Meshless analysis of three-dimensional steady-state heat conduction problems

Transcription

1 Cin. Pys. B ol. 19, No. 9 (010) Mesless analysis of tree-dimensional steady-state eat conduction problems Ceng Rong-Jun( 程荣军 ) a) and Ge Hong-Xia( 葛红霞 ) b) a) Ningbo Institute of ecnology, Zejiang University, Ningbo , Cina b) Faculty of Science, Ningbo University, Ningbo 31511, Cina (Received November 009; revised manuscript received 16 January 010) Steady-state eat conduction problems arisen in connection wit various pysical and engineering problems te functions satisfy a given partial differential equation and particular boundary conditions, ave attracted muc attention and researc recently. ese problems are independent of time and involve only space coordinates, as in Poisson s equation or te Laplace equation wit Diriclet, Neuman, or mixed conditions. Wen te problems are too complex, it is difficult to find an analytical solution, te only coice left is an approximate numerical solution. is paper deals wit te numerical solution of tree-dimensional steady-state eat conduction problems using te mesless reproducing kernel particle metod (RKPM). A variational metod is used to obtain te discrete equations. e essential boundary conditions are enforced by te penalty metod. e effectiveness of RKPM for tree-dimensional steady-state eat conduction problems is investigated by two numerical examples. Keywords: reproducing kernel particle metod, mesless metod, steady-state eat conduction problem PACC: 000, 060, Introduction Partial differential equations arise in connection wit various pysical and geometrical problems, te involved functions depend on two or more independent variables, usually on time t and on one or several space variables. [1] A steady-state eat conduction equation is one of te most important partial differential equations in engineering matematics, because it occurs in connection wit gravitational fields, electrostatics fields, incompressible fluid flow, and oter areas. [1] Matematically, te problem is a function wic satisfies a given partial differential equation and particular boundary conditions (boundary-value problems). Pysically speaking, te problem is independent of time and involves only space coordinates, just as initial-value problems are associated wit yperbolic partial differential equations of te elliptic type. [,3] Wen steady-state eat conduction problems are too complex, it is difficult to find an analytical solution for suc problems, so te only coice left is an approximate numerical solution. In tis paper, te reproducing kernel particle metod (RKPM) is used to obtain te numerical solution of a tree-dimensional steady-state eat conduction problem. In recent years mesless tecniques ave attracted te attention of researcers. In a mesless (mes-free) metod, a set of scattered nodes are used instead of mesing te domain of te problem. For many engineering problems, suc as large deformation and crack growt, it is necessary to deal wit extremely large deformations or fractures of te mes wit te remesing tecnique. e mesless metod is a new and interesting numerical tecnique wic can solve many engineering problems unsuited for conventional numerical metods wit a minimum of mesing or no mesing at all. [4] Some mesless metods ave been developed, suc as smoot particle ydrodynamics (SPH) metods, [5] radial basis function (RBF), [6] element free Galerkin (EFG) metod, [7 9] mesless local Petrov Galerkin (MLPG) metod, [10] RKPM, [11,1] finite point metod (FPM), [13,14] mesless metod based on local boundary integral equation, [15] boundary element-free metod (BEFM), [16,17] mesless metod wit complex variables, [18 1] and so on. e eat conduction problems arising in many Project supported by te Natural Science Foundation of Ningbo, Cina (Grant Nos. 009A and 009A610154) and te Natural Science Foundation of Zejiang Province, Cina (Grant No. Y ). Corresponding autor. cengrongjun76@16.com c 010 Cinese Pysical Society and IOP Publising Ltd ttp:// ttp://cpb.ipy.ac.cn

2 Cin. Pys. B ol. 19, No. 9 (010) kinds of engineering fields ave tus attracted muc attention and researc recently. It is difficult to find an analytical solution for suc problems, so it is important to find te numerical solution. Cen uses te RBF mesless metod to solve a simple eat conduction problem, [] Sing et al. ave done muc work on steady, unsteady, nonlinear eat transfer problems based on EFG metod, [3 7] Arefmanes et al. put forward a mesless metod for solving eat conduction problem based on MLPG, [8,9] Liu et al. use a mesless least-squares metod for solving te steadystate eat conduction problem, [30,31] Sladek et al. use a mesless local boundary integral equation metod (LBIE) for eat conduction analysis, [3,33] Ceng and Liew use RKPM for two-dimensional unsteady eat conduction problem and obtain te convergence analysis of tis metod. [34] e RKPM is greatly developed and widely applied in various fields. o te best knowledge of te autors, tis metod as not been used for a treedimensional (3D) steady-state eat conduction problem. In tis paper, RKPM is first used to obtain te numerical solution of a 3D steady-state eat conduction problem. In tis metod, te function over te solution domain needs only a set of nodes, it does not require element connectivity. e variational metod is used to obtain te corresponding discrete equations and te essential boundary conditions are enforced by a penalty metod. e numerical results are presented in comparison wit te exact results.. e RKPM [10] In RKPM, we let u (x) = Ω u(x ) w(x x )dx, (1) w(x x ) is te correction kernel function w(x x ) = c(x; x x )w(x x ), () c(x; x x ) is te correction function wic can be te combination of te polynomial basis. Let c(x; x x ) = m p i (x x )b i (x) i=1 = p (x x )b(x), (x Ω), (3) m is te number of terms in te basis, p i (x x ) are te monomial basis functions, and b i (x) are coefficients of te monomial basis functions. Generally, te basis can be cosen as follows: Linear basis: p = (1, x 1 x 1, x x ), (D) (4) p = (1, x 1 x 1, x x, x 3 x 3). (3D) (5) Quadratic basis: p = ( 1, x 1 x 1, (x 1 x 1) ), (1D) (6) p = ( 1, x 1 x 1, x x, (x 1 x 1), (x 1 x 1)(x x ), (x x ) ). (D) (7) Corresponding to te kernel approximation of Eq. (1), we can obtain te discretization approximation of Eq. (1) by te trapezoidal integral rule as follows: u (x) = w(x x I )u(x I ) I = c(x; x x I )w(x x I )u I I, (8) is related to te nodes x I, I =. (9) For a 3D problem, means te total volume of te problem domain. Equation (8) can be rewritten in te vector form u (x) = C(x)W (x) u, (10) w(x x I ) are te weigting functions wit property of compact support, x I are nodes in te domain of te node x, I is te domain measure wic u = (u 1, u,..., u n ), (11)

3 w(x x 1 ) w(x x )... 0 W (x) = = ten Let Cin. Pys. B ol. 19, No. 9 (010) w(x x n ) n, (1). (13) c I (x) = c(x; x x I ), (14) C(x) = (c 1 (x), c (x),..., c n (x)) = b (x)p, (15) p 1 (x x 1 ) p 1 (x x )... p 1 (x x n ) p (x x 1 ) p (x x )... p (x x n ) P =..,.... p m (x x 1 ) p m (x x )... p m (x x n ) (16) b (x) = (b 1 (x), b (x),..., b m (x)), (17) te coefficients b i (x) are determined by te reproducing conditions of te approximation function. In order to obtain te accuracy of n order, we obtain as M(x) = M(x) = M(x)b(x) = H, (18) p(x x I )p (x x I ) w(x x I ) I, (19) H = (1, 0,..., 0), (0) b(x) = M 1 (x)h. (1) e approximation u (x) can ten be expressed u (x) = Φ I (x)u I = Φ(x)u, () te sape function Φ(x) is given by Φ(x) = [Φ 1 (x), Φ (x),..., Φ n (x)] = C(x)W (x). (3) 3. e discretion and numerical implementation A tree-dimensional steady-state eat conduction equation in isotropic materials wit termal properties independent of temperature is governed by ( ) k x + y + z + Q = 0, (on ) (4a) wit boundary conditions: at surface x = 0(S 1 ), = s1, (4b) at surface x = L(S ), = s, (4c) at surface y = 0(S 3 ), at surface y = W (S 4 ), at surface z = 0(S 5 ), at surface z = H(S 6 ), k y n y = ( ), (4d) k y n y = ( ), (4e) k z n z = ( ), (4f) k z n z = ( ), (4g) is te temperature, k denotes te coefficient of termal conductivity and ρ represents te density of te material, c is specific eat of te material, Q denotes internal eat generation per unit volume. is convective eat transfer coefficient, denotes surrounding fluid temperature, is te 3D domain. L, W and H denote te lengt of domain in te direction of x, y and z, respectively. e weigted integral form of Eq. (4a) is obtained as follows [ ( ) ] w k x + y + z + Q d = 0. (5) e weak form of Eq. (5) is [ k w w Q ] ( d wk S x n x + x n y + ) x n z ds = 0. (6) e functional Π ( ) wic denotes te total potential energy can be written as

4 Π ( ) = 1 + Cin. Pys. B ol. 19, No. 9 (010) (k )d S 4 ( ) ds + Qd + S 5 ( ) S 3 ds 3 ( ) ( ) ds + ds. (7) Enforcing essential boundary conditions using a penalty metod, te modified functional Π ( ) can be obtained as Π ( ) = 1 ( ) (k )d Qd + S 3 ds ( ) ( ) ( ) + S 4 ds + S 5 ds + S 6 ds + α ( S1 ) ds + α ( S ) ds. (8) S 1 S aking variation, we ave δπ ( ) = (kδ )d δ Qd + δ ( )ds S 3 S 4 S 5 S 6 + δ α( S1 )ds + δ α( S )ds, (9) S 1 S α is te penalty factor. By Eq. (), we can obtain te approximation of te temperature function (x, t) = S 6 Φ I (x) I (t) = Φ(x), (30) Φ(x) = (Φ 1 (x), Φ (x),..., Φ n (x)), (31) = ( 1 (t), (t),..., n (t)), (3) x = (x, y, z). (33) Substituting Eq. (30) into Eq. (9) yields δπ ( ) = (Φ(x) )(kδ (Φ(x) )d [δ(φ(x) ) Qd + δ(φ(x) )(Φ(x) )ds + δ(φ(x) )α(φ(x) S1 )ds S 3 S 4 S 5 S 6 S 1 + δ(φ(x) )α(φ(x) S )ds. (34) S Since δπ ( ) = 0 and δ is arbitrary in Eq. (34), te following equations can be obtained K = F, (35) K = Φ (x)k Φ(x)d + Φ (x)αφ(x)ds + Φ (x)φ(x)ds, (36) S 1 S S 3 S 4 S 5 S 6 K IJ = ΦI (x)ρc Φ J (x)d + ΦI (x)αφ J (x)ds + ΦI (x)φ J (x)dω, (37) S 1 S S 3 S 4 S 5 S 6 F I = Φ I (x) Qd + αφ I (x) S1 ds + αφ I (x) S ds + Φ I (x) dγ. (38) S 1 S S 3 S 4 S 5 S

5 4. Numerical example o test te efficiency of te metod of te reproducing kernel particle approximation on a 3D steadystate eat conduction problem, a numerical analysis of te model sown in Fig. 1 as been carried out by using a linear basis function. e results obtained by te RKPM are also compared wit analytical metods. Cin. Pys. B ol. 19, No. 9 (010) is plotted in comparison wit numerical results wen x = 0.75 and y = 1.6. From Fig. 3, we can conclude tat numerical result obtained by RKPM metod is in good agreement wit te exact solution. Fig.. Nodes arrangement for example. Fig. 1. e 3D model (L = 1 m, W = m, H = 3 m, k = 1 W/mK, = 0 W/m K, = 0 C, S1 = 0 C, S = 0 C). Consider te following problem x + y + z = f(x, y, z), 0 x a, 0 y b, 0 z c, (39) wit boundary conditions (0, y, z) = (a, y, z) = 0, (x, 0, z) = (x, b, z) = 0, (x, y, 0) = (x, y, c) = 0, f(x, y, z) = π ( 1 a + 1 b + 1 c ) (40a) (40b) (40c) sin πx πy πz sin sin a b c. (41) e exact solution of Eq. (39) is (x, y, z) = sin πx πy πz sin sin a b c. (4) We use te RKPM to solve te above eat conduction problem wit penalty factor α = and scaling parameter d max = 1.13 and a = 1, b =, c = 3. e back-ground mes is adopted for integration in tis metod. Figure sows uniform nodes arrangement for example. e weigt function is cosen to be Gauss function and te bases are cosen to be linear. In Fig. 3, te exact solution (x, y, z) = sin πx a sin πy b sin πz c Fig. 3. Numerical solution and exact solution at location x = 0.75 and y = Conclusion e RKPM was used for solving te 3D steadystate eat conduction problem. e mesless property of RKPM is te most important advantage of tis sceme over te traditional mes dependent tecniques suc as te finite difference metod, finite element metod and boundary element metod. e mesless nature allows us to solve problems wit nonregular geometry. Our numerical results sow tat te tecnique is accurate and efficient. e RKPM can be extended for 3D unsteady-state eat conduction problems and nonlinear eat conduction problems

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