Determination of an unknown shear force in cantilever Kirchhoff-Love plate from measured final data with application to atomic force microscope

Hasanov A., Kawano A.

INVERSE PROBLEMS, cilt.38, sa.6, 2022 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 38 Sayı: 6
  • Basım Tarihi: 2022
  • Doi Numarası: 10.1088/1361-6420/ac6283
  • Dergi Adı: INVERSE PROBLEMS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Compendex, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH, Civil Engineering Abstracts
  • Anahtar Kelimeler: inverse source problem, damped Kirchhoff-Love plate, atomic force microscope, measured boundary displacement, uniqueness, existence of a quasi-solution, Frechet differentiability
  • Kocaeli Üniversitesi Adresli: Evet

Özet

We present a new and feasible formulation for the inverse problem of identifying the unknown shear force acting on the inaccessible tip of the nonhomogeneous micro-cantilever plate which is a key component in most atomic force microscopes. The mathematical modelling of this phenomena leads to the inverse problem of determining the shear force g(x(2), t) acting on the inaccessible boundary x(1) = l(1) in a system governed by the variable coefficient damped Kirchhoff-Love plate equation rho(x)h(x)u(tt)+mu(x)u(t) + (D(x)(u(x1x1)+nu u(x2x2)))(x1x1)+ D(x)(nu u(x1x1) + u(x2x2))) + 2(1 - nu)(D(x)u(x1x2))(x1x2)= F(x, t), where (x,t) is an element of Omega(T) := Omega x (0,T), x := (x(1), x(2)) and Omega := (0, l(1)) x (0, l(2)) subset of R-2, subject to clamped-free boundary conditions, from final time measured output (displacement) u(T)(x):= u(x, T). We prove that under appropriate conditions, this inverse problem has a unique solution. We introduce the input-output operator and prove that it is a linear compact and Lipschitz continuous operator. Based on these results, the existence of a quasisolution of the inverse problem, as a solution of the minimization problem for the Tikhonov functional, is proved. Furthermore, an explicit gradient formula for the Frechet derivative of the Tikhonov functional is derived. Moreover, it is proved that the Frechet derivative is Lipschitz continuous. These results provide a mathematical basis for gradient based computational algorithm.