# Intrinsically locking-free formulations

Research project

A study on geometric and material locking-free finite element formulations

## Overview

• Applicability of hierarchic and mixed-displacement formulations for structural problems
• Hierarchic reparameterization techniques for membrane locking
• Eliminating volumetric locking effects intrinsically

## Project description

### Background

Locking results in an inferior rate of convergence in the pre-asymptotic range for problems in structural mechanics. Locking effects depend on certain critical parameters, for instance the slenderness of a shell. This numerical pathology results in underestimated displacement values and oscillating, parasitic stresses. Locking exists irrespective of the discretization scheme (like the finite element method (FEM), isogeometric analysis (IGA), mesh-free methods or collocation methods) used to solve the underlying physical equations. Even though many remedies for locking (like enhanced and assumed strain methods, reduced integration, hybrid methods, discrete strain gap method) were developed over the years, they are mostly specific to one discretization method (e.g. the FEM) and not directly transferable to other discretization schemes. However, the origin of locking does not arise from the discretization scheme but from the differential equations governing the physical problem itself. This leads to the motivation to unfold the too stiff behavior on a theoretical level.

The philosophy is to remove the cause of locking instead of fighting its symptoms.

### Hierarchic and mixed-displacement formulations

The objective is to obtain an intrinsically locking-free problem which could be solved by using any standard discretization scheme and obtaining coarse mesh accuracy independent of problem parameters without special tricks. On that note in the last decade, a set of hierarchic formulations and a Mixed-Displacement (MD) formulation were developed. The former was obtained by reparametrizing the governing kinematic equations and the latter includes a specifically designed differential operator, which is applied to new independent variables to create a better approximation space. The basic validity of these approaches is already confirmed with several numerical examples. Nevertheless, these methods carry their own challenges.

Shear stress resultant (Q_xz) for a simply supported plate (10x10) with uniformly distributed load and slenderness L/t=10000
Shear stress resultant (Q_xz) for a simply supported plate with uniformly distributed load and slenderness L/t=10000 at the mid-section (y=5.0)

### Challenges and research focus

For these novel formulations, application of boundary conditions is not straightforward. For instance, reparameterization results in choosing the total shear strain or other newly introduced shear displacements as independent variables. This makes it complicated to apply rotational Dirichlet boundary conditions. These new independent variables also create a parasitic null space of the stiffness matrix, which has to be removed by establishing certain additional constraints. The newly introduced independent variables in the mixed-displacement formulations require of similar constraints. In addition to the above, hierarchic formulations are more suitable to alleviate only shear locking, while the mixed-displacement formulation can be used to avoid both shear and membrane locking.

The current research work focuses on studying the additional constraints required with the motivation of more general applicability of these formulations. Reparameterization techniques are explored further to diminish also membrane locking effects. The research also emphasizes on investigating the possibility to develop similar techniques to attenuate volumetric locking effects.

## Project data

Project title:
Intrinsically locking-free formulations for problems in structural mechanics
Funding:

German Research Foundation (DFG), Research Grant BI 722/12-1, GEPRIS project number 452589815

### Publications

1. Oesterle, B., Thierer, R., Krauß, L.-M., & Bischoff, M. (2024). Hierarchische Formulierungen für statische und dynamische Analysen von Flächentragwerken. In B. Oesterle, A. Bögle, W. Weber, & L. Striefler (Eds.), Berichte der Fachtagung Baustatik – Baupraxis 15, 04. und 05. März 2024, Hamburg (pp. 357--364). https://doi.org/10.15480/882.9247
2. Bieber, S., Oesterle, B., Bischoff, M., & Ramm, E. (2022). Strategy for Preventing Membrane Locking Through Reparametrization. In F. Aldakheel, B. Hudobivnik, M. Soleimani, H. Wessels, C. Weißenfels, & M. Marino (Eds.), Current Trends and Open Problems in Computational Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-87312-7_7
3. Bieber, S., Oesterle, B., Ramm, E., & Bischoff, M. (2018). A variational method to avoid locking – independent of the discretization scheme. International Journal for Numerical Methods in Engineering, 114, 801–827. https://doi.org/10.1002/nme.5766
4. Oesterle, B., Bieber, S., Sachse, R., Ramm, E., & Bischoff, M. (2018). Intrinsically locking-free formulations for isogeometric beam, plate and shell analysis. Proc. Appl. Math. Mech., 18. https://doi.org/10.1002/pamm.201800399
5. Oesterle, B., Sachse, R., Ramm, E., & Bischoff, M. (2017). Hierarchic isogeometric large rotation shell elements including linearized transverse shear parametrization. Computer Methods in Applied Mechanics and Engineering, 321, 383–405. https://doi.org/10.1016/j.cma.2017.03.031
6. Oesterle, B., Ramm, E., & Bischoff, M. (2016). A shear deformable, rotation-free isogeometric shell formulation. Computer Methods in Applied Mechanics and Engineering, 307, 235–255. https://doi.org/10.1016/j.cma.2016.04.015
7. Echter, R., Oesterle, B., & Bischoff, M. (2013). A hierarchic family of isogeometric shell finite elements. Computer Methods in Applied Mechanics and Engineering, 254, 170–180. https://doi.org/10.1016/j.cma.2012.10.018

M. Sc.

Scientific Staff