Two-field formulations for isogeometric Reissner–Mindlin plates and shells with global and local condensation

Abstract

In this paper, mixed formulations are presented in the framework of isogeometric Reissner–Mindlin plates and shells with the aim of alleviating membrane and shear locking. The formulations are based on the Hellinger-Reissner functional and use the stress resultants as additional unknowns, which have to be interpolated in appropriate approximation spaces. The additional unknowns can be eliminated by static condensation. In the framework of isogeometric analysis static condensation is performed globally on the patch level, which leads to a high computational cost. Thus, two additional local approaches to the existing continuous method are presented, an approach with discontinuous stress resultant fields at the element boundaries and a reconstructed approach which is blending the local control variables by using weights in order to compute the global ones. Both approaches allow for a static condensation on the element level instead of the patch level. Various numerical examples are investigated in order to verify the accuracy and effectiveness of the different approaches and a comparison to existing elements that include mechanisms against locking is carried out.

On the use of mixed basis function degrees within a convective isogeometric element formulation

“…In order to counteract the different locking phenomena that can occur while solving linear elasticity problems using common finite element formulations, various methods can be employed [1,2]. Particularly, this comprises higher order formulations or mixed methods [3][4][5]. As a result, the computational effort is increased.…”

Extending CAS elements to remove shear and membrane locking from quadratic NURBS‐based discretizations of linear plane Timoshenko rods

“…However, when applied to structural theories that take into account transverse shear deformations, spline-based discretizations suffer from the same types of locking as conventional FEA discretizations based on Lagrange polynomials. [19][20][21] Thus, spline-based discretizations of curved Timoshenko rods [22][23][24][25][26] and Reissner-Mindlin shells [27][28][29][30][31] are adversely affected by shear and membrane locking. Shear and membrane locking lead to displacements, rotations, and bending moments with smaller values than expected and shear and membrane forces with large-amplitude spurious oscillations.…”

High‐fidelity tensor‐decomposition based matrix formation for isogeometric buckling analysis of laminated shells with solid‐shell formulation

“…The researchers have shown that the problem can be solved or alleviated by using higher-order finite-element methods or adding special locking treatments to lower-order methods. The typical locking treatment approaches, which may not be restricted to solid-shell elements, include assumed natural strains (ANS), 18,19 enhanced assumed strain (EAS), 20 discrete strain gap (DSG), 21 B-method, [22][23][24][25] mixed formulations, 22,[26][27][28] selective/reduced integration, 26,[29][30][31] and projection techniques based on the moving least square (MLS). 32 Extensive literatures have reported that shells are studied by kinds of numerical methods: the finite element method (FEM), [33][34][35] the meshfree method, [36][37][38][39] and analytical 40,41 or semianalytical 42,43 methods and so forth.…”

“…The researchers have shown that the problem can be solved or alleviated by using higher‐order finite‐element methods or adding special locking treatments to lower‐order methods. The typical locking treatment approaches, which may not be restricted to solid‐shell elements, include assumed natural strains (ANS), 18,19 enhanced assumed strain (EAS), 20 discrete strain gap (DSG), 21 ‐method, 22–25 mixed formulations, 22,26–28 selective/reduced integration, 26,29–31 and projection techniques based on the moving least square (MLS) 32 …”