The stability of stars of triangular equilibrium plate elements

Abstract

SUMMARYEquilibrium models for finite element analyses are becoming increasingly important in complementary roles to those from conventional conforming models, but when formulating equilibrium models questions of stability, or admissibility of loads, are of major concern. This paper addresses these questions in the context of flat plates modelled with triangular hybrid elements involving membrane and/or flexural actions. Patches of elements that share a common vertex are considered, and such patches are termed stars. Stars may be used in global analyses as assemblies of elements forming macro-elements, or in local analyses. The conditions for stability, or the existence and number of spurious kinematic modes, are determined in a general algebraic procedure for any degree of the interpolation polynomials and for any geometric configuration. The procedure involves the determination of the rank of a compatibility matrix by its transformation to row echelon form. Examples are presented to illustrate some of the characteristics of spurious kinematic modes when they exist in stars with open or closed links.

Recovery of equilibrium on star patches from conforming finite elements with a linear basis

“…It is similar to the spurious kinematic mode at the vertices of an element described in [20]. However, for an admissible load, it is never excited and it can 1510 E. A. W. MAUNDER AND J. P. M. DE ALMEIDA be blocked by inserting a non-zero term in the lower diagonal block of the system, at the position that corresponds to the rotation of the node.…”

Recovery of equilibrium on star patches using a partition of unity technique

“…But we demonstrate that when this basis is linear it is necessary to solve a global problem, before solving the local ones. We also show how the results in [18] can be used to characterize the problems that are thus obtained, in particular to prove that the local problems are well-posed. This paper consists of nine parts, starting with this introduction and a brief review of equilibrium equations and some definitions related to star patches.…”

“…The perception that the results recently derived for patches of equilibrium elements [18] could be used in this context leads to the development of the variant flux-free technique presented in this paper. Each star patch is subject to a self-balanced system of loads, with homogeneous boundary tractions, which is derived without using the traction jumps from the compatible solution.…”

“…The conditions for stability of stars in two-dimensional continuum problems have been determined by the authors [18]; • the enforcement of compatibility is stronger, while preserving the local nature of the problem.…”

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

“…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]. Note that whatever the approach, the numerical cost is never negligible.…”