Regularization Methods And Their Applications In Dynamic Load Identification

Load identification not only analyzes the structure of system in which it isdiffcult or impossible to directly measure the load, but also accurately and properlyreffects the practical dynamics characteristics of engineering structure, so it willgain more practical experience for our future study work. However, it is generallydiffcult or impossible to directly measure the load in lots of engineering problems,and the theories and methods of inverse problems are usually exploited to indirectlyidentify the load acting on the engineering structure. When we study the inverseproblems of load identification in natural sciences and engineering, there existsthe solution of many ill-posed problems. Regularization methods can solve thediffculty of the solution of ill-conditioned inverse problems by the approximation ofan ill-posed problem by a family of neighbouring well-posed problems. So we studythe regularization methods to increase the effciency and precision of the solutionof inverse problems and provide their application in load identification, which is ofapplicable value in engineering and important scientific research. This dissertationconducts a systematical research for the regularization theoretical methods,andaims at contributing some useful researches and trials on themselves and theirpractical algorithms.As a result,the following studies are carried out in thisdissertation:Based on the idea of regularization and by the corresponding knowledge ofsingular system theory of compact operator, we establish a new regularizationmethod based on an improved Tikhonov regularization operator. We prove the va-lidity, stability, the convergence of the present method and obtain the more optimalasymptotic convergence rate of the regularized solution than traditional Tikhonovregularization method. Meanwhile, the results of numerical example test demon-strate the stability and effectiveness of the proposed regularization method of thischapter. Finally, we apply them to the problems of dynamic load identification onthe structure of composite laminated cylindrical shell and simply supported plate.The computational results of two engineering examples show that the proposedregularization method basing on the improved Tikhonov regularization operatorin this chapter is very stable and effective, and it has the antinoise ability in theengineering problems of dynamic load identification.In terms of singular system theory of compact operator, we propose a newfamily of regularizing filter operators, and construct a new regularization method. We strictly prove its stability and perform the error analysis of regularized solu-tion from the view of the theory, and prove that the error bound of the regularizedsolution has the optimal asymptotic convergence order by choosing the priori ap-propriate regularization parameter. Finally, we apply it to the problem of dynamicload identification on the engineering structure. The computational results of en-gineering examples show that the proposed method of this chapter based on thefilter regularization operators is feasible and stable method, and also gives verygood identified results.For the usually so-called linear ill-posed problems, we propose a new fastconvergence iteration regularization method for solving the first kind of opera-tor equation based on the idea of traditional Landweber iteration regularizationmethod. From the view of mathematical theory, we prove its stability and validity,and also that we can obtain the optimum asymptotic convergence order of theregularized solution by choosing a posterior regularization parameter based on theidea of the Morozov's discrepancy principle. Meanwhile, we also prove that thepresent method can obtain the more optimal asymptotic order estimate than theLandweber iteration method. The performances of numerical test show that theproposed method reduces the amount of calculations, and quickens the speed ofconvergence of the regularized solutions. Finally, we apply it to the problem ofdynamic load identification on the structure of plate. The computational resultsshow that the present method is very effective and accurate in solving the dynamicload identification problems of engineering.Based on the idea of homotopy and perturbation theory, we propose a stableand effective modification, and obtain a new regularization method. This methodis different from the traditional perturbation theory, and does not rely on smallparameter. It establishes the equation containing the inserted parameter by usingthe technology of homotopy, and then obtains a new regularization method byexploiting this inserted parameter. It is also an extension of tradition homotopyperturbation method. The present method can overcome the lack of traditionhomotopy perturbation method, and its execution is also very simple, effectiveand precise. The computational results of numerical test and engineering examplesshow that the proposed method of this chapter can give very good results in solvingthe dynamic load identification problems of practical engineering.