysis of homogenous and isotropic shells is presented here. The formulation of the model proceeds in several stages, described in the following chapters. First, a theory for thick spherical shells is developed, providing a set of shell constitutive equations. These equations incorporate the effects of transverse shear deformation, initial curvature and radial stresses. The proposed shell equations are conveniently used in finite element analysis. A simple C0 quadrilateral, doubly curved shell element is developed. By means of a quasi-conforming technique shear and membrane locking are prevented. The element stiffness matrix is given explicitly which makes this formulation computationally very efficient. The elasto-plastic behavior of thick shells and plates is represented by means of the non-layered model, with an Updated Lagrangian method used to describe a small strain geometric non-linearity. In the treatment of material non-linearities an Iliushin?s yield function expressed in terms of stress resultants is adopted, with isotropic and kinematic hardening rules. Finally, the damage effects modeled through the evolution of porosity are incorporated into the yield function, giving a generalized and convenient yield surface expressed in terms of the stress resultants. Since the elastic stiffness matrix is derived explicitly, and a non-layered model is employed in which integration through the thickness is not necessary, the current stiffness matrix is also given explicitly and numerical integration is not performed at any stage during the analysis. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power

A finite element fundamental to thin shell theory

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This work is intended to contribute to the search for an explanation of the way a really good finite element should behave during analysis of a general shell problem. The finite element analysis of thin shells is currently receiving much attention in the international literature. Indeed it seems that in almost every new issue of the engineering journals there is a proposal for a new and more efficient shell element. The reason for this is of course the underlying complexity of the shell problem and, more specifically the difficulty of taking bending into correct account. In order to elicit an understanding of the use of the finite element method in shell problems an in-depth study is presented of the behaviour of a vehicle shell finite element. This element is the very simple combined constant membrane stress and constant bending moment flat triangle. The examination of its behaviour reveals that the characteristics of an assembly of these elements are such as to enable recovery, in a remarkable way, of each of the types of deformation identified by the classical first approximation theory. Recovery of rigid body movement, inextensional. bending, membrane action and edge effect- is achieved to an accuracy consistent with the order of magnitude of inherent errors of the classical theory. Thus, the element is seen to hold a position of fundamental importance with regard to the numerical analysis of thin shells. Special attention is given to the sensitive low energy bending response. This reveals that there are two quite different roles for the element bending freedoms. One role concerns inextensional bending movements which extend over the whole model. The other role concerns local rotational movements which accompany the curvature changes of inextensional bending and edge effect. Extensive numerical comparisons are made against solutions obtained from the classical theory for shells which are very deep with strongly negative Gaussian curvature and which are considered to provide very severe tests. Investigation of edge effect concerns a cantilevered circular cylinder under edge moment. To complete this examination of bending details are given of a matrix procedure which is intended to assess thin shell finite element models in their response to inextensional bending. To conclude this work the results of a preliminary study of the mathematical details of convergence of the vehicle element are presented. This investigation is specific to the geometry of a circular cylinder and clamped boundary conditions. It is shown that, despite the highly nonconforming nature of the element, O(h) asymptotic convergence in the energy norm is achieved and in this respect is similar in behaviour to the Clough-Johnson flat plate shell finite element.Financial support was provided by the Procurement Executive of the Ministry of Defence

Membrane locking in the finite element computation of very thin elastic shells

International audienc

Simple and extensible plate and shell finite element models through automatic code generation tools

A large number of advanced finite element shell formulations have been developed, but their adoption is hindered by complexities of transforming mathematical formulations into computer code. Furthermore, it is often not straightforward to adapt existing implementations to emerging frontier problems in thin structural mechanics including nonlinear material behaviour, complex microstructures, multi-physical couplings, or active materials. We show that by using a high-level mathematical modelling strategy and automatic code generation tools, a wide range of advanced plate and shell finite element models can be generated easily and efficiently, including: the linear and non-linear geometrically exact Naghdi shell models, the Marguerre-von K ́arm ́an shallow shell model, and the Reissner-Mindlin plate model. To solve shear and membrane-locking issues, we use: a novel re-interpretation of the Mixed Interpolation of Tensorial Component (MITC) procedure as a mixed-hybridisable finite element method, and a high polynomial order Partial Selective Reduced Integration (PSRI) method. The effectiveness of these approaches and the ease of writing solvers is illustrated through a large set of verification tests and demo codes, collected in an open-source library, FEniCS-Shells, that extends the FEniCS Project finite element problem solving environment

Fundamental considerations for the finite element analysis of shell structures

International audienceThe objective in this paper is to present fundamental considerations regarding the finite element analysis of shell structures. First, we review some well-known results regarding the asymptotic behaviour of a shell mathematical model. When the thickness becomes small, the shell behaviour falls into one of two dramatically different categories; namely, the membrane-dominated and bending-dominated cases. The shell geometry and boundary conditions decide into which category the shell structure falls, and a seemingly small change in these conditions can result into a change of category and hence into a dramatically different shell behaviour. An effective finite element scheme should be applicable to both categories of shell behaviour and the rate of convergence in either case should be optimal and independent of the shell thickness. Such a finite element scheme is difficult to achieve but it is important that existing procedures be analyzed and measured with due regard to these considerations. To this end, we present theoretical considerations and we propose appropriate shell analysis test cases for numerical evaluations

Membrane locking in discrete shell theories

This work is concerned with the study of thin structures in Computational Mechanics. This field is particularly interesting, since together with traditional finite elements methods (FEM), the last years have seen the development of a new approach, called discrete differential geometry (DDG). The idea of FEM is to approximate smooth solutions using polynomials, providing error estimates that establish convergence in the limit of mesh refinement. The natural language of this field has been found in the formalism of functional analysis. On the contrary, DDG considers discrete entities, e.g., the mesh, as the only physical system to be studied and discrete theories are being formulated from first principles. In particular, DDG is concerned with the preservation of smooth properties that break down in the discrete setting with FEM. While the core of traditional FEM is based on function interpolation, usually in Hilbert spaces, discrete theories have an intrinsic physical interpretation, independently from the smooth solutions they converge to. This approach is related to flexible multibody dynamics and finite volumes. In this work, we focus on the phenomenon of membrane locking, which produces a severe artificial rigidity in discrete thin structures. In the case of FEM, locking arises from a poor choice of finite subspaces where to look for solutions, while in the DDG case, it arises from arbitrary definitions of discrete geometric quantities. In particular, we underline that a given mesh, or a given finite subspace, are not the physical system of interest, but a representation of it, out of infinitely many. In this work, we use this observation and combine tools from FEM and DDG, in order to build a novel discrete shell theory, free of membrane locking