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An analysis of the TDNNS method using natural norms
Astrid S Pechstein1, Joachim Schöberl2
11Institute of Technical Mechanics, Johannes Kepler University Linz, Altenbergerstr. 69, 4040 Linz, Austria.
Numerische Mathematik
|April 21, 2018
Summary
The tangential-displacement normal-normal-stress (TDNNS) method offers robust finite element analysis for elasticity problems. This study provides a rigorous mathematical framework and error estimates for the TDNNS method, enhancing its applicability in structural mechanics.
Area of Science:
- Computational Mechanics
- Solid Mechanics
- Numerical Analysis
Background:
- The tangential-displacement normal-normal-stress (TDNNS) method is a finite element technique for mixed elasticity.
- Previous work demonstrated its optimal convergence and robustness against locking, but lacked analysis in natural norms.
- A rigorous mathematical theory for the infinite-dimensional problem was missing.
Purpose of the Study:
- To present a sound mathematical theory for the infinite-dimensional TDNNS problem.
- To define appropriate function spaces for displacement and stress.
- To establish stability and derive optimal a-priori error estimates.
Main Methods:
- Utilized Sobolev spaces [Formula: see text] for displacement and [Formula: see text] for stresses.
- Defined trace operators for the normal-normal stress component.
- Demonstrated stability with respect to [Formula: see text] and discrete [Formula: see text] norms.
Main Results:
- Established a rigorous mathematical foundation for the TDNNS method.
- Proved stability of the finite element problem in relevant norms.
- Obtained a-priori error estimates of optimal order.
Conclusions:
- The presented analysis validates the TDNNS method for elasticity.
- The numerical techniques serve as a basis for advanced structural mechanics models.
- This work fills a gap in the theoretical understanding of the TDNNS method.
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