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S. Wolff, C. Bucher:
"A finite element method based on C0-continuous assumed gradients";
International Journal for Numerical Methods in Engineering, 86 (2011), 7; 876 - 914.

This article presents an alternative approach to assumed gradient methods in FEM applied to three-dimensional elasticity.
Starting from nodal integration (NI), a general C0-continuous assumed interpolation of the deformation gradient is formulated. The assumed gradient is incorporated using the principle of Hu-Washizu. By dual Lagrange multiplier spaces the functional is reduced to the displacements as the only unknowns. An integration scheme is proposed where the integration points coincide with the support points of the interpolation. Requirements for regular finite element meshes are explained. Using this interpretation of NI, instabilities (appearance of spurious modes) can be explained. The article discusses and classifies available strategies to stabilize NI such as penalty methods, SCNI, $\alpha$-FEM. Related approaches, as the smoothed finite element method, are presented and discussed. New stabilization techniques for NI are presented being entirely based on the choice of the assumed gradient interpolation, i.e. nodal-bubble support, edge-based support and support using tensor-product interpolations. A strategy is presented how the interpolation functions can be derived for various element types. Interpolation functions for the 1st order hexahedral element, the 1st order and the 2nd order tetrahedral element are given. Numerous examples illustrate the strengths and limitations of the new schemes.

nodal integration; SFEM; assumed gradient; continuous strain; stabilization; dual multipliers

http://dx.doi.org/10.1002/nme.3082

Project Head Christian Bucher:
Formulierung und Test eines glatten zeitadaptiven Raum-Zeit-Integrators