[...]... conventional finite elements and propose the smoothed finite element method (SFEM) for 2D elastic problems [49] -Kenji Uchino (2009) [45 ], in which the research presents piezoelectric material and structure of piezoelectric motors 12 Introduction -Liu G R, Nguyen-Thoi T and Lam K Y, 200 9, An edge-based smoothed finite element method (ES-FEM) for static, free and force vibration analyses of solids, J Sound... development of adaptive piezoelectric finite elements [14] -Sze K Y and Yao L Q 200 0, Modelling smart structures with segmented piezoelectric sensors and actuators, In this paper, a number of finite element models have been developed for comprehensive modelling of smart structures with segmented piezoelectric sensing and actuating patches [90] -Vincent Piefort , Finite Element Modeling of Piezoelectric. .. all, describe the piezoelectric effect and details the basic formula in the case of unidimentional piezoelectricity to show the nature of the electromechanical coupling phenomena The next is the behavior of PZT and PVDF material also considered under electric field In chapter 2 concerns with mathematical modeling of piezoelectricity, standard finite element method (FEM) and smoothed finite element method. ..   e j   and h  max{diameter of element} 31 Numerical method for piezoelectric material We can formulate the following discrete weak form of the problems (2-36) and (2-37) : Find u h  ( V h ) n and  h  ( V h ) such that a ( u h , v h )  b ( h , v h )  v f  , vh s f , vh (2-38)  v h  V h ,0 ; u b ( h , u h )  c ( h ,   q s , )  h (2-39) h  V h ,0 ; v h Where (V h 0 ,  u ) 2 ... introduce the following bilinear forms  c iEk l u j  a (u , v )  , v  (H e k ij , k v i , j d  1 0 ( )) 2 (2-29) ,  v  (H vi, jd  k ,l 1 0 ( )) 2 (2-30)  b ( , v )   b ( , u )   e i k l u k , l ,i ,i ,   H d  d 1 0 ( ) (2-31)   s ik  , k f c ( , )  v ,   H 1 0 ( ) (2-32)  And let   ,v (2-33) fivvid   f  s ,v   0 u q s ,      fisvid      ... [76 ], Finite Element Analysis of a Synthetically Loaded Stator for a Piezoelectrically Driven Ultrasonic Traveling ware Motor, Worcester Polytechnic Institute of Massachusetts, USA, 1996 In which, the research build elements in FEM to simulate traveling ware ultrasonic motor -J A Christman, R R Woolcott, A I Kingon, and R J Nemanich, 199 8, Piezoelectric measurements with atomic force microscopy, American... corresponds to the dilation of piezoelectric material when one applies an electric field to it The material then plays the role of a piezoelectric actuator From a mathematical point of view, the piezoelectric field is governed by the Maxwell's equations in the material and the general equations of elasticity In this section, mathematical modeling of piezoelectricity is perform, which author formulated the variational... Insteading equations (2-23 ), (2-24 ), (2-25) and (2-26) into (2-21 ), (2-22) we obtain 29 Numerical method for piezoelectric material  c iEk l u k , l v i , j d   j   e k ij , k v i , j d    c iEk l u k , l v i n j d  j  e k ij , k v i n j d         e i k l u k , l  ,i   e i k l v k , l n i d  (2-28)   s ik  , k d  ,i (2-27)    fivvid  s ik   , k n i d   0   ... method (PIM) is presented for static and mode-frequency analysis of two-dimensional piezoelectric structures [50] -Liu G R, Dai K Y, Lim K M and Gu Y T 200 3, A radial point interpolation method for simulation of two-dimensional piezoelectric structures, Smart Mater Struct A meshfree, radial point interpolation method (RPIM) is presented for the analysis of piezoelectric structures, in which the fundamental... equations (2-29 ), (2-30 ), (2-31 ), (2-32 ), (2-33 ), (2-34) and (2-35) into equations (2-27) and (2-28) we get the weak form (variational problem ), find u  (H 1)2 ,   H 1 such that a ( u , v )  b ( , v )   v  (H f 1 0 v  ,v s (2-36) ,v 2 )u ( ) 1 0 (2-37) q s , b ( , u )  c (  ,  )     (H f )v ( ) where H H 1 0 ; 1 0 ; v u v  (  ) : (  ) :  H 1 ( ),  H 1 ( ), v | u = 0