Parametric identification of nonlinear mathematical models of longitudinal waves propagation in materials

Аuthors

Alifanov O. M.^1*, Nenarokomov A. V.^2**, Nenarokomov K. A.¹, Titov D. M.^1***

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. ,

*e-mail: o.alifanov@yandex.ru
**e-mail: nenarokomovav@mai.ru
***e-mail: d.titov@mai.ru

Abstract

The purpose of this research was to develop a computational-experimental technique for remote (non- contact) diagnostics of structural defects in elastic materials. The performance of equipment is based on nonlinear interaction of acoustic beam of finite amplitude and analyzed material. Elastic properties of flaws differ from ones of base medium. Gradient of elastics properties on the border of flaws results in the appearance of non-classic nonlinearity in the domain. Such nonlinearity significantly exceeds the physical nonlinearity of a medium under diagnostics. Determination of spatial distribution of structural acoustic nonlinearity in the specimen allows to spot the defects.
The most promising direction in further development of non-destructive diagnostics (research methods) for the elastic composite materials is to use the procedure of inverse problems for estimation. Such problems are of great practical importance in the study of properties of composite materials used as non-destructive elastic surface coating in objects of space technology, power engineering etc. The developed method could be applied for determination of materials properties; the availability of corresponded experimental facilities allows us to provide uniqueness of the solution.
The iterative regularization method is outlined as one of the most efficient and universal ones for solving the ill-posed inverse problems, which arise in the course of the diagnostics. Therefore an extreme method of solving of ill-posed problems of wave propagation is developed. This method is based on the iterative regularization principle. The stability of the solution is achieved by timing the number of iterations in the gradient methods of the first order with the error of initial data. In the beginning rigorous mathematical results are stated in brief substantiating of this approach with respect to the most widely spread gradient algorithms, i.e., steepest descent and conjugate gradients, namely, information is supplied on regularization of the mentioned methods by the number of iterations and legitimacy of using or residual criterion to choose the number of the last iteration or restart the iterative process. To solve the inverse problems there have been developed iterative computational algorithms, implementing a minimization through gradient methods of residual functional, characterizing the square deviation of pressure values computed and measured by means of the assumed mathematical model. Here, the unknown functions are approximated by the cubic B-splines and the inverse problem is thus reduced to a definition of the unknown parameter vector. The gradient of residual functional is computed using the solution of boundary-value problem for the adjoin variable. In the approach under consideration all experimental information is used to estimate material characteristics as pressure functions on the all pressure interval of interest.

Keywords:

wave prorogation, iterative regularization, nonlinear acoustics, inverse problems, defects in elastic materials

References

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