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Boundary Knot Method for Heat Conduction in Nonlinear Functionally Graded Material

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Boundary Knot Method for Heat Conduction in Nonlinear Functionally Graded Material Engineering Analysis with Boundary Elements 35 (2011) 729–734 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound Boundary knot method for heat conduction in nonlinear functionally graded material Zhuo-Jia Fu a,b, Wen Chen a,n, Qing-Hua Qin b a Center for Numerical Simulation Software in Engineering and Sciences, Department of Engineering Mechanics, Hohai University, Nanjing, Jiangsu, PR China b School of Engineering, Building 32, Australian National University, Canberra ACT 0200, Australia article info abstract Article history: This paper firstly derives the nonsingular general solution of heat conduction in nonlinear functionally Received 13 August 2010 graded materials (FGMs), and then presents boundary knot method (BKM) in conjunction with Kirchhoff Accepted 11 December 2010 transformation and various variable transformations in the solution of nonlinear FGM problems. The proposed BKM is mathematically simple, easy-to-program, meshless, high accurate and integration-free, Keywords: and avoids the controversial fictitious boundary in the method of fundamental solution (MFS). Numerical Boundary knot method experiments demonstrate the efficiency and accuracy of the present scheme in the solution of heat Kirchhoff transformation conduction in two different types of nonlinear FGMs. Nonlinear functionally graded material & 2010 Elsevier Ltd. All rights reserved. Heat conduction Meshless 1. Introduction MLPG belong to the category of weak-formulation, and MFS belongs to the category of strong-formulation. Functionally graded materials (FGMs) are a new generation of This study focuses on strong-formulation meshless methods composite materials whose microstructure varies from one mate- due to their inherent merits on easy-to-program and integration- rial to another with a specific gradient. In particular, ‘‘a smooth free. The MFS distributes the boundary knots on fictitious boundary transition region between a pure ceramic and pure metal would [19] outside the physical domain to avoid the singularities of result in a material that combines the desirable high temperature fundamental solutions, and selecting the appropriate fictitious properties and thermal resistance of a ceramic, with the fracture boundary plays a vital role for the accuracy and reliability of the toughness of a metal’’ [1]. In virtue of their excellent behaviors, MFS solution, however, it is still arbitrary and tricky task, largely FGMs have become more and more popular in material engineering based on experiences. and have featured in a wide range of engineering applications Later, Chen and Tanaka [20] develops an improved method, (e.g., thermal barrier materials [2], optical materials [3], electronic boundary knot method (BKM), which used the nonsingular general materials [4] and ever biomaterials [5]) solution instead of the singular fundamental solution and thus During the past decades extensive studies have been carried out circumvents the controversial artificial boundary in the MFS. This on developing numerical methods for analyzing the thermal beha- study first derives the nonsingular general solution of heat con- vior of FGMs, for example, the finite element method (FEM) [6],the duction in FGM, and then applies the BKM in conjunction with the boundary element method (BEM) [7,8], the meshless local boundary Kirchhoff transformation to heat transfer problems with nonlinear integral equation method (LBIE) [9], the meshless local Petrov– thermal conductivity. A brief outline of the paper is as follows: Galerkin method (MLPG) [10–13] and the method of fundamental Section 2 describes the BKM coupled with Kirchhoff transformation solution (MFS) [14–16]. However, the conventional FEM is inefficient for heat conduction in nonlinear FGM, followed by Section 3 to for handling materials whose physical property varies continuously; present and discuss the numerical efficiency and accuracy of the BEM needs to treat the singular or hyper-singular integrals [17,18], proposed approach in two typical examples. Finally some conclu- which is mathematically complex and requires additional comput- sions are summarized in Section 4. ing costs. It is worth noting that, with the exception of mesh-based FEM and BEM, the other above-mentioned methods are classified to the meshless method. Among these meshless methods, LBIE and 2. Boundary knot method for nonlinear functionally graded material n Corresponding author. Consider a heat conduction problem in an anisotropic hetero- E-mail address: [email protected] (W. Chen). geneous nonlinear FGM, occupying a 2D arbitrary shaped region 0955-7997/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2010.11.013 730 Z.-J. Fu et al. / Engineering Analysis with Boundary Elements 35 (2011) 729–734 O R2 bounded by its boundary G, and in the absence of heat Step 2: To transform the anisotropic Eq. (8) into isotropic one, we set 0 1 sources. Its governing differential equation is stated as ! pffiffiffiffiffiffiffiffi ! 1= K 0 y1 B 11qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi C x1 X2 @ @TðxÞ ¼ @ A ð9Þ K ðÞx,T ¼ 0, xAO ð1Þ y2 ÀK12= K11D K11= D x2 @x ij @x K K i,j ¼ 1 i j ÀÁ 2 where D ¼ det K ¼ K K ÀK 40. with the following boundary conditions. K 11 22 12 It follows from Eq. (8) that Dirichlet/essential condition: ! X2 @2CðÞy TðxÞ¼T, xAG ð2aÞ Àl2CðÞy ¼ 0, yAO ð10Þ D @y @y i ¼ 1 i i Neumann/natural condition: Therefore, Eq. (10) is the isotropic modified Helmholtz equation, X2 @TðxÞ the corresponding nonsingular solution can be found in [20]. Then A qðxÞ¼ Kij niðxÞ¼q, x GN ð2bÞ the nonsingular solution of Eq. (8) can be obtained by using inverse @xj i,j ¼ 1 transformation (9), Robin/convective condition: 1 uGðx,sÞ¼ qffiffiffiffiffiffiffi I0ðlRÞð11Þ qðxÞ¼heðTðxÞT1Þ, xAGR ð2cÞ 2p D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK P P where T is the temperature, G ¼ G þG þG and K ¼ 2 2 À1 D N R in which R ¼ i ¼ 1 j ¼ 1 riKij rj,r1 ¼ x1Às1,r2 ¼ x2Às2, where- fK ðx,TÞg denotes the thermal conductivity matrix which ij 1 r i, j r 2 x,sare collocation points and source points, respectively, and I0 satisfies the symmetry (K12 ¼ K21) and positive-definite denotes the zero-order modified Bessel function of first kind. 2 (DK ¼ detðKÞ¼K11K22ÀK 40) conditions. fgni the outward unit Finally,P by implementing the variable transformation 12 2 A À biðÞxi þ si normal vector at boundary x @O,he the heat conduction coefficient FT ¼ Ce i ¼ 1 , the nonsingular solution of Eq. (5) is in and T1 the environmental temperature. the following form: In this study, we assume the heat conductor is an exponentially P I ðlRÞ 2 0 À biðÞxi þ si functionally graded material such that its thermal conductivity can uGðx,sÞ¼ qffiffiffiffiffiffiffi e i ¼ 1 ð12Þ be expressed by 2p DK P 2 2bixi It is worth noting that the source points are placed on the KijðÞ¼x,T aTðÞKije i ¼ 1 , x ¼ ðÞx1,x2 AO ð3Þ physical boundary by using the present nonsingular general in which aTðÞ40,K ¼fK g is a symmetric positive definite solution u . ij 1 r i,j r 2 G matrix, and the values are all real constants. b1 and b2 denote In the boundary knot method, the solution of Eqs. (5) and (6) is constants of material property characteristics. approximated by a linear combination of general solutions with the By employing the Kirchhoff transformation unknown expansion coefficients as shown below: Z XN ð Þ¼ ð Þ ð Þ f T a T dT 4 FðÞ¼x aiuGðÞx,si ð13Þ i ¼ 1 Eqs. (1) and (2) can be reduced as the following form: where faig are the unknown coefficients determined by boundary 0 1 !P2 conditions. After FðxÞ is obtained, the temperature solution T to Eqs. X2 2 2b x @ @ FT ðxÞ @FT ðxÞ A i i (1) and (2) can be obtained using Eq. (7). K þ2b K ei ¼ 1 ¼ 0, xAO ð5Þ ij @x @x i ij @x i,j ¼ 1 i j j The heat flux can then be given by XN qðxÞ¼ aiQðx,siÞð14Þ FT ðxÞ¼fðTÞ, xAGD ð6aÞ i ¼ 1 P X2 2 X2 in which @TðxÞ 2b x @FT ðxÞ i ¼ 1 i i A qðxÞ¼ Kij niðxÞ¼e Kij niðxÞ¼q, x GN P @xj @xj X2 2 i,j ¼ 1 i,j ¼ 1 @uGðx,siÞ 2b x Qðx,s Þ¼ K n ðxÞe i ¼ 1 i i i ij @x i ð6bÞ i,j ¼ 1 j P 0 1 2 ÀÁ 2b r X2 X2 X2 A e i ¼ 1 i i l qðxÞ¼he FT ðxÞjðT1Þ , x GR ð6cÞ ¼ qffiffiffiffiffiffiffi @À I ðlRÞ n ðxÞr þI ðlRÞ n ðxÞK b A R 1 i i 0 i ij j 2p DK i ¼ 1 i ¼ 1 j ¼ 1 where FT ðxÞ¼jðTðxÞÞand the inverse Kirchhoff transformation ð15Þ À1 TðxÞ¼j ðÞFT ðxÞ ð7Þ And then we derive the nonsingular general solution of Eq. (5) where I1 denotes the first-order modified Bessel function of by two-step variable transformations: first kind. Step 1: To simplify the expression of Eqs. (5), let F ¼ P T In view of the general solution satisfying the governing Eq. (5), a 2 À b ðxi þ siÞ priori, the presented method only needs boundary discretization to Ce i ¼ 1 i . Eq. (5) can then be rewritten as follows: 0 1 satisfy boundary conditions P X2 2 @ @CðxÞ 2 A b ðÞx þ s Aa ¼ b ð16Þ K Àl CðxÞ e i ¼ 1 i i i ¼ 0, xAO ð8Þ ij @x @x i,j ¼ 1 i j in which qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 0 1 P P 2 u ðx ,s Þ 2 2 biðÞxi þ si G j i where l ¼ i ¼ 1 j ¼ 1 biKijbj. Since e i ¼ 1 40. The Trefftz B C A ¼ @ Qðxj,siÞ A ð17aÞ functions of Eq. (8) are equal to those of anisotropic modified Qðx ,s Þh uðx ,s Þ Helmholtz equation.
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