A guide to properly select the defocusing distance for accurate solution of transport of intensity equation while testing aspheric surfaces

Background: The determination of the mechanical properties of biological samples using Atomic Force Microscopy (AFM) at the nanoscale is usually performed using basic models arising from the contact mechanics theory. In particular, the Hertz model is the most frequently used theoretical tool for data processing. However, the Hertz model requires several assumptions such as homogeneous and isotropic samples and indenters with perfectly spherical or conical shapes. As it is widely known, none of these requirements are 100 % fulfilled for the case of indentation experiments at the nanoscale. As a result, significant errors arise in the Young’s modulus calculation. At the same time, an analytical model that could account complexities of soft biomaterials, such as nonlinear behavior, anisotropy, and heterogeneity, may be far-reaching. In addition, this hypothetical model would be ‘too difficult’ to be applied in real clinical activities since it would require very heavy workload and highly specialized personnel. Objective: In this paper a simple solution is provided to the aforementioned dead-end. A new approach is introduced in order to provide a simple and accurate method for the mechanical characterization at the nanoscale. Method: The ratio of the work done by the indenter on the sample of interest to the work done by the indenter on a reference sample is introduced as a new physical quantity that does not require homogeneous, isotropic samples or perfect indenters. Results: The proposed approach, not only provides an accurate solution from a physical perspective but also a simpler solution which does not require activities such as the determination of the cantilever’s spring constant and the dimensions of the AFM tip. Conclusion: The proposed, by this opinion paper, solution aims to provide a significant opportunity to overcome the existing limitations provided by Hertzian mechanics and apply AFM techniques in real clinical activities.

Download Full-text

The Indentation Rolling Resistance in Belt Conveyors: A Model for the Viscoelastic Friction

Lubricants ◽

10.3390/lubricants7070058 ◽

2019 ◽

Vol 7 (7) ◽

pp. 58 ◽

Cited By ~ 1

Author(s):

Nicola Menga ◽

Francesco Bottiglione ◽

Giuseppe Carbone

Keyword(s):

Contact Problem ◽

Finite Thickness ◽

Numerical Procedure ◽

Contact Problems ◽

Accurate Solution ◽

Rolling Contact ◽

Viscoelastic Layer ◽

Force Coefficient ◽

Belt Conveyors ◽

Energy Dissipated

In this paper, we study the steady-state rolling contact of a linear viscoelastic layer of finite thickness and a rigid indenter made of a periodic array of equally spaced rigid cylinders. The viscoelastic contact model is derived by means of Green’s function approach, which allows solving the contact problem with the sliding velocity as a control parameter. The contact problem is solved by means of an accurate numerical procedure developed for general two-dimensional contact geometries. The effect of geometrical quantities (layer thickness, cylinders radii, and cylinders spacing), material properties (viscoelastic moduli, relaxation time) and operative conditions (load, velocity) are all investigated. Physical quantities typical of contact problems (contact areas, deformed profiles, etc.) are calculated and discussed. Special emphasis is dedicated to the viscoelastic friction force coefficient and to the energy dissipated per unit time. The discussion is focused on the role played by the deformation localized at the contact spots and the one in the bulk of the thin layer, due to layer bending. The model is proposed as an accurate solution for engineering applications such as belt conveyors, in which the energy dissipated on the rolling contact of idle rollers can, in some cases, be by far the most important contribution to their energy consumption.

Download Full-text

An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space

Advances in Difference Equations ◽

10.1186/s13662-021-03428-3 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Mohammed Al-Smadi ◽

Nadir Djeddi ◽

Shaher Momani ◽

Shrideh Al-Omari ◽

Serkan Araci

Keyword(s):

Differential Equation ◽

Hilbert Space ◽

Reproducing Kernel ◽

Numerical Approach ◽

Accurate Solution ◽

Stability And Convergence ◽

Type Model ◽

Schmidt Orthogonalization ◽

And Performance ◽

Abel Differential Equation

AbstractOur aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space $\mathcal{H}^{2}[a,b]$ H 2 [ a , b ] . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.

Download Full-text

Accurate solution of linear algebraic systems

10.1145/1465482.1465533 ◽

1967 ◽

Cited By ~ 2

Author(s):

Cleve B. Moler

Keyword(s):

Accurate Solution ◽

Algebraic Systems ◽

Linear Algebraic Systems

Download Full-text

Tilt angle correction based forming algorithm for high steepness aspheric surfaces polishing

Optik ◽

10.1016/j.ijleo.2021.167082 ◽

2021 ◽

pp. 167082

Author(s):

Changjun Jiao ◽

Yong Shu ◽

Zhen Zhang ◽

Bo Wang ◽

Feihai Gao

Keyword(s):

Tilt Angle ◽

Aspheric Surfaces ◽

Angle Correction

Download Full-text

Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

Advances in Difference Equations ◽

10.1186/s13662-021-03288-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Ji Lin ◽

Yuhui Zhang ◽

Chein-Shan Liu

Keyword(s):

Boundary Conditions ◽

Boundary Value Problems ◽

Initial Conditions ◽

Boundary Value ◽

Accurate Solution ◽

Point Boundary ◽

Third Order ◽

Boundary Shape ◽

Free Function ◽

New Algorithms

AbstractFor nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

Download Full-text

Computational Method for Wavefront Sensing Based on Transport-Of-Intensity Equation

Photonics ◽

10.3390/photonics8060177 ◽

2021 ◽

Vol 8 (6) ◽

pp. 177

Author(s):

Iliya Gritsenko ◽

Michael Kovalev ◽

George Krasin ◽

Matvey Konoplyov ◽

Nikita Stsepuro

Keyword(s):

A Priori ◽

Spherical Wave ◽

Computational Method ◽

Optical Systems ◽

Radius Of Curvature ◽

Imaging Method ◽

Wavefront Sensing ◽

Phase Imaging ◽

Transport Of Intensity Equation ◽

Priori Information

Recently the transport-of-intensity equation as a phase imaging method turned out as an effective microscopy method that does not require the use of high-resolution optical systems and a priori information about the object. In this paper we propose a mathematical model that adapts the transport-of-intensity equation for the purpose of wavefront sensing of the given light wave. The analysis of the influence of the longitudinal displacement z and the step between intensity distributions measurements on the error in determining the wavefront radius of curvature of a spherical wave is carried out. The proposed method is compared with the traditional Shack–Hartmann method and the method based on computer-generated Fourier holograms. Numerical simulation showed that the proposed method allows measurement of the wavefront radius of curvature with radius of 40 mm and with accuracy of ~200 μm.

Download Full-text