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On the stability of hybrid equilibrium and Trefftz finite element models for plate bending problems

Maunder E. A. W.

Computer Assisted Mechanics and Engineering Sciences

2008

Vol. 15, No. 3/4

279-288

This paper is concerned with hybrid stress elements in the context of modelling the behaviour of plates subject to out of plane loading and based on Reissner-Mindlin assumptions. These elements are considered as equilibrium elements with statically admissible stress fields of which Trefftz fields form a special case. The existence of spurious kinematic modes in star patches of triangular elements is reviewed when boundary displacement fields are defined independently for each side. It is shown that for elements of moment degree > 1, the spurious modes for stars only exist at specific locations and/or for certain configurations. The kinematic properties of these modes are used to define sufficient conditions for the stability of a complete mesh of triangular elements. A method is proposed to check mesh stability, and introduce local modifications to ensure overall stability.

Trefftz in translation

Maunder E. A. W.

Computer Assisted Mechanics and Engineering Sciences

2003

Vol. 10, No. 4

545-563

This paper reviews the important concepts presented by Trefftz in 1926 regarding bounds to solutions, error estimation, and hybrid fields for use with domain decomposition. Observations are offered from the perspective of today's relatively mature state of the art in finite element methods. The numerical examples presented by Trefftz are also reviewed with the benefit of `exact' solutions made available from current commercial finite element methods. The accuracies of the solutions given by Trefftz are quantified and compared, and the effectivity indices of Trefftz's proposed error estimates are also quantified. An English translation of the original German version of Trefftz's paper is included for reference in an Appendix.

Hybrid equilibrium plate elements of high degree

Maunder E. A. W.

Computer Assisted Mechanics and Engineering Sciences

2003

Vol. 10, No. 4

531-543

This paper concerns the modelling of plate bending problems governed by Reissner-Mindlin theory when hybrid equilibrium elements of high polynomial degree are used. The fields of statically admissible stress-resultants are categorised into three types according to the nature of their incompatibilities, i.e. pure Trefftz or strongly compatible, weakly compatible, and hyperstatic or strongly incompatible. The effects of this categorisation are reflected in the element formulation. Incompatibilities are quantified in terms of local discontinuities which also account for transverse twist terms. The construction of bases for the three corresponding subspaces of stress-resultants by numerical and/or algebraic means is reviewed. The potential use of a reformulated element is considered in the context of glass plate structures where residual or hyperstatic stresses play an important role.