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. Bishay, S. N. Atluri

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26 Scopus citations


Higher-order two-dimensional as well as low and higher-order threedimensional 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 threedimensional 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)]. The same idea of the simple collocation is used in developing a general threedimensional Voronoi cell finite element, denoted as VCFEM-RBF-W, based on radial basis functions (RBF) as the interior displacement fields and the WachspressBarycentric linear functions as the boundary surface displacement field, for modeling micro-mechanics of solids. The compatibility between the interior and boundary displacements in the present VCFEM-RBF-W element is enforced using two methods: the first by collocation at some carefully chosen points at the boundaries of the VCFEM-RBF-W element, and the second by using the least squares method which can be considered as the limiting case of the collocation method when the number of collocation points increases to infinity. The developed 3D Voronoi cell finite element has an arbitrary number of faces, and each face has an arbitrary number of sides or edges. Some numerical experiments are presented to evaluate the performance of this new element. The VCFEM-RBF-W element is then used in a micro-mechanical application of determining the effective material properties of functionally graded materials (FGM), and the results are found to be in agreement with those of the experiments, and are better than those determined by other models used in the literature. The new VCFEM-RBF-W element formulation is much simpler and efficient, as compared to the VCFEM-HS developed by Ghosh and coworkers (1991-2011), based on Pian's hybrid stress method. The new elements are suitable for extension to dynamical, geometrically nonlinear, elastic-plastic, and fracture analyses.

Original languageEnglish
Pages (from-to)41-97
Number of pages57
JournalCMES - Computer Modeling in Engineering and Sciences
Issue number1
StatePublished - 2012


  • Collocation
  • Efficiency
  • Functionally graded materials
  • Higher order
  • Hybrid/Mixed finite elements
  • Invariance
  • Least square
  • Locking
  • Radial basis functions
  • Three-dimensional finite elements
  • Voronoi cells
  • Wachspress Barycentric coordinates


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