Open Access

ARTICLE

A Simple Procedure to Develop Efficient & Stable Hybrid/Mixed Elements, and Voronoi Cell Finite Elements for Macro- & Micromechanics

L. Dong¹, S. N. Atluri²

1 Department of Civil & Environmental Engineering, University of California, Irvine, Email: leit-ingd@uci.edu
2 Center for Aerospace Research & Education, University of California, Irvine

Computers, Materials & Continua 2011, 24(1), 61-104. https://doi.org/10.3970/cmc.2011.024.061

Abstract

A simple procedure to formulate efficient and stable hybrid/mixed finite elements is developed, for applications in macro- as well as micromechanics. In this method, the strain and displacement field are independently assumed. Instead of using two-field variational principles to enforce both equilibrium and compatibility conditions in a variational sense, the independently assumed element strains are related to the strains derived from the independently assumed element displacements, at a finite number of collocation points within the element. The element stiffness matrix is therefore derived, by simply using the principle of minimum potential energy. Taking the four-node plane isoparametric element as an example, different hybrid/mixed elements are derived, by adopting different element strain field assumptions, and using different collocation points. These elements are guaranteed to be stable. Moreover, the computational efficiency of these elements is far better than that for traditional hybrid/mixed elements, or even better than primal finite elements, because the strain field is expressed analytically as simple polynomials (whereas, in isoparametric displacement-based element, the strain field is far more complicated), with nodal displacements as unknowns. The essential idea is thereafter extended to Voronoi cell finite elements, for micromechanical analysis of materials. Neither these four-node hybrid/mixed elements nor the Vonroni cell finite elements need to satisfy the equilibrium conditions a priori, making them suitable for extension to geometrically nonlinear and dynamical analyses. Various numerical experiments are conducted using these new elements, and the results are compared to those obtained by using traditional hybrid/mixed elements and primal finite elements. Performances of the different elements are compared in terms of efficiency, stability, invariance, locking, sensitivity to mesh distortion, and convergence rates.

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