本文使用了無網格徑向點差值法做分析，此法有著計算成本低，本質邊界條件處理簡便，添加物理參數容易等優點，是傳統有網格之有限元素之所不及。因此本文以無網格法進行新興材料石墨烯的振動特性以及實際應用之分析，討論實際應用時石墨烯剛性變化。
　　為了達到石墨烯振動強度之提升，本文使用無網格法將系統離散，並利用Eringen提出的非局域彈性理論來建構系統之統御方程。在分析中先將問題場域以節點離散並以無網格徑向點插值法構成每個節點之形狀函數，接著以非局域理論構成統御方程，以前述之形狀函數做離散，建構整體的剛性以及質量矩陣，求得物體的自然頻率，並且觀察加入其他物理因素後所得自然頻率的變化情形，提出強化剛性之設計。為了驗證本文所提理論之可行性與實際之應用，本研究提出了五個數值範例以進行分析。
一. 石墨烯平板之振動分析與比對。為了驗證方法之可行性，以無網格法分析平板振動，並與解析解做比對。無網格法分析得到的自然頻率與解析解相比誤差並不超過百分之二，而振動情況符合物理意義，可知此法應用於平板振動是可行的。
二. 奈米尺度結構振動分析。為了分析非局域性對自然頻率的影響，在此以石墨烯薄板做分析材料，當考慮非局域模數時，剛性下降，而非局域模數升高，剛性下降的幅度降低。
三. 強度增加分析設計。為了分析外層彈性體對材料剛性提升狀況，將上述所得結果加入外層彈性體因素分析。在局域自然頻率部份上升了大約四成，而非局域部分上升可達到八倍之多，可知外層彈性體的影響顯著。
四. 致動器吸附電壓。為了之後分析彈性體對致動器吸附電壓之影響，在此以石墨烯為基材做分析，分別加入電壓、凡德瓦力、卡西米爾效應做分析並比對。分析結果與文獻比對誤差大約百分之一，可知此數學模型以及計算流程可行且精準的。
五. 彈性體對吸附電壓影響。最後為了討論彈性體之影響，將上述所得結果加入外層彈性體因素分析，可發現吸附電壓上升了約半成多，所得結果與第三個數值模型結果吻合。由前述總合之結果提出奈米尺度材料之剛性強度提升之設計。
關鍵字：徑向點插值法，無網格，石墨烯，非局域。

　　This article uses meshless radial point interpolation method for analysis, this method is low-cost, great ability on handling essential boundary conditions and adding physical parameters easily compare to the finite element method. Therefore, this article choose this method to analyze the vibration characteristics and the practical application of graphene sheets.
　　In order to achieve the lifting of graphene vibration intensity, we construct the governing equation with nonlocal elasticity theory. Then, discrete the system with meshless method. Constructing the stiffness matrix and the mess matrix, calculate the nature frequency of object. And observe the changes of the nature frequency after adding other physical factors. Then, propose the design of strengthen the rigidity. To prove the feasibility of the method in this paper. This study presents five numerical examples for analysis.
I. Graphene plate of vibration analysis and comparison.
II. Nanoscale structural vibration analysis.
III. Strength increased analysis and design.
IV. The pull-in voltage of actuator.
V. Effect of elastic medium to the pull-in voltage
Key words: radial point interpolation method, meshless method, graphene, nonlocal.

[1] Liu, G.R. and Gu, Y.T., “An introduction to meshfree methods and their programming,” Springer Science and Business Media, 2005.
[2] Gopalakrishnan, S. and Narendar, S., “Wave propagation in nanostructures,” Switzerland, Springer, 2013.
[3] Liu, G.R., Zhang, G.Y., Gu, Y. and Wang, Y.Y., “A meshfree radial point interpolation method (RPIM) for three-dimensional solids,” Computational Mechanics, 36(6), 421-430, 2005.
[4] Chen, J.T., Chen, I.L., Chen, K.H., Lee, Y.T. and Yeh, Y.T., “A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function,” Engineering Analysis with Boundary Elements, 28(5), 535-545, 2004.
[5] Dehghan, M. and Shokri, A., “A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions,” Numerical Algorithms, 52(3), 461-477, 2009.
[6] Bui, T.Q., Nguyen, M.N. and Zhang, C., “An efficient meshfree method for vibration analysis of laminated composite plates,” Computational Mechanics, 48(2), 175-193, 2011.
[7] Wu, Z., “Compactly supported positive definite radial functions,” Advances in Computational Mathematics, 4(1), 283-292, 1995.
[8] Wendland, H., “Error estimates for interpolation by compactly supported radial basis functions of minimal degree,” Journal of approximation theory, 93(2), 258-272, 1998.

[9] Belytschko, T., Krongauz, Y., Organ, D., Fleming, M. and Krysl, P., “Meshless methods: an overview and recent developments,” Computer methods in applied mechanics and engineering, 139(1), 3-47, 1996.
[10] Bhat, R.B., “Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method,” Journal of Sound and Vibration, 102(4), 493-499, 1985.
[11] Itao, K. and Crandall, S.H., “Natural modes and natural frequencies of uniform, circular, free-edge plates,” Journal of Applied Mechanics, 46(2), 448-453, 1979.
[12] Liu, G.R., Yan, L., Wang, J.G. and Gu, Y.T., “Point interpolation method based on local residual formulation using radial basis functions,” Structural Engineering and Mechanics, 14(6), 713-732, 2002.
[13] Vasques, C.M.A. and Rodrigues, J.D. (Eds.)., “Vibration and Structural Acoustics Analysis: Current Research and Related Technologies,” Springer Science and Business Media, 2011.
[14] Yuqiu, L. and Yiqiang, Z., “Technical note: Calculation of stress intensity factors in plane problems by the sub-region mixed finite element method,” Advances in Engineering Software (1978), 7(1), 32-35, 1985.
[15] Zhu, T., “Meshless methods in computational mechanics,” 1998.
[16] Liu, G.R., Dai, K.Y. and Lim, K.M., “Static and vibration control of composite laminates integrated with piezoelectric sensors and actuators using the radial point interpolation method,” Smart materials and structures, 13(6), 1438, 2004.

[17] Liu, G.R., Zhao, X., Dai, K.Y., Zhong, Z.H., Li, G.Y. and Han, X., “Static and free vibration analysis of laminated composite plates using the conforming radial point interpolation method,” Composites Science and Technology, 68(2), 354-366, 2008.
[18] Leissa, A.W. and Narita, Y., “Natural frequencies of simply supported circular plates,” Journal of Sound and Vibration, 70(2), 221-229, 1980.
[19] Mohammadi, M., Ghayour, M. and Farajpour, A., “Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model,” Composites Part B: Engineering, 45(1), 32-42, 2013.
[20] Kukla, S. and Szewczyk, M., “Free vibration of annular and circular plates of stepped thickness. Application of Green's function method,” Scientific Research of the Institute of Mathematics and Computer Science, 5(1), 46-54, 2006.
[21] Chang, W.J. and Lee, H.L., “Free Asymmetric Transverse Vibration of Circilar Monolayer Graphene,” NANO, 9(07), 1450072, 2014.
[22] Behfar, K. and Naghdabadi, R., “Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium,” Composites science and technology, 65(7), 1159-1164, 2005.
[23] Anjomshoa, A., “Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory,” Meccanica, 48(6), 1337-1353, 2013.
[24] Babaei, H. and Shahidi, A.R. “Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method”, Archive of Applied Mechanics, 81(8), 1051-1062, 2011.
[25] Shi, J.X., Ni, Q.Q., Lei, X.W. and Natsuki, T., “Nonlocal vibration analysis of nanomechanical systems resonators using circular double-layer graphene sheets,” Applied Physics A, 115(1), 213-219, 2014.
[26] Malekzadeh, P. and Farajpour, A., “Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium,” Acta Mechanica, 223(11), 2311-2330, 2012.
[27] Arash, B. and Wang, Q., “Vibration of single-and double-layered graphene sheets,” Journal of Nanotechnology in Engineering and Medicine, 2(1), 011012, 2011.
[28] Pradhan, S.C. and Phadikar, J.K., “Nonlocal elasticity theory for vibration of nanoplates,” Journal of Sound and Vibration, 325(1), 206-223, 2009.
[29] Alibeigloo, A., “Free vibration analysis of nano-plate using three-dimensional theory of elasticity,” Acta mechanica, 222(1-2), 149-159, 2011.
[30] Murmu, T. and Pradhan, S.C., “Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory,” Journal of Applied Physics, 105(6), 064319, 2009.

[31] Dehghan, M. and Ghesmati, A., “Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM),” Computer Physics Communications, 181(4), 772-786, 2010.
[32] Naderi, A. and Baradaran, G.H., “Element free Galerkin method for static analysis of thin micro/nanoscale plates based on the nonlocal plate theory,” International Journal of Engineering, 26, 795-806, 2013.
[33] Pradhan, S.C. and Phadikar, J.K., “Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models,” Physics Letters A, 373(11), 1062-1069, 2009.
[34] Kumar, R.P. and Dodagoudar, G.R., “Two-dimensional modelling of contaminant transport through saturated porous media using the radial point interpolation method (RPIM),” Hydrogeology journal, 16(8), 1497-1505, 2008.
[35] Liew, K.M., Han, J.B. and Xiao, Z.M., “Vibration analysis of circular Mindlin plates using the differential quadrature method,” Journal of Sound and Vibration, 205(5), 617-630, 1997.
[36] Yoon, H.J., Yang, J.H., Zhou,Z., Yang, S.S. and Cheng, M. M. C., “Carbon dioxide gas sensor using a graphene sheet,” Sensors and Actuators B: Chemical, 157(1), 310-313, 2011.
[37] Pradhan, S.C. and Murmu, T., “Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory,” Physica E: Low-dimensional Systems and Nanostructures, 42(5), 1293-1301, 2010.
[38] Ansari, R., Arash, B. and Rouhi, H., “Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity,” Composite Structures, 93(9), 2419-2429, 2011.
[39] Adali, S., “Variational principles for nonlocal continuum model of orthotropic graphene sheets embedded in an elastic medium,” Acta Mathematica Scientia, 32(1), 325-338, 2012.
[40] Bardell, N.S., “Free vibration analysis of a flat plate using the hierarchical finite element method,” Journal of sound and vibration, 151(2), 263-289, 1991.
[41] Soroush, R., Koochi, A., Kazemi, A.S., Noghrehabadi, A., Haddadpour, H. and Abadyan, M., “Investigating the effect of Casimir and van der Waals attractions on the electrostatic pull-in instability of nano-actuators,” Physica scripta, 82(4), 045801, 2010.
[42] Sakhaee-Pour, A., Ahmadian, M.T. and Naghdabadi, R., “Vibrational analysis of single-layered graphene sheets,” Nanotechnology, 19(8), 085702, 2008.
[43] 楊建軍 鄭建龍. “無網格法求解軸對稱彈性力學問題,” 工程力學, 29(8), 8-13, 2012.
[44] 鄭長良. “非局部彈性直桿振動特徵及Eringen 常數的一個上限,” 力學學報,37(6), 796-798, 2005.
[45] 顧元通 丁樺. “無網格法及其最新進展.” 力學進展,35(3), 323-337, 2005.
[46] 李頂河 卿光輝 徐建新. “基於徑向基點插值函數的彈性力學Hamilton 正則方程無網格法,” 工程力學, 10, 007, 2011.