Determination of unknown shear force in transverse dynamic force microscopy from measured final data
Journal of Inverse and Ill-Posed Problems, vol.32, no.2, pp.243-260, 2024 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 32 Issue: 2
- Publication Date: 2024
- Doi Number: 10.1515/jiip-2023-0021
- Journal Name: Journal of Inverse and Ill-Posed Problems
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Compendex, INSPEC, MathSciNet, zbMATH, Civil Engineering Abstracts
- Page Numbers: pp.243-260
- Keywords: damped Euler–Bernoulli beam, Fréchet gradient, inverse boundary value problem, Shear force identification, solvability of the inverse problem, transverse dynamic force microscopy cantilever
- Kocaeli University Affiliated: Yes
Abstract
In this paper, we present a new methodology, based on the inverse problem approach, for the determination of an unknown shear force acting on the inaccessible tip of the microcantilever, which is a key component of transverse dynamic force microscopy (TDFM). The mathematical modelling of this phenomenon leads to the inverse problem of determining the shear force g(t) acting on the inaccessible boundary x = ? in a system governed by the variable coefficient Euler Bernoulli equation Formula Prensented subject to the homogeneous initial conditions and the boundary conditions from the final time measured output (displacement) uT (x) := u(x, T). We introduce the input-output map (?g)(x) := u(x, T; g), g ? G, and prove that it is a compact and Lipschitz continuous linear operator. Then we introduce the Tikhonov functional and prove the existence of a quasi-solution of the inverse problem.We derive a gradient formula for the Fréchet gradient of the Tikhonov functional through the corresponding adjoint problem solution and prove that it is a Lipschitz continuous functional. The results of the numerical experiments clearly illustrate the effectiveness and feasibility of the proposed approach.