DSpace at VNU: A stabilized finite element method for certified solution with bounds in static and frequency analyses of...
Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 Contents lists available at SciVerse ScienceDirect Comput Methods Appl Mech Engrg journal homepage: www.elsevier.com/locate/cma A stabilized finite element method for certified solution with bounds in static and frequency analyses of piezoelectric structures L Chen a,⇑, Y.W Zhang a, G.R Liu b, H Nguyen-Xuan c, Z.Q Zhang d a Department of Engineering Mechanics, Institute of High Performance Computing, Fusionopolis Way #16-16 Connexis, Singapore 138632, Singapore School of Aerospace Systems, University of Cincinnati, Cincinnati, OH 45221-0070, USA c Faculty of Mathematics and Computer Science, University of Science, Vietnam National University-HCM, Viet Nam d Singapore-MIT Alliance (SMA), E4-04-10, Engineering Drive 3, Singapore 117576, Singapore b a r t i c l e i n f o Article history: Received 14 January 2012 Received in revised form 21 May 2012 Accepted 22 May 2012 Available online 29 May 2012 Keywords: Numerical methods Piezoelectric structures NS-FEM Solution bound Frequency Stabilization a b s t r a c t This paper develops a stabilization procedure in piezoelectric media to ensure the temporal stability of node-based smoothed finite element method (NS-FEM), and applies it to obtain certified solution with bounds in both static and frequency analyses of piezoelectric structures using three-node triangular elements For such stabilized NS-FEM, two stabilization terms corresponding to squared-residuals of two equilibrium equations, i.e., mechanical stress equilibrium and electric displacement equilibrium, are added into the smoothed potential energy functional of the original NS-FEM A gradient smoothing operation is then performed on second-order derivatives of shape functions to achieve the stabilization terms Due to the use of divergence theory, the smoothing operation relaxes the requirement of shape functions, so that the square-residuals can be evaluated using linear elements The effectiveness of the present stabilized NS-FEM is demonstrated via numerical examples Ó 2012 Elsevier B.V All rights reserved Introduction Piezoelectric materials showing an ability of transformation between mechanical energy and electric energy have been widely used in various applications, where they serve as sensors, actuators, transducers or active damping devices These applications range from sub-millimeter length scales in micro-electro-mechanical systems up to large scales in the design of smart electromechanical structures However, analytical solutions are limited for solving practical problems of complicated geometry, for which we have to resort to numerical methods when analyzing and designing piezoelectric structures, such as the finite element method (FEM) [1,2], the bubble/incompatible displacement method [3], the mixed and hybrid formulations [4–6] and the piezoelectric finite element with drilling degrees of freedom [7] Several meshless methods [8] have also been used to analyze piezoelectric structures such as the meshless point collocation method (PCM) [9], the point interpolation method (PIM) [10], the radial point interpolation method (RPIM) [11], and the moving Kriging (MK) interpolation-based meshless method [12] In practical engineering, the upper and lower bound analyses [13] or the so-called dual analyses [14] have been an important ⇑ Corresponding author Tel.: +65 64191246 E-mail address: chenl@ihpc.a-star.edu.sg (L Chen) 0045-7825/$ - see front matter Ó 2012 Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.cma.2012.05.018 mean for safety and reliability assessments of piezoelectric structural properties In order to implement these analyses, two numerical models are usually used: one gives a lower bound and the other gives an upper bound to the unknown exact solution The most popular models giving lower bounds to the exact strain energy and electric energy (or upper bounds to the exact natural frequencies) are the FEM models in which displacement and electric potential both satisfy fully compatibility conditions, which are widely used in solving complicated engineering problems The models that give upper bounds in energy norm to the exact solutions (or lower bounds to the exact natural frequencies) can be one of the following models: (1) the stress equilibrium FEM models [15]; (2) the recovery models using a statically admissible stress field from displacement FEM solutions [16,17]; (3) the hybrid equilibrium FEM models [18] These three models, however, are known to have the following two common disadvantages: (1) the formulation and numerical implementation are complicated and expensive computationally; (2) there exist spurious modes in the hybrid models or the spurious modes often occur due to the simple fact that tractions cannot be equilibrated by the stress approximation field Due to those drawbacks, these three models are not widely used in practical applications, and are still very much confined in the area of academic research On another front of computational mechanics, a strain smoothing technique [19,20] was introduced by Chen et al [19] for spatial stabilization of nodal integrated meshfree methods, and later 66 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 extended by Yoo and Moran to the natural element method (NEM) [20] More recently, Liu [21] has generalized this gradient smoothing technique in order to weaken the consistence requirement for the field functions, allowing the use of certain types of discontinuous displacement functions Based on this generalization, a G space theory and a generalized smoothed Galerkin (GS-Galerkin) weak form have been developed [22], leading to the so-called weakened weak ðW2 Þ foundations of a family of numerical methods Among them, a cell-based smoothed finite element method (SFEM or CSFEM) [23] was first formulated by introducing the gradient smoothing technique to (compatible) FEM settings In such SFEM, the elements are divided into smoothing domains (SD) over which the strain is smoothed In addition, uniquely conceived, an edgebased smoothed finite element method (ES-FEM) constructs smoothing domains based on the element edges [24] It is found that such unique technique gives the ES-FEM remarkable and superior convergence properties, computational accuracy and efficiency, and spatial and temporal stability These attractive properties have also led to the applications of ES-FEM to both static and frequency analyses of piezoelectric structures [25,26] However, the ES-FEM usually produces a lower bound to the exact solution in energy norm as the standard fully compatible FEM [27] A node-based smoothed finite element (NS-FEM) [28] was also formulated using smoothing domains associated with nodes in FEM settings The most important property of NS-FEM explored and proven by Liu and coworkers [29,30] is that the NS-FEM is a general method producing an upper bound solution in energy norm to the exact solutions of the force-driven elasticity problems, A simple explanation of the upper-bound property of NS-FEM is the underestimation of the system stiffness (in a monotonic fashion [31,32]), in contrast to the well known overestimation of the system stiffness for the displacement-based fully compatible finite element model The overestimation behavior of FEM results in the upper bounds to the exact natural frequencies In contrast, the stiffness underestimation behavior of the NS-FEM models can lead to lower-bound solutions in natural frequencies of free vibrating solids and structures However, similar to other nodal integrated methods [31–33], the NS-FEM suffers from the temporal instability due to its ‘‘overly soft’’ feature rooted at the use of a relatively small number of SDs in relation to the nodes The temporal instability is defined to have spurious non-zero eigen modes Such models are spatially stable (with positive coercivity constants), and will not have zero-energy modes However, when they are excited at (strictly non-zero) higher energy level, it can behave unphysically To eliminate these spurious modes, one possible method is to employ the Lagrangian kernels as Rabczuk et al [34,35] proposed Also, Beissel and Belytschko [36] have developed a scheme to stabilize these nodal integrated methods by the addition to the potential energy functional a stabilization term, which contains the square of the residual of the equilibrium equation Further, the latter has recently been applied to the NS-FEM by adding the stabilization term over the problem domain regulated by a stabilization parameter to the corresponding smoothed potential energy functional [37] However, both of these were only limited to mechanical effects, and did not consider a coupling between mechanical and electrical variables This paper further extends the stabilization technique in [36] to the piezoelectric media to cure the temporal instability of NS-FEM, by means of adding to the smoothed potential energy functional of the original NSFEM two stabilization terms corresponding to squared-residuals of two equilibrium equations, i.e., mechanical stress equilibrium and electric displacement equilibrium These squared-residual terms can be regarded as an additional constraint of the system, and be used to cure the ‘‘overly soft’’ behavior of NS-FEM, for which the spurious non-zero energy modes can be removed In order to realize these two stabilization terms, the gradient smoothing technique is extended to the second-order derivatives, so that only the first-order derivatives of the shape function are needed in our formulation Therefore, the present squared-residual stabilization technique works very well for linear elements, such as 3-node triangular elements, and suits ideally in many ways to the NS-FEM models Further, the stabilized NSFEM is applied to obtain certified solution with bounds in both static and frequency analyses of piezoelectric structures Intensive benchmark numerical examples are presented to demonstrate the interesting properties of the proposed method It is found that upper bound in energy norm to the exact solutions of static piezoelectric problems and lower bound natural frequencies in vibration analyses can be achieved using a proper stabilization parameter Basic piezoelectric formulations 2.1 Governing equations Consider a 2D piezoelectric solid governed by the equilibrium equation in the domain X R2 bounded by CðC ¼ Cu ỵ Ct ; Cu \ Ct ẳ 0ị as ' div r ỵ b ẳ on X; div D ỵ qs ẳ 1ị where r is the Cauchy stress tensor, b represents the vector of body force applied in the problem domain, D denotes the electric displacement and qs is the free point charge density For dynamics problems of linear electroelastic solids, the strong form of the governing equation is ỵ gu_ on X; div r ỵ b ẳ qu 2ị where q is the density of the mass, and g is a set of viscosity parameters The strain e and the electric field E are, respectively, derived from the displacement u and the electric potential u, and could be written by the vector form e ¼ rs u; ð3Þ E ¼ Àgrad u; where rs is the symmetric gradient operator, " rs ¼ @ @x @ @y @ @y @ @x #T ð4Þ : Writing the stress tensor r as the vector form, the constitutive equations have the following form r D ! ¼ cE ÀeT e jS ! e E ! ; ð5Þ where cE denotes the elastic matrix measured at constant electric field, jS is the dielectric matrix at constant mechanical strain, and e is the piezoelectric matrix These tensors are known experimentally for various kinds of piezoelectric materials They are usually not isotropic To be specific, Eq (5) can also be written in a component form for the 2D plane piezoelectric problem rxx c11 c12 c13 Àe11 Àe21 32 exx 76 6 ryy c21 c22 c23 Àe12 Àe22 76 eyy 7 76 7 76 6 rxy ¼ c31 c32 c33 Àe13 Àe23 76 cxy 7: 76 7 D 6e j12 54 Ex x 11 e12 e13 j11 Dy Ey e21 e22 e23 j21 j22 ð6Þ On the other hand, Eq (7) in the following can be recast into a matrix form in contrast to Eq (5) as 67 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 e E " ! ¼ sE T Àd # ! ð7Þ D eS d r Taking Hamilton’s variational principle yields to dpuị ẳ ẳ in which use is made of the relationships d ¼ ecÀ1 E ; sE ¼ ecE , and T eS ¼ jS þ ecÀ1 e , where, e is the dielectric matrix measured at conS E stant stress, d stands for the piezoelectric strain matrix and sE is the elastic compliance matrix At part Cu , the essential boundary condition is given by u ¼ uC on Cu ; u ¼ uC ð8Þ where uC is the vector of the prescribed displacement, and uC denotes the prescribed electrical potential, whereas at the part Ct , the natural boundary condition is given by r Á n ¼ tC on Ct ; D Á n ẳ qC 9ị where tC is the vector of prescribed tractions, qC denotes the surface change on Ct , and nj is the surface outward normal of the boundary Ct 2.2 Galerkin weak form and finite element formulation Z Z T qu_ u_ ỵ eT ðuÞrðuÞ À uT b dX À uT tC dC X Ct |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ð10Þ electric part Consider the domain discretized into N e of non-overlapping and e non-gap elements and N n nodes, such that X ¼ [Nm¼1 Xem and e e Xi \ Xj ¼ 0; 8i – j, the approximation of displacement field uh and electric potential for a 2D electroelastic problem is given by: uðxÞ ¼ X Ni ðxÞui ; i2nen X uðxÞ ¼ Ni ðxÞui ; ð11Þ i2nen where nen is the set of nodes of the element containing T x; ui ẳ ẵ uxi uyi is the vector of nodal displacements, respectively, in x axis and y axis, ui is the nodal electric potential, and Ni is a matrix of shape functions Ni xị ẳ Ni xị 0 Ni xị T ð16Þ electric part and then substituting Eqs (11)–(13) into Eq (10), we have a set of piezoelectric dynamic equations !& m 0 € u € u " ' þ # kuu & u ' kuu T kuu kuu u ẳ & ' f ; q 17ị where mẳ Z qNT NdX; ð18Þ ðBu ÞT cE Bu dX; ð19Þ Bu ịT eT Bu dX; 20ị X kuu ẳ Z X Z kuu ¼ À f¼ Z X q¼À Z Bu ịT jS Bu dX; X NT xịbdX ỵ Z Ct Z NT ðxÞqs dX À ð21Þ NT ðxÞtC dC; Z NT ðxÞqC dC: ð22Þ ð23Þ Ct X NS-FEM for the piezoelectricity mechanical part Z Z T D uịEuị uqs dX ỵ uqC dC X Ct |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} mechanical part Z dD uịEuị duqs dX ỵ duqC dC X Ct | {z } X puị ẳ Z T ỵ deT uịruị duT b dX du qu duT tC dC X Ct |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Z kuu ¼ In this section, a finite element formulation for piezoelectricity is established via a variational formulation, in which the following general energy functional p is used to express a summation of the following two parts: (1) mechanical contribution including kinetic energy, strain energy, and mechanical external work, and (2) electrical contribution involving dielectric energy and electric external work [38] Z Detailed formulations of the NS-FEM have been proposed in the previous work [28] Here, we mainly focus on the extension of the NS-FEM to the piezoelectric problem using a basic mesh for 3-node linear triangular elements 3.1 Gradient smoothing The strain smoothing method was proposed by Chen et al in [19], and later generalized by Liu to form the basis of G space theory [19,20] Consider the 2D domain X discretized into N s nonoverlapping smoothing domains as shown in Fig 1, the smoothing operation on the gradient of a field / for a point xk in a smoothing domain Xsk is given as follows kị ẳ /x Z Xsk /ðxÞWðx; x À xk ÞdX; ð24Þ ! ð12Þ in which Ni ðxÞ is the shape function for node i Substituting the approximations of Eq (11) into Eq (3), we obtain e ¼ rs u ¼ X u B i ui ; 13ị i2nen X E ẳ grad u ¼ À Bu i ui ; ð14Þ i2nen where Ni;x u Bi ¼ N i;y Ni;y 5; Ni;x Bu i ¼ Ni;x Ni;y ! ð15Þ Fig Division of problem domain X into non-overlapping smoothing domains Xsk for xk The smoothing domain is also used as basis for integration 68 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 ically for problems with complicated geometry, which is also employed as the mesh platform in this work Consider the domain X discretized into N e non-overlapping and non-gap triangular elements and N n nodes, the local smoothing domains in the NS-FEM are constructed with respect to the nodes of n triangular elements, such that X ¼ [Nk¼1 Xsk and Xsi \ Xsj ¼ 0; 8i – j, in which N n is the total number of nodes in the element mesh In this case, the number of smoothing domains are the same as the number of nodes:N s ¼ N n For the triangular elements, the smoothing domain Xsk for node k is created by connecting sequentially the mid-edge-points and the centroids of the surrounding triangles of the node as shown in Fig Fig Construction of node-based strain smoothing domains based on 3-node triangular elements 3.3 Smoothed Galerkin weak form and discrete equations where Wðx; x À xk Þ is a smoothing function that generally satisfies the following properties [19]: Because a smoothed Galerkin weak form with smoothed gradient over smoothing domains is variationally consistent as proven in [22], using this smoothed or weakened weak form with displacement field and electrical potential satisfying the essential boundary conditions, we have Wðx; x À xk Þ P 0; and Z Xsk Wx; x xk ịdX ẳ 1: ð25Þ The Heaviside-type piecewise constant function is employed in this research: ( Wx; x xk ị ẳ 1=Ask x Xsk ; x R Xsk ; Z Xsk Ask /;j xịWx; x xk ịdX ẳ Z /xịnj dC; Csk 27ị ek xk ị ẳ ẳ Xsk Ask ek xịWx; x xk ịdX ẳ Z Xsk Z Xsk ð28Þ where Ln is the matrix of unit outward normal which can be expressed as nx Ln ¼ ny ny 5: ð29Þ Xsk EðxÞWðx; x À xk ÞdX Z ¼ Àu;j ðxÞWðx; x À xk ÞdX ¼ À s Ak Xsk ð31Þ electric part Employing the strain smoothing operation, the smoothed strain ek over Xsk from the displacement approximation in Eq (3) can be written in the following matrix form ek ẳ X Bui xk ịui : 32ị i2nsk Likewise, the matrix form of the smoothed electric field can be expressed by Eẳ X Bu i xk ịui ; ð33Þ where nsk is the set of nodes associated with the smoothing domain Xsk Bui ðxk Þ is the smoothed strain matrix for the mechanical part, and Bu i ðxk Þ corresponds to the electric part, i.e., the smoothed electric field matrix Those two matrix operations can be written as follows 2 bix ðxk Þ iy x ị Bui xk ị ẳ b k 5; biy ðxk Þ bix ðxk Þ x ị ẳ b ih k Ask Similarly, the smoothed electric field can be expressed by Eðxk Þ ¼ T " Bu i ðxk Þ ¼ # ðx Þ b ix k ðx Þ ; b iy 34ị k x ị; h ẳ x; y, is computed by where b ih k nx Z mechanical part À i2nsk rs uWðx; x À xk ÞdX Ld uðxÞdX; À Z Á À dD ðuÞEðuÞ À duqs dX þ duqC dC: X Ct |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ð26Þ where Csk is the segments of boundary of the smoothing domain Xsk i;j in Eq (24) to the followSubstituting the smoothed gradient / ing smoothing operation of strain vector e in Eq (3) yields the smoothed strain as follows Z Z ỵ deT uịr uị duT b dX À duT qu duT tC dC X Ct |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Z where Ask is the area of the smoothing domain Xsk Assuming /;j exists (the assumed field / is continuous), and introducing divergence theorem to Eq (24), we shall have ;j xk ị ẳ / Z dps uị ¼ ¼ Z Csk Z Csk nh ðxÞNi ðxÞdC: ð35Þ Using the Gaussian integration along the segments of boundary uðxÞnj dC: ð30Þ Csk , we have: "N # Nseg X gau 1X wm;n Ni ðxm;n Þnh ðxm;n Þ h ẳ x; yị; bih ẳ s Ak mẳ1 nẳ1 ð36Þ 3.2 Construction of smoothing domains Smoothing domain of the NS-FEM is constructed based on the nodes of elements, as illustrated in Fig 2, and the elements used can be 3-node triangular element, 4-node quadrilateral element, and n-side polygonal element The only requirement of the smoothing domain is non-overlap, and not required to be convex In order to simplify meshing, the NS-FEM generally relies on the 3-node triangular elements that can be usually generated automat- where Nseg is the number of segments of the boundary Csk ; Ngau is the number of Gaussian points used in each segment, wm;n is the corresponding weight of Gaussian points, nh is the outward unit normal corresponding to each segment on the smoothing domain boundary and xm;n is the nth Gaussian point on the mth segment of the boundary Csk Substituting the approximated displacements and electric potential in Eq (3), and the smoothed strains and electric field, L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 respectively, from Eqs (32) and (33) into the smoothed Galerkin weak form leads to the following equation Z ðBu ÞT cE Bu udX þ duT ðBu ÞT eT Bu udX X X X Z Z u T u u T u ỵ du ðB Þ eB udX À du ðB Þ js B udX X ZX Z Z À duT NT bdX À duT NT tC dC ỵ du NT qs dX X Ct X Z T ỵ du N qC dC ẳ 0: 37ị duT Z qNT Nu dX ỵ duT Z Ct Eliminating du and du yields the following two discrete equilibrium equations m 0 !& € u € u " ' ỵ uu k kT uu # uu & u ' & f ' k ¼ ; uu u q k ð38Þ where f and q are computed similarly by Eqs (22) and (23), respectively, and the mass matrix m adopts a consistent mass matrix, thus can be calculated in the same way as Eq (18) The stiffness matrix is then assembled by uu ¼ k Ns Z X Xsk k¼1 ¼ k uu ðBu ÞT cE Bu dX; Ns Z X u T T Xsk kẳ1 u B ị e B dX; Ns Z X uu ¼ À k ðBu ÞT jS Bu dX: Xsk k¼1 gard, the gradient smoothing based on divergence theorem has been proposed to eliminate the spatial instability in the nodal integrated methods, such as EFG, NEM and NS-FEM This technique produces the smoothed derivatives of shape functions using only the shape function values and does not need to calculate the derivatives of shape functions To be specific, the NS-FEM has been proven spatially stable [29] On the other hand, the ‘‘overly-soft’’ property of NS-FEM leads to spurious non-zero eigen modes, that is, temporal instability This kind of instability does not influence the calculation of the static problems, but, it affects the time-dependent analyses (e.g., dynamics problems, transient analyses, and so on) One approach to cure this temporal instability in the NS-FEM is to use a scheme of Beissel and Belytschko [36], in which a modified potential energy functional is constructed by adding a smoothed squared-residual stabilization term into the smoothed potential energy functional [37] In this regard, we extend the stabilization technique in [36] to the piezoelectric media, by means of adding two stabilization terms corresponding to squared-residuals of two equilibrium equations into the smoothed potential energy functional of the original NSFEM ps uị ẳ 39ị uu ẳ k uu ẳ k Bu ÞT cE Bu Ask ; k¼1 Ns X ðBu ÞT eT Bu Ask ; k¼1 Ns X ¼À k uu ðBu ÞT jS Bu Ask : smoothed potential functionalÀelectric part Z al2 ỵ div r ỵ bịT div r þ bÞdX À c ðdiv D þ qs Þ2 dX ; j X ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} E X |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ð41Þ ð42Þ al2c ð45Þ Eliminating the u yields the following equation where a is the dimensionless, real, finite and non-negative stabilization parameter; lc is the characteristic length of the elements in the mesh that is determined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi areaðXÞ ; Ne ð49Þ where areaðXÞ is the area of the problem domain, and N e is the number of elements E is the effective Young’s modulus of the material, and the average diagonal value in the elastic matrix is employed for anisotropic material as E ẳ c11 ỵ c22 ỵ c33 ị=3 Likewise, j is the effective dielectric coefficient of the material as j ẳ j11 ỵ j22 ị=2 It is clear from Eq (48) that the stabilization terms consist of the squared-residuals of mechanical stress equilibrium and electric displacement equilibrium in Eq (1) It is constructed by considering ð46Þ uu Therewhere G denotes the Moore–Penrose pseudoinverse of k fore the natural frequency x and mode k can be computed by solving the following eigenvalue problem h i uu k uu Gk T x2 ẵm ỵ k fkg ẳ 0: uu 48ị lc ẳ 44ị Z smoothed squared residualÀmechanical part smoothed squared residualÀelectric part ð43Þ In this work, modal analysis of the system is analyzed for dynamics problems of linear electroelastic solids Hence, Eq (38) reduces to the following equation without damping and forcing terms uu À k uu Gk T fug ¼ 0; g ỵ ẵk ẵmfu uu smoothed potential functionalmechanical part Z T À D ðuÞEðuÞ À uqs dX ỵ uqC dC X Ct |{z } 40ị kẳ1 uu fug ỵ ẵk uu fug ẳ 0; g ỵ ẵk ẵmfu T ẵkuu fug ỵ ẵkuu fug ẳ 0: Z Z T qu_ u_ ỵ eT uịr uị uT b dX À uT tC dC X Ct |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} Z All entries in matrix B in Eq (39) are constants over each smoothing domain, the stiffness matrix in Eq (39) can therefore be rewritten as Ns X 69 ð47Þ Stabilization of NS-FEM 4.1 Governing equations and variational principle With regard to the nodal integrated methods, direct nodal integration leads to a numerically spatial instability in meshfree settings (spurious zero-energy modes exist) due to vanishing derivatives of shape functions at the nodes during integration [29,32] In this re- (i) while a ! 0, the functional in Eq (48) converges to the original smoothed potential energy functional; (ii) while lc ! 0, the functional in Eq (48) also converges to the original smoothed potential energy functional, for any finite a; (iii) for a finite model (lc is finite positive constant), the strongform equilibrium system equation is better enforced by using a larger a, and the weakened weak form [22] is better enforced using a smaller a Therefore, adjusting the stabilization parameter a suppresses the ‘‘overly soft’’ effect of original NS-FEM models, thereby achieving a desired stability In this work, we prefer to use a possible a to obtain desired number of smallest eigen-modes for a given 2D solids, so that we 70 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 can obtain the upper bounds in energy norm to the exact solution (or lower bounds to the exact natural frequency) Because of the known fact that a fully compatible FEM model can give lower bounds to the exact strain energy and electric energy (or upper bounds to the exact natural frequency), the use of our stabilized NS-FEM and FEM can bound the solutions from both sides with complicated geometry as long as a triangular element mesh can be generated Taking variation and applying stationary condition to Eq (48) yields dps ðuÞ ¼ ¼ Z Z Á ðuÞ À duT b dX du_ T qu_ ỵ deT uịr duT tC dC X Ct |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} À À Á Z X1 Z Ni;m ðxÞnn dCui : As s i2ns k Ck Z Csk u;m ðxÞnn dC ð52Þ From Eqs (51) and (52), it is clear that smoothing operation relaxes the requirement of field function Consequently, the smoothed second-order derivative only requires C continuity, so that the squareresiduals can be evaluated using linear elements Next, the smoothing operation is applied to the divergence of the Cauchy stress tensor r In this regard, one can arrive at the smoothed divergence of the Cauchy stress tensor that can be expressed in the following vector form: & ' r xx;x ỵ r xy;y : r yy;y ỵ r yx;x 53ị Substituting the constitutive relation of Eq (6), on defining Csm smoothed potential functionalÀelectric part as Z Z 2alc 2alc ỵ div drịT div rịdX ỵ div drịT bdX E E X X |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Csm ¼ smoothed square residualÀmechanical part Z Z 2alc 2alc À ðdiv dDÞðdiv DÞdX À ðdiv dDÞqs dX: j X j X |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} Ask u;mn xịWx; x xk ịdX ẳ k div r ẳ dDT uịEuị duqs dX ỵ duqC dC X Ct |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Xsk ¼ smoothed potential functionalÀmechanical part Z Z ;mn xk ị ẳ u c11 c12 c13 c31 c32 c33 e11 e21 e13 e23 c31 c32 c33 c21 c22 c23 e13 e23 e12 e22 ! ; ð54Þ we have 50ị div r ẳ & smoothed square residualelectric part r xx;x ỵ r yx;x r yy;y ỵ r xy;y ' ẳ Csm Ks ; 55ị where Ks is defined as  ÃT xy;x exx;y eyy;y c xy;y ÀEx;x ÀEy;x ÀEx;y ÀEy;y : Ks ¼ exx;x eyy;x c 4.2 Discretization In the proposed variational principle in Eq (50), the discretization of smoothed total potential functional manifested by the first two terms can directly use the procedure in Section 3.3 Therefore, we now construct specifically the discretization on the last two stabilization terms of Eq (50), which consist of the square-residuals of mechanical stress equilibrium and electric displacement equilibrium 4.2.1 Square-residual of mechanical stress equilibrium Obviously, the stabilization term regarding to the square-residual of mechanical stress equilibrium includes the second-derivatives of the displacements, whereas the assumed displacement fields used in this work not have the second-order derivatives over the whole problem domain Further, the use of T3 element leads to zero second order derivatives of shape function, i.e., N ;mn xị ẳ m; n ẳ x; yị, hence, the stabilization term will have no contribution for the stabilization if N ;mn ðxÞ is calculated directly using the FEM shape functions In order to realize the stabilization term, the present work performs the gradient smoothing technique on the second-order derivatives u;mn (m; n ẳ x; yị of the displacement fields (the assumed displacement fields u and its gradient u;m are continuous) Letting /ðxÞ in Eq (27) be u;m , i.e., /xị ẳ u;m , we have the smoothed the second-order derivatives of displacements in a smoothing domain Xsk ;mn xk ị ẳ u ẳ Z Xsk u;mn xịWx; x xk ịdX ẳ Z Ask Csk @ 1 Ask Z Csk Employing the strain–displacement and electric field-potential relationships in Eq (5), then y;xx u y;xy u y;yx u x;yx ỵ u x;xy u y;yy u x;yy ỵ u x;xx u ;xx u ;yx u ;xy u ;yy T : Ks ẳ ẵ u ð57Þ s Substituting Eqs (51) and (52) into K leads to the smoothed the smoothed divergence of the Cauchy stress tensor over the smoothing domain Xsk div r ¼ & r xx;x ỵ r yx;x r yy;y ỵ r xy;y ' ¼ Csm Ks ¼ Csm X Bsi di ; ð58Þ i2nsk where nsk is the set of nodes associated with the smoothing domain Xsk ; di ẳ ẵ ui ui T , and Bsi is expressed by " Bsi ¼ Ni;xx Ni;yx Ni;xy Ni;yy Ni;yx Ni;xx Ni;yy Ni;xy #T 59ị ; where Ni;mn ẳ Ask Z Csk Ni;m xịnn dC; m; n ẳ x; y ð60Þ 4.2.2 Square-residual of electric displacement equilibrium Performing the similar smoothing operation as before, the smoothed divergence of electric displacements can be expressed in the form u;m ðxÞnn dC X Ni;m xịui Ann dC ( div D ẳ i2nsk X1 Z ẳ Ni;m xịnn dCui : As s i2ns k Ck ð56Þ ð51Þ k In the same way, the smoothed second-order derivatives of ;ij over the smoothing domain Xsk can be exelectric potential, u pressed by: Dx;x Dy;y ) ¼ Cse Ks ¼ Cse X Bsi di ; 61ị i2nsk where Cse is dened by Cse ẳ ẵ e11 e12 e13 e21 e22 e23 Àj11 Àj12 Àj21 Àj22 : ð62Þ Also, it is worth noting that Bsi is the same as that given in Eq (59) 71 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 Substituting these equations to Eq (50) leads to the following equation duT Z qNT Nu dX ỵ duT Z X Z Bu ịT cE Bu udX ỵ duT X Bu ịT eT Bu udX ỵ du X À du Z ðBu ÞT eBu udX X Z ðBu ÞT js Bu udX À duT Z X À duT NT bdX X Z NT tC dC ỵ du Ct Z NT q s d X ỵ d u Z Ct X NT q C d C ỵ E T dd Z À ÁT À ÁT 2alc T Bs ịT Csm Csm Bs d ỵ Bs ịT Csm bdX À dd j X Z À ÁT À ÁT Bs ịT Cse Cse Bs d ỵ Bs ịT Cse qs dX ẳ 0: 2alc 63ị X !& u u " ' ỵ uu k kT 0 uu & ' f ẳ ỵ ff m g ỵ ff e g q uu k kuu # s s ỵ k m ỵ ke !& u u ' 64ị s ; k s ; fm k m e in which and f e are the newly introduced matrices in the discretized algebraic equations of system that are then assembled by 2X Ns Z s ẳ 2alc k Bs ịT Csm ÞT Csm Bs dX m E k¼1 Xsk Ns À ÁT 2alc X ðBs ÞT Csm Csm Bs Ask ; E k¼1 2X Ns Z À ÁT s ¼ À 2alc ðBs ÞT Cse Cse Bs dX k e s j X ẳ kẳ1 65ị k Ns lc X 2a ẳ j T Bs ịT Cse Cse Bs Ask ; 66ị kẳ1 Ns Z T 2alc X Bs ịT Csm bdX; fm ẳ E k¼1 Xsk Ns Z À ÁT 2alc X Bs ịT Cse qs dX: fe ẳ s j X k¼1 (7) Numerical examples Eliminating du and du yields m (4) (5) (6) e compute the smoothed matrix for the divergence of stress Bs ðxk Þ by Eq (59), and obtain the stabilized stiff s and k s from the square-residuals, ness matrices k m e respectively, of mechanical stress equilibrium and electric displacement equilibrium, using Eqs (65) and (66); f evaluate the contribution of load vector over the current smoothing domain; g assemble the contribution of the current smoothing domain to form the global system stiffness matrix and load vector; calculate the consistent mass matrix m; implement essential boundary conditions; solve the linear system of equations to obtain the nodal displacements and electric potentials (static analysis); and eigenmodes and frequencies (eigenvalue problems); post-processing of desired results Benchmark problems are examined to demonstrate the validity of the proposed stabilization scheme within the framework of NSFEM for the piezoelectricity The strain energy used in this research is defined as EXị ẳ Z eT DedX: 69ị X Numerical errors are then calculated by the following equations Relative displacement error in L2 norm : eu vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Á2 uPNn À exact u u À unumerical i ; ¼ t i¼1 Pi N À Á n exact i¼1 ui Relative electric potential error in L2 norm : eu vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uPNn À exact Á2 u u À unumerical i ¼ t i¼1 PiN À Á n exact i¼1 ui ð70Þ ð71Þ ð67Þ where the superscript exact denotes the exact solution (if the exact solution does not exists, exact denotes the reference solutions), and numerical denotes the numerical solution obtained using a numerical method ð68Þ 6.1 Patch test k Numerical implementation The numerical procedure for the stabilized NS-FEM is outlined as follows: (1) divide the problem domain into a set of elements and obtain information on node coordinates and element connectivity; (2) create the smoothing domains using the rule given in Section 3.2; (3) loop over smoothing domains a determine the node connecting information of the smoothing domain Xsk associated with node k; b calculate the outward unit normal for each boundary segment of the smoothing domains Xsk ; c compute the smoothed strain matrix Bu ðxk Þ and the smoothed electric field matrix Bu ðxk Þ by using Eq (34); uu for the d evaluate the smoothed stiffness matrix k uu for mechanical field, kuu for the electric field, and k the mechanical-electric coupling field over the current smoothing domain by using Eqs (39)–(41); A standard patch test is first considered, whose nodal distribution and geometry are presented in Fig The piezoelectric material PZT-4 as listed in Table is employed in this patch test The boundary conditions for the mechanical displacements and the electric potential are assumed to be [5] ux ¼ s11 r0 x; uy ¼ s13 r0 y; u ¼ g 31 r0 y; Fig Nodal setting and geometry of piezoelectric patch test ð72Þ 72 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 Table Piezoelectric material properties of PZT-4 and PVDF PZT-4 [6] q ¼ 7500 kg=m3 PVDF [9] q ¼ 1800 kg=m3 3 2:18 6:33 cE ¼ 6:33 2:18 GPa 0 0:775 139 74:3 cE ¼ 74:3 113 GPa 0 25:6 e¼ jS ¼ ! 0 13:44 Coulomb=m2 À6:98 13:84 ! 6:00  10À9 F=m 5:47 u v / T xx T yy T xy Dx Dy ! 0 Coulomb=m2 0:046 0:046 jS ¼ ! 0:1062  10À9 F=m 0:1062 that the desired results gained by the present method with stabilization parameter a ¼ 0:05 match the exact solutions (other parameters are found to match as well, however, the corresponding results are not listed here due to the length limit), and hence the method successfully passes the patch test Table The results of patch test Variable eẳ Results Exact sNS-FEM a ẳ 0:05ị 2.376547556249814e006 1.818789953339896e007 1.066703050081747e009 1.0 0 0 2.376547526243648eÀ006 À1.818789951339893eÀ007 À1.066712350081748eÀ009 1.0 À6.753880495917718eÀ015 À3.8934368989602721eÀ016 À8.139613253960852eÀ014 4.599293089164516eÀ013 6.2 Single-layer piezoelectric strip In order to examine the accuracy of the present method, a piezoelectric strip of L  2h ¼ mm  mm undergoing a shear deformation condition as depicted in Fig is considered The piezoelectric material is polarized along the thickness, i.e., along the y direction, and is assumed to be transversely isotropic The strip is subjected to a uniform stress r0 ¼ 1:0 Pa in the y direction on its top and bottom boundaries and an applied voltage V ¼ 1000 V to the left and right boundaries as shown in Fig The piezoelectric material PZT-5 is taken and its related parameters are provided as follows 16:4 sE ¼ À7:22 À7:22  10À6 mm2 =N; 0 47:5 ! 0 584 d¼  10À9 N=V; À172 374 ! 1:53105 eS ¼  10À8 N=V2 : 15:05 18:8 ð74Þ Due to the acting compressive stress together with an applied electric field perpendicular to the direction of the polarization, a shear strain is consequently generated in the y direction and expanded slightly in the x direction because of the Poisson effect The overall deformation is a superposition of the deformation due to the shear strain and the compressive loading [7] The mechanical and electrical boundary conditions are prescribed to the edges of the strip Fig Piezo-strip under a uniform stress and an applied voltage where r0 is an arbitrary stress parameter For the given boundary conditions, the corresponding analytical solutions for the stresses r and for the electric displacements D are obtained as T xx ¼ r0 ; T xy ¼ T yy ¼ Dx ¼ Dy ¼ 0: ð73Þ In this patch test, the mechanical displacements and the electric potential are prescribed on all boundaries by the given boundary conditions with linear functions presented in Eq (72) Satisfaction of the patch test then requires that the mechanical displacements and the electric potential of any interior nodes inside the patch follow ‘‘exactly’’ (to machine precision) the same linear function of the imposed boundary conditions of the patch It shows in Table /;y x; y ẳ ặhị ẳ 0; T yy x; y ẳ ặhị ẳ r0 ; T xy x; y ẳ ặhị ẳ 0; /x ¼ L; yÞ ¼ ÀV ; /ðx ¼ 0; yị ẳ ỵV ; ux x ẳ 0; yị ¼ 0; T xy ðx ¼ L; yÞ ¼ 0; T xx x ẳ L; yị ẳ 0; uy x ¼ 0; y ¼ 0Þ ¼ 0: ð75Þ The analytical solutions for this problem are given by Ohs and Aluru [9] ux ¼ s13 r0 x; uy ¼ d15 V x ỵ s33 r0 y; h x / ¼ V0 À : L ð76Þ The numerical simulations of the proposed stabilized NS-FEM are carried out using a regular mesh with nodal distribution of  as shown in Fig 5(a) The mechanical displacements and the electric potential at the central line y ẳ 0ị with stabilization parameters a ¼ 0:05 as depicted in Figs 6–8 are compared directly 73 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 (a) (b) Fig (a) Regular; and (b) irregular meshes for piezo-strip under a uniform stress and an applied voltage x 10 -5 x 10 -6 stabilized NS-FEM α=0.05 Exact 0.8 3.5 0.6 Electric potential(GV) ux displacement (mm) 2.5 1.5 0.4 0.2 -0.2 -0.4 -0.6 -0.8 0.5 -1 stabilized NS-FEM α=0.05 Exact 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fig Variation of horizontal displacement u at the central line ðy ¼ 0Þ of the single-layer piezoelectric strip -3 x 10 uy displacement (mm) 0.8 0.6 0.2 0.2 stabilized NS-FEM α=0.05 Exact 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x (mm) Fig Variation of vertical displacement layer piezoelectric strip 0.4 0.5 0.6 0.7 0.8 0.9 with the analytical solutions available in [9] It is evident that the computed results show an excellent agreement with those of the exact solutions Additionally, different stabilization parameters a are employed for the numerical simulations The numerical errors in displacement and electric potential solutions calculated by Eqs (70) and (71) respectively are presented in Table 3, which demonstrates that the stabilized NS-FEM using different stabilization parameters a in a proper interval can reproduce the linear behavior of the exact solutions accurately within round-off errors In order to illustrate the robustness of the present method, this shear problem of the piezoelectric strip is also tested using the mesh with irregular nodal distribution whose coordinates are generated in the following fashion y0 ¼ y ỵ Dy r c air ; 0.3 Fig Variation of electric potential / at the central line y ẳ 0ị of the single-layer piezoelectric strip x0 ẳ x ỵ Dx rc air ; 0.4 0.1 x (mm) x (mm) 1.2 v at the central line y ẳ 0ị of the single- ð77Þ where Dx and Dy are initial regular element sizes in x and y directions, respectively r c is a computer-generated random number between À1.0 and 1.0, and air is a prescribed irregularity factor whose value is chosen between 0.5 in this research (see Fig 5(b)) Also, the numerical errors in displacement and electric potential solutions are listed in Table 3, and it is found that the stabilized NS-FEM are in excellent agreement with the linear exact solutions within machine precision, regardless of the element shape 74 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 Table Single-layer piezoelectric strip: numerical errors in displacement and electric potential solutions Stabilization parameter a 0.0 0.01 0.03 0.05 0.1 0.3 1.0 eu Regular mesh Irregular mesh 1.36EÀ12 1.30EÀ12 7.56EÀ13 5.86EÀ13 4.41EÀ13 1.15EÀ12 4.54EÀ13 5.39EÀ13 3.36EÀ13 2.33EÀ12 2.33EÀ12 4.03EÀ12 3.61EÀ12 4.02EÀ12 eu Regular mesh Irregular mesh 1.15EÀ14 1.46EÀ14 8.12EÀ15 1.23EÀ14 6.65EÀ15 3.04EÀ14 1.29EÀ14 8.37EÀ15 1.38EÀ14 3.76EÀ14 4.39EÀ14 1.71EÀ14 8.04EÀ14 4.36EÀ14 x 10 -8 2.6 2.4 Electric potential 2.2 1.8 NS-FEM-T3 sNS-FEM-T3 α=0.01 sNS-FEM-T3 α=0.03 sNS-FEM-T3 α=0.05 sNS-FEM-T3 α=0.1 sNS-FEM-T3 α=0.3 sNS-FEM-T3 α=1.0 FEM-T6 FEM-Q4 FEM-T3 Reference solu 1.6 1.4 1.2 0.8 0.6 500 1000 1500 2000 2500 3000 DOF Fig 11 Comparison of electric potential at point A of Cook’s membrane Fig Geometry and boundary conditions of Cook’s membrane 9.5 x 10 2.2 8.5 2.1 1.9 NS-FEM-T3 sNS-FEM-T3 α=0.01 sNS-FEM-T3 α=0.03 sNS-FEM-T3 α=0.05 sNS-FEM-T3 α=0.1 sNS-FEM-T3 α=0.3 sNS-FEM-T3 α=1.0 FEM-T6 FEM-Q4 FEM-T3 Reference solu 1.8 1.7 1.6 1.5 1.4 -5 500 1000 1500 2000 2500 3000 DOF Fig 10 Comparison of vertical displacement at point A of Cook’s membrane Upper bound solution is obtained using the NS-FEM-T3 and the sNS-FEM-T3 (a ẵ0:0; 0:1ị The lower bound solution is obtained using the FEM-T3 and the FEM-Q4 6.3 Cook’s membrane A benchmark problem, shown in Fig 9, a clamped tapered panel subjected to a distributed tip load F ¼ N, resulting in deformation Strain energy Vertical displacement v 2.3 x 10 -4 7.5 NS-FEM-T3 sNS-FEM-T3 α=0.01 sNS-FEM-T3 α=0.03 sNS-FEM-T3 α=0.05 sNS-FEM-T3 α=0.1 sNS-FEM-T3 α=0.3 sNS-FEM-T3 α=1.0 FEM-T6 FEM-Q4 FEM-T3 Reference solu 6.5 5.5 500 1000 1500 2000 2500 3000 DOF Fig 12 Comparison of strain energy of Cook’s membrane dominated by a bending response, is then analyzed The piezoelectric material PZT-4 whose parameters listed in Table is employed The mechanical boundary conditions are similar to the popular Cook’s membrane [38] The electric boundary condition of the lower surface is prescribed by zero voltage (0 V) The geometry, loading and boundary conditions can be referred to Fig Four discretizations (3-node triangular elements) with uniform nodal distribution: (5  5;  9; 17  17, and 33  33Þ, are used for the present stabilized NS-FEM (sNS-FEM-T3) For comparison, such 75 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 Table Cook’s membrane: comparisons of vertical displacement and electric potential at point A, and strain energy in the whole problem domain using different numerical methods À4 Vertical displacement 2:109ð10 mm) Electric potential 1:732ð10À8 GV) Strain energy 8:5405ð10À5 J=mm2 Þ Mesh (nodes)  (% error)  (% error) 17  17 (% error) 33  33 (% error) FEM-T3 FEM-Q4 FEM-T6 NS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 a ¼ 0:01 a ¼ 0:03 a ¼ 0:05 a ¼ 0:10 a ¼ 0:30 a ¼ 1:00 1.0356 1.6617 1.9917 2.2630 2.2012 2.1700 2.1564 2.1406 2.0528 2.0011 (À50.8) (À21.2) (À5.6) (7.3) (4.4) (2.9) (2.2) (1.5) (À2.7) (À5.1) 1.5383 2.0179 2.0739 2.1688 2.1597 2.1544 2.1508 2.1452 2.1349 2.1230 (À27.1) (À4.3) (À1.7) (2.8) (2.4) (2.2) (2.0) (1.7) (1.2) (0.7) 1.8990 2.0897 2.1057 2.1227 2.1225 2.1217 2.1211 2.1201 2.1184 2.1166 (À10.0) (À0.9) (À0.2) (0.6) (0.6) (0.6) (0.6) (0.5) (0.4) (0.4) 2.0470 2.1034 2.1079 2.1133 2.1126 2.1122 2.1120 2.1116 2.1111 2.1105 (À2.9) (À0.3) (À0.1) (0.2) (0.2) (0.2) (0.1) (0.1) (0.1) (0.1) a ¼ 0:01 a ¼ 0:03 a ¼ 0:05 a ¼ 0:10 a ¼ 0:30 a ¼ 1:00 0.6427 0.8348 1.2221 2.6646 1.9751 1.9077 1.8630 1.8240 1.7007 1.5849 (À62.9) (À51.8) (À29.4) (53.8) (14.0) (10.1) (7.6) (5.3) (À1.8) (À8.5) 1.1629 1.4555 1.5301 1.9597 1.9337 1.9001 1.8817 1.8554 1.8169 1.7893 (À32.9) (À16.0) (À11.7) (13.1) (11.6) (9.7) (8.6) (7.1) (4.9) (3.3) 1.5186 1.6608 1.6808 1.8177 1.7864 1.7801 1.7774 1.7743 1.7703 1.7672 (À12.3) (À4.1) (À2.9) (4.9) (3.1) (2.8) (2.6) (2.4) (2.2) (2.0) 1.6608 1.7037 1.7087 1.7511 1.7434 1.7396 1.7380 1.7363 1.7349 1.7346 (À4.1) (À1.6) (À1.3) (1.1) (0.7) (0.4) (0.3) (0.3) (0.2) (0.1) a ¼ 0:01 a ¼ 0:03 a ¼ 0:05 a ¼ 0:10 a ¼ 0:30 a ¼ 1:00 4.5180 7.1399 7.8401 9.4103 9.2359 9.0455 8.9201 8.6980 8.1011 6.3611 (À47.1) (À16.4) (À8.2) (10.2) (8.1) (5.9) (4.4) (1.8) (À5.1) (À25.5) 6.6113 8.3230 8.4232 8.8599 8.8333 8.8029 8.7792 8.7289 8.5599 8.0346 (À22.6) (À2.5) (À1.4) (3.7) (3.4) (3.1) (2.8) (2.2) (0.2) (À5.9) 7.8917 8.5148 8.5191 8.6383 8.6345 8.6290 8.6237 8.6114 8.5663 8.4167 (À7.6) (À0.3) (À0.3) (1.1) (1.1) (1.0) (1.0) (0.8) (0.3) (À1.5) 8.3607 8.5392 8.5395 8.5643 8.5653 8.5648 8.5638 8.5610 8.5490 8.5072 (À2.1) (À0.02) (À0.02) (0.3) (0.3) (0.3) (0.3) (0.2) (0.1) (À0.4) FEM-T3 FEM-Q4 FEM-T6 NS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 FEM-T3 FEM-Q4 FEM-T6 NS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 sNS-FEM-T3 four models are also computed using the standard FEM-T3 and the standard NS-FEM-T3 In addition, the 4-node quadrilateral elements with the same number of nodes are employed in the standard FEM-Q4 using  Gaussian points Also, the six-node triangular meshes are used by the FEM for comparison, and the DOFs of these quadratic meshes are the same as those of the linear meshes that other methods use for fairness The reference value of the vertical displacement and the electric potential at the center tip (A) are 2:109  10À4 mm and 1:732  10À8 GV [7] It is noted that the analytical solutions for this problem is unknown, therefore, the FEM-Q4 with a very fine mesh (10,505 nodes) is employed to provide the reference solution of strain energy in the whole domain Figs 10 and 11, respectively, show the convergence status of vertical displacement and electric potential at point A (see Fig 9) against the increase of Degree of Freedom (DOF) using different numerical methods mentioned above Also, we compare the variation of strain energy in the whole domain, as mesh refines, and the results are plotted in Fig 12 Additionally, the corresponding data of Figs 10–12 are tabulated in Table in detail for reference From the table and figures, it can be first observed that varying the stabilization parameter a changes the properties of sNS-FEM (1) Either decreasing the value of a or reducing the element size lc makes the solutions calculated by the sNS-FEM converge to the original NS-FEM ones (sNS-FEM: a ẳ 0:0ị; (2) with a relatively small value of a a ẵ0:0; 0:1ị, the sNSFEM is found overly-soft and obtains upper bound solutions, whereas the FEM give lower bound solutions Thus we can bound the exact solution from both sides with complicated geometry as long as an element mesh could be generated; (3) it is also worth noting that increasing the stabilization parameter a in this interval with small values can improve the numerical accuracy of the method, because the least- -1 Log 10(displacement error) Ref solution -1.5 -2 -2.5 -3 -3.5 -1 NS-FEM-T3 sNS-FEM-T3 α=0.01 sNS-FEM-T3 α=0.03 sNS-FEM-T3 α=0.05 sNS-FEM-T3 α=0.1 sNS-FEM-T3 α=0.3 sNS-FEM-T3 α=1.0 FEM-T6 -0.5 0.5 1.5 Log10(t) Fig 13 Comparison of computational efficiency in term of displacement error for Cook’s membrane squared stabilization term introduces ‘‘stiffening’’ effects into the ‘‘overly soft’’ NS-FEM-T3, which makes the sNSFEM model more close to the exact models At the same time, such stiffening effects in sNS-FEM eliminate temporal instability existed in the standard NS-FEM, which will be demonstrated later; (4) by gradually increasing a, the additional constraint of the squared-residual of the equilibrium equation takes more effects, leading to a stiffer system However, the solution bound property is numerically found uncertain Consequently, the a in this range is not recommended; 76 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 (5) while a is larger than a certain value, sNS-FEM can find the lower bound solution similarly to ‘‘overly stiff’’ fully compatible FEM Nonetheless, the accuracy with this range of a values is not guaranteed; (6) in order to find the upper bound solution in energy norm (or lower bound solution in natural frequency) and to stabilize the system, the value of a can be selected as 0.03–0.1 In addition to the above, the computational efficiency in terms of the displacement error against computational time(s) is compared for the present sNS-FEM-T3 and the FEM-T6, and the results are plotted in Fig 13 It can be observed that the sNS-FEM-T3 is obviously less efficient compared to the FEM-T6 Two factors associated with computational cost lead to this performance for our proposed sNS-FEM-T3: (1) the ‘‘overhead’’ cost for all operations until the stiffness matrix is formed (including mapping, Gauss integration and etc.), and (2) the solver time to solve the resultant system equations Firstly, in our sNS-FEM models, the overhead time includes the smoothing operations that need the construction of smoothing domains and thus relatively complicated data storage structures Also, the forming penalty, i.e., two stabilization terms, requires computations that the FEM does not Therefore, the overhead cost of the present sNS-FEM models is apparently increased The second factor is attributed to the bandwidth of the global stiffness matrix As we know, each element of FEM-T6 definitely requires nodes to assemble the global stiffness matrix On the other hand, each integration (node-based smoothing) domain in the sNS-FEM models needs information from all the nodes belonging to the elements that include the current node (called as influence nodal set here) for assembling For a specific smoothing domain, this influence nodal set arises from the surrounding discretization of the current node, and generally, about nodes are included in this influence nodal set as shown in Fig Consequently, this difference leads to a little larger bandwidth, and the sNS-FEM-T3 is, accordingly, a little more expensive for solving the resultant system equations Based on these two factors, the ‘‘overall’’ computational cost of the present sNS-FEM is thus increased compared to the FEM-T6 However, the important feature of the sNS-FEM-T3 lies in that an upper bound to the exact solutions in energy norm (or a lower bound in natural frequencies) can be achieved with a proper parameter, while the FEM-T3 produces the lower bound in energy norm (or the upper bound in natural frequencies) Therefore, we now can use our stabilized NS-FEM together with the FEM to Fig 15 Construction of strain smoothing domains for the nodes falling on the bimaterial interface numerically obtain both upper and lower bounds of solutions for piezoelectric problems as long as a triangular element mesh can be generated 6.4 Bimorph MEMS device The purpose of this problem is to simulate the linear tilt angle of the reflected light through a mirror of a MEMS device The device is constructed from two parallel bimorphs made of PVDF material connected by a mirror as shown in Fig 14(a) Each bimorph with length L ¼ 10 lm and height H ¼ lm is assumed The bimorph beam is divided into the top and bottom layers as show in Fig 14(b) The following boundary conditions are applied to layer of bimorph beam /1ị x; y ẳ 0ị ẳ V; T 1ị yy x; y ẳ 0ị ẳ 0; 1ị T xy x; y ẳ 0ị ẳ 0; /1ị x; y ẳ hị ẳ 0; 2ị T 1ị yy x; y ẳ hị ẳ T yy x; y ẳ hị; 2ị T 1ị xy x; y ẳ hị ẳ T xy x; y ẳ hị; /;x1ị x /;x1ị x 78ị ẳ 0; yị ẳ 0; u x ẳ 0; yị ẳ 0; v ẳ L; yị ẳ 0; T 1ị xx x T ð1Þ xy ðx ð1Þ (a) (b) Fig 14 (a) Bimorph MEMS device; (b) parallel bimorph geometry ẳ L; yị ¼ 0; ð1Þ ðx ¼ 0; yÞ ¼ 0; ¼ L; yị ẳ 77 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 7.5 x 10 -3 NS-FEM-T3 sNS-FEM-T3 α=0.1 FEM-T6 FEM-Q4 FEM-T3 Reference solu Tip deflections (um) 6.5 5.5 electroded surfaces 4.5 1000 2000 3000 4000 5000 6000 7000 8000 DOF Fig 16 Convergence of bimorph tip deflection in the MEMS device Upper bound solution is obtained using the NS-FEM-T3 and the sNS-FEM-T3, whereas lower bound solution is obtained using the FEM-T3, the FEM-Q4 and the FEM-T6 Fig 17 Schematic of a transducer Boundary conditions for layer are /2ị x; y ẳ hị ẳ /1ị x; y ẳ hị; v 2ị x; y ẳ hị ẳ v 1ị T 2ị yy x; y ẳ 2hị ẳ 0; x; y ẳ hị; u2ị x; y ẳ hị ẳ u1ị x; y ẳ hị; 2ị / x; y ẳ 2hị ¼ V; ð2Þ T xy ðx; y ¼ 2hÞ ¼ 0; /2ị ;x x ẳ 0; yị ẳ 0; u2ị x ẳ 0; yị ẳ 0; v 2ị x ẳ 0; yị ẳ 0; /2ị ;x x ẳ L; yị ¼ 0; ð2Þ T xx ðx ¼ L; yÞ ¼ 0; 2ị T xy x ẳ L; yị ẳ 0: ð79Þ The centers of bimorphs are connected by a lm long mirror Linear elasticity is assumed to the mirror When a voltage is applied across the thickness, the bimorphs vertically displace in opposite directions and rotate the mirror with the tilt angle As a result, the direction of the reflected light can change when the various voltages are applied In the simulation, the bimorph is discretized into three triangular meshes with uniform nodal distribution of 41  5; 81  9, and 161  17 Two smoothing domains are constructed for the nodes falling on the bi-material interface as shown in Fig 15, in order to avoid the discontinuity within one integration domain as the standard FEM does Since the analytical solutions of this problem are not available, the reference solutions are calculated using the FEM-Q4 with a very fine mesh (92,345 nodes) in this study In Fig 16, the convergence process of the bimorph tip deflection with the nodal distribution refined using different numerical methods are compared Note that the stabilization parameter a is selected as 0.05 that falls in the suggested interval in Section 6.3 From the figure, it can be found again that the tip deflection of the FEM-T3, Q4 and T6 are no-more than the exact solutions and converge from below, the corresponding values of the stabilized sNS-FEM-T3 and the NS-FEM-T3 are no-less than the exact ones and converge from above Thus, again, we also can bound the exact or reference solutions from two directions Further, obviously, the stabilized NS-FEM-T3 with a proper parameter improves significantly the accuracy in comparison with the standard NS-FEM-T3 Meanwhile, the proposed sNS-FEM-T3 is again found slightly less accurate than the quadratic FEM-T6 with the same DOFs We also varied the voltage linearly, and Table lists the tip deflection for different applied voltages with nodal distribution of 161  17 Further, the tilt angle of mirror could be calculated by normal deflection of beam at point connection between mirror and beam in which the horizontal displacement of this point is Table Bimorph MEMS device: comparisons of bimorph tip deflection and mirror tilt angle using different numerical methods Applied voltage (V) Tip deflection (lmÞ 1.00 2.00 5.00 10.00 20.00 50.00 Tilt angle (°) 1.00 2.00 5.00 10.00 20.00 50.00 FEM-T3 0.004821 0.009643 0.024107 0.048214 0.096427 0.241068 0.5525 1.1050 2.7635 5.5335 11.1194 28.8250 FEM-T6 0.004890 0.009780 0.024449 0.048898 0.097797 0.244492 0.5603 1.1207 2.8028 5.6123 11.2794 29.2738 FEM-Q4 0.004868 0.009737 0.024342 0.048684 0.097369 0.243422 0.5579 1.1158 2.7905 5.5877 11.2294 29.1333 NS-FEM-T3 0.005283 0.010566 0.026416 0.052831 0.105663 0.264157 0.6054 1.2109 3.0284 6.0653 12.2000 31.8916 Stabilized NS-FEM-T3 (a ẳ 0:05ị 0.004914 0.009829 0.024572 0.049144 0.098289 0.245722 0.5632 1.1264 2.8169 5.6406 11.3369 29.4356 Ref solution 0.004901 0.009802 0.024505 0.049010 0.098020 0.245050 0.5616 1.1233 2.8092 5.6252 11.3055 29.3472 78 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 Model Model Model Model Model Model Model Model Model Model Model Model Fig 18 Mode shapes of piezoelectric transducer solved by (a) the NS-FEM-T3 with 369 nodes; (b) the stabilized NS-FEM-T3 with parameter a ¼ 0:05 using 369 nodes; and (c) the FEM-Q4 with 8241 nodes supposed to be zero, and the corresponding results are tabulated in Table as well It is observed that the tip displacements of the bimorphs and the tilt angle of the mirror for both FEM and NSFEM models vary linearly with applied voltages, which is very similar to the finding by Ohs and Aluru [9] using the meshless point collocation method (PCM) More importantly, the results of the proposed sNS-FEM are found in good agreement with the reference value with a fraction of percent error All of these indicate the proposed sNS-FEM can solve the piezoelectric problems effectively 6.5 Eigenvalue analysis of a piezoelectric transducer A cylindrical piezoelectric transducer is considered as the final example for the eigenvalue analysis using a piezoelectric ceramic wall of the material PZT-4 of which the material parameters are given in Table Brass end caps are placed on the top and the bot- tom of the transducer as depicted in Fig 17 The material parameters of the brass are as follows: E = 105 GPa, v = 0.37 and q = 8500 kg/m3 In this model, the inner and outer covering surfaces of the structure are designed with electrodes The transducer is particularly modeled as an axisymmetric structure [39] All the models are constrained under open-circuit voltages and in all cases the potentials on the inner surface are restrained to zero, and therefore, the frequencies correspond to those for antiresonance Eigenvalue analysis of the piezoelectric transducer is performed using the sNS-FEM-T3 to demonstrate the proposed stabilization scheme on curing the temporal instability presented in the standard NS-FEM-T3 Fig 18 illustrates the mode shapes solved by the sNS-FEM (a ¼ 0:05Þ and the NS-FEM-T3 with one regular mesh setting of 369 nodes In addition, the FEM-Q4 with a very fine mesh (8,241 nodes) is employed to provide the reference solutions as is 79 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 Model Model Model Model Model Model Model 13 Model 13 Model 13 (b) Model 19 (c) Model 19 (a) Model 19 Fig 18 (continued) Table Piezoelectric transducer: comparisons of natural frequencies (kHz) using different numerical methods.a a Methods FEM-T3 (% error) FEM-T6 (% error) NS-FEM T3 (% error) sNS-FEM-T3 a ẳ 0:05ị (% error) Ref solution Experiment [39] Mode Mode Mode Mode Mode Mode Mode Mode 18.93 (2.0) 40.7 (0.9) 59.19 (2.8) 64.94 (1.2) 90.62 (3.5) 161.18 (2.9) 277.74 (2.4) 408.54 (1.2) 18.71 40.52 58.24 64.40 89.02 158.80 274.55 406.21 18.19 39.33 55.24 61.07 83.48 148.18 260.52 391.37 18.39 39.87 56.24 62.38 85.10 153.70 266.79 400.32 18.56 40.32 57.55 64.14 87.53 156.61 271.29 403.79 18.6 35.4 54.2 63.3 88.8 13 19 (0.8) (0.5) (1.2) (0.4) (1.7) (1.4) (1.2) (0.6) (À2.0) (À2.5) (À4.0) (À4.8) (À4.6) (À5.4) (À4.0) (À3.1) (À0.9) (À1.1) (À2.3) (À2.8) (À2.8) (À1.9) (À1.7) (À0.9) Upper bound of natural frequencies is obtained using the FEM-T3 and the FEM-T6, whereas lower bound solution is obtained using the NS-FEM-T3 and the sNS-FEM-T3 done before One can clearly find spurious non-zero energy modes (e.g., mode 13 and mode 19) in Fig 18(a) if no stabilization is used, which indicates the temporal instability of the original NS-FEM (a ẳ 0:00ị Employing the stabilization scheme with a proper a value yields the changes of the natural modes, and the vanish of spurious modes as plotted in Fig 18(b) 80 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 Additionally, the corresponding natural frequency solutions solved by the numerical methods mentioned above are listed in Table The relative error of natural frequency is given by e¼ xnum À xref  100%: xref ð80Þ From Eq (80), it is clear that the negative relative error means that the numerical solution is smaller than the reference value, and vice versa It is observed that all the results by the stabilized NS-FEM-T3 give the negative relative errors in natural frequency with the corresponding reference solutions, which contrast with results by the FEM-T3 It means again that the stabilized NS-FEM-T3 produces a lower bound solution in natural frequency, while the FEM-T3 produces the upper bound In that case, we can still bound the exact solution from both sides, although the direction is opposite with regard to the solution in energy norm It is also noted that the remarkable improvement of accuracy can be achieved by the present stabilized sNS-FEM-T3 comparing to the standard NS-FEM-T3, with the relative errors being less than percent, although it is again found less accurate compared to the FEM-T6 under the same Dofs Noted that this problem was also investigated by the experimental work described in Mercer et al [39] and the experimental result is thus given in Table for comparison Also, the solutions of our sNSFEM-T3 are found in excellent agreement with the corresponding experimental values Conclusions In this work, a stabilization technique stemming from [36] is extended to piezoelectric media to cure temporal instability of node-based smoothed finite element method (NS-FEM) In this stabilization scheme, two stabilization terms corresponding to squared-residuals of two equilibrium equations, i.e., mechanical stress equilibrium and electric displacement equilibrium, are added to the smoothed potential energy functional of the original NSFEM Through the formulations and numerical examples, the following conclusions can be drawn: The gradient smoothing technique is performed on the second-order derivatives to realize two formulated stabilization terms, and only the first order derivatives of the shape function are needed Therefore, the present squared-residual stabilization technique works very well for 3-node triangular linear elements, and suits ideally in many ways to the NS-FEM models The sNS-FEM can always pass the standard patch test and thus converge to the exact solution with the mesh refinement With a small stabilization parameter, the stabilized NSFEM (sNS-FEM) behaves ‘‘overly soft’’ as the original NS-FEM does, which, however, is found more accurate than the original NSFEM with a properly small a In that case, the sNS-FEM can be used to find an upper bound to the exact solutions in energy norm (or a lower bound in natural frequencies), which contracts with the bound properties of the FEM solutions Therefore, we now can use our stabilized NS-FEM together with the FEM to numerically obtain both upper and lower bounds of solutions for piezoelectric problems as long as a triangular element mesh can be generated Temporal instability and spurious modes in the original NS-FEM can be eliminated by the added two stabilization terms This is because the least-squared stabilization term introduces stiffening effects into the ‘‘overly soft’’ NS-FEM models, which makes the sNS-FEM models temporally stable Increasing a to a certain value, both the accuracy and bound property can not be guaranteed for sNS-FEM Numerical examples indicate that a sNS-FEM-T3 model with a stabilization parameter a ½0:03; 0:1 can be made to achieve both temporal stability and certified solution with bounds References [1] A Benjeddou, Advances in piezoelectric finite element modeling of adaptive structural elements: a survey, Comput Struct 76 (2000) 347–363 [2] H Allik, T.J.R Hughes, Finite element method for piezo-electric vibration, Int J Numer Methods Engrg (1970) 151–157 [3] H.S Tzou, C.I Tseng, Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a finite element approach, J Sound Vib 138 (1990) 17–34 [4] A.A Cannarozzi, F Ubertini, Some hybrid variational methods for linear electroelasticity problems, Int J Solids Struct 38 (2001) 2573–2596 [5] K.Y Sze, Y.S Pan, Hybrid finite element models for piezoelectric materials, J Sound Vib 226 (1999) 519–547 [6] K.Y Sze, X.M Yang, L.Q Yao, Stabilized plane and axisymmetric piezoelectric finite element models, Finite Elem Anal Des 40 (2004) 1105–1122 [7] C.S Long, P.W Loveday, A.A Groenwold, Planar four-node piezoelectric with drilling degrees of freedom, Int J Numer Methods Engrg 65 (2006) 1802– 1830 [8] V.P Nguyen, T Rabczuk, S.BordasM Duflot, Meshless methods: a review and computer implementation aspects, Math Comput Simul 79 (2008) 763–813 [9] R.R Ohs, N.R Aluru, Meshless analysis of piezoelectric devices, Comput Mech 27 (2001) 23–36 [10] G.R Liu, K.Y Dai, K.M Lim, Y.T Gu, A point interpolation meshfree method for static and frequency analysis of two-dimensional piezoelectric structures, Comput Mech 29 (2002) 510–519 [11] G.R Liu, K.Y Dai, K.M Lim, Y.T Gu, A radial point interpolation method for simulation of two-dimensional piezoelectric structures, Smart Mater Struct 12 (2003) 171–180 [12] T.Q Bui, M.N Nguyen, C.Z Zhang, D.A.K Pham, An efficient meshfree method for analysis of two-dimensional piezoelectric structures, Smart Mater Struct 20 (2011) 065016 [13] Z.R Li, C.W Lim, C.C Wu, Bound theorem and implementation of dual finite elements for fracture assessment of piezoelectric materials, Comput Mech 36 (2005) 205–216 [14] C.C Wu, Q.Z Xiao, G Yagawa, Dual analysis for path integrals and bounds for crack parameter, Int J Solids Struct 35 (1998) 1635–1652 [15] B.F Veubeke, Displacement and equilibrium models in the finite element method, in: O.C Zienkiewicz, G.S Holister (Eds.), Stress Analysis, Wiley, London, 1965 [16] P Ladeveze, J.P Pelle, P Rougeot, Error estimation and mesh optimization for classical finite elements, Engrg Comput (1991) 69–80 [17] P Coorevits, P Ladevezex, J.P Pelle, An automatic procedure with a control of accuracy for finite element analysis in 2D elasticity, Comput Methods Appl Mech Engrg 121 (1995) 91–121 [18] J.P.M Almeida, J.P.M Freitasx, Alternative approach to the formulation of hybrid equilibrium finite elements, Comput Struct 40 (1991) 1043–1047 [19] J.S Chen, C.T Wu, Y Yoon, A stabilized conforming nodal integration for Galerkin mesh-free methods, Int J Numer Methods Engrg 50 (2001) 435– 466 [20] J.W Yoo, B Moran, J.S Chen, Stabilized conforming nodal integration in the natural-element method, Int J Numer Methods Engrg (2004) 61–90 [21] G.R Liu, A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods, Int J Comput Methods (2) (2008) 199–236 [22] G.R Liu, A G space theory and weakened weak ðW2 Þ form for a unified formulation of compatible and incompatible methods: Part I Theory, Int J Numer Methods Engrg 81 (2009) 1093–1126 [23] G.R Liu, K.Y Dai, T.T Nguyen, A smoothed finite element method for mechanics problems, Comput Mech 39 (2007) 859–877 [24] G.R Liu, T Nguyen-Thoi, K.Y Lam, An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analysis, J Sound Vib 320 (2009) 1100–1130 [25] H Nguyen-Xuan, G.R Liu, T Nguyen-Thoi, C Nguyen-Tran, An edge-based smoothed finite element method for analysis of two-dimensional piezoelectric structures, Smart Mater Struct 18 (2009) 065015 [26] H Nguyen-Van, N Mai-Duy, T Tran-Cong, A smoothed four-node piezoelectric element for analysis of two-dimensional smart structures, CMES-Comput Model Engrg Sci 23 (3) (2008) 209–222 [27] L Chen, H Nguyen-Xuan, T Nguyen-Thoi, K.Y Zeng, S.C Wu, Assessment of smoothed point interpolation methods for elastic mechanics, Commun Numer Methods Engrg 89 (2010) 1635–1655 [28] G.R Liu, T Nguyen-Thoi, K.Y Lam, A node-based smoothed finite element method for upper bound solution to solid problems (NS-FEM), Comput Struct 87 (2008) 14–26 L Chen et al / Comput Methods Appl Mech Engrg 241–244 (2012) 65–81 [29] G.R Liu, G.Y Zhang, Upper bound solution to elasticity problems: a unique property of the linearly conforming point interpolation method (LC-PIM), Int J Numer Methods Engrg 74 (2008) 1128–1161 [30] G.R Liu, L Chen, T Nguyen-Thoi, K.Y Zeng, G.Y Zhang, A novel singular nodebased smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems, Int J Numer Methods Engrg 83 (2010) 1466–1497 [31] M.A Puso, J.S Chen, E Zywicz, W Elmer, Meshfree and finite element nodal integration methods, Int J Numer Methods Engrg 74 (2008) 416–446 [32] M.A Puso, J Solberg, A stabilized nodally integrated tetrahedral, Int J Numer Methods Engrg 67 (2006) 841–867 [33] C.R Dohrmann, M.W Heinstein, J Jung, S.W Key, W.R Witkowski, Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes, Int J Numer Methods Engrg 47 (2000) 1549–1568 [34] T Rabczuk, T Belytschko, Q Xiao, Stable particle methods based on lagrangian kernels, Comput Methods Appl Mech Engrg 193 (2004) 1035–1063 81 [35] T Rabczuk, T Belytschko, A three dimensional large deformation meshfree method for arbitrary evolving cracks, Comput Methods Appl Mech Engrg 196 (2007) 2777–2797 [36] S Beissel, T Belytschko, Nodal integration of the element-free Galerkin method, Comput Methods Appl Mech Engrg 139 (1996) 49–74 [37] Z.Q Zhang, G.R Liu, Temporal stabilization of the node-based smoothed finite element method and solution bound of linear elastostatics and vibration problem, Comput Mech 46 (2010) 229–246 [38] R Cook, Improved two-dimensional finite element, J Struct Division ASCE 100 (ST6) (1974) 1851–1863 [39] C.D Mercer, B.D Reddy, R.A Eve, Finite Element Method for Piezoelectric Media Applied Mechanics Research Unit Technical Report No 92, University of Cape Town/CSIR, 1987 ... of our stabilized NS-FEM and FEM can bound the solutions from both sides with complicated geometry as long as a triangular element mesh can be generated Taking variation and applying stationary... 4.1 Governing equations and variational principle With regard to the nodal integrated methods, direct nodal integration leads to a numerically spatial instability in meshfree settings (spurious... problems and lower bound natural frequencies in vibration analyses can be achieved using a proper stabilization parameter Basic piezoelectric formulations 2.1 Governing equations Consider a 2D piezoelectric