Abstract
We present the five field mixed finite element formulation introduced by Armero and extend it to 3D problems. It combines the nonlinear mixed pressure element with an enhanced assumed strain (EAS) method employing the transposed Wilson modes. The well-known mixed pressure element arises from a Hu-Washizu-like variational principle, where dilatation and pressure are independent variables. This functional is further modified using the EAS framework to get the mixed formulation presented in this work. The element is compared to several mixed pressure and EAS element formulations showing its great performance in alleviating volumetric and shear locking in large deformation problems. The main focus of the present work is spurious hourglassing of mixed finite elements that arise in hyperelastic and elasto-plastic simulations.
This article is dedicated to Professor Peter Wriggers who fostered the successful development of Computational Mechanics and who has made significant scientific contributions to the field. This includes mixed finite element methods for large deformation problems, a topic also addressed in the present work.
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Notes
- 1.
Note that all terms in (6) refer to the spatial configuration except for terms containing \(\mathbf {P}\) or \(\delta \mathbf {P}\) since these terms vanish on discrete level due to the orthogonality condition given below.
- 2.
We also tested the same element with all nine transposed Wilson Modes for the enhanced field. That element is even softer than the element of Armero, which is too soft in pure bending problems with undistorted meshes. Thus, the element with all 9 enhanced modes is not taken into account in subsequent investigations.
- 3.
- 4.
- 5.
Elements with MIP method need even slightly more iterations in this example, which shows that there are cases where this method is not advantageous. Nevertheless, it improves convergence in many cases (see Pfefferkorn et al. [9]).
- 6.
Due to the non-linearity of this expression a solver e.g. Newton’s method might be required to solve for \(\lambda _2\). However, for the considered material models, an analytic solution is possible.
- 7.
Q1/S5 has a different behavior in tension than all other elements which might emerge from problems with the Legendre transformation.
- 8.
We use the Newton tolerance \(||R||<1\cdot 10^{-8}\) as well as a maximum of 20 numerical iterations per load step.
- 9.
The convergence test of a circular ring (see e.g. Pfefferkorn et al. [9]) provides a good example of the poor behavior of H1/P0ET6 in shell-like problems.
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Hille, M., Pfefferkorn, R., Betsch, P. (2022). Locking-Free Mixed Finite Element Methods and Their Spurious Hourglassing Patterns. In: Aldakheel, F., Hudobivnik, B., Soleimani, M., Wessels, H., Weißenfels, C., Marino, M. (eds) Current Trends and Open Problems in Computational Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-87312-7_19
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