Pure equilibrium tetrahedral finite elements for global error estimation by dual analysis

Abstract

SUMMARYThis study presents a general procedure of creating pure equilibrium tetrahedral finite elements for use under the elastostatic hypothesis. These pure equilibrium elements are of the Fraeijs de Veubeke type and the degree of the polynomial approximation functions of their internal stress field is the parameter generating this new elements family. The spurious kinematic modes (SKM), inherent in the equilibrium approach, are eliminated at the element level by converting each tetrahedron into a super-element defined as an assembly of four tetrahedral primitive elements. A mathematical discussion on the number of SKM of the primitive elements as well as their elimination by the super-element technique has been carried out. The development of first and second degree elements is presented here in detail and their efficiency in the frame of global error estimation by dual analysis is emphasized by two numerical applications. The main attribute of the error estimation by dual analysis is that it provides an upper bound on the global discretization error.

The stability of stars of triangular equilibrium plate elements

“…The total number of independent stress fields in element e, including any hyperstatic fields, will be denoted by n e s . Although it is a simple matter to check this condition by counting the appropriate quantities [24], it is not in general a sufficient condition. It fails to allow for the existence of hyperstatic stress fields within a star, which may exist as combinations of stress fields within individual elements.…”

The stability of stars of simplicial hybrid equilibrium finite elements for solid mechanics

“…In 2010, Kempeneers et al , extended their work in as regards a detailed study of pure equilibrium tetrahedral elements and macro‐elements formed from four tetrahedra. An important result, the number of spurious modes for a single element of general degree, was proved.…”

Improved recovery of admissible stress in domain decomposition methods — application to heterogeneous structures and new error bounds for FETI‐DP

“…Statically admissible stress field can be obtained by separate dual analysis [14], or by a postprocessing of the finite element displacement. Among these post-processors, we can cite the Element Equilibration Technique (EET) [18,21] and its recent variant [17,28,29], and flux-free techniques based on partition of unity [4,25,26,9,23,24].…”