Procedia Engineering 15 (2011) 3562 – 3566

1877-7058 © 2011 Published by Elsevier Ltd.doi:10.1016/j.proeng.2011.08.667

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Procedia Engineering 00 (2010) 000–000

ProcediaEngineering

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Advanced in Control Engineering and Information Science

NURBS-based Isogeometric Finite Element Method for Analysis of Two-dimensional Piezoelectric Device

CHEN Tao∗, MO Rong, WAN Neng

The key laboratory of Contemporary Design and integrated Manufacturing Technology, Northwestern Polytechnical University, Xi’an, 710072, China

Abstract

A novel numerical simulation method based on NURBS geometric description, first proposed by Hughes et al. is applied to analyze the two-dimensional piezoelectric structure. It was reported recently to have some remarkable advantages, such as the exact and unified geometry representation, high-order continuous elements and superior accuracy for per degree-of-freedom. Regrettably, the NURBS basis functions don’t verify the Kronecker delta property at the control points, which would lead to an awkward implementation for imposing the essential boundary conditions. A variational form with the constraints was proposed in order to remedy this issue, and the essential boundary conditions are enforced by penalty methods. Two advantages can be observed for presented method: (i) symmetric and positive definite matrix; (ii) the dimension of the linear system is not increased. The numerical results demonstrate the efficiency and robustness of the present method. © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [CEIS 2011] Keywords: isogeoemtric analysis; piezoelectric structure; penalty method; essential bondary condition

1. Introduction

The piezoelectric material has been widely used in the domain of micro-eletrico-mechanical system (MEMS), and can be used for designing the smart structures such as sensors and actuators. Various numerical approaches have been applied for analyzing such materials and structures over the past decades, such as finite element method[1], meshfree method[2] and boundary element method[3]; more recently, smoothed finite element method[4] and moving kriging method[5]. To ensure the high accuracy results in the analysis, the exact models should be considered in the numerical method. Isogeometric Analysis, proposed firstly by Hughes and his colleagues[6], is a new numerical simulation method that integrates the finite element approaches and Computer-aided design into a

∗ Corresponding author. E-mail address: chentaonwpu@163.com.

3563CHEN Tao et al. / Procedia Engineering 15 (2011) 3562 – 3566 Chen Tao et al / Procedia Engineering 00 (2011) 000–000

unify framework. Complied with the isoparametric concept, the finite element meshes are replaced with the NURBS patches. The precise and unified mesh representation is a competing trait than traditional approaches.

Regrettably, the NURBS basis functions don’t interpolate at the control points as similar as the meshfree method, thus imposing the displacement boundary conditions is not trivial as in the finite element. Some experienced methods were proposed in the domain of the meshfree methods, and refer to the article for a review[7]. These techniques can be divided roughly into three categories: (a) the modified variational principle; (b) imposing strongly into the nodes; (c) coupled with finite element interpolation. The penalty method fall into the first group and the weak form is constructed with the additional penalty terms. It has two clear advantages: (i) the dimension of the linear system is not increased; (ii) the symmetric and positive definite matrix when the stiffness matrix is symmetric and the penalty parameters are large enough.

The paper is outlined as follows. The aim of section 2 is to present a brief introduction of the governing equations of the piezoelectric problem and Non-Uniform Rational B-Spline. Followed by the main framework of isogeometric analysis, the weak forms and discreted linear system are represented in section 3. Numerical results are investigated in section 4 for verifying the performances of the proposed method.

2. Preliminary

2.1. Governing equations

The constitutive equations for two-dimensional piezoelectric material can be expressed by the relation of the stress, strain, and electric field in the stress-charge form:

T

E

S

⎧ = −⎪⎨

= −⎪⎩

σ c ε e ED eε k E

(1)

For two-dimension domain, the general stress tensor σ and strain tensor ε can be represented in the x-z plane as { , , }x z xzσ σ τ=σ and { , , }x z xzε ε γ=ε , respectively. There, D denotes the electric displacement vector { , }x zD D=D and E the electric field vector { , }x zE E=E . Ec , e and Sk are the elastic stiffness matrix for constant

electric filed, the piezoelectric matrix and the dielectric constant matrix for constant structural strain, respectively. Consider the mechanical and electrical equilibrium equations:

, 0 ( ) 0x zi j i

D Df and div g gx z

σ ∂ ∂+ = − = + − =∂ ∂

D (2)

, which corresponding to Dirichlet and Neumann boundary conditions

( )1,, x x xz z xz x z z z tx y u

x x z z q

n n t n n t onu u w u onand

D n D n q onon φ

σ τ τ σφ φ

+ = + = Γ= = Γ ⎧⎧ ⎪⎨ ⎨ − + = Γ= Γ ⎪⎩ ⎩

(3)

There, u denotes the displacement vector and φ the scalar electric potential field. The strain is related to the displacement by , ,( ) / 2ij i j j iu uε = + , which can be expressed for two-dimensional domain:

, ,x z xzu w u wx z z x

ε ε γ∂ ∂ ∂ ∂= = = +∂ ∂ ∂ ∂

(4)

The electric field E is related to electric potential field by ,x zφ φϕ ∂ ∂⎛ ⎞= −∇ = − −⎜ ⎟∂ ∂⎝ ⎠

E

2.2. NURBS basis functions.

NURBS are piecewise parametric curves/surfaces that have the ability of describing the complex shape exactly, which are formulated as the linear combination of the NURBS basis functions and control points. In the isogeometric analysis, the geometries in plane are described by the tensor product NURBS surfaces: ,

,1 1( , ) ( , )n m p q

i j iji jS N dξ η ξ η

= == ∑ ∑ (5)

3564 CHEN Tao et al. / Procedia Engineering 15 (2011) 3562 – 3566 Chen Tao et al / Procedia Engineering 00 (2011) 000–000

, where ijd denote control points and ,,p q

i jN tensor product NURBS basis functions. It’s convenient to reformulate the NURBS surfaces Eq. (5) to be general form: : ( ) ( )i ii

X F N dξ ξ∈

= = ∑ I (6)

, where F represents a mapping that maps any point in the parametric domain to the physical domain. It’s similar to the meshfree method, where the NURBS basis functions don’t generally verify the Kronecker

delta property ( )i j ijN ξ δ= .The non-interpolatory property makes the control points not locate within the physical domain. The most immediate results are not trivial to impose inhomogeneous essential boundary into the control points as in the conventional finite element method.

3. NURBS-based isogeometric FEM

3.1. Main framework

The main framework of the isogeometric analysis is summarized in the following [6]: In contrast to the conventional finite element method, the geometries are exactly described in terms of NURBS

surfaces or volumes rather than developed the approximated meshes with the piecewise polynomials. The isoparametric concept is invoked, so the field variable is formulated as the linear combination of the field value on control points and the basis functions describing exactly geometries. The approximated solution is formulated by: 1

1 1( ) ( )np np

h i i i ii iu N u N F x uξ −

= == =∑ ∑ o (7)

Let F to be the parameterization of the physical domain, as given by(6), and assume that the Jacobian matrix of F is invertible. Thus, the space of NURBS functions on the physical domain is defined as the span of the push-forward of the basis functions in the parametric domain, i.e. 1: { } { ( ), 1,..., }h i iV span span N F x i npψ −= = =o

Neumann boundary condition is imposed directly into the weak form of the boundary value problem, thus the implementation is trivial.

3.2. Weak forms

The solution of the problem(2) can be acquired by minimizing the energy functional in the context of the conventional finite element method, and the essential boundary condition is easily imposed into the trial function space. However the NURBS basis functions don’t verify the Kronecker delta properties in the isogeometric analysis, imposing the displacement and electric potential constraints are not as straightforward as in the usual manner. We borrow the notion of penalty from the meshfree method, and construct a minimization problem with boundary constrains. The procedure is shown as follows:

Considering the elastic equilibrium equation, the mechanical functional can be obtained by summing the strain energy with the potential energy of the body force and external tension:

1( ; )2 t

T T Tu u f d u tdsσ ϕ ε σΩ Γ

⎛ ⎞Π = − Ω −⎜ ⎟⎝ ⎠∫ ∫ (8)

And the electric functional can also obtained by summing the dielectric energy and the potential energy:

1( ; )2 q

T T TD u E D gd qdsϕ φ φ

Ω Ω ΓΠ = − Ω + Ω +∫ ∫ ∫ (9)

The coupled general functional is obtained by the summation of Eq. (8) and Eq. (9) with the displacement and electric potential constraints, which denotes as

2 2( ; ) ( ; ) ( ; ) ( ) ( )2 2u

Du u u u u ds dsφ

σα βϕ ϕ ϕ φ φ

Γ ΓΠ = Π + Π + − − −∫ ∫ (10)

, where the penalty parameters α and β are positive scalar constants. In accordance with variational principle, the weak form of the problem (10) is represented as

3565CHEN Tao et al. / Procedia Engineering 15 (2011) 3562 – 3566 Chen Tao et al / Procedia Engineering 00 (2011) 000–000

( )( ; , ; ) ( ; )

( ; , ; )

( ; )u

t q u

T T T

T T T T T

a u u l u

a u u E D d u uds ds

l u u fd gd u tds qds u uds dsφ

φ

δ δφ φ δ δφ

δ δφ φ δε σ δ α δ β δφφ

δ δφ δ δφ δ δφ α δ β δφ φ

Ω Γ Γ

Ω Ω Γ Γ Γ Γ

=

= − Ω + −

= Ω − Ω + − + −

∫ ∫ ∫∫ ∫ ∫ ∫ ∫ ∫

(11)

3.3. Discretization Equations

The solutions of the Eq. (11) are projected into the NURBS basis function space Vh, and the approximation of the displacement and electric potential field can be expressed as

1 1( ) ( ) ( )n ne e

h i i h i ii ix x xψ φ ψ φ

= == =∑ ∑u u (12)

, where eu and eφ are the displacement and electric potential values located in the control points. And the approximated elastic stain and electric potential gradient are represented as

1 1 1 1

n n n ne u e e e

h S i i i i h i i i ii i i i

N u B u E N Bφφ φ= = = =

= ∇ ⋅ = ⋅ = − ∇ = −∑ ∑ ∑ ∑ε (13)

, where ▽denotes the gradient operator and S∇ is defined as strain differential operator. Substituting the Eq. (12)and Eq.(13) into the equation(11), we get the linear system:

uu u u

u

K K FuK K F

φ

φ φφ φφ⎡ ⎤ ⎧ ⎫⎧ ⎫

=⎨ ⎬ ⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦ ⎩ ⎭

(14)

, where

u

T T T Tuu u E u u u u u

T T Tu u

K B c B d N N ds K B e B d

K B kB d N N ds K B eB dφ

φ φ

φφ φ φ φ φ φ φ

α

βΩ Γ Ω

Ω Γ Ω

= Ω + = Ω

= − Ω − = Ω

∫ ∫ ∫∫ ∫ ∫

t u

T T T T T Tu u u u q

F N fd N tds N uds F N gd N qds N dsφ

φ φ φ φα β φΩ Γ Γ Ω Γ Γ

= Ω + + = − Ω − −∫ ∫ ∫ ∫ ∫ ∫

4. Numerical experiment

A two-dimensional example is investigated for verifying the program code and comparing the performance of the presented method. It is worthwhile to mention that the L2-norm errors are employed to measure the departure of the approximated solution uh to the exact solution u of the problem.

Problem definition. Consider a problem to be defined within a ring domain (only quarter), the exact solutions for this problem are given as sin( )sin( )u w x zπ π= = and exp( )x zφ = + . Supposed the material parameters to be set artificially with the suppositive configuration, which may be unreasonable in the physical world

11 13

13 33 2

55

0 1 00 0 1 1 0

0 1 0 , ,1 1 0 0 11

0 0 0 0 0.5(1 )E S

c cEc c c e k

c

μμ

μμ

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦

, where Young’s modulus E=1 and Poisson rate 0.3μ = .The essential boundary conditions are applied to all four edges of the domain, and the body force and source term in Eq. (2) are given as

( )( )

2

2

2 2

(( 1)cos( ( )) 2cos( ( )))2 1

g= 2 cos( )cos( ) sin( )sin( )

x y

x z

E x z x zf f

e x z x z

π μ π πμ

π π π π π π+

− − + += =−

+−−

Numerical results. The numerical experiments were performed for the different NURBS meshes. Mesh refinement strategies are developed by the combination of the knot insertion and order elevation algorithms. Fig.1-(a) showed the contour plots of the approximated solution of displacement component u and its L2-norm errors (Fig.1-(b)). The contour plots of approximated electric potential field φ and its L2-norm errors are listed in Fig.1-(c) and -

3566 CHEN Tao et al. / Procedia Engineering 15 (2011) 3562 – 3566 Chen Tao et al / Procedia Engineering 00 (2011) 000–000

(d). It is observed in Table 1 and Fig.1-(e) that the numerical results excellently agree with the analytical solutions, and the high-order splines usually mean the high convergence speed.

Fig. 1 Contour plots of approximated displacement component u (a) and its L2-norm errors (b); Contour plots of approximated electric potential field φ and its L2-norm errors; (e) Comparison of rate-of-convergence with different NURBS meshes.

Table 1 the L2-norm error for the different-order NURBS meshes hmax 1.7246 0.9188 0.4721 0.2391 0.1203 0.0603 p=2 6.1131 1.0454 0.0726 0.0080 9.7.34e-04 1.2047e-04 p=3 1.4493 0.2135 0.0147 5.2393-e04 2.8251e-05 1.7072e-06 p=4 0.8615 0.2046 0.0050 7.1715e-05 1.6702e-06 1.0334e-07

5. Conclusion

The main contribution of the present paper is to develop a unified and exact mesh representation for two-dimensional piezoelectric structure. For the sake of remedying the deficiency caused by the non-interpolatory property of the NURBS basis functions, the essential boundary conditions are enforced by the penalty method. The numerical experiments demonstrate the high-accurate results can be acquired with the presented methods.

Acknowledgements

The study was supported by the National High-Tech. R&D Program (863 Program), China (No. 2007AA04Z184).

References

[1] Lerch, R. Simulation of piezoelectric devices by two-and three-dimensional finite elements. IEEE Trans. Sonics Ultrason 1990;233:37-3. [2] Ohs, R.R., and Aluru, N.R. Meshless analysis of piezoelectric devices. Comput Mech 2001;23:27-1 [3] Peng, M., and Cheng, Y. A boundary element-free method (BEFM) for two-dimensional potential problems. Eng Anal Bound Elem 2009;77: 33-1 [4] Nguyen-Van, H., Mai-Duy, N., and Tran-Cong, T. A smoothed four-node piezoelectric element for analysis of two-dimensional smart structures. Computer Modeling in Engineering and Sciences 2008;209:23-3 [5] Bui, T.Q., Nguyen, M.N., Zhang, C.,Pham, D.A.K. An efficient meshfree method for analysis of two-dimensional piezoelectric structures. Smart Materials and Structures 2011;650:20- [6] Hughes, T., Cottrell, J.A., Bazilevs, Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Method Appl M 2005;4135:194-39-41 [7] Fernandez-Mendez, S., Huerta, A. Imposing essential boundary conditions in mesh-free methods. Comput Method Appl M 2004;1257:193-12-14