NUMERICAL SOLUTION OF TEMPERATURE ...

Numerical Heat Transfer, Part A, 50: 125–145, 2006 Copyright # Taylor & Francis Group, LLC ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/10407780500507111

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NUMERICAL SOLUTION OF TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY PROBLEMS USING A MESHLESS METHOD Akhilendra Singh Department of Mechanical Engineering, Birla Institute of Technology and Science, Pilani, Rajasthan, India

Indra Vir Singh Department of Mechanical Systems Engineering, Shinshu University, Wakasato, Nagano, Japan

Ravi Prakash Department of Mechanical Engineering, Birla Institute of Technology and Science, Pilani, Rajasthan, India In this article, the meshless element-free Galerkin (EFG) method is extended to obtain numerical solution of nonlinear heat conduction problems with temperature-dependent thermal conductivity. The thermal conductivity of the material is assumed to vary linearly with temperature. A quasi-linearization scheme is adopted to avoid the iteration for nonlinear solution, and time integration is performed by the backward difference method. The essential boundary conditions are enforced by Lagrange multiplier technique. Meshless formulations are presented for one- and two-dimensional nonlinear heat conduction problems. MATLAB codes have been developed to obtain the EFG results. The results obtained by the EFG method are compared with those obtained by finite-element and analytical methods.

1. INTRODUCTION A large number of numerical techniques are available to solve nonlinear heat conduction problems. Vujanovic [1] analyzed three numerical techniques—generalized integral, iterative, and variational techniques—to find approximate solutions for one-dimensional nonlinear transient heat conduction problems. A finite-element method (FEM)-based iterative method has been used by Donea and Giuliani [2] to solve steady-state nonlinear heat transfer problems in two-dimensional (2-D) structures with temperature-dependent thermal conductivities and rediative heat transfer. Bathe and Khoshgoftaar [3] have given an FEM-based formulation for the analysis of nonlinear steady-state and transient heat transfer problems in which they considered the convection and radiation boundary conditions. The explicit stable method of Saulev has been applied to nonlinear finite-element heat Received 23 March 2005; accepted 11 November 2005. Address correspondence to Akhilendra Singh, Department of Mechanical Engineering, Birla Institute of Technology and Science, Pilani, Rajasthan 333031, India. E-mail: [email protected]

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NOMENCLATURE aj ðxÞ c dmax k0


kðTÞ m n n NI ðsÞ pj ðxÞ _ Q r

nonconstant coefficients specific heat of the material, J=kgC scaling parameter reference thermal conductivity, W=mC coefficient of thermal conductivity, W=mC number of terms in the basis number of nodes in the domain of influence number of iterations Lagrange interpolent monomial basis function rate of internal heat generation= volume, W=m3 normalized radius

t Dt T_ T h ðxÞ w wðx xI Þ b Ci k q UI ðxÞ X

time, s time-step size, s ¼ qT qt moving least-square (MLS) approximant weighting function used in weak form weight function used in MLS approximation coefficient of thermal conductivity variation, C 1 boundary of the domain Lagrangian multiplier density of the material, kg=m3 shape function two-dimensional domain

conduction problems by Trujillo and Busby [4], and they solved several nonlinear example problems with temperature-varying material properties and radiation boundary conditions. Ling and Surana [5] used the p-version least-square finiteelement formulation for axisymmetric heat conduction problems with temperaturedependent thermal conductivities. Yang [6] developed an FEM-based time integration algorithm for the solution of nonlinear heat transfer problems. The general boundary-element method has been used by Liao [7] to solve nonlinear heat transfer problems governed by a hyperbolic heat conduction equation. Further use of the general boundary-element method has been made by Liao and Chwang [8] to solve strongly nonlinear heat transfer problems. An efficient algorithm has been proposed by Chan [9] to solve steady-state nonlinear heat conduction equations based on the boundary-element method (BEM). Chan considered nonlinearity in the heat conduction equation due to nonlinear boundary conditions and temperature dependence of thermal conductivity of the material. A low-order spectral method has been used by Siddique and Khayat [10] to solve nonlinear heat conduction problems with periodic boundary conditions and periodic geometry. Transient heat conduction and radiation heat transfer problems with variable thermal conductivity have been solved by Talukdar and Mishra [11] using discrete transfer and implicit schemes. Finite-element perturbation analysis of nonlinear heat conduction problems with random field parameters has been done by Nicolaı¨ and Baerdemaeker [12]. The homotophy analysis method has been improved and applied by Liao [13] to solve strongly nonlinear heat transfer problems. The precise time integration (PTI) method has been introduced and applied to linear and nonlinear transient heat conduction problems with temperature-dependent thermal conductivity [14], and the predictor–corrector algorithm has been employed to solve the nonlinear equations, etc. Of all the numerical methods developed so far, the finite-element method has been found to be the most general method, not only to solve the problems of heat transfer but also to solve various problems in different areas of engineering and science. Despite its numerous advantages and unparalleled success, it is not well suited for


THERMAL CONDUCTIVITY USING A MESHLESS METHOD

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certain classes of problems such as crack propagation and moving discontinuities, moving phase boundaries, phase transformation, plate bending, large deformations, solution of higher-order partial differential equations, modeling of multiscale phenomena, nonlinear thermal analysis, and dynamic impact problems. Therefore, the thrust to find new numerical methods continues. To overcome the problems of the FEM, a number of meshless methods have been developed in the last decade. The essential feature of these meshless methods is that they require only a set of nodes to construct the approximation functions. In contrast to the conventional finite-element method, these techniques save the tedious job of mesh generation, as no element is required in the entire model. Furthermore, re-meshing appears to be easier, because nodes can be easily added or removed in the analysis domain. Among the meshless methods developed so far, the element-free Galerkin (EFG) method has achieved popularity due to its successful applicability in different areas such as fracture mechanics [15, 16], wave and crack propagation [17, 18], plates and shells [19, 20], nondestructive testing (NDT) [21], electromagnetic fields [22], metal forming [23], vibration analysis [24], linear heat transfer [25–28], etc. In the present work, the EFG method is extended to obtain the numerical solution of nonlinear heat conduction problems with temperature-dependent thermal conductivity. The Lagrange multiplier method is used to enforce the essential boundary conditions. The discrete equations are obtained using the variational method. Meshless formulations are given for nonlinear heat conduction problems. Code has been written in MATLAB to obtain the EFG results. Two model problems are solved, and the results obtained by the EFG method are compared with those obtained by finite-element [29] and analytical methods [30, 31]. 2. THE ELEMENT-FREE GALERKIN METHOD The EFG method utilizes the moving least-square (MLS) approximants, which are constructed in terms of nodes only. The MLS approximation consists of three components: a basis function, a weight function associated with each node, and a set of coefficients that depends on node position [25, 26]. 2.1. Moving Least-Square Approximants In this article, the MLS approximation scheme is used to develop mesh-free shape functions. The unknown function TðxÞ is approximated by MLS approximants T h ðxÞ over the computational domain [25, 26]. The local approximation is given as T h ðxÞ ¼

m X

pj ðxÞaj ðxÞ pT ðxÞaðxÞ

ð1Þ

j¼1

where pðxÞ is a vector of complete basis functions (usually polynomial), given as pT ðxÞ ¼ ½1; x; y

ð2Þ

aT ðxÞ ¼ ½a1 ðxÞ; a2 ðxÞ; a3 ðxÞ

ð3Þ

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Here m is the number of terms in the basis. The unknown coefficients aðxÞ at any given point x are determined by minimizing the weighted least-square sum J, J¼

n X

wðx xI Þ½pT ðxÞaðxÞ TI 2

ð4Þ


I¼1

where TI is the nodal parameter at x ¼ xI , but these are not nodal values of T h ðx ¼ xI Þ because T h ðxÞ is an approximant and not an interpolant; wðx xI Þ is a nonzero weight function of node I at xi and n is the number of nodes in the domain of influence of x for which wðx xI Þ 6¼ 0. The stationary value of J in Eq. (4) with respect to aðxÞ leads to the following set of linear equations: AðxÞaðxÞ ¼ BðxÞT

ð5Þ

where A and B are given as A¼

n X

wðx xI ÞpðxI ÞpT ðxI Þ

ð6Þ

I¼1

BðxÞ ¼ fwðx x1 Þpðx1 Þ; wðx x2 Þpðx2 Þ; . . . ; wðx xn Þpðxn Þg

ð7Þ

Substituting aðxÞ in Eq. (1), the MLS approximant is obtained as T h ðxÞ ¼

n X

UI ðxÞTI ¼ UT ðxÞT

ð8Þ

I¼1

where UT ðxÞ ¼ fU1 ðxÞ; U2 ðxÞ; U3 ðxÞ; . . . ; Un ðxÞg

ð9Þ

TT ¼ ½T1 ; T2 ; T3 ; . . . ; Tn

ð10Þ

The mesh-free shape function UI ðxÞ is defined as UI ðxÞ ¼

m X

pj ðxÞ½A1 ðxÞBðxÞjI ¼ pT A1 BI

ð11Þ

j¼1

2.2. Weight Function Description The choice of weight function wðx xI Þ affects the resulting approximation T h ðxI Þ in the meshless EFG method, and the smoothness as well as the continuity of the shape function UI depend on smoothness and continuity of the weight function wðx xI Þ. Therefore, the selection of an appropriate weight function is essential in meshless methods. The weight function used in the present analysis can be written


129

as a function of normalized radius r as the cubic spline (C.S.) weight function [28],


8 2 2 3 > > > > 3 4r þ 4r < 4 wðx xI Þ wðrÞ ¼ 4 > 4r þ 4r2 r3 > > 3 > :3 0

r

1 2

9 > > > > =

1 < r 1> > > 2 > ; r > 1

ð12Þ

where rI ¼ jjx xI jj=dmI ; where kx xI k is the distance from a sampling point x to a node xI and dmI is the domain of influence of node I. ðrx ÞI ¼ kx xI k=dmxI ; ðry ÞI ¼ kx xI k=dmyI ; dmxI ¼ dmax cxI ; dmyI ¼ dmax cyI ; dmax ¼ scaling parameter. cxI and cyI at node I are the distances to the nearest neighbors. dmxI and dmyI are chosen such that the matrix is nonsingular everywhere in the domain. The weight function at any given point is obtained as wðx xI Þ ¼ wðrx Þwðry Þ ¼ wx wy

ð13Þ

where wðrx Þ and wðry Þ can be calculated by replacing r by rx and ry in the expression for wðrÞ. 2.3. Enforcement of Essential Boundary Conditions The EFG shape functions do not satisfy the Kronecker delta property, i.e., UI ðxJ Þ 6¼ dIJ . The lack of Kronecker delta property in EFG shape functions UI poses some difficulty in the imposition of essential boundary conditions. Various numerical techniques have been proposed to enforce the essential boundary conditions, such as the Lagrange multiplier method [25], a modified variational principle approach [32], coupling with the finite-element method [33], a penalty approach [34], full transformation technique [35], etc. In the present work, the Lagrange multiplier method has been used because of its accuracy. In two dimensions, Lagrange multiplier k is expressed as kðxÞ ¼ NI ðsÞkI dkðxÞ ¼ NI ðsÞdkI

x2C x2C

ð14aÞ ð14bÞ

where NI ðsÞ is a Lagrange interpolant and s is the length along the essential boundary conditions. 3. NUMERICAL IMPLEMENTATION 3.1. One-Dimensional Formulation The energy equation for one-dimensional (1-D) unsteady-state heat transfer in an isotropic material with thermal conductivity dependent on temperature is given as q qT _ ¼ qc qT kðTÞ þQ qx qx qt where kðTÞ ¼ k0 ð1 bTÞ.

ð15Þ

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The initial and boundary conditions are At t ¼ 0; At x ¼ 0; At x ¼ L;

9 T ¼ Tini= T ¼ TL ; T ¼ TR


The weak form of Eq. (15) is obtained as Z qw qT _ _ kðTÞ þ wQ q cwT dC ¼ 0 qx C qx The functional IðTÞ can be written as Z Z Z 1 qT qT _ dC þ q c T_ dC kðTÞ IðTÞ ¼ dC T Q qx C 2 qx C C

ð16Þ

ð17Þ

Enforcing essential boundary conditions using Lagrange multipliers, the functional I ðTÞ is obtained as I ðTÞ ¼

Z C

Z Z 1 qT qT _ dC þ q c T_ TdC þ k1 ðT TL Þj kðTÞ dC T Q x¼0 2 qx qx C C

þ k2 ðT TR Þjx¼L

ð18Þ

Taking variation, i.e., dI ðTÞ of Eq. (18), it reduces to

dI ðTÞ ¼

Z C

Z Z qT qT _ kðTÞd dC QdTdC þ qcT_ dTdC qx qx C C

þ dk1 ðT TL Þjx¼0 þ k1 dTjx¼0 þ dk2 ðT TR Þjx¼L þ kR dTjx¼L

ð19Þ

Since dk1 , dk2 , and dT are arbitrary in the preceding equation, the following relations are obtained by using Eq. (8): ½KðTÞfTg þ ½C T_ þ ½G1fk1 g þ ½G2fk2 g ¼ ff g

ð20aÞ

G1T fTg ¼ fTL g

ð20bÞ

G2T fTg ¼ fTR g

ð20cÞ

# Z " qUI T qUJ KIJ ðTÞ ¼ kðTÞ dC qx qx C

ð21aÞ

where

CIJ ¼

Z C

UTI q c UJ dC

ð21bÞ


Z

131

_ UI dC Q

ð21cÞ

G1IK ¼ UK jx¼0

ð21dÞ

G2IK ¼ UK jx¼L

ð21eÞ

fI ¼

C


Quasi-linearizing of Eq. (20a), the unconditionally stable implicit backward difference method [29] for time approximation can be written as 2 ðTÞ nþ1 þ C K 4 G1T G2T

G1 0 0

9 38 9 8 nþ1 G2