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Chapter 4 Trefftz Method for Piezoelectricity

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Chapter 4 Trefftz Method for Piezoelectricity Chapter 4 Trefftz Method for Piezoelectricity In Chapter 3, theoretical solutions for problems of PFC pull-out and push-out are presented. The solutions are, however, restricted to axi-symmetric problems. To remove this restriction, Trefftz numerical methods are presented for solving various engineering problems involved in piezoelectric materials in this chapter. Trefftz methods discussed here include the Trefftz FEM, Trefftz boundary element methods, and the Trefftz boundary-collocation method. 4.1 Introduction Over the past decades the Trefftz approach, introduced in 1926 [1], has been considerably improved and has now become a highly efficient computational tool for the solution of complex boundary value problems. Particularly, Trefftz FEM has been successfully applied to problems of elasticity [2], Kirchhoff plates [3], moderately thick Reissner-Mindlin plates [4], thick plates [5], general 3-D solid mechanics [6], potential problems [7], elastodynamic problems [8], transient heat conduction analysis [9], geometrically nonlinear plates [10], materially nonlinear elasticity [11], and contact problems [12]. Recently, Qin [13-16] extended this method to the case of piezoelectric materials. Wang et al. [17] analyzed singular electromechanical stress fields in piezoelectrics by combining the eigensolution approach and Trefftz FE models. As well as Trefftz FEM, Wang et al. [18] presented a Trefftz boundary element method for anti-plane piezoelectric problems. Sheng et al. [19] developed a Trefftz boundary collocation method for solving piezoelectric problems. The multi-region boundary element method [20] and the Trefftz indirect method [21] were also recently applied to electroelastic problems. This chapter, however, focuses on the results presented in [13-16,18,19,22]. 4.2 Trefftz FEM for generalized plane problems In this section, discussion is based on the formulation presented in [14]. Essentially, a family of variational formulations is presented for deriving Trefftz-FEs of generalized plane piezoelectric problems. It is based on four free energy densities, each with two kinds of independent variables as basic independent variables, i.e., (, D ), (, E ), (, D ), and (,) E . 4.2.1 Basic field equations and boundary conditions Consider a linear piezoelectric material, in which the differential governing equa- 4.2 Trefftz FEM for generalized plane problems 109 tions and the corresponding boundary conditions in the Cartesian coordinates xi (i=1, 2, 3) are defined by Eqs. (1.10)-(1.12) (Chapter 1), and the relation between the strain tensor and the displacement, ui, is governed by Eq. (1.2). For an anisotropic piezoelectric material, the constitutive relation is defined in Table 1.1 for (, E )as basic variables [14], HH(,DD )D (, ) ij s ijkl kl gD kij k, E i g ikl kl ik D k (4.1) ijD i for (, D ) as basic variables, HH(,DD )D (, ) ijchDE ijkl kl kij k, i h ikl kl ik D k (4.2) ijD i for (, D ) as basic variables, and HH(,σ E )E (,σ E ) ij s ijkl kl dD kij k, D i d ikl kl ik E k (4.3) ijE i for (, E )as basic variables, and 11 H (, D )sDDgDD (4.4) 22ijkl ij kl ij i j kij ij k 11 H (,ε D )cDDhDD (4.5) 22ijkl ij kl ij i j kij ij k 11 H (, E )sEEdEE (4.6) 22ijkl ij kl ij i j kij ij k E DED and H (, E ) is defined in Eq. (1.1), where ccijkl, ijkl and ss ijkl, ijkl are the stiff- ness and compliance coefficient tensor for E=0 or D=0, ij, ij and ij, ij are the permittivity matrix and the conversion of the permittivity constant matrix for =0 or =0. Moreover, in the Trefftz FE form, Eqs. (4.1)-(4.6) should be completed by the following inter-element continuity requirements: uuie if, e f (on ef , conformity) (4.7) ttie if0, D ne D nf 0 (on ef , reciprocity) (4.8) where “e” and “f ” stand for any two neighboring elements. Eqs. (1.1), (1.2), (1.5), (1.10)-(1.12), and (4.1)-(4.8) are taken as the basis to establish the modified variational principle for Trefftz FE analysis of piezoelectric materials. 110 Chapter 4 Trefftz Method for Piezoelectricity 4.2.2 Assumed fields The main objective of the Trefftz FEM is to establish an FE formulation whereby the intra-element continuity is enforced on a non-conforming internal displacement field chosen so as to a priori satisfy the governing differential equation of the prob- lem under consideration [23,24]. In other words, as an obvious alternative to the Rayleigh-Ritz method as a basis for an FE formulation, the model here is based on the method of Trefftz [1]. With this method the solution domain is subdivided into elements, and over each element, the assumed intra-element fields are uu11N1 uu22N2 ucuNcuNc (4.9) uuN jj 33 3 j1 N4 T where ci stands for undetermined coefficient, and u (={,uuu123 , ,} ) and N are known functions. If the governing differential equation (1.10) is rewritten in a gen- eral form ux( ) bx ( ) 0 ( x e ) (4.10) where stands for the differential operator matrix for Eq. (1.10), x the position T vector, b ={,f123ffQ , ,} the known right-hand side term, the overhead bar indi- cates the imposed quantities and e stands for the eth element sub-domain, then uux () and NNx () in Eq. (4.9) have to be chosen so that ub0 and N 0 (4.11) everywhere in e . A complete system of homogeneous solutions Nj can be gen- erated by way of the solution in Stroh formalism uA 2Re{fz ( ) c } (4.12) where “Re” stands for the real part of a complex number, A is the material eigen- vector matrix which is well defined in the literature (see pp. 17-18 of [25]), f ()zfzfzfzfz diag[()1234 () () ()] is a diagonal 44 matrix, while f ()zi is an arbitrary function with argument zxpxii12. pi (i=1–4) are the material eigenvalues. Of particular interest is a complete set of polynomial solutions which may be generated by setting in Eq. (4.12) in turn fz() zk (k 1,2, ) (4.13) k fz() iz 4.2 Trefftz FEM for generalized plane problems 111 where i 1 . This leads, for Nj of Eq. (4.9), to the following sequence: j NA2 j 2Re{z } (4.14) j NA21j 2Re{iz } (4.15) The unknown coefficient c may be calculated from the conditions on the exter- nal boundary and/or the continuity conditions on the inter-element boundary. Thus various Trefftz element models can be obtained by using different approaches to enforce these conditions. In the majority of cases a hybrid technique is used, whereby the elements are linked through an auxiliary conforming displacement frame which has the same form as in the conventional FE method. This means that in the Trefftz FE approach, a conforming electric potential and a displacement (EPD) field should be independently defined on the element boundary to enforce the field continuity between elements and also to link the coefficient c, appearing in Eq. (4.9), with nodal EPD d (={d}). The frame is defined as u1 N1 u2 N ux( )2 d Nd ( x ) (4.16) e u3 N3 N4 where the symbol “~” is used to specify that the field is defined on the element boundary only, d=d(c) stands for the vector of the nodal displacements which are the final unknowns of the problem, e represents the boundary of element e, and N is a matrix of the corresponding shape functions which are the same as those in conventional FE formulation. For example, along the side A-O-B of a particular element (see Fig. 4.1), a simple interpolation of the frame displacement and electric potential can be given in the form u1 u2 d A ux( ) N N ( x ) (4.17) A Be u3 dB where NNABdiag[NN1111 Ν NNN] diag[2222 Ν N ] (4.18) TT ddAAAAA{uuu123 } BBBBB{ uuu 123 } (4.19) with 112 Chapter 4 Trefftz Method for Piezoelectricity 11 NN (4.20) 1222 Fig. 4.1 A quadrilateral element for a generalized two-dimensional problem. Using the above definitions the generalized boundary forces and electric dis- placements can be derived from Eq. (1.11) and (4.9), and denoted by t111 jjn t1 Q t22 2 jjn t2 Q TcTQc (4.21) t n t Q 333 jj 3 Dn Dn jj Dn Q4 where tDin and are derived from u . 4.2.3 Modified variational principle The Trefftz FE equation for piezoelectric materials can be established by the varia- tional approach [23]. Since the stationary conditions of the traditional potential and complementary variational functional cannot satisfy the inter-element continuity condition which is required in Trefftz FE analysis, some new variational functionals need to be developed. Following the procedure given in [24], the functional corre- sponding to the problem defined in Eqs. (1.1), (1.2), (1.5), (1.10)-(1.12), and (4.1)-(4.8) is constructed as mme (4.22) e where me e()d Ds n ( u i uts i )d i (4.23) ee with eiibiin[(,)H E bu q ]d tu d s D d s (4.24) eteDe 4.2 Trefftz FEM for generalized plane problems 113 in which Eq. (1.10) is assumed to be satisfied, a priori and Dqns . The bound- ary e of a particular element consists of the following parts: eueteIeeDeIe (4.25) where ue u e, te t e, e e, De D e (4.26) and Ie is the inter-element boundary of the element “e”. We now show that the stationary condition of the functional (4.23) leads to Eqs.
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