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Asymmetric Shell Elements Based on a Corrected Updated-Lagrangian Approach

P. Areias1,2,3, T. Rabczuk4, D. Dias-da-Costa5,6

Corresponding Author. ISI search: areias p*; email:,, Ph.: +351 96 3496307, Fax: +351 266745394, URL:
Physics Department, University of Évora, Colégio Luís António Verney, Rua Romão Ramalho, 59, 7002-554 Évora, Portugal
Institute of Structural Mechanics, Bauhaus-University Weimar, Marienstraße 15, 99423 Weimar, Germany
INESC Coimbra, Rua Antero de Quental 199, 3000-033 Coimbra
Civil Eng. Dept., University of Coimbra, Rua Luís Reis Santos, 3030-788 Coimbra, Portugal

Computer Modeling in Engineering & Sciences 2012, 88(6), 475-506.


Surprisingly good displacement results are obtained by using the Petrov-Galerkin method with assumed and enhanced metric components in the test functions and enhanced metric components in the trial functions. Cartesian trial functions are required to ensure completeness and assumed/enhanced metric components are introduced to ensure high coarse-mesh accuracy. In the trial functions, the original incompatible-mode in-plane Q6 element by Wilson et al. can be used without violating the patch test. As a beneficial side-effect, Newton-Raphson convergence behavior for non-linear problems is improved. Transverse-shear and in-plane patch tests are satisfied while distorted-mesh performance is better than with symmetric formulations due to the absence of coordinate transformation. Covariant coordinates are used to calculate the (mixed) test metric and a combination of Cartesian coordinates and quadratic terms in the metric are used for the trial functions. Classical test functions with assumed metric components are required for compatibility reasons. Verification tests are performed with very good performance being observed in all of them. Applications to large displacement elasticity and finite strain plasticity are shown with both low sensitivity to mesh distortion and high accuracy. A equilibrium-consistent (and consistently linearized) updated-Lagrangian algorithm is proposed and tested. Concerning the time-step dependency, it was found that the consistent updated-Lagrangian algorithm is nearly time-step independent and can replace the multiplicative plasticity approach if only moderate elastic strains are present.

Cite This Article

Areias, P., Rabczuk, T., Dias-da-Costa, D. (2012). Asymmetric Shell Elements Based on a Corrected Updated-Lagrangian Approach. CMES-Computer Modeling in Engineering & Sciences, 88(6), 475–506.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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