Finite strain quadrilateral shell using least-squares fit of relative Lagrangian in-plane strains

P. Areias, Timon Rabczuk, J. M. César De Sá, J. E. Garção

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

This work presents a finite strain quadrilateral element with least-squares assumed in-plane shear strains (in covariant/contravariant coordinates) and classical transverse shear assumed strains. It is an alternative to enhanced-assumed-strain (EAS) formulation and, in contrast to this, produces an element satisfying ab initio the Patch-test. No additional degrees-of-freedom are present, unlike EAS. Least-squares fit allows the derivation of invariant finite strain elements which are both in-plane and out-of-plane shear-locking free and amenable to standardization in commercial codes. With that goal, we use automatically generated code produced by AceGen and Mathematica to obtain novel finite element formulations. The corresponding exact linearization of the internal forces was, until recently, a insurmountable task. We use the tangent modulus in the least-squares fit to ensure that stress modes are obtained from a five-parameter strain fitting. This reproduces exactly the in-plane bending modes. The discrete equations are obtained by establishing a four-field variational principle (a direct extension of the Hu-Washizu variational principle). The main achieved goal is coarse-mesh accuracy for distorted meshes, which is adequate for being used in crack propagation problems. In addition, as an alternative to spherical interpolation, a consistent director normalization is performed. Metric components are fully deduced and exact linearization of the shell element is performed. Full linear and nonlinear assessment of the element is performed, showing similar performance to more costly approaches, often on-par with the best available shell elements.

Original languageEnglish
Pages (from-to)26-40
Number of pages15
JournalFinite Elements in Analysis and Design
Volume98
DOIs
Publication statusPublished - 2015
Externally publishedYes

Fingerprint

Finite Strain
Plane Strain
Least Squares
Shell
Shell Element
Shear strain
Linearization
Variational Principle
Mesh
Locking-free
Shear Locking
Patch Test
Quadrilateral Element
Formulation
Alternatives
Mathematica
Degrees of freedom (mechanics)
Crack Propagation
Discrete Equations
Standardization

Keywords

  • Finite strain plasticity
  • Finite strains
  • Least-square assumed strain
  • Pian-Sumihara stress modes
  • Shell elements

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Engineering(all)
  • Computer Graphics and Computer-Aided Design

Cite this

Finite strain quadrilateral shell using least-squares fit of relative Lagrangian in-plane strains. / Areias, P.; Rabczuk, Timon; César De Sá, J. M.; Garção, J. E.

In: Finite Elements in Analysis and Design, Vol. 98, 2015, p. 26-40.

Research output: Contribution to journalArticle

Areias, P. ; Rabczuk, Timon ; César De Sá, J. M. ; Garção, J. E. / Finite strain quadrilateral shell using least-squares fit of relative Lagrangian in-plane strains. In: Finite Elements in Analysis and Design. 2015 ; Vol. 98. pp. 26-40.
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