[en] Finite elements ; patch test ; variational principles

[en] The patch test is shown to be contained in the variational formulations of the finite element methods at the assembling level, all of which require the vanishing of the virtual work of interface connexion loads. By a systematic introduction of stress generating functions, attention is drawn to the fact that any given finite element model can be assembled in two different ways: either by identiification of a set of boundary displacements (leading to the direct stiffness method), or by identification of a set of local stress function values (leading to the dual direct flexibility method). Looking at any conjugate couple (generalized displacement - generalized surface traction) at an interface, one is strongly transmitted, the other weakly. Discretization of the zero virtual work condition at an interface of plate bending models, by means of Legendre polynomials expansions, allows q systematic construction of so-called "non-conforming" elements that pass the patch test. They are in fact identified with weakly conforming, but strongly diffusive , htbrids, and the lowest degree element (quadratic) is in fact the Morley constant-moment element. Examples are given for higher displacement fields. The case of plate stretching elements can be handles by duality, the difficulties being here associated with the requirements for diffusivity. Non diffusive elements that pass the zero interface virtual worktest can be constructed systematically and are identifiable with weakly diffusive, but strongly conforming, hybrids, of the type first proposed by T.H.H. Pian.