Shape and topology optimization for tailoring the ratio between two flexural eigenfrequencies of atomic force microscopy cantilever probe

Abstract

In an operation mode of atomic force microscopy that uses a higher eigenmode to determine the physical properties of material surface, the ratio between the eigenfrequency of a higher flexural eigenmode and that of the first flexural eigenmode was identified as an important parameter that affects the sensitivity and accessibility. Structure features such as cut-out are often used to tune the ratio of eigenfrequencies and to enhance the performance. However, there lacks a systematic and automatic method for tailoring the ratio. In order to deal with this issue, a shape and topology optimization problem is formulated, where the ratio between two eigenfrequencies is defined as a constraint and the area of the cantilever is maximized. The optimization problem is solved via the level set based method.

Keywords

atomic force microscopy / cantilever probe / eigenfrequency / optimization

References

[1] StarkR W, DrobekT, HecklW M. Tapping-mode atomic force microscopy and phase imaging in higher eigenmodes. Applied Physics Letters, 1999, 74(22): 3296

[2] HillenbrandR, StarkM, GuckenbergerR. Higher-harmonic generation in tapping-mode atomic-force microscopy: Insights into the tip–sample interaction. Applied Physics Letters, 2000, 76(23): 3478

[3] SahinO, QuateC F, SolgaardO, AtalarA. Resonant harmonic response in tapping-mode atomic force microscopy. Physical Review B: Condensed Matter and Materials Physics, 2004, 69(16): 165416

[4] SahinO, YaraliogluG, GrowR, ZappeS F, AtalarA, QuateC, SolgaardO. High-resolution imaging of elastic properties using harmonic cantilevers. Sensors and Actuators A: Physical, 2004, 114(2-3): 183-190

[5] LiH, ChenY, DaiL. Concentrated-mass cantilever enhances multiple harmonics in tapping-mode atomic force microscopy. Applied Physics Letters, 2008, 92(15): 151903

[6] SethianJ A, WiegmannA. Structural boundary design via level set and immersed interface methods. Journal of Computational Physics, 2000, 163(2): 489-528

[7] OsherS, SantosaF. Level-set methods for optimization problems involving geometry and constraints: Frequencies of a two-density inhomogeneous drum. Journal of Computational Physics, 2001, 171(1): 272-288

[8] AllaireG, JouveF, ToaderA M. A level-set method for shape optimization. Comptes Rendus Mathematique, 2002, 334(12): 1125-1130

[9] AllaireG, JouveF, ToaderA M. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 2004, 194(1): 363-393

[10] WangM Y, WangX M, GuoD M. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1-2): 227-246

[11] PedersenN L. Maximization of eigenvalue using topology optimization. Structural and Multidisciplinary Optimization, 2000, 20(1): 2-11

[12] PedersenN L. Design of cantilever probes for atomic force microscopy (AFM). Engineering Optimization, 2000, 32(3): 373-392

[13] ChenK N. Model updating and optimum designs for V-shaped atomic force microscope probes. Engineering Optimization, 2006, 38(7): 755-770

[14] DíaazA, KikuchiN. Solution to shape and topology eigenvalue optimization problems using a homogenization method. International Journal for Numerical Methods in Engineering, 1992, 35(7): 1487-1502

[15] MaZ D, ChengH C, KikuchiN. Structural design for obtaining desired eigenfrequencies 23 by using the topology and shape optimizationg method. Computing Systems in Engineering, 1994, 5(1): 77-89

[16] KosakaI, SwanC C. A symmetry reduction method for continuum structural topology optimization. Computers & Structures, 1999, 70(1): 47-61

[17] AllaireG, JouveF. A level–set method for vibration and multiple loads structural optimization. Computer Methods in Applied Mechanics and Engineering, 2005, 194(30-33): 3269-3290

[18] DuJ B, OlhoffN. Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Structural and Multidisciplinary Optimization, 2007, 34(2): 91-110

[19] XiaQ, ShiT, WangM Y. A level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration. Structural and Multidisciplinary Optimization, 2011, 43(4): 473-485

[20] HopcroftM A, NixW D, KennyT W. What is the Young’s modulus of silicon? Journal of Microelectromechanical Systems, 2010, 19(2): 229-238

[21] ZienkiewiczO C, TaylorR L. The Finite Element Method (5th Edition), Vol. 2. Butterworth-Heinemann, 2000

[22] LiewK M, WangC M, XiangY, KitipornchaiS. Vibration of Mindlin Plates: Programming the P-Version Ritz Method. Elsevier, 1998

[23] XingY, LiuB. Closed form solutions for free vibrations of rectangular Mendelian plates. Acta Mechanica Sinica, 2009, 25(5): 689-698

[24] ChoiK K, KimN H. Structural Sensitivity Analysis and Optimization. Springer, 2005

[25] HaugE J, ChoiK K, KomkovV. Design Sensitivity Analysis of Structural Systems. Academic Press, 1986

[26] NocedalJ, WrightS J. Numerical Optimization. Springer, 1999

[27] WangX M, WangM Y, GuoD M. Structural shape and topology optimization in a level-set-based framework of region representation. Structural and Multidisciplinary Optimization, 2004, 27(1-2): 1-19

[28] MeiY, WangX. A level set method for structural topology optimization and its applications. Advances in Engineering Software, 2004, 35(7): 415-441

[29] XiaQ, ShiT L, WangM Y, LiuS Y. A level set based method for the optimization of cast part. Structural and Multidisciplinary Optimization, 2010, 41(5): 735-747

[30] XiaQ, ShiT, LiuS, WangM Y. A level set solution to the stress-based structural shape and topology optimization. Computers & Structures, 2012, 90-91: 55-64

[31] SethianJ A. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. 2nd Edition. Cambridge Monographs on Applied and Computational Mathematics. Cambridge, UK:Cambridge University Press, 1999

[32] OsherS, FedkiwR. Level Set Methods and Dynamic Implicit Surfaces. New York: Springer-Verlag, 2002

[33] TcherniakD. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering, 2002

Acknowledgements

This research work was partly supported by the National Natural Science Foundation of China (Grant No. 51105159), Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110142120091), which the authors gratefully acknowledge.

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg