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

Abstract

Summary In this paper, we define the spurious kinematic modes of hybrid equilibrium 2D and 3D simplicial elements of general degree and present the results of studies on the stability of star patches of such elements. The approach used in these studies is based on first establishing the kinematic properties of a pair of elements that share an interface. This approach is introduced by its application to star patches of hybrid equilibrium triangular plate elements for modelling membrane and plate bending problems and then generalised to 3D continua. It is then shown how the existence of spurious kinematic modes depends on the topological and geometrical properties of a patch, as well as on the degree of the polynomial approximation functions of stress and displacement. Copyright © 2015 John Wiley & Sons, Ltd.

A superconvergent hybridisable discontinuous Galerkin method for linear elasticity

“…Eventually, other high‐order formulations based on a description of the displacement field by means of a hybrid variable and a strong enforcement of the symmetry of the stress tensor have been discussed in other works . For a comprehensive presentation of the dual hybrid approaches known as equilibrium formulations , the interested reader is referred to the monograph .…”

“…Eventually, other high-order formulations based on a description of the displacement field by means of a hybrid variable and a strong enforcement of the symmetry of the stress tensor have been discussed in other works. [44][45][46] For a comprehensive presentation of the dual hybrid approaches known as equilibrium formulations, the interested reader is referred to the monograph. 47 Inspired by the works of Arnold and Brezzi 48 and Stenberg 18 on mixed methods, Cockburn et al have investigated several procedures to construct a superconvergent postprocessed solution by exploiting both the optimally convergent primal and mixed variables.…”

Equilibrium finite element and error estimation for frictional contact problem based on linear complementarity problem formulation

“…In order to circumvent these difficulties, the hybrid equilibrium element is subsequently proposed. [20][21][22][23][24][25][26][27][28] It constructs the equilibrated stress field by polynomial basis functions in each element, and introduces auxiliary weighting functions, also known as elementary boundary displacements, to guarantee the interelement traction continuity on each internal element boundary. 29 As is noteworthy, the hybrid equilibrium element is plagued by spurious kinematic modes (SKMs).…”

“…However, it is difficult to deal with multiply‐connected domains and in linear elasticity problems, the stress functions should be $$$ {C}^1 $$$‐continuous, turning into another difficult interpolation problem. In order to circumvent these difficulties, the hybrid equilibrium element is subsequently proposed 20‐28 . It constructs the equilibrated stress field by polynomial basis functions in each element, and introduces auxiliary weighting functions, also known as elementary boundary displacements, to guarantee the interelement traction continuity on each internal element boundary 29 .…”

“…To eliminate the SKMs, the macro-element/super-element technique 20,29 and the restraining approach 25 have been developed. Up till now, the hybrid equilibrium element method has been successfully applied to many problems, including elastostatics, 28,29 plates, 22,26 delamination/crack propagation, 23 elastodynamics, 24 and folded plates/shells. 25 Recently, an alternative traction-based equilibrium element 30,31 is proposed for the EFEM.…”