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Stable linear traction‐based equilibrium elements for elastostatics: Direct access to linear statically admissible stresses and quadratic kinematically admissible displacements for dual analysis
Author(s) -
Wang Li,
Zhong Hongzhi
Publication year - 2015
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.4828
Subject(s) - traction (geology) , quadrilateral , lagrange multiplier , linear elasticity , mathematics , quadratic equation , finite element method , mathematical analysis , geometry , mathematical optimization , structural engineering , engineering , mechanical engineering
Summary This paper establishes the basic framework for the traction‐based equilibrium finite element method (traction‐based EFEM). Stable linear traction‐based equilibrium elements are formulated using the macro‐element technique. An arbitrary internal macro‐point renders a linear triangular element stable, while a stable linear quadrilateral element requires the macro‐point to locate at the intersection of diagonals. Then, a Lagrangian formulation is utilized to minimize the complementary energy under equilibrium constraints, and consequently, tractions as well as additional Lagrange multipliers are obtained. Linear statically admissible (SA) stresses are thereafter acquired from tractions. As for Lagrange multipliers, they turn out to coincide well with rigid‐body displacements in each element after simple modifications. With rigid‐body displacements and linear tractions known, quadratic displacements and the kinematically admissible (KA) counterpart thereof by recovery are obtainable. The knowledge of both SA stresses and KA displacements renders dual analysis directly applicable. That is to say, the traction‐based EFEM is featured with direct access to strict upper and lower bounds of strain energy and other quantities of interest. Copyright © 2014 John Wiley & Sons, Ltd.