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A variational r ‐adaption and shape‐optimization method for finite‐deformation elasticity
Author(s) -
Thoutireddy P.,
Ortiz M.
Publication year - 2004
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.1052
Subject(s) - triangulation , linear elasticity , energy minimization , shape optimization , energy functional , convergence (economics) , mathematics , variational principle , minification , deformation (meteorology) , elasticity (physics) , finite element method , domain (mathematical analysis) , anisotropy , elastic energy , mathematical optimization , mathematical analysis , geometry , physics , structural engineering , engineering , quantum mechanics , meteorology , economics , thermodynamics , economic growth
This paper is concerned with the formulation of a variational r ‐adaption method for finite‐deformation elastostatic problems. The distinguishing characteristic of the method is that the variational principle simultaneously supplies the solution, the optimal mesh and, in problems of shape optimization, the equilibrium shapes of the system. This is accomplished by minimizing the energy functional with respect to the nodal field values as well as with respect to the triangulation of the domain of analysis. Energy minimization with respect to the referential nodal positions has the effect of equilibrating the energetic or configurational forces acting on the nodes. We derive general expressions for the configurational forces for isoparametric elements and non‐linear, possibly anisotropic, materials under general loading. We illustrate the versatility and convergence characteristics of the method by way of selected numerical tests and applications, including the problem of a semi‐infinite crack in linear and non‐linear elastic bodies; and the optimization of the shape of elastic inclusions. Copyright © 2004 John Wiley & Sons, Ltd.