02 Fakultät Bau- und Umweltingenieurwissenschaften

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Author: search.filters.author.Bieber, Simon

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    Artificial instabilities of finite elements for nonlinear elasticity : analysis and remedies
    (2023) Bieber, Simon; Auricchio, Ferdinando; Reali, Alessandro; Bischoff, Manfred
    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.
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    Investigation and elimination of nonlinear Poisson stiffening in 3d and solid shell finite elements
    (2022) Willmann, Tobias; Bieber, Simon; Bischoff, Manfred
    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.
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    Locking and hourglassing in nonlinear finite element technology
    (Stuttgart : Institut für Baustatik und Baudynamik, Universität Stuttgart, 2024) Bieber, Simon; Bischoff, Manfred (Prof. Dr.-Ing. habil.)
    This thesis deals with locking and hourglassing issues that arise in nonlinear finite element analyses of problems in mechanics. The major focus lies on the analysis of these numerical deficiencies, the design of suitable benchmarks and the development of novel remedies. A new nonlinear locking phenomenon is described. It is caused by parasitic nonlinear strain terms and it is particularly pronounced for large element deformations in combination with higher-order integration and a critical parameter, such as the element aspect ratio or the Poisson's ratio. To avoid this problem within the popular class of enhanced assumed strain formulations, novel strain enhancements are presented. An analytical solution of a tailored finite bending problem is used to benchmark the newly proposed element formulations. Further, the problem of hourglassing in both compression and tension of solid bodies is analysed. It is shown that the underlying causes of hourglassing can be explained by geometry-induced and material-induced trigger mechanisms of structural instabilities. Crucial for understanding as well as benchmarking is the analytical in-depth analysis of a large strain bifurcation problem. Based on these insights, an obvious remedy for the geometric hourglassing phenomenon is presented. The last part of this thesis is devoted to the efficient algorithmic treatment of the computation of instability points. The difficulties in choosing a suitable load-stepping approach with methods from the literature are discussed and a methodological idea of an adaptive load-stepping scheme is presented. Efficiency and practicability are demonstrated for several benchmarks.
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    Improving efficiency and robustness of enhanced assumed strain elements for nonlinear problems
    (2021) Pfefferkorn, Robin; Bieber, Simon; Oesterle, Bastian; Bischoff, Manfred; Betsch, Peter
    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 nonlinear analyses is their lack of robustness in the Newton-Raphson scheme, which is characterized by the necessity of small load increments and large number 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.