Simon Bieber

Zusammenfassung

Within the framework of plane strain nonlinear elasticity, we present a discussion on the stability properties of various Enhanced Assumed Strain (EAS) finite element formulations with respect to physical and artificial (hourglassing) instabilities. By means of a linearized buckling analysis we analyze the influence of element formulations on the geometric stiffness and provide new mechanical insights into the hourglassing phenomenon. Based on these findings, a simple strategy to avoid hourglassing for compression problems is proposed. It is based on a modification of the discrete Green-Lagrange strain, simple to implement and generally applicable. The stabilization concept is tested for various popular element formulations (namely EAS elements and the assumed stress element by Pian and Sumihara). A further aspect of the present contribution is a discussion on proper benchmarking of finite elements in the context of hourglassing. We propose a simple bifurcation problem for which analytical solutions are readily available in the literature. It is tailored for an in-depth stability analysis of finite elements and allows a reliable assessment of its stability properties.

Zusammenfassung

We show that most geometrically nonlinear three-dimensional shell elements and solid shell elements suffer from a previously unknown artificial stiffening effect that only appears in geometrically nonlinear problems, in particular in the presence of large bending deformations. It can be interpreted as a nonlinear variant of the well-known Poisson thickness locking effect. We explain why and under which circumstances this phenomenon appears and propose concepts to avoid it.

Zusammenfassung

The enhanced assumed strain (EAS) method is one of the most frequently used methods to avoid locking in solid and structural finite elements. One issue of EAS elements in the context of geometrically non‐linear analyses is their lack of robustness in the Newton‐Raphson scheme, which is characterized by the necessity of small load increments and large numbers of iterations. In the present work we extend the recently proposed mixed integration point (MIP) method to EAS elements in order to overcome this drawback in numerous applications. Furthermore, the MIP method is generalized to generic material models, which makes this simple method easily applicable for a broad class of problems. In the numerical simulations in this work, we compare standard strain based EAS elements and their MIP improved versions to elements based on the assumed stress method in order to explain when and why the MIP method allows to improve robustness. A further novelty in the present work is an inverse stress‐strain relation for a Neo‐Hookean material model.

Zusammenfassung

We present a variational method for problems in solid and structural mechanics that is designed to be intrinsically free from locking when using equal order interpolation for all involved fields. The specific feature of the formulation is that it avoids all geometrical locking effects (as opposed to material locking effects, e.g. Poisson locking) for any type of structural or solid model, independent of the underlying discretization scheme. The possibility to employ equal order interpolation for all involved fields circumvents the task of finding particular function spaces to remove locking and avoid artificial stress oscillations. This is particularly attractive for instance for isogeometric analysis using unstructured meshes or T-splines. Comprehensive numerical tests underline the promising behaviour of the proposed method for geometrically linear and non-linear problems in terms of displacements and stress resultants using standard finite elements, isogeometric finite elements and a meshless method.

Zusammenfassung

In this contribution a class of formulations for beams, plates and shells is presented, which intrinsically avoids locking, independent of the utilized discretization scheme. The key idea is the reparametrization of the kinematic equations to avoid locking on theory level – prior to discretization. Thus, the resulting formulations are locking‐free for any equal‐order interpolation. As demonstrator, we present both mixed and primal concepts for Timoshenko beams in both weak and strong form, as well as their theoretical relationships. Besides a weak form Galerkin‐type solution using B‐Splines, we show the generality of the presented concepts by employing isogeometric collocation based on the corresponding Euler‐Lagrange equations of the boundary value problem. The quality of both stress resultants and displacements is investigated. Although the underlying concept addresses beams, plates and shells, the present contribution illustrates the methodology for the Timoshenko beam.

Zusammenfassung

The higher inter-element continuity of the Isogeometric Analysis (IGA) applying NURBS functions for geometry as well as mechanics opens up new possibilities in the analysis of thin-walled structures, i.e. beams, plates and shells. An important advantage is the straightforward implementation of classical theories, which require C1-continuity, e. g. Kirchhoff-Love shell formulations. Based on these "simplest" models shear deformable theories, introducing Timoshenko and Reissner-Mindlin kinematics, are formulated in a hierarchic manner. In contrast to using total rotations the present formulations introduce incremental transverse shear rotations as primary variables. This reparametrization of the kinematic equations can be established in several different fashions, which is addressed in this contribution. All parametrizations have in common, that they lead to shear deformable beam, plate and shell formulations being intrinsically free from transverse shear locking. This is a remarkable feature since the spectrum of different smooth discretization schemes in IGA and other fields of computational shell analysis is continuously growing and the proposed hierarchic shell formulations are locking-free, independent of the underlying discretization.

Zusammenfassung

A modified local NURBS discrete strain gap (DSG) approach to eliminate membrane locking for geometrically linear isogeometric shell finite elements (Kirchhoff-Love type) is presented. The DSG method from Echter and Bischoff (2010) is modified in order to improve the efficiency by means of a local projection method, leading to sparse structured global stiffness matrices. The ability to eliminate locking effects for thin shell finite elements is preserved. However, the locality is partially accompanied by a suboptimal convergence of displacements. The efficiency of the formulation is further improved by modifying the in-plane shear gap integration. Additional an hybrid-mixed formulation based on a modified Hellinger-Reissner variational principle is proposed. Inspired by the DSG method, the basic idea is to solely use displacement(-like) primary variables, which may offer certain advantages compared to classical mixed approaches, since no lower ansatz space has to be constructed. The efficiency and quality of the proposed methods are assessed and compared with other methods. An intense focus lies on the assessment of stress resultants.

Zusammenfassung

Die vorliegende Bachelorarbeit beschäftigt sich mit der Vermeidung geometrischer Lockingeffekte bei eindimensionalen isogeometrischen finiten Elementen für gekrümmte Balkenstrukturen. Das Augenmerk liegt hierbei auf dem, für die isogeometrische Analyse prädestinierten, schubstarren Bernoullibalken und dessen schubweiche Erweiterung zum hierarchischen Timoschenkobalken. Es werden auftretende Lockingeffekte untersucht und analog zu bereits bestehenden Schalenformulierung hybrid gemischete Spannungsformulierungen erfolgreich an numerischen Beispielen angewandt. Ein weiterer Aspekt ist die Untersuchung von Randbedingungsformulierungen und oszillierenden Schubkräften beim hierarchischen Timoschenkobalken.