Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 290301, 11 pages
http://dx.doi.org/10.1155/2015/290301
Research Article

Identification of Flexural Rigidity in a Kirchhoff Plates Model Using a Convex Objective and Continuous Newton Method

1Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USA
2Department of Mathematics and Computer Science, University of Catania, 95125 Catania, Italy
3Institute of Mathematics, Martin Luther University of Halle-Wittenberg, 06120 Halle (Saale), Germany

Received 29 March 2015; Revised 17 August 2015; Accepted 20 August 2015

Academic Editor: Evangelos J. Sapountzakis

Copyright © 2015 B. Jadamba et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. G. Ciarlet and P. Rabier, Les équations de von Kármán, vol. 826 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1980.
  2. L. W. White, “Estimation of elastic parameters in a nonlinear elliptic model of a plate,” Applied Mathematics and Computation, vol. 42, no. 2, pp. 139–187, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. D. Lesnic, L. Elliott, and D. B. Ingham, “Analysis of coefficient identification problems associated to the inverse Euler-Bernoulli beam theory,” IMA Journal of Applied Mathematics, vol. 62, no. 2, pp. 101–116, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. T. T. Marinov and R. S. Marinova, “Coefficient identification in Euler-Bernoulli equation from over-posed data,” Journal of Computational and Applied Mathematics, vol. 235, no. 2, pp. 450–459, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. T. T. Marinov and R. S. Marinova, “An inverse problem for estimation of bending stiffness in Kirchhoff-Love plates,” Computers & Mathematics with Applications, vol. 65, no. 3, pp. 512–519, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. R. E. Ewing, T. Lin, and Y. Lin, “A mixed least-squares method for an inverse problem of a nonlinear beam equation,” Inverse Problems, vol. 15, no. 1, pp. 19–32, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. S. Manservisi and M. Gunzburger, “A variational inequality formulation of an inverse elasticity problem,” Applied Numerical Mathematics, vol. 34, no. 1, pp. 99–126, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. H. W. Engl and P. Kügler, “The influence of the equation type on iterative parameter identification problems which are elliptic or hyperbolic in the parameter,” European Journal of Applied Mathematics, vol. 14, no. 2, pp. 129–163, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. P. Kügler, “A parameter identification problem of mixed type related to the manufacture of car windshields,” SIAM Journal on Applied Mathematics, vol. 64, no. 3, pp. 858–877, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. D. Salazar and R. Westbrook, “Inverse problems of mixed type in linear plate theory,” European Journal of Applied Mathematics, vol. 15, no. 2, pp. 129–146, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. P. d. Lopes, A. B. Jorge, and J. Sebastião Cunha, “Detection of holes in a plate using global optimization and parameter identification techniques,” Inverse Problems in Science and Engineering, vol. 18, no. 4, pp. 439–463, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. S. Kim and K. L. Kreider, “Parameter identification for nonlinear elastic and viscoelastic plates,” Applied Numerical Mathematics, vol. 56, no. 12, pp. 1538–1554, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. G. Yuan and M. Yamamoto, “Lipschitz stability in inverse problems for a Kirchhoff plate equation,” Asymptotic Analysis, vol. 53, no. 1-2, pp. 29–60, 2007. View at Google Scholar · View at MathSciNet · View at Scopus
  14. M. S. Gockenbach, B. Jadamba, and A. A. Khan, “Equation error approach for elliptic inverse problems with an application to the identification of Lamé parameters,” Inverse Problems in Science and Engineering, vol. 16, no. 3, pp. 349–367, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. M. S. Gockenbach and A. A. Khan, “Identification of Lamé parameters in linear elasticity: a fixed point approach,” Journal of Industrial and Management Optimization, vol. 1, no. 4, pp. 487–497, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  16. M. S. Gockenbach and A. A. Khan, “An abstract framework for elliptic inverse problems. II. An augmented Lagrangian approach,” Mathematics and Mechanics of Solids, vol. 14, no. 6, pp. 517–539, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. L. W. White, “Estimation of elastic parameters in beams and certain plates: H1 regularization,” Journal of Optimization Theory and Applications, vol. 60, no. 2, pp. 305–326, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. L. W. White, “Identification of flexural rigidity in a dynamic plate model,” Journal of Mathematical Analysis and Applications, vol. 144, no. 1, pp. 275–303, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. B. Jadamba, A. A. Khan, and M. Sama, “Inverse problems of parameter identification in partial differential equations,” in Mathematics in Science and Technology, pp. 228–258, World Science Publisher, Hackensack, NJ, USA, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  20. M. S. Gockenbach and A. A. Khan, “An abstract framework for elliptic inverse problems. I. An output least-squares approach,” Mathematics and Mechanics of Solids, vol. 12, no. 3, pp. 259–276, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. B. Jadamba, A. A. Khan, G. Rus, M. Sama, and B. Winkler, “A new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location,” SIAM Journal on Applied Mathematics, vol. 74, no. 5, pp. 1486–1510, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. L.-H. Zhang, C. T. Kelley, and L.-Z. Liao, “A continuous Newton-type method for unconstrained optimization,” Pacific Journal of Optimization, vol. 4, no. 2, pp. 259–277, 2008. View at Google Scholar · View at Scopus
  23. H. Attouch and M. Teboulle, “Regularized Lotka-Volterra dynamical system as continuous proximal-like method in optimization,” Journal of Optimization Theory and Applications, vol. 121, no. 3, pp. 541–570, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. A. A. Brown and M. C. Bartholomew-Biggs, “ODE versus SQP methods for constrained optimization,” Journal of Optimization Theory and Applications, vol. 62, no. 3, pp. 371–386, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. A. A. Brown and M. C. Bartholomew-Biggs, “Some effective methods for unconstrained optimization based on the solution of systems of ordinary differential equations,” Journal of Optimization Theory and Applications, vol. 62, no. 2, pp. 211–224, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. L.-Z. Liao, L. Qi, and H. W. Tam, “A gradient-based continuous method for large-scale optimization problems,” Journal of Global Optimization, vol. 31, no. 2, pp. 271–286, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus