Shear deformable shell elements for large strains and rotations

Abstract

Well-known finite element concepts like the Assumed Natural Strain (ANS) and the Enhanced Assumed Strain (EAS) techniques are combined to derive efficient and reliable finite elements for continuum based shell formulations. In the present study two aspects are covered:The first aspect focuses on the classical 5-parameter shell formulation with Reissner-Mindlin kinematics. The above-mentioned combinations, already discussed by Andelfinger and Ramm for the linear case of a four-node shell element, are extended to geometrical non-linearities. In addition a nine-node quadrilateral variant is presented. A geometrically non-linear version of the EAS-approach is applied which is based on the enhancement of the Green-Lagrange strains instead of the displacement gradient as originally proposed by Simo and Armero.In the second part elements are derived in a similar way for a higher order, so-called 7-parameter non-linear shell formulation which includes the thickness stretch of the shell (Bu¨chter and Ramm). In order to avoid artificial stiffening caused by the three dimensional displacement field and termed 'thickness locking', special provisions for the thickness stretch have to be introduced.

Fluid–structure interaction in lower airways of CT‐based lung geometries

“…The generalized minimal residual iterative solver with ILU preconditioner, which is available within the open-source package AZTEC [56], is used in the fluid domain. Spatial discretization of the airway walls is done through mixed/hybrid triangular elements based on a 7-parameter shell formulation [46,47]. This system is solved using the non-linear version of the 'generalized-method' of Chung and Hulbert [57] along with consistent linearization and a Newton-Raphson iterative scheme.…”

“…Owing to the complex geometries considered here, tetrahedral finite elements based on an arbitrary Lagrangian-Eulerian (ALE) description of motion are used. In the solid domain, we employ 7-parametric triangular shell elements, Bischoff and Ramm [46] and Bischoff et al [47], which represent the airway walls.…”

A 4-node 3D-shell element to model shell surface tractions and incompressible behavior

“…This behavior can be alleviated using the MITC approach applied to the normal strain [12,25,26]. To fulfill the condition that the normal strain be zero throughout the element when this strain is zero at the element nodes, we simply interpolate the normal strain bilinearly over the element using the nodal values directly calculated from the displacement assumptions.…”

“…To model the effects of surface tractions, it appears that 3D shell-solid elements are most appropriate, see [12][13][14][15][16] and the references therein. In the formulation of these elements, the top and bottom surfaces of the shell are represented geometrically and their positions are updated through the displacement degrees of freedom, just like in a fully 3D analysis of solids but with only one element layer through the shell thickness [1,2].…”

“…To improve the element behavior and computational effectiveness, instead of displacement degrees of freedom, enhanced strains have been used, see for example Refs. [12,19,20], and the references therein. However, enhanced strain formulations can be unstable, see Refs.…”