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

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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

Abstract
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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 and 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 "Proceedings of the Workshop on Optimization and Optimal Control" (Jyväskylä, 1992), Report 58, Univ. Jyväskylä, Jyväskylä, (1993), 5-16.
[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), 48-51; translation in Dokl. Phys., 44 (1999), 446-449.
[3]	H. T. Banks and K. Kunisch, "Estimation Techniques for Distributed Parameter Systems," Syst. & Contr: Found. and Appl., 1, Birkhäuser Boston, Inc., Boston, MA, 1989.
[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, Mt. Holyoke College, South Hadley, MA, July 1992, Society for Industrial and Applied Mathematics, Philadelphia, 1993.
[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), 27-53. doi: 10.1007/s10444-007-9035-6.
[6]	P. R. Gill, W. Murray and M. H. Wright, The Levenberg-Marquardt Method, S 4.7.3, in "Practical Optimization," Academic Press, London, 1981, 136-137.
[7]	T. T. Marinov, R. S. Marinova and C. I. Christov, Coefficient identification in elliptic partial differential equation, in "Large-scale Scientific Computing," Lecture Notes in Comput. Sci., 3743, Springer, Berlin, 2006, 372-379.
[8]	Z. Mróz and G. E. Stavroulakis, "Parameter Identification of Materials and Structures," Springer Verlag, Vienna, New York, 2005.
[9]	D. L. Russell, Structural parameter optimization in linear elastic systems, Commun. Pure & Appl. Anal., 10 (2011), 1517-1536.
[10]	D. L. Russell, Gauss-Newton and inverse Gauss-Newton methods for coefficient identification in linear elastic systems, to appear in Acta Applicandae Mathematicae.
[11]	D. L. Russell, Some methods for parameter identification in Elliptic Systems, to appear.
[12]	J. Schoukens and R. Pintelon, "Identification of Linear Systems. A Practical Guideline to Accurate Modeling," Pergamon Press, Oxford, 1991.
[13]	U. Tautenhahn, A fast iterative method for solving regularized parameter identification problems in elliptic boundary value problem, Computing, 43 (1989), 47-58. doi: 10.1007/BF02243805.

show all references

References:

[1]	A. I. Artjukh and N. V. Banichuk, Application of optimization methods to identification problems, in "Proceedings of the Workshop on Optimization and Optimal Control" (Jyväskylä, 1992), Report 58, Univ. Jyväskylä, Jyväskylä, (1993), 5-16.
[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), 48-51; translation in Dokl. Phys., 44 (1999), 446-449.
[3]	H. T. Banks and K. Kunisch, "Estimation Techniques for Distributed Parameter Systems," Syst. & Contr: Found. and Appl., 1, Birkhäuser Boston, Inc., Boston, MA, 1989.
[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, Mt. Holyoke College, South Hadley, MA, July 1992, Society for Industrial and Applied Mathematics, Philadelphia, 1993.
[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), 27-53. doi: 10.1007/s10444-007-9035-6.
[6]	P. R. Gill, W. Murray and M. H. Wright, The Levenberg-Marquardt Method, S 4.7.3, in "Practical Optimization," Academic Press, London, 1981, 136-137.
[7]	T. T. Marinov, R. S. Marinova and C. I. Christov, Coefficient identification in elliptic partial differential equation, in "Large-scale Scientific Computing," Lecture Notes in Comput. Sci., 3743, Springer, Berlin, 2006, 372-379.
[8]	Z. Mróz and G. E. Stavroulakis, "Parameter Identification of Materials and Structures," Springer Verlag, Vienna, New York, 2005.
[9]	D. L. Russell, Structural parameter optimization in linear elastic systems, Commun. Pure & Appl. Anal., 10 (2011), 1517-1536.
[10]	D. L. Russell, Gauss-Newton and inverse Gauss-Newton methods for coefficient identification in linear elastic systems, to appear in Acta Applicandae Mathematicae.
[11]	D. L. Russell, Some methods for parameter identification in Elliptic Systems, to appear.
[12]	J. Schoukens and R. Pintelon, "Identification of Linear Systems. A Practical Guideline to Accurate Modeling," Pergamon Press, Oxford, 1991.
[13]	U. Tautenhahn, A fast iterative method for solving regularized parameter identification problems in elliptic boundary value problem, Computing, 43 (1989), 47-58. doi: 10.1007/BF02243805.

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