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Assumed strain finite element methods for conserving temporal integrations in non‐linear solid dynamics
Author(s) -
Armero F.
Publication year - 2007
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.2233
Subject(s) - conservation law , mathematics , operator (biology) , finite element method , discretization , dissipation , mathematical analysis , conserved quantity , classical mechanics , physics , mathematical physics , biochemistry , chemistry , repressor , transcription factor , gene , thermodynamics
This paper presents a new class of assumed strain finite elements to use in combination with general energy‐momentum‐conserving time‐stepping algorithms so that these conservation properties in time are preserved by the fully discretized system in space and time. The case of interest corresponds to nearly incompressible material responses, in the fully non‐linear finite strain elastic and elastoplastic ranges. The new elements consider the classical scaling of the deformation gradient with an assumed Jacobian (its determinant) defined locally through a weighted averaging procedure at the element level. The key aspect of the newly proposed formulation is the definition of the associated linearized strain operator or B‐bar operator. The developments presented here start by identifying the conditions that this discrete operator must satisfy for the fully discrete system in time and space to inherit exactly the conservation laws of linear and angular momenta, and the conservation/dissipation law of energy for elastic and inelastic problems, respectively. Care is also taken of the preservation of the relative equilibria and the corresponding group motions associated with the momentum conservation laws, and characterized by purely rotational and translational motions superimposed to the equilibrium deformed configuration. With these developments at hand, a new general B‐bar operator is introduced that satisfies these conditions. The new operator not only accounts for the spatial interpolations (e.g. bilinear displacements with piece‐wise constant volume) but also depends on the discrete structure of the equations in time. The aforementioned conservation/dissipation properties of energy and momenta are then proven to hold rigorously for the final numerical schemes, unconditionally of the time step size and the material model (elastic or elastoplastic). Different finite elements are considered in this framework, including quadrilateral and triangular elements for plane problems and brick elements for three‐dimensional problems. Several representative numerical simulations are presented involving, in particular, the use of energy‐dissipating momentum‐conserving time‐stepping schemes recently developed by the author and co‐workers for general finite strain elastoplasticity in order to illustrate the properties of the new finite elements, including these conservation/dissipation properties in time and their locking‐free response in the quasi‐incompressible case. Copyright © 2007 John Wiley & Sons, Ltd.