Abstract
The problem of differentiating non-smooth functions of specimen displacements, which are measured during the material removal, is discussed. This problem arises when employing the layer removal method, namely a method of rings and strips, for residual stress depth profiling. It is shown that this problem is ill-posed and special solution methods are required in order to obtain a stable solution. The stability of the solution affects to a high extent the resulting accuracy of the residual stress evaluation in the investigated material. The presented study discusses a numerical approach to solving such ill-posed problems. The proposed approach, which is based on the Tikhonov regularization and a regularized finite difference method, provides a stable approximate solution, including its pointwise error estimation. The advantage of this approach is that it does not require any knowledge about the unknown exact solution; the pointwise error estimation of the measured data is the only prior information that must be available. In addition, this approach provides a convergence of the approximate solution to the unknown exact one when the perturbation of the initial data approaches zero.
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Acknowledgments
This research was supported by the Government of Russian Federation (the Ministry of Education and Science), Project No. 2012-218-03-120. The author is deeply grateful to his late colleague Prof. Zamashchikov for the insightful discussions and the valuable research assistance.
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Tolstikhin, K. An approach to differentiation of non-smooth functions obtained during residual stress measurements by layer-removal method. J Eng Math 103, 87–95 (2017). https://doi.org/10.1007/s10665-016-9862-x
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DOI: https://doi.org/10.1007/s10665-016-9862-x