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Adaptive finite element methods for the identification of elastic constants
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, United Kingdom.
2006 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 26, no 2, 217-235 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, the elastic constants of a material are recovered from measured displacements where the model is the equilibrium equations for the orthotropic case. The finite element method is used for the discretization of the state equation and the Gauss–Newton method is used to solve the nonlinear least squares problem attained from the parameter estimation problem. A posteriori error estimators are derived and used to improve the accuracy by an appropriate mesh refinement. A numerical experiment is presented to show the applicability of the approach.

Place, publisher, year, edition, pages
2006. Vol. 26, no 2, 217-235 p.
Keyword [en]
Parameter estimation, finite element approximation, adaptive finite element methods, least squares, Gauss–Newton
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-3242DOI: 10.1007/s10915-004-4935-9ISI: 000235445800004Scopus ID: 2-s2.0-32944461738Local ID: 3196OAI: oai:DiVA.org:miun-3242DiVA: diva2:28274
Available from: 2008-09-30 Created: 2008-09-30 Last updated: 2016-09-27Bibliographically approved
In thesis
1. Adaptive finite element methods for parameter estimation problems in partial differential equations
Open this publication in new window or tab >>Adaptive finite element methods for parameter estimation problems in partial differential equations
2005 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Physical and chemical phenomena are often described by a system of partial di®erential equations. These equations usually involve unknown parameters, which cannot be measured directly but which can be adjusted to make the model predictions match the observed data. The process of ¯tting these para- meters to laboratory or plant data is called parameter estimation. In order to recover these parameters, the well-known output least squares formulation is of- ten utilized. To solve the optimization problem governed by partial di®erential equations, the in¯nite-dimensional problem must be approximated by introduc- ing discretizations such as a ¯nite elements or di®erences. It is clear that the e±ciency of the numerical methods dealt with here will be in°uenced by the discretization scheme. The goal of this thesis is to develop e±cient numerical methods for the parameter estimation problems governed by partial di®erential equations, based on adaptive ¯nite element methods. This work was initiated by an investigation into an a posteriori error esti- mator of residual type for parameter estimation problems with a ¯nite number of unknown parameters. It appears that an adaptive ¯nite element algorithm guided by the derived a posteriori error estimator produces a sequence of eco- nomical, locally re¯ned meshes. The methods are then applied to the identi¯ca- tion of elastic constants in paper from measured displacements. Further, some a posteriori error estimators of gradient recovery type are derived for the error in parameters due to the discretization. The main advantages of using error estimators of this type are the simplicity of their implementation and their cost e®ectiveness. Often, the unknown parameters are functions, which also need to be dis- cretized. Adaptive ¯nite element method is developed for the estimation of distributed parameters in elliptic equations with multi-mesh techniques. Finally, a goal-oriented adaptive method, dual weighted residual methods (DWR methods) are employed determining the elastic constants in paper from measured displacements.

Place, publisher, year, edition, pages
Sundsvall: Mittuniversitetet, 2005. 30 p.
Series
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 4
Keyword
parameter estimation, ¯nite element approximation, adaptive ¯nite element methods, a posteriori error estimates, least squares.
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-8866 (URN)
Public defence
2005-10-22, 00:00 (English)
Available from: 2009-05-06 Created: 2009-05-06 Last updated: 2009-09-21Bibliographically approved

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