American Institute of Mathematical Sciences

Advanced
September  2011, 1(3): 391-411. doi: 10.3934/mcrf.2011.1.391

Coefficient identification and fault detection in linear elastic systems; one dimensional problems

David L. Russell 1,

1. 

Department of Mathematics, 460 McBryde Hall, Virginia Tech, Blacksburg, VA 24060, United States

Received  November 2010 Revised  January 2011 Published  September 2011

The determination of parameter distributions, including fault detection, in elastic structures is a subject of great importance in structural engineering and related areas of applied mathematics. In this article we explore, in both continuous and discrete settings, some methods for approximate solution of such identification problems in a one dimensional linear elasticity framework. Methods for related optimization problems based on the matrix trace norm are described. The main objective of the paper is to introduce a method, believed new with this article, for which we suggest the names adjoint null space method or complementary projection method. Computational results for sample problems based on this technique are presented.
Keywords: linear elastic systems., Parameter identification, inverse problems.
Mathematics Subject Classification: Primary: 65L09; Secondary: 74B05, 74G7.
Citation: David L. Russell. Coefficient identification and fault detection in linear elastic systems; one dimensional problems. Mathematical Control & Related Fields, 2011, 1 (3) : 391-411. doi: 10.3934/mcrf.2011.1.391
References:
[1]

A. I. Artjukh and N. V. Banichuk, Application of optimization methods to identification problems,, in, (1993), 5.   Google Scholar

[2]

N. V. Banichuk, Optimization formulation and decomposition of the problem of the identification of the distributed parameters of elastic constructions,, (Russian) Dokl. Akad. Nauk, 367 (1999), 48.   Google Scholar

[3]

H. T. Banks and K. Kunisch, "Estimation Techniques for Distributed Parameter Systems,", Syst. & Contr: Found. and Appl., 1 (1989).   Google Scholar

[4]

H. T. Banks, R. H. Fabiano and K. Ito, eds., "Identification and Control in Systems Governed by Partial Differential Equations,", Proc. AMS-IMS-SIAM Joint Sumr. Res. Conf. on Control and Identification of Partial Differential Equations, (1992).   Google Scholar

[5]

T. Feng, N. Yan and W. Liu, Adaptive finite element methods for the identification of distributed parameters in elliptic equation,, Adv. Comput. Math., 29 (2008), 27.  doi: 10.1007/s10444-007-9035-6.  Google Scholar

[6]

P. R. Gill, W. Murray and M. H. Wright, The Levenberg-Marquardt Method,, S 4.7.3, (1981), 136.   Google Scholar

[7]

T. T. Marinov, R. S. Marinova and C. I. Christov, Coefficient identification in elliptic partial differential equation,, in, 3743 (2006), 372.   Google Scholar

[8]

Z. Mróz and G. E. Stavroulakis, "Parameter Identification of Materials and Structures,", Springer Verlag, (2005).   Google Scholar

[9]

D. L. Russell, Structural parameter optimization in linear elastic systems,, Commun. Pure & Appl. Anal., 10 (2011), 1517.   Google Scholar

[10]

D. L. Russell, Gauss-Newton and inverse Gauss-Newton methods for coefficient identification in linear elastic systems,, to appear in Acta Applicandae Mathematicae., ().   Google Scholar

[11]

D. L. Russell, Some methods for parameter identification in Elliptic Systems,, to appear., ().   Google Scholar

[12]

J. Schoukens and R. Pintelon, "Identification of Linear Systems. A Practical Guideline to Accurate Modeling,", Pergamon Press, (1991).   Google Scholar

[13]

U. Tautenhahn, A fast iterative method for solving regularized parameter identification problems in elliptic boundary value problem,, Computing, 43 (1989), 47.  doi: 10.1007/BF02243805.  Google Scholar

show all references

References:
[1]

A. I. Artjukh and N. V. Banichuk, Application of optimization methods to identification problems,, in, (1993), 5.   Google Scholar

[2]

N. V. Banichuk, Optimization formulation and decomposition of the problem of the identification of the distributed parameters of elastic constructions,, (Russian) Dokl. Akad. Nauk, 367 (1999), 48.   Google Scholar

[3]

H. T. Banks and K. Kunisch, "Estimation Techniques for Distributed Parameter Systems,", Syst. & Contr: Found. and Appl., 1 (1989).   Google Scholar

[4]

H. T. Banks, R. H. Fabiano and K. Ito, eds., "Identification and Control in Systems Governed by Partial Differential Equations,", Proc. AMS-IMS-SIAM Joint Sumr. Res. Conf. on Control and Identification of Partial Differential Equations, (1992).   Google Scholar

[5]

T. Feng, N. Yan and W. Liu, Adaptive finite element methods for the identification of distributed parameters in elliptic equation,, Adv. Comput. Math., 29 (2008), 27.  doi: 10.1007/s10444-007-9035-6.  Google Scholar

[6]

P. R. Gill, W. Murray and M. H. Wright, The Levenberg-Marquardt Method,, S 4.7.3, (1981), 136.   Google Scholar

[7]

T. T. Marinov, R. S. Marinova and C. I. Christov, Coefficient identification in elliptic partial differential equation,, in, 3743 (2006), 372.   Google Scholar

[8]

Z. Mróz and G. E. Stavroulakis, "Parameter Identification of Materials and Structures,", Springer Verlag, (2005).   Google Scholar

[9]

D. L. Russell, Structural parameter optimization in linear elastic systems,, Commun. Pure & Appl. Anal., 10 (2011), 1517.   Google Scholar

[10]

D. L. Russell, Gauss-Newton and inverse Gauss-Newton methods for coefficient identification in linear elastic systems,, to appear in Acta Applicandae Mathematicae., ().   Google Scholar

[11]

D. L. Russell, Some methods for parameter identification in Elliptic Systems,, to appear., ().   Google Scholar

[12]

J. Schoukens and R. Pintelon, "Identification of Linear Systems. A Practical Guideline to Accurate Modeling,", Pergamon Press, (1991).   Google Scholar

[13]

U. Tautenhahn, A fast iterative method for solving regularized parameter identification problems in elliptic boundary value problem,, Computing, 43 (1989), 47.  doi: 10.1007/BF02243805.  Google Scholar

[1]

David Russell. Structural parameter optimization of linear elastic systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1517-1536. doi: 10.3934/cpaa.2011.10.1517
[2]

Davide Guidetti. Some inverse problems of identification for integrodifferential parabolic systems with a boundary memory term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 749-756. doi: 10.3934/dcdss.2015.8.749
[3]

Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial & Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471
[4]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020
[5]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19
[6]

Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005
[7]

David L. Russell. Trace properties of certain damped linear elastic systems. Evolution Equations & Control Theory, 2013, 2 (4) : 711-721. doi: 10.3934/eect.2013.2.711
[8]

Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020181
[9]

Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems & Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371
[10]

Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems & Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163
[11]

Andrea Braides, Margherita Solci, Enrico Vitali. A derivation of linear elastic energies from pair-interaction atomistic systems. Networks & Heterogeneous Media, 2007, 2 (3) : 551-567. doi: 10.3934/nhm.2007.2.551
[12]

Harry L. Johnson, David Russell. Transfer function approach to output specification in certain linear distributed parameter systems. Conference Publications, 2003, 2003 (Special) : 449-458. doi: 10.3934/proc.2003.2003.449
[13]

Guanghui Hu, Andreas Kirsch, Tao Yin. Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves. Inverse Problems & Imaging, 2016, 10 (1) : 103-129. doi: 10.3934/ipi.2016.10.103
[14]

Yuepeng Wang, Yue Cheng, I. Michael Navon, Yuanhong Guan. Parameter identification techniques applied to an environmental pollution model. Journal of Industrial & Management Optimization, 2018, 14 (2) : 817-831. doi: 10.3934/jimo.2017077
[15]

Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025
[16]

Ingrid Daubechies, Gerd Teschke, Luminita Vese. Iteratively solving linear inverse problems under general convex constraints. Inverse Problems & Imaging, 2007, 1 (1) : 29-46. doi: 10.3934/ipi.2007.1.29
[17]

Shiyun Wang, Yong-Jin Liu, Yong Jiang. A majorized penalty approach to inverse linear second order cone programming problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 965-976. doi: 10.3934/jimo.2014.10.965
[18]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1
[19]

Kewei Zhang. On equality of relaxations for linear elastic strains. Communications on Pure & Applied Analysis, 2002, 1 (4) : 565-573. doi: 10.3934/cpaa.2002.1.565
[20]

Peijun Li, Xiaokai Yuan. Inverse obstacle scattering for elastic waves in three dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 545-573. doi: 10.3934/ipi.2019026

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences