On evaluating the inf–sup condition for plate bending elements

Abstract

This paper addresses the evaluation of the inf-sup condition for Reissner-Mindlin plate bending elements. This fundamental condition for stability and optimality of a mixed ÿnite element scheme is, in general, very di cult to evaluate analytically, considering for example distorted meshes. Therefore, we develop a numerical test methodology. To demonstrate the test methodology and to obtain speciÿc results, we apply it to standard displacement-based elements and elements of the MITC family. Whereas the displacement-based elements fail to satisfy the inf-sup condition, we ÿnd that the MITC elements pass our numerical test for uniform meshes and a sequence of distorted meshes. ?

Measuring the convergence behavior of shell analysis schemes

“…Indeed, even for 'just' plate bending problems, analytical investigations regarding the inf-sup condition are limited [9,10]. However, numerical tests can be performed for plates and shells [12][13][14], albeit always for only a given geometry, boundary conditions and mesh sequences used. Hence these numerical inf-sup tests only give limited information, and indeed such insight into whether the inf-sup condition may be generally satisfied can be more directly obtained by convergence tests as we perform below [3,15,16].…”

The inf–sup condition and its evaluation for mixed finite element methods

“…Let V h B h  W h be the ®nite element interpolation space of the plate section rotations b h (in B h ) and transverse displacement w h (in W h ), with V being the space of the continuous problem, and let G h be the ®nite element interpolation space of the transverse shear strains times lkat 2 . Then the ®nite element problem is [4]:…”

“…The reduction operator R h is introduced to``weaken'' the shear strain constraint in order to not have shear locking in the formulation. The operator must be chosen judiciously to preserve consistency and to satisfy the ellipticity and inf±sup conditions [1,4]. Note that the formulation in Eq.…”

“…(40) and (41), the vectors W h and V h contain the ®nite element nodal displacement and rotation variables. The inf±sup value c h is given by [4] c h P c In the second part, we need to show that the C H h -and C H -norms are equivalent so that in fact the conclusions reached using Eq. (38) with the C H h -norm are also applicable when the C H -norm is used.…”

The three‐field formulation for elasticity problems

“…This cannot however be easily done with our original choice for the norms in Φ and in M (see e.g. [29]). We introduce therefore some different norms to be used instead of the original ones.…”