For full functionality of this site it is necessary to enable JavaScript. Here are the instructions how to enable JavaScript in your web browser.

Position within the page tree

Home
Research
Current reseach projects
Geometric and material hourglassing

Geometric and material hourglassing in large deformations

Research project

Development and application of stable, locking-free finite elements for large deformations

Overview

Geometric Hourglassing
Material Hourglassing

Project description

The finite element method is used for many problems in mechanics, such as the deformation analysis of solids with elastic or elastoplastic material behavior. This often leads to locking effects, which can result in a significantly too stiff system response.

Various approaches exist for avoiding locking in geometrically linear and non-linear finite elements. However, many questions remain unanswered, particularly for non-linear elements. Due to their general applicability, this research project focuses on enhanced assumed strain (EAS) elements.

A central problem of EAS elements in large deformations is so-called hourglassing, an undesirable numerical instability. A distinction is made between geometric and material hourglassing.

Comparison of physically occurring instabilities (shown here: buckling modes) and numerical instabilities (hourglassing).

Geometric Hourglassing

Geometric hourglassing occurs due to instabilities in the geometric stiffness under compressive loads.

These effects were already investigated in detail in the previous project. This led to the development of modified EAS elements that prevent instabilities in many cases. An important point here is that physically occurring instabilities can still be represented. This project now examines further open questions, in particular the occurrence of hourglassing in inhomogeneous deformations and distorted meshes.

Material Hourglassing

Material hourglassing, on the other hand, is triggered by instabilities arising from material stiffness. This often occurs during material softening, such as in the elasto-plastic regime.

The first step in this project is to systematically investigate the causes of this numerical phenomenon. Based on these findings, stabilized finite elements will then be developed. A key objective here is to continue to be able to accurately represent the physically correct material response.

Elasto-plastic tensile test: simulated with locking standard elements (no hourglassing).

Elasto-plastic tensile test: simulated with EAS elements, whereby numerical instabilities occur in the form of hourglassing.

Project data

Project title:
Nonlinear finite element technology for stable and locking-free analysis of large deformation problems
Funding:
Deutsche Forschungsgemeinschaft (DFG), Sachbeihilfe BI 722/11-2, GEPRIS-Projektnummer 299369509

Publications

Bieber, S. (2024). Locking and hourglassing in nonlinear finite element technology [Doktorarbeit, Bericht Nr. 76. Institut für Baustatik und Baudynamik der Universität Stuttgart]. https://doi.org/10.18419/opus-14214
- Abstract
Abstract
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.
Bieber, S., Auricchio, F., Reali, A., & Bischoff, M. (2023). Artificial instabilities of finite elements for nonlinear elasticity: Analysis and remedies. International Journal for Numerical Methods in Engineering. https://doi.org/10.1002/nme.7224
- Abstract
Abstract
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.
Pfefferkorn, R., Bieber, S., Oesterle, B., Bischoff, M., & Betsch, P. (2020). Improving Efficiency and Robustness of EAS Elements for Nonlinear Problems. International Journal for Numerical Methods in Engineering, 122, Article 8. https://doi.org/10.1002/nme.6605
- Abstract
Abstract
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.

Researcher:

Geometric and material hourglassing in large deformations

Overview

Project description

Geometric Hourglassing

Material Hourglassing

Project data

Publications

Abstract

Abstract

Abstract

Researcher:

Henrik Jakob

Audience

Formalities

Services

Organization

Geometric and material hourglassing in large deformations

Overview

Project description

Geometric Hourglassing

Material Hourglassing

Project data

Publications

Abstract

Abstract

Abstract

Researcher:

Henrik Jakob

Here you can reach us

Audience

Formalities

Services

Organization