Title: Reconstruction of shear force in Atomic Force Microscopy from measured displacement of the cone-shaped cantilever tip Show Chinese title

Title: 基于锥形悬臂梁尖端测得位移的原子力显微镜剪切力重建

Abstract: 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 $\rho_A(x)u_{tt}$ $+\mu(x)u_{t}+(r(x)u_{xx}+\kappa(x)u_{xxt})_{xx}=0$, $(x,t)\in (0,\ell)\times (0,T)$, subject to the following initial, $u(x,0)=0$, $u_t(x,0)=0$ and boundary, $u(0,t)=0$, $u_{x}(0,t)=0$, $\left (r(x)u_{xx}(x,t)+\kappa(x)u_{xxt} \right )_{x=\ell}=M(t)$, $\left (-(r(x)u_{xx}+\kappa(x)u_{xxt})_x\right )_{x=\ell}=g(t)$ conditions, where $M(t):=2h\cos \theta\,g(t)/\pi$ is the momentum generated by the transverse shear force $g(t)$. For the reconstruction of $g(t)$ the measured displacement $\nu(t):=u(\ell,t)$ is used as an additional data. The least square functional $J(F)=\frac{1}{2}\Vert u(\ell,\cdot)-\nu \Vert_{L^2(0,T)}^2$ is introduced and an explicit gradient formula for the Fr\'echet 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 $\nu (t)$. 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. Show Chinese abstract

Abstract: 本文提出了一种动态模型，用于重建原子力显微镜（AFM）悬臂尖端-样品相互作用中的剪切力$g(t)$。 锥形悬臂尖端与试样（样品）表面的相互作用通过阻尼Euler-Bernoulli梁方程$\rho_A(x)u_{tt}$ $+\mu(x)u_{t}+(r(x)u_{xx}+\kappa(x)u_{xxt})_{xx}=0$ ,$(x,t)\in (0,\ell)\times (0,T)$模拟，受到以下初始条件$u(x,0)=0$,$u_t(x,0)=0$和边界条件$u(0,t)=0$,$u_{x}(0,t)=0$,$\left (r(x)u_{xx}(x,t)+\kappa(x)u_{xxt} \right )_{x=\ell}=M(t)$,$\left (-(r(x)u_{xx}+\kappa(x)u_{xxt})_x\right )_{x=\ell}=g(t)$的约束，其中$M(t):=2h\cos \theta\,g(t)/\pi$是由横向剪切力$g(t)$产生的动量。 对于$g(t)$的重构，使用了测量得到的位移$\nu(t):=u(\ell,t)$作为附加数据。引入了最小二乘泛函$J(F)=\frac{1}{2}\Vert u(\ell,\cdot)-\nu \Vert_{L^2(0,T)}^2$，并通过求解伴随问题得到了关于 Fréchet 导数的显式梯度公式。这使得可以构建一个基于梯度的数值算法，用于从无噪声以及随机噪声测量的输出$\nu (t)$中重建剪切力。计算实验表明，所提出的算法非常快速且稳健。这使得能够开发出一个数值“工具”，用于通用 AFM 的计算实验。