Abstract
The primary aim of the paper is to identify the loading source of infinite beams on an elastic foundation from given information of vertical deflection of infinite beams. An integral equation is obtained for the relationship between loading distribution and vertical deflection. It is shown that the inverse identification of a loading source is one-to-one but ill-posed. Because of ill-posedness, the usual numerical schemes produce arbitrarily large errors. A method for the solution is proposed by using Tikhonov’s regularization. L-curve criterion is introduced for the determination of optimal regularization parameter. Numerical experiments show that the present methodology is accurate and robust in the inverse determination of loading source.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. A. Avdonin, N. G. Medhin and T. L. Sheronova, Identification of a piecewise constant coefficient in the beam equation, Journal of Computational and Applied Mathematics, 114(1) (2000) 11–21.
R. Wolfgang, Identification of the load of a partially breaking beam from inclination measurements, Inverse Problems, 15(4) (1999) 1003–1020.
E. Lucchinetti and E. Stüssi, Measuring the flexural rigidity in non-uniform beams using an inverse problem approach, Inverse Problems, 18(3) (2002) 837–857.
J. Guo, P. Le Masson, S. Rouquette, T. Loulou and E. Artioukhine, Estimation of a source term in a two-dimensional heat transfer problem: application to an electron beam welding, theoretical and experimental validations, Inverse Problems in Science and Engineering, 15(7) (2007) 743–763.
M. Kashiwagi, Research on hydroelastic responses of VLFS: Recent progress and future work, International Journal of Offshore and Polar Engineering, 10(2) (2000) 81–90.
J. W. Kim and R. C. Ertekin, Hydroelastic response of mat-type VLFS: Effects of non-zero draft and mass assumptions, Proceedings of OCEANS 2000 (MTS/IEEE Conference and Exhibition), 1 (2000) 541–547.
S. Ohmatsu, Overview: Research on wave loading and responses of VLFS, Marine Structures, 18(2) (2005) 149–168.
M. H. Meylan and V. A. Squire, The response of ice floes to ocean waves, Journal of Geophysical Research: JGR. C: Oceans 99(C1), (1994) 891–900.
A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, (1996)
A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method. Sov. Doklady, 4 (1963) 1035–1038.
W. J. Kammerer and M. Z. Nashed, Iterative methods for best approximate solutions of integral equations of the first and second kinds. J. Math. Anal. Appl., 40 (1972) 547–573.
C. W. Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg, (1993).
V. Fridman, A method of successive approximations for fredholm integral equations of the first kind. uspeki mat. Nauk., 11 (1965) 233–234.
V. Isakov, Inverse Problems for Partial Differential Equations, Springer, (1998).
T. S. Jang and T. Kinoshita, An Ill-posed inverse problem of a wing with locally given velocity data and its analysis, Journal of Marine Science and Technology, 5(1) (2000) 16–20.
T. S. Jang, H. S. Choi and T. Kinoshita, Numerical experiments on an ill-posed inverse problem for a given velocity around a hydrofoil by iterative and noniterative regularizations. Journal of Marine Science and Technology, 5(3) (2000a) 107–111.
T. S. Jang, H. S. Choi and T. Kinoshita, Solution of an unstable inverse problem: wave source evaluation from observation of velocity distribution. Journal of Marine Science and Technology, 5(4) (2000b) 181–188.
T. S. Jang, S. H. Kwon and B. J. Kim, Solution of an unstable axisymmetric cauchy-poisson problem of dispersive water waves for a spectrum with compact support, Ocean Engineering, 34 (2007) 676–684.
T. S. Jang and S. L. Han, Application of tikhonov regularization to an unstable water waves of the two-dimensional fluid flow: spectrum with compact support, Ships and Offshore Structures, 3(1) (2008) 41–47.
S. Timoshenko, Strength of materials, Part 1, D. Van Nostrand, Princeton, N. J (1955).
M. D. Greenberg, Foundations of Applied Mathematics, Prentice-Hall Inc. (1978).
H. Hochstadt, Integral Equations, A Wiley Interscience Publication. New York. (1973).
P. Roman, Some Modern Mathematics for Physicists and Other Outsiders, 1, 270–290, Pergamon Press Inc. New York. (1975).
P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992) 561–580.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper was recommended for publication in revised form by Associate Editor Joo Ho Choi
T. S. Jang, the corresponding author of the paper, is by birth a Korean, with Naval Architecture and Ocean Engineering Ph.D degrees from Seoul National University, who worked at the department of Naval Architecture and Ocean Engineering in Pusan National University from 2003 until now. His main field of research has been the optimization theory, water wave motion and inverse problem with special focus on ocean-related fields
Rights and permissions
About this article
Cite this article
Jang, T.S., Sung, H.G., Han, S.L. et al. Inverse determination of the loading source of the infinite beam on elastic foundation. J Mech Sci Technol 22, 2350–2356 (2008). https://doi.org/10.1007/s12206-008-0822-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12206-008-0822-x