Open Access

ARTICLE

High-Performance 3D Hybrid/Mixed, and Simple 3D Voronoi Cell Finite Elements, for Macro- & Micro-mechanical Modeling of Solids, Without Using Multi-field Variational Principles

P. L. Bishay¹, S.N. Atluri¹

1 Center for Aerospace Research & Education, University of California, Irvine, CA, USA.

Computer Modeling in Engineering & Sciences 2012, 84(1), 41-98. https://doi.org/10.3970/cmes.2012.084.041

Abstract

Higher-order two-dimensional as well as low and higher-order three-dimensional new Hybrid/Mixed (H/M) finite elements based on independently assumed displacement, and judiciously chosen strain fields, denoted by HMFEM-2, are developed here for applications in macro-mechanics. The idea of these new H/M finite elements is based on collocating the components of the independent strain field, with those derived from the independently assumed displacement fields at judiciously and cleverly chosen collocation points inside the element. This is unlike the other techniques used in older H/M finite elements where a two-field variational principle was used in order to enforce both equilibrium and compatibility conditions in a variational sense. The eight- and nine-node quadrilateral iso-parametric elements are used as examples of higher order two-dimensional elements; the eight-node brick element is used as an example of a low order three-dimensional element, while the twenty-node brick element is used as an example of higher order three-dimensional element. The performance of these new elements are compared with those of the primal (displacement-based) finite elements in terms of stability, efficiency, invariance, locking, and sensitivity to mesh distortion in various numerical experiments. All these new H/M elements proved to be stable, invariant, less sensitive to mesh distortion and experience no locking. The superiority of these new HMFEM-2 elements over the displacement-based elements is very much more significant for the low order elements than that for the higher order ones. The performance and efficiency of these new H/M finite elements are much better than that of many other H/M elements in the literature [Pian and co-workers (1964-1984), and Atluri and co-workers (1975-1984)].

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