Isogeometric analysis of two–dimensional piezoelectric structures

Vietnam Journal of Mechanics, VAST, Vol. 35, No. 1 (2013), pp. 79 – 91 ISOGEOMETRIC ANALYSIS OF TWO–DIMENSIONAL PIEZOELECTRIC STRUCTURES Hoang H. Truong1, Chien H. Thai2, H. Nguyen-Xuan2,3 1 University of Technical Education of Ho Chi Minh City, Vietnam 2 Ton Duc Thang University, Ho Chi Minh City, Vietnam 3 University of Science Ho Chi Minh City, VNU-HCM, Vietnam Abstract. The isogeometric analysis (IGA) that integrates Computer Aided Design (CAD) and Computer Aided Engineering (CAE) is found so far the effectively numer- ical tool for the analysis of a variety of practical problems. In this paper, we develop further the NURBS based isogeometric analysis framework for piezoelectric structures. The method employs the NURBS basis functions in both geometry representation and analysis. The main advantages of the present method are capable of handling the exact geometry of conic sections and making the flexibility of refinement and degree elevation with an arbitrary continuity of basic functions. These features results in high accuracy of approximate solutions for practical applications, especially piezoelectric problems. Three numerical examples are provided to validate excellent performance of the present method. Keywords: NURBS, isogeometric analysis, piezoelectric materials, smart materials. 1. INTRODUCTION In recent years, the use of smart materials has become widespread and almost com- monplace. The technology employed in piezoelectric applications in particular, has reached a mature level, and piezoelectric materials are frequently used in engineering applications. Piezoelectric materials transfer electric energy to mechanical energy and vice versa, and can therefore be used as either actuators or sensors, or both. Applications include ultra- sonic transducers for sonar and medical purposes, compact piezoelectric motors, struc- tural monitoring or active damping elements, and even ignition systems [1,2]. Analytical solutions which are, however, very useful as benchmark problems to problems involving piezoelectric materials are often difficult to find unless some geometries and boundary conditions are relatively simple. Numerical methods have been devised to find the approx- imate solution of these piezoelectric problems. Among them, the finite element method has become a standard modelling utility for various physical processes, including piezo- electricity. In development of advanced computational methodologies, Hughes et al. [3] have recently proposed a NURBS-based isogeometric analysis to bridge the gap between Com- puter Aided Design (CAD) and Finite Element Analysis (FEA). In contrast to the standard FEM with Lagrange polynomial basis, isogeometric approach utilized more general basis 80 Hoang H. Truong, Chien H. Thai, H. Nguyen-Xuan functions such as Non-Uniform Rational B-splines (NURBS) that are common in CAD approaches. Isogeometric analysis is thus very promising because it can directly use CAD data to describe both exact geometry and approximate solution. For structural mechan- ics, isogeometric analysis has been extensively studied for nearly incompressible linear and non-linear elasticity and plasticity problem [4], structural vibrations [5], the compos- ite Reissner-Mindlin plates [6], the Reissner-Mindlin shells [7], Kirchhoff-Love shells [8-10], the large deformation with rotation-free [11] and structural shape optimization [12], etc. In this paper, a NURBS-based isogeometric analysis formulation is presented for piezoelec- tric material structures. The isogeometric stiffness matrices are constructed for quadratic, cubic and quartic elements. Several numerical examples are illustrated to demonstrate the effectiveness of the present method. The paper is arranged as follows: a brief of the B-spline and NURBS surface is described in section 2. Section 3 describes an isogeometric approximation for piezoelectric materials. The numerical examples are illustrated in section 4. Finally, we close our paper with some concluding remarks. 2. NURBS-BASED ISOGEOMETRIC ANALYSIS FUNDAMENTALS 2.1. Knot Vectors and Basis Functions In one-dimensional problems, a knot vector Ξ is the set of coordinates in the para- metric space as Ξ = {ξ1, ξ2, . . . , ξn+p+1} (1) where p, n are the order of the B-Spline and the number of basis functions associated with control points, respectively. The interval [ ξ1 ξn+p+1 ] is called a patch. Given a knot vector, the B-spline basis functions Ni,p(ξ) of order p = 0 are defined recursively on the corresponding knot vector as follows Ni,0 (ξ) = { 1 if ξi < ξ < ξi+1 0 otherwise (2) The basis functions of p > 1 are defined by the following recursion formula Ni,p (ξ) = ξ − ξi ξi+p − ξi Ni,p−1 (ξ) + ξi+p+1 − ξ ξi+p+1 − ξi+1 Ni+1,p−1 (ξ) with p > 1 (3) 2.2. NURBS Surface The B-spline curve is defined as C (ξ) = n∑ i=1 Ni,p (ξ)Pi (4) where Pi are the control points andNi,p(ξ) is the p th-degree B-spline basis function defined on the open knot vector. Fig. 1 illustrates a set of cubic B-splines curves and cubic B-spline basis functions for open uniform knot vectors Ξ = {0, 0, 0, 0, 1/4, 1/2, 3/4, 1, 1, 1, 1}. The B-spline surfaces are defined by the tensor product of basis functions in two parametric dimensions ξ and η with two knot vectors Ξ = {ξ1, ξ2, ..., ξn+p+1} and H = Isogeometric analysis of two–dimensional piezoelectric structures 81 {η1, η2, ..., ηm+q+1} are expressed as follows S (ξ, η) = p∑ i=1 q∑ j=1 Ni,p (ξ)Mj,q (η)Pi,j (5) where Pi,j is the bidirectional control net, Ni,p(ξ) and Mj,q(η) are the B-spline basis functions defined on the knot vectors over an n ×m net of control points Pi,j. To have (a) (b) Fig. 1. An illustration of cubic B-splines curves: a) Cubic B-spline curves; b) basis functions the same notation as the finite element method, we identify the logical coordinates (i, j) of the B-spline surface with the traditional notation of a node A. Eq. (5) is now rewritten as S (ξ, η) = nm∑ A NA (ξ, η)PA (6) where NA(ξ, η) = Ni,p(ξ)Mj,q(η) is the shape function associated with node A. Similar to B-Splines, a NURBS surface is defined as S (ξ, η) = nm∑ A=1 RA (ξ, η)PA (7) where RA = NAwA nm∑ A NAwA and wA are the rational basis functions and the weight functions, respectively. 82 Hoang H. Truong, Chien H. Thai, H. Nguyen-Xuan 3. AN ISOGEOMETRIC ANALYSIS FORMULATION OF 2D PIEZOELECTRIC PROBLEMS The piezoelectric constitutive equations for a two-dimensional can be expressed under the form as [1]  TxTz Txz   =  c11 c13 0c13 c33 0 0 0 c55     SxSz Sxz  −   0 e310 e33 e15 0  [Ex Ez ] [ Dx Dz ] = [ 0 0 e15 e31 e33 0 ] SxSz Sxz  − [ ε11 0 0 ε33 ] [ Ex Ez ] (8) Eq. (8) also can be written matrix form as T = cES− eTE D = eS+ εSE (9) where T, S, E and D are the stress vector, the strain vector, the electric field and the electric displacement, respectively. cE is the elastic coefficients at constant E, εS is the dielectric coefficients at constant S and e is the piezoelectric coupling coefficients. The strain displacement and electric field potential relationships are expressed by S = Lu (10) E = −gradφ (11) where L is the symmetric gradient operator defined such as L =   ∂ ∂x 0 ∂ ∂y 0 ∂ ∂y ∂ ∂x   T (12) A weak form of the dynamic model for 2D piezoelectric can be briefly expressed as [1]∫ Ω δSTTdΩ + ∫ Ω δuTρu¨dΩ− ∫ Ω δETDdΩ− ∫ Ω δuT f¯dΩ− ∫ Γ δuT t¯dΓ + ∫ Γ δψTqsdΓ = 0 (13) Using the NURBS basis functions, the variables are the displacement and the electric potential at all control points, which can be expressed as u = nm∑ A=1 [ RA 0 0 RA ]{ uA vA } = nm∑ A=1 RAqA and φ = nm∑ A=1 RAφA (14) where n×m is the number basis functions. RA, qA = [ uA vA ]T and φA are rational basic functions, the degrees of freedom of u and the degrees of freedom of Φ associated with a control point A, respectively. Substituting the approximations Eq. (14) into equations Eqs. (10) and (11), we obtain Isogeometric analysis of two–dimensional piezoelectric structures 83 S = nm∑ A=1 BuAqA and E = nm∑ A=1 BφAφA (15) where BuA =  RA,x 00 RA,y RA,y RA,x   and BφA = [ RA,x RA,y ] (16) Substituting Eqs. (15) and (16) into (13), we have a set of piezoelectric static equa- tions Muuu¨+Kuuu+KuφΦ = f (17) Kφuu+KφφΦ = g (18) or in matrix form [ Muu 0 0 0 ] [ u¨ Φ¨ ] + [ Kuu Kuφ Kφu Kφφ ] [ u Φ ] = [ f g ] (19) where Muu = ∫ Ω ρRTARA dΩ; Kuu = ∫ Ω BuTA c EBuA dΩ; Kuφ = ∫ Ω B φT A e TB φ A dΩ; Kφφ = − ∫ Ω B φT A ε SB φ A dΩ; Kφu = K T uφ; f = ∫ Ω RTA f¯ dΩ + ∫ Γ RTA t¯ dΓ and g = ∫ Γ RTAqs dΓ. (20) 4. NUMERICAL RESULTS 4.1. Infinite piezoelectric plate with a circular hole Consider a piezoelectric plate with a central circular cavity subjected to a uniform uniaxial far-field stress σ∞ = 10 in the y direction as shown in Fig. 2. This example is used to show the efficiency of the developed elements in predicting stresses in a stress concentration problem. The reference solution can be found in [2]. In this example we Fig. 2. An infinite piezo-plate with a circular hole subjected to the far-field stress 84 Hoang H. Truong, Chien H. Thai, H. Nguyen-Xuan used the PZT-4 material with its properties are given in Tab. 1. Table 1. The PZT-4 material c11 = 12.6e4 N/mm 2 e15 = 12.7e6 pC/mm 2 c13 = 7.43e4 N/mm 2 e31 = −5.2e6 pC/mm 2 c12 = 7.78e4 N/mm 2 e33 = 15.1e6 pC/mm 2 c33 = 11.5e4 N/mm 2 ε11 = 6.464e9 pC/GVmm c55 = 2.56e4 N/mm 2 ε33 = 5.622e9 pC/GVmm Table 2. Control net for the plate with a circular hole i Pi,1 Pi,2 Pi,3 Pi,4 1 (0, 1) (0, 3.4278) (0, 7.75) (0, 10) 2 (0.4142, 1) (0.5954, 3.4278) (5.375, 7.75) (10, 10) 3 (1, 0.4142) (3.4278, 0.5954) (7.75, 5.3750) (10, 10) 4 (1, 0) (3.4278, 0) (7.75, 0) (10, 0) Table 3. Weights for the plate with a circular hole i Pi,1 Pi,2 Pi,3 Pi,4 1 1 1 1 1 2 0.8536 1 1 1 3 0.8536 1 1 1 4 1 1 1 1 (a) (b) (c) Fig. 3. Coarse mesh and control net for the infinite piezo-plate with a circular hole: a) quadratic; b) cubic and c) quartic elements. Isogeometric analysis of two–dimensional piezoelectric structures 85 Due to its symmetry, one fourth of the plate is modeled. A circular hole is represented the exact by a rational quadratic basis. The coarsest mesh, E×H, is defined by the knot vectors E = {0 0 0 0.5 1 1 1} and H = {0 0 0 0.5 1 1 1}. The exact geometry is represented with only four elements based on 16 control points, as shown in Fig. 3. The geometric data are given in Tabs. 2 and 3. Fig. 4 illustrates the first three meshes of an infinite piezo-plate. Fig. 4. NURBS meshes produced by h-refinement (knot insertion) The results obtained from NURBS are compared with the reference solution by Y. Weian and H. Wang [2]. Fig. 5 shows the distribution of σr and σθ along the line θ = 0 (x axis 1 2 3 4 5 6 7 8 9 10 11 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 s r r (q=0) Ref solu. IGA(p=2,q=2) IGA(p=3,q=3) IGA(p=4,q=4) 1 2 3 4 5 6 7 8 9 10 11 5 10 15 20 25 30 s q r (q=0) Ref solu. IGA(p=2,q=2) IGA(p=3,q=3) IGA(p=4,q=4) Fig. 5. Distribution of σr and σθ along the line θ = 0 1 2 3 4 5 6 7 8 9 10 11 -2 0 2 4 6 8 10 12 s r r (q=90) Ref solu. IGA(p=2,q=2) IGA(p=3,q=3) IGA(p=4,q=4) 1 2 3 4 5 6 7 8 9 10 11 -14 -12 -10 -8 -6 -4 -2 0 2 s q r (q=0) Ref solu. IGA(p=2,q=2) IGA(p=3,q=3) IGA(p=4,q=4) Fig. 6. Distribution of σr and σθ along the line θ = pi/2 86 Hoang H. Truong, Chien H. Thai, H. Nguyen-Xuan in Fig. 2). It can be seen from Fig. 5 that σθ reaches maximum value at the intersection of the hole and the x axis. Fig. 6 describes the distribution of σr and σθ along the line θ = pi/2 (y axis in Fig. 2). The minimum value of σθ is obtained at the position where the hole intersects the y axis. The obtained result from present method matches well with the reference solution [2]. 4.2. Single-layer piezoelectric strip in shear deformation Next we consider the shear deformation of a 1 × 1 mm single-layer square strip polarized in the y direction. The strip is subjected to a combined loading of pressure σ0 in the y direction and an applied voltage V0 as depicted on Fig. 7. The material PZT-5 is Fig. 7. The PZT-5 material Table 4. Weights for the plate with a circular hole s11 = 16.4e− 4 mm 2/N d31 = −172e− 9 mm/V s13 = −7.22e− 6 mm 2/N d33 = 374e− 9 mm/V s33 = 18.8e− 6 mm 2/N d15 = 584e− 9 mm/V s55 = 47.5e− 6 mm 2/N L = 1 mm; h = 0.5 mm g11 = 1.53105e− 8 N/V 2 σ0 = −5 N/mm 2; V0 = 1e− 6 V g33 = 1.505e− 8 N/V 2 used and its properties are summarized in Tab. 4. For this problem, the elastic coefficients, the dielectric coefficients and the piezoelectric coupling coefficients are unavailable and then they can be calculated as [1] cE =  s11 s13 0s13 s33 0 0 0 s55   −1 , eT = cEdT and εS = εT − dcEdT (21) where εT = [ g11 0 0 g22 ] and d = [ d11 d13 d15 d31 d33 0 ] (22) Isogeometric analysis of two–dimensional piezoelectric structures 87 Mechanical boundary conditions are applied to the upper and lower sides of the strip: Tyy(x, y = ±h) = σ0, Txy(x = L, y) = 0, Txy(x, y = ±h) = 0, Txx(x = L, y) = 0, u(x = 0, y) = 0, v(x = 0, y = 0) = 0, and electrical boundary conditions is applied to the left and right sides of the strip : ϕ(x = 0, y) = +V0, ϕ(x = L, y) = −V0, ϕy(x, y = ±h) = 0, The analytical solution for this problem is given in Ohs et al. [1] u = s13σ0x; v = d15V0x h + s33σ0y and φ = V0 ( 1− 2 x L ) The horizontal displacement and vertical displacement at the central line (y = 0) of the single-layer piezoelectric strip are shown in Fig. 8. The results are compared with the analytical solution in Ohs et al. [1]. It can be observed that the results of the present method are in excellent agreement with the analytical solutions. Fig. 9 shows the electric potential at the central line (y = 0) of the single-layer piezoelectric strip. Again, the obtained results match well with the analytical solutions given in [1]. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 -5 x (mm) u d is p la c e m e n t (m m ) Exact solu. IGA(p=2,q=2) IGA(p=3,q=3) IGA(p=4,q=4) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2 0 2 4 6 8 10 12 x 10 -4 x (mm) v d is p la c e m e n t (m m ) Exact solu. IGA(p=2,q=2) IGA(p=3,q=3) IGA(p=4,q=4) Fig. 8. Variation of horizontal displacement u and vertical displacement v at the central line (y = 0) of the single-layer piezoelectric strip in IGA quadratic, cubic and quartic elements 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5 -1 -0.5 0 0.5 1 1.5 x 10 -6 x (mm) E le ct ric po te nt ia l ( G V ) Exact solu. IGA(p=2,q=2) IGA(p=3,q=3) IGA(p=4,q=4) Fig. 9. Variation of electric potential φ at the central line (y = 0) of the single- layer piezoelectric strip in IGA quadratic, cubic and quartic elements 88 Hoang H. Truong, Chien H. Thai, H. Nguyen-Xuan 4.3. An extension to free vibration problem This example is an eigenvalue analysis of a piezoelectric transducer consisting of a piezoelectric wall made of PZT4 material with brass end caps as shown in Fig. 10. The piezoelectric material is electroded on both the inner and outer surfaces. This problem has been investigated numerically by Liu et al. (2003) [13] and experimentally by Mercer et al. (1987) [14]. It is also a typical example described in Section 5.1.1 of ABAQUS Example Problems Manual [15]. The transducer is modeled as an axisymmetric problem. Fig. 10. Representative sketch and domain discretization control net of a piezoelectric transducer The material properties of PZT4 are as ρ = 7500 kgm−3 c =   115.4 74.28 74.28 0 0 0 74.28 139.0 77.84 0 0 0 74.28 77.84 139.0 0 0 0 0 0 0 25.64 0 0 0 0 0 0 25.64 0 0 0 0 0 0 25.64   GPa e =  15.08 −5.207 −5.207 0 0 00 0 0 12.71 0 0 0 0 0 0 12.74 0   Cm−2 g =  5.872 0 00 6.752 0 0 0 6.752  × 10−9 Fm−1 Isogeometric analysis of two–dimensional piezoelectric structures 89 And the material properties of brass are ρ = 8500 kgm−3; E = 10.4× 1010 Pa; v = 0.37 To illustrate we evaluate the performance of the present method using only the quadratic NURBS element. Tab. 5 shows the first five frequencies, and the relative error percentages compared with experimental results are given in parentheses. Fig. 11 depicts five modes given in Table using the quadratic NURBS element. It is again seen that the present method outperforms with other published approaches. Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Fig. 11. Eigenmodes for the piezoelectric transducer Table 5. Eigenvalues (kHz) of the piezoelectric transducer Element type Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 T3 19.98 43.31 62.78 67.78 94.23 (272 elements) (7.42%) (22.34%) (15.83%) (7.08%) (6.12%) Q4 19.7 42.9 61.1 66.7 92.2 (136 elements) (5.91%) (21.19%) (12.73%) (5.37%) (3.83%) IGA 18.73 39.45 61.44 67.82 87.69 (58 elements) (0.69%) (11.4%) (13.3%) (7.14%) (–1.25%) CAX4E 18.6 40.3 57.8 64.2 88.1 (320 elements) CAX8RE 18.6 40.3 57.6 64.2 87.6 (80 elements) Experimental 18.6 35.4 54.2 63.3 88.8 where CAX4E and CAX8RE are ABAQUS 4-node axisymmetric elements and 8-node axisymmetric elements respectively [15]. 90 Hoang H. Truong, Chien H. Thai, H. Nguyen-Xuan 5. CONCLUSIONS The isogeometric analysis formulation has been developed for 2D-piezoelectric struc- tures. The quadratic, cubic and quartic elements are utilized and their results are well compared with those of several existing methods. Main advantages of the present method are to maintain the exact geometry of problems containing conic sections and to provide a flexible way to make refinement, and degree elevation. It allows us to easily achieve the smoothness with arbitrary continuity order compared with the traditional FEM. The method is thus very useful to apply for analyzing piezoelectric structures. ACKNOWLEDGEMENT This research is funded by Vietnam National University Ho Chi Minh City (VNU- HCM). REFERENCES [1] R. R. Ohs and N. R. Aluru, Meshless analysis of piezoelectric devices, Computational Me- chanics, 27, (2001), 23-36. [2] Y. Weian and H. Wang, Virtual boundary element integral method for 2-D piezoelectric media, Finite Elements in Analysis and Design, 41, (2005), 875–891. 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Hughes, A large deformation, rotation-free, isogeometric shell, Computer Methods in Applied Mechanics and Engineering, 200, (2011), 1367–1378. Isogeometric analysis of two–dimensional piezoelectric structures 91 [12] W. A. Wall, M. A. Frenzel, and C. Cyron, Isogeometric structural shape optimization, Com- puter Methods in Applied Mechanics and Engineering, 197, (2008), 2976–2988. [13] G. R. Liu, K. Y. Dai, K. M. Lim and Y. T. Gu, A radial point interpolation method for simulation of two-dimensional piezoelectric structures, Smart Mater. Struct. 12, (2003), 171– 180. [14] Mercer C. D., Reddy B. D. and Eve R. A., Finite element method for piezoelectric media, Applied Mechanics Research Unit Technical Report No 92 University of Cape Town/CSIR, (1987). [15] ABAQUS Example Problems Manual (2008). Received January 02, 2012 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 35, N. 1, 2013 CONTENTS Pages 1. Dao Huy Bich, Nguyen Xuan Nguyen, Hoang Van Tung, Postbuckling of functionally graded cylindrical shells based on improved Donnell equations. 1 2. Bui Thi Hien, Tran Ich Thinh, Nguyen Manh Cuong, Numerical analysis of free vibration of cross-ply thick laminated composite cylindrical shells by continuous element method. 17 3. Tran Ich Thinh, Bui Van Binh, Tran Minh Tu, Static and dynamic analyses of stiffended folded laminate composite plate. 31 4. Nguyen Dinh Kien, Trinh Thanh Huong, Le Thi Ha, A co-rotational beam element for geometrically nonlinear analysis of plane frames. 51 5. Nguyen Chien Thang, Qian Xudong, Ton That Hoang Lan, Fatigue perfor- mance of tubular X-joints: Numberical investigation. 67 6. Hoang H. Truong, Chien H. Thai, H. Nguyen-Xuan, Isogeometric analysis of two–dimensional piezoelectric structures. 79 7. Pham Chi Vinh, Do Xuan Tung, Explicit homogenized equations of the piezo- electricity theory in a two-dimensional domain with a very rough interface of comb-type. 93