Levenberg-Marquardt iterative regularization for the pulse-type impact-force reconstruction

In this paper, we study the Levenberg–Marquardt algorithm with the trust region strategy to iteratively solve the ill-posed impact-force reconstruction problem. This particular problem is interesting because the impact-force particularly those of the pulse-type are difficult to be measured directly. In the present approach, the necessity of regularization is enforced by means of the trust region approach. This study mainly contributes a systematic approach to locate the optimal solution by tracking the Levenberg–Marquardt parameter. The proposed method is evaluated by solving two typical problem existed in the inverse problem of the impact-force reconstruction. Reasonable accurate impact-forces were produced from the both evaluations.

[1]  C. W. Groetsch,et al.  The theory of Tikhonov regularization for Fredholm equations of the first kind , 1984 .

[2]  James F. Doyle,et al.  Force identification from dynamic responses of a bimaterial beam , 1993 .

[4]  S. Rizal,et al.  Dimple fracture under short pulse loading , 2000 .

[5]  Hirotsugu Inoue,et al.  Regularization of Numerical Inversion of the Laplace Transform for the Inverse Analysis of Impact Force , 1998 .

[6]  F. Gunawan,et al.  Two-step B-splines regularization method for solving an ill-posed problem of impact-force reconstruction , 2006 .

[7]  F. Gunawan,et al.  Impact-Force Estimation by Quadratic Spline Approximation , 2008 .

[8]  F. Gunawan,et al.  Efficient Iterative Solution for Large Elasto-Dynamic Inverse Problems , 2004 .

[9]  E. O. Brigham,et al.  The Fast Fourier Transform , 1967, IEEE Transactions on Systems, Man, and Cybernetics.

[10]  Fergyanto E. Gunawan,et al.  Time and Frequency Domains Iterative Regularization for Inverse Analysis of an Instrumented One-Point Bend Specimen , 2006 .

[11]  Yi Liu,et al.  Dynamic force identification based on enhanced least squares and total least-squares schemes in the frequency domain , 2005 .

[12]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series , 1964 .

[13]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[14]  Per Christian Hansen,et al.  Rank-Deficient and Discrete Ill-Posed Problems , 1996 .

[15]  Xingxing Jiang,et al.  Reconstruction of distributed dynamic loads on an Euler beam via mode-selection and consistent spatial expression , 2008 .

[16]  S. Aoki,et al.  A Simple Formula for the Dynamic Stress Intensity Factor of an Impacted Freely-Supported Bend Specimen , 1990 .

[17]  Hirotsugu Inoue,et al.  Estimation of impact load by inverse analysis. Optimal transfer function for inverse analysis , 1992 .

[18]  A. Nowak,et al.  Inverse thermal problems , 1995 .

[19]  Per Christian Hansen,et al.  Deconvolution and Regularization with Toeplitz Matrices , 2002, Numerical Algorithms.

[20]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

[21]  M. Hanke.  A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems , 1997 .

[22]  Quentin Leclere,et al.  Vibration source identification using corrected finite difference schemes , 2012 .

[23]  E. Jacquelin,et al.  Force reconstruction: analysis and regularization of a deconvolution problem , 2003 .

[24]  James G. Nagy,et al.  A total least squares method for Toeplitz systems of equations , 1998 .

[25]  Taek Soo Jang,et al.  Indirect measurement of the impulsive load to a nonlinear system from dynamic responses: Inverse problem formulation , 2010 .

[26]  Gene H. Golub,et al.  Matrix computations , 1983 .

[27]  Misha Elena Kilmer,et al.  Choosing Regularization Parameters in Iterative Methods for Ill-Posed Problems , 2000, SIAM J. Matrix Anal. Appl..

[28]  X. Y. Li,et al.  Adaptive Tikhonov regularization for damage detection based on nonlinear model updating , 2010 .

[29]  James W. Dally,et al.  Strain-gage methods for measuring the opening-mode stress-intensity factor,KI , 1987 .

[30]  Kenneth Levenberg.  A METHOD FOR THE SOLUTION OF CERTAIN NON – LINEAR PROBLEMS IN LEAST SQUARES , 1944 .

[31]  Carl Tim Kelley,et al.  Iterative methods for optimization , 1999, Frontiers in applied mathematics.

[32]  Hirotsugu Inoue,et al.  Measurement of Impact Load by using an Inverse Analysis Technique : Comparison of Methods for Estimating the Transfer Function and its Application to the Instrumented Charpy Impact Test , 1991 .

[33]  J. F. Doyle,et al.  Impact force identification from wave propagation responses , 1996 .