Dual‐mixed p and hp finite elements for elastic membrane problems

Abstract

This paper is dedicated to Professor Barna A. Szabà o on the occasion of his 65th birthday SUMMARY A complementary energy-based, dimensionally reduced plate model using a two-ÿeld dual-mixed variational principle of non-symmetric stresses and rotations is derived. Both the membrane and bending equilibrium equations, expressed in terms of non-symmetric mid-surface stress components, are satisÿed a priori introducing ÿrst-order stress functions. It is pointed out that (i) the membrane-, shear-and bending energies of the plate written in terms of ÿrst-order stress functions are decoupled, (ii) although unmodiÿed 3-D constitutive equations are applied, the energy parts do not contain the 1=(1 − 2 ) term for isotropic, linearly elastic materials. These facts mean that the ÿnite element formulation based on the present plate model should be free from shear locking when the thickness tends to zero and free from incompressibility locking when the Poisson ratio converges to 0.5, irrespective of low-order h-, or higher-order p elements are used.Curvilinear dual-mixed hp ÿnite elements with higher-order stress approximation and continuous surface tractions are developed and presented for the membrane (2-D elasticity) problem. In this case the formulation requires the approximation of three independent variables: two components of a ÿrst-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dualmixed elements is presented and compared to displacement-based hp ÿnite elements when the Poisson ratio converges to the incompressible limit of 0.5. The numerical results show that, as expected, the dual-mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, for both h-and p-extensions.

A Cook-féle membrán feladat megoldása p-verziós végeselem-modellekkel

“…dual-mixed végeselem-modelleket fejlesztett ki, többek között, lineáris sík-rugalmasságtani feladatokra [8][9] [5], nemlineáris síkbeli feladatokra [10][11] [12] és rugalmas-képlékeny feladatokra [13] [14]. Síkrugalmasságtani problémák és lemezfeladatok megoldására alkalmas p-és hp-verziós végeselem-modellek mutat be [15] és [16].…”

“…Ez a cikk a [15,16] munkákban ismertetett, feszültségfüggvényeken és forgásokon alapuló, p-verziós dual-mixed végeselem-modellt veti össze az elmozdulásmezőn alapuló p-verziós végeselem-modellel egy szakirodalomból jól ismert sík-rugalmasságtani peremérték-feladat, a Cook-féle membrán probléma numerikus megoldásán keresztül. A 2. fejezet ismerteti a Fraeijs de Veubeke-féle variációs elvet, majd származtatja annak sík-rugalmasságtani feladatokra érvényes alakját.…”

A higher‐order equilibrium finite element method

“…Remark The use of rotation ω as a Lagrange multiplier to enforce the rotational equilibrium condition was already discussed by Fraeijs de Veubeke. () See also the work of Bertoti for this approach. The use of vorticity as an independent unknown in flow problems is addressed in the work of Palha and Gerritsma …”

Fraeijs de Veubeke variational principle‐based cylindrical shell finite element model

“…The developments of the new dimensionally reduced cylindrical shell model and related dual-mixed finite elements are motivated by the results of the first-order stress function-based finite element formulations addressed in [1,2]. Cylindrical shell models using the equilibrium hypothesis (the definition is given in Section 3) have been developed and investigated assuming axisymmetric problems in [3][4][5].…”