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Adjoint‐weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data
Author(s) -
Barbone Paul E.,
Rivas Carlos E.,
Harari Isaac,
Albocher Uri,
Oberai Assad A.,
Zhang Yixiao
Publication year - 2009
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.2760
Subject(s) - mathematics , discretization , isotropy , mathematical analysis , compressibility , inverse , elasticity (physics) , scalar (mathematics) , inverse problem , geometry , physics , quantum mechanics , thermodynamics
We describe a novel variational formulation of inverse elasticity problems given interior data. The class of problems considered is rather general and includes, as special cases, plane deformations, compressibility and incompressiblity in isotropic materials, 3 D deformations, and anisotropy. The strong form of this problem is governed by equations of pure advective transport. The variational formulation is based on a generalization of the adjoint‐weighted variational equation (AWE) formulation, originally developed for flow of a passive scalar. We describe how to apply AWE to various cases, and prove several properties. We prove that the Galerkin discretization of the AWE formulation leads to a stable, convergent numerical method, and prove optimal rates of convergence. The numerical examples demonstrate optimal convergence of the method with mesh refinement for multiple unknown material parameters, graceful performance in the presence of noise, and robust behavior of the method when the target solution is C ∞ , C 0 , or discontinuous. Copyright © 2009 John Wiley & Sons, Ltd.