https://hal.archives-ouvertes.fr/hal-01513346/document

Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement

Meshless methods: An overview and recent developments

A new four‐node (non‐flat) general quadrilateral shell element for geometric and material non‐linear analysis is presented. The element is formulated using three‐dimensional continuum mechanics theory and it is applicable to the analysis of thin and thick shells. The formulation of the element and the solutions to various test and demonstrative example problems are presented and discussed.

http://web.mit.edu/kjb/www/Publications_Prior_to_1998/A_Continuum_Mechanics_Based_Four-Node_Shell_Element_for_General_Nonlinear_Analysis.pdf

A continuum mechanics based four‐node shell element for general non‐linear analysis

The variational formulation and computational aspects of a three-dimensional finite-strain rod model, considered in Part I, are presented. A particular parametrization is employed that bypasses the singularity typically associated with the use of Euler angles. As in the classical Kirchhoff-Love model, rotations have the standard interpretation of orthogonal, generally noncommutative, transformations. This is in contrast with alternative formulations proposed by Argyris et al. [5–8], based on the notion of semitangential rotation. Emphasis is placed on a geometric approach, which proves essential in the formulation of algorithms. In particular, the configuration update procedure becomes the algorithmic counterpart of the exponential map. The computational implementation relies on the formula for the exponential of a skew-symmetric matrix. Consistent linearization procedures are employed to obtain linearized weak forms of the balance equations. The geometric stiffness then becomes generally nonsymmetric as a result of the non-Euclidean character of the configuration space. However, complete symmetry is recovered at an equilibrium configuration, provided that the loading is conservative. An explicit condition for this to be the case is obtained. Numerical simulations including postbuckling behavior and nonconservative loading are also presented. Details pertaining to the implementation of the present formulation are also discussed.

A three-dimensional finite-strain rod model. Part II: Computational aspects

https://hal.archives-ouvertes.fr/hal-01517950/document

Isogeometric analysis using T-splines

Stress projection for membrane and shear locking in shell finite elements

On developpe un element de plaque triangulaire du type de Mindlin qui utilise des champs lineaires pour les rotations et les deflexions transversales. L'element ne necessite qu'un seul point de quadrature ce qui est tres interessant pour l'analyse non lineaire

A C0 triangular plate element with one‐point quadrature

An 18 degrees-of-freedom flat triangular shell element is formulated using the DKT bending element and a new membrane stress formulation. The membrane stress formulation is based on a degenerate linear strain triangle utilizing in-plane rotations at each node. This new formulation is very simple and satisfies the plane stress patch test. Based on the numerical results from a rigorous set of test problems, the new formulation is shown to be an improvement over earlier attempts to improve the membrane stress in the shell element.

A flat triangular shell element with improved membrane interpolation

A simple triangular shell element which incorporates the effects of coupling between membrane and flexural behaviour and avoids membrane locking is described. The element uses a discrete Kirchhoff bending formulation and a constant strain membrane element. For the purpose of permitting inextensional modes and thus avoiding membrane locking, a decomposition technique, which can also be viewed as a strain projection method, is used. The method is illustrated first for a beam element and then for a triangular shell element. Results are presented for a variety of linear static problems to illustrate its accuracy and some highly non‐linear problems to indicate its applicability to collapse analysis.

Nicholas Carpenter

Papers

Stress projection for membrane and shear locking in shell finite elements

A C0 triangular plate element with one‐point quadrature

A flat triangular shell element with improved membrane interpolation

A simple triangular curved shell element

Locking and shear scaling factors in C° bending elements