arXiv2023
In this paper, a dynamic model of reconstruction of the shear force $g(t)$ in the Atomic Force Microscopy (AFM) cantilever tip-sample interaction is proposed. The interaction of the cone-shaped cantilever tip with the surface of the specimen (sample) is modeled by the damped Euler-Bernoulli beam equation $ρ_A(x)u_{tt}$ $+μ(x)u_{t}+(r(x)u_{xx}+κ(x)u_{xxt})_{xx}=0$, $(x,t)\in (0,\ell)\times (0,T)$, subject to the following initial, $u(x,0)=0$, $u_t(x,0)=0$ and boundary, $u(0,t)=0$, $u_{x}(0,t)=0$, $\left (r(x)u_{xx}(x,t)+κ(x)u_{xxt} \right )_{x=\ell}=M(t)$, $\left (-(r(x)u_{xx}+κ(x)u_{xxt})_x\right )_{x=\ell}=g(t)$ conditions, where $M(t):=2h\cos θ\,g(t)/π$ is the momentum generated by the transverse shear force $g(t)$. For the reconstruction of $g(t)$ the measured displacement $ν(t):=u(\ell,t)$ is used as an additional data. The least square functional $J(F)=\frac{1}{2}\Vert u(\ell,\cdot)-ν\Vert_{L^2(0,T)}^2$ is introduced and an explicit gradient formula for the Fréchet derivative through the solution of the adjoint problem is derived. This allows to construct a gradient based numerical algorithm for the reconstructions of the shear force from noise free as well as from random noisy measured output $ν(t)$. Computational experiments show that the proposed algorithm is very fast and robust. This allows to develop a numerical "gadget" for computational experiments of generic AFMs.