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Publication . Article . 2007

A two-scale model for an array of AFM’s cantilever in the static case

Michel Lenczner; Ralph C. Smith;
Open Access
Published: 01 Sep 2007 Journal: Mathematical and Computer Modelling, volume 46, issue 5-6, pages 776-805 (issn: 0895-7177, Copyright policy )
Publisher: Elsevier BV
Abstract

The primary objective of this paper is to present a simplified model for an array of Atomic Force Microscopes (AFMs) operating in static mode. Its derivation is based on the asymptotic theory of thin plates initiated by P. Ciarlet and P. Destuynder and on the two-scale convergence introduced by M. Lenczner which generalizes the theory of G. Nguetseng and G. Allaire. As an example, we investigate in full detail a particular configuration, which leads to a very simple model for the array. Aspects of the theory for this configuration are illustrated through simulation results. Finally the formulation of our theory of two-scale convergence is fully revisited. All the proofs are reformulated in a significantly simpler manner.

Subjects by Vocabulary

Microsoft Academic Graph classification: Mathematics Theoretical physics Homogenization (chemistry) Cantilever Atomic force microscopy Scale model Multiscale modeling Mathematical proof Mathematical analysis Static mode Plate theory

Subjects

Modelling and Simulation, Computer Science Applications, Modeling and Simulation

61 references, page 1 of 7

[1] P. Destuynder, M. Salaun, Mathematical Analysis of Thin Plate Models, Springer, Berlin, 1996.

[2] P.G. Ciarlet, Mathematical Elasticity, vol. II, North-Holland, Amsterdam, 1997.

[3] E. Sanchez-Palencia, Nonhomogeneous Media; Vibration Theory, in: Lecture Notes in Phys., vol. 127, Springer, Berlin, 1980.

[4] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.

[5] G. Allaire, SIAM Journal on Mathematical Analysis 23 (6) (1992) 1482-1518.

[6] E. Canon, M. Lenczner, Mathematical and Computer Modelling 26 (5) (1997) 79-106.

[7] G. Nguetseng, SIAM Journal on Mathematical Analysis 20 (3) (1989) 608-623.

[8] D. Cioranescu, A. Damlamian, G. Griso, Comptes Rendus Mathe´matique Acade´mic des Sciences Paris 335 (1) (2002) 99-104.

[9] J. Casado-Diaz, Two-scale convergence for nonlinear Dirichlet problems in perforated domains, Proceedings of the Royal Society of Edinburgh. Section A 130 (2) (2000) 249-276.

[10] M. Lenczner, D. Mercier, Multiscale Modeling & Simulation 2 (3) (2004) 359-397.

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