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NEW IMPLEMENTATION OF MLBIE METHOD FOR HEATCONDUCTION ANALYSIS IN FUNCTIONALLY GRADED
MATERIALS
DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
Abstract. The meshless local boundary integral equation (MLBIE) method
with an efficient technique to deal with the time variable are presented in this
article to analyze the transient heat conduction in continuously nonhomoge-
neous functionally graded materials (FGMs). In space, the method is based
on the local boundary integral equations and the moving least squares (MLS)
approximation of the temperature and heat flux. In time, again the MLS
approximates the equivalent Volterra integral equation derived from the heat
conduction problem. It means that, the MLS is used for approximation in both
time and space domains, and we avoid using the finite difference discretiza-
tion or Laplace transform methods to overcome the time variable. Finally
the method leads to a single generalized Sylvester equation rather than some
(many) linear systems of equations. The method is computationally attrac-
tive, which is shown in couple of numerical examples for a finite strip and a
hollow cylinder with an exponential spatial variation of material parameters.
Keywords. Meshless local boundary integral equation (MLBIE) method; Mov-
ing least squares (MLS) approximation; Heat conduction problem; Function-
ally graded materials (FGM).
1. Introduction
Functionally graded materials (FGMs) are those in which the composition andthe volume fraction of FGM constituents vary gradually, giving a non-uniform mi-crostructure with continuously graded macro-properties (e.g. specific heat, conduc-tivity, density). For instance, one face of a structural component (plate or cylinder)may be an engineering ceramic that can resist severe thermal loading, and the otherface may be a metal to maintain structural strength and toughness [1, 2]. In recent
Date: October 16, 2011.∗Corresponding author.
Email address: [email protected] (D. Mirzaei).
Email address: [email protected] (M. Dehghan) .†Department of Mathematics, University of Isfahan, 81745-163, Isfahan, Iran. Tel: +98
3117934642.‡Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirk-
abir University of Technology, No. 424, Hafez Ave.,15914, Tehran, Iran.
1
2 DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
years FGMs have drawn considerable interest [3]. These FGMs are expected to behighly heat resistant materials that can be used under high temperature and hightemperature gradient conditions. In this case, it is important to investigate thetemperature distributions in the FGMs.
Due to the high mathematical complexity of the initial boundary value problems,analytical approaches for the thermo-mechanics of FGMs are restricted to simplegeometry and boundary conditions. Thus, the transient heat conduction analysisin FGM demands accurate and efficient numerical methods [4].
There are plenty of papers considered the heat conduction problem in FGMs.Here we present some of those papers which are more related to the meshlessmethods. Sutradhar et.al [1] presented the Galerkin boundary element methodwith Laplace transform for heat conduction problem in homogeneous and nonho-mogeneous FGMs. J. Sladek et.al [5] proposed a meshless method for transientheat conduction analysis in FGMs using local boundary integral equation. In [6]they used this technique for heat conduction with nonlinear source term in FGMs.V. Sladek et.al [7] developed a local boundary integral method for transient heatconduction in anisotropic and functionally graded media. Again, Sladek brothersand their collaborates employed meshless local Petrov-Galerkin (MLPG) methodto analyze heat conduction in FGMs [8, 4]. Wang et.al [9] developed a meshlessnumerical method based on RBF interpolation and time discretization to analyzethis problem. Hamza-Cherif et.al [10] solved transient temperature fields in ax-isymmetric FGM cylinders under various boundary conditions using an h-p finiteelement method.
There are also many works on meshless methods for solving some other time-dependent problems. For example see [11, 12, 13, 14, 15] and etc.
Meshless methods are based on approximation in terms of scattered data [16].These methods are becoming popular, due to their high adaptivity and a low costto prepare input data for numerical analysis. The moving least squares (MLS) isone of the scattered data approximation methods, which has been used successfullyto approximate the trial space in many meshless methods. For instance we canmention the MLBIE method [17, 18]. This method is based on local sub-domains,rather than a global domain, and requires neither domain elements nor backgroundcells in either the approximation or the integration. Hence it is a truly weak-basedmeshless method. During these recent years, MLBIE methods have been widelyused to solve many steady-state and time-dependent engineering problems.
The authors of all above-mentioned papers employed either the Laplace trans-form or the time discretization method to overcome the time variable. In thispaper, we propose an interesting meshless local boundary integral equation (ML-BIE) method which employs the MLS to approximate the quantities in both time
12 3
and space domains. Although the technique is presented for the MLBIE method,it can be easily extended to other MLS based meshless methods.
This paper is organized as follows. In Section 2, MLS approximation is discussed.In Section 3, the governing equations of heat conduction problem in FGMs areconverted to an algebraic system of equations using MLBIE method. In Section 4,numerical results for a finite stirp and a hallow cylinder are presented.
2. MLS approximation
The MLS as an approximation method has been introduced by Lancaster andSalkauskas [19] inspired by the pioneer work of Shepard [20]. Since the numericalapproximations of MLS are based on a cluster of scattered nodes instead of interpo-lation on elements, there have many meshless methods based on the MLS methodfor the numerical solution of differential equations in recent years.
Given data values ujNj=1 at centers x1, ..., xN, the MLS method produces afunction u that approximates u in a weighted square sense. Let Pdm be the space ofpolynomial of degree m in Rd, and let B = p1, p2, ..., pQ be any polynomial basisof Pdm, where Q =
(m+dd
). The MLS approximation u(x) of u(x), ∀x ∈ Ω, can be
defined as
u(x) = pT (x)a(x), ∀x ∈ Ω,
where pT (x) =[p1(x), p2(x), ..., pQ(x)
]and a(x) is a vector with components aj(x),
j = 1, 2, ..., Q, which are functions of the space coordinates x. In this paper wesuppose that the basis of Pdm is a complete monomial basis of order Q.
The coefficient vector a(x) is determined by minimizing a weight discrete squarenorm, which is defined as
I(x) =N∑j=1
w(x, xj)(pT (xj)a(x)− uj
)2, (2.1)
where w(x, xj) is the weight function associated with the node j. In MLS, we areinterested in local continuous weight function w. To be more precise we choose acontinuous function φ : [0,∞)→ [0,∞) with
• φ(r) > 0, 0 ≤ r < 1,• φ(r) = 0, r ≥ 1,
and define
wδ(x, y) = φ
(‖x− y‖2
δ
),
for δ > 0 as a weight function. We further define the set of indices
J(x) = J(x, δ,X) = j ∈ 1, 2, ..., N : ‖x− xj‖2 ≤ δ,
of centers contained in the interior of closed ball B(x, δ) of radius δ around x.
4 DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
If we define the matrices P and W as
P :=(p`(xj)
), 1 ≤ ` ≤ Q, j ∈ J(x),
W (x) :=(δjkwδ(x, xk)
)j,k∈J(x)
,
the stationarity point of I, in equation (2.1), with respect to a(x) leads to
ϕ(x) = p(x)A−1B, (2.2)
and
u(x) =N∑j=1
ϕj(x)u(xj), (2.3)
where A(x) = PW (x)PT , B(x) = PW (x) and ϕ(x) = [ϕ1(x), ..., ϕN (x)], providingϕj(x) = 0 for j /∈ J(x). The functions ϕj(x) are called the shape functions of theMLS approximation, corresponding to nodal point xj . If w(x, xj) ∈ Ck(Ω), thenϕj(x) ∈ Ck(Ω) [21, 22].
The matrix A is of size Q×Q. The MLS approximation is well-defined only whenthe moment matrix A is non-singular. It can be seen that this is the case if and onlyif the rank of P equals Q (P should be a full-rank matrix). A necessary conditionis |J(x)| ≥ Q, i.e. there should be at least Q points in the support of weightfunction wδ (the domain of definition of MLS approximation), but this condition isnot sufficient. A necessary and sufficient condition is the Pdm-unisolvency conditionwhich is defined on set x1, ..., xN and depends on the geometric shape of Ω. Formore precise details see [21] and Chapters 3 and 4 of [22].
The partial derivatives of u(x) are obtained as:
u,i(x) =N∑j=1
ϕj,i(x)u(xj), x ∈ Ω, (2.4)
where
ϕj,i(x) =Q∑k=1
(pk,i[A−1B]kj + pk[A−1B,i +A−1
,i B]kj),
where A−1,i = (A−1),i represents the derivative of the inverse of A with respect to
i-th component of x, which is given by
A−1,i = −A−1A,iA
−1.
In the MLS and other scattered data approximation methods two quantities filldistance and separation distance are important to measure the quality of centersand derive the rate of convergence. For a set of points X = x1, x2, ..., xN in abounded domain Ω ⊆ Rd the fill distance is defined to be
hX,Ω = supx∈Ω
min1≤j≤N
‖x− xj‖2,
12 5
and the separation distance is defined by
qX =12
mini 6=j‖xi − xj‖2.
A set X of data sites is said to be quasi-uniform with respect to a constant cqu > 0if
qX ≤ hX,Ω ≤ cquqX . (2.5)
The error analysis of MLS approximation can be found in [23, 21, 22]. If Ω satisfiesan interior cone condition, then there exist constants C, such that for all u ∈Cm+1(Ω∗) and all quasi-uniform X ⊂ Ω, we have
‖u− u‖∞ ≤ Chm+1X,Ω |u|Cm+1(Ω∗), (2.6)
where Ω∗ is the closure of⋃x∈ΩB(x,Ch0) with hX,Ω ≤ h0 and h0 depends on Ω.
The semi-norm in the right hand side is defined by
|u|Ck(Ω∗) := max|β|=k
‖Dβu‖L∞(Ω).
The order of convergence of MLS approximation mainly depends on the polynomialbasis function B and the quality of point set X. It dose not depend on the weightfunction. The weight function is used to make a moving, local and consequentlystable approximation. Here, the following Gaussian weight function is used:
wδ(x, xj) =
exp[−(dj/c)
2]−exp[−(δ/c)2]1−exp[−(δ/c)2] , 0 ≤ dj ≤ δ,
0, dj > δ,
where dj = ‖x− xj‖2, c = c0hX,Ω is a constant controlling the shape of the weightfunction and δ = δ0hX,Ω is the size of the support domain, with constants c0 andδ0.
3. Transient heat conduction in FGM
The thermal heat conduction problem in the FGM is described by the followinggoverning equation [5]
1α(x)
∂u
∂t(x, t) = ∆u(x, t) +
1κ(x)
∇κ(x) · ∇u(x, t) +1
κ(x)f(x, t), (3.1)
where x ∈ Ω and 0 ≤ t ≤ tF denote the time and the space variables, respectively,and tF is the finial time. The initial and boundary conditions are
u(x, 0) = u0(x), x ∈ Ω, (3.2)
u(x, t) = u(x, t), x ∈ Γu, 0 ≤ t ≤ tF , (3.3)
κ(x)∂u
∂n(x, t) = q(x, t), x ∈ Γq, 0 ≤ t ≤ tF . (3.4)
In (3.1)-(3.4), ∆ represents the Laplacian operator, u(x, t) is the temperature field,κ(x) and α(x) stand for the thermal conductivity and diffusivity, respectively, andf(x, t) is the density of the body heat sources. Moreover n is the unit outward
6 DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
normal to the boundary Γ, u and q are specified values on the Dirichlet boundaryΓu and Neumann boundary Γq where Γ = Γu ∪ Γq.
To employ the MLS approximation for the time domain, we do the following:from (3.1) and (3.2) we have
1αu(x, t) =
∫ t
0
[∆u(x, τ) +
1κ∇κ · ∇u(x, τ)
]dτ +
1κ
∫ t
0
f(x, τ)dτ +1αu0(x), (3.5)
keeping in mind α := α(x) and κ := κ(x) are some functions of spatial variable.First the MLS approximation is written in respect to the time variable t. The ideacomes up from the MLS based method for Fredholm and Volterra integral equationspresented by Mirzaei and Dehghan [24]. Here the Volterra type is compatible.
Consider F distinct points T = t1, t2, ..., tF in the time domain [0, tF ], withthe fill distance hT,[0,tF ]. It is better tF be included because in various cases weneed the solution at the final time. According to (2.3), replacing x by t and N
by F , the univariate MLS approximation to equation (3.5) in respect to the timevariable t, after imposing at t = tk for 1 ≤ k ≤ F , is
1α
F∑`=1
ϕ`(tk)u(x, t`)−F∑`=1
(∫ tk
0
ϕ`(τ)dτ)(
∆u(x, t`) +1κ∇κ · ∇u(x, t`)
)=
1κ
∫ tk
0
f(x, τ)dτ +1αu0(x).
The integrations over [0, tk] can be done by converting this interval to interval [0, 1]using the following linear transformation
τ(t, θ) = tθ. (3.6)
Therefore, if we set
Ek,` = ϕ`(tk), 1 ≤ k, ` ≤ F,
Gk,` =∫ tk
0
ϕ`(τ)dτ = tk
∫ 1
0
ϕ`(τ(tk, θ))dθ
= tk
M∑j=1
ϕ`(tkθj)ωj , 1 ≤ k, ` ≤ F,
u(x) = [u(x, t1), ..., u(x, tF )],
f(x) = [f1(x), ..., fF (x)],
fk(x) = tk
M∑j=1
f(x, tkθj)ωj , 1 ≤ k ≤ F,
u0(x) = [u0(x), ..., u0(x)]1×F ,
(3.7)
where θj , ωjMj=1 is a M -point Gauss quadrature (or others) formula in [0, 1], thenwe have
1αEuT (x)−G
(∆uT (x) +
1κ
[∇κ · ∇u(x)
]T) =1κfT (x) +
1αuT0 (x). (3.8)
12 7
In addition from Dirichlet boundary condition (3.3), we have
F∑`=1
ϕ`(tk)u(x, t`) = u(x, tk), 1 ≤ k ≤ F, x ∈ Γu.
If we set u(x) = [u(x, t1), ..., u(x, tF )], then
EuT (x) = uT (x), x ∈ Γu. (3.9)
Moreover, from the Neumann boundary condition (3.4), we have
F∑`=0
ϕ`(tk)∂u
∂n(x, t`) =
1κq(x, tk), 1 ≤ k ≤ F, x ∈ Γq
and if we set q(x) = [q(x, t1), ..., q(x, tF )], then
E∂uT
∂n(x) =
1κqT (x), x ∈ Γq. (3.10)
Equations (3.8)-(3.10) construct a time-free system of boundary value problemswhich can be solved by a proper method. We are trying to apply the MLBIEmethod, presented by Zhu et.al [17], to this system of equations. Thus the MLSapproximation in respect to the space variable x should be applied. Accordingto the MLBIE process, first, the local weak form is written on local sub-domainsΩyσ ⊂ Ω where ∂Ωyσ = Lyσ∪Γyσ, and Γyσ is a part of sub-domain’s boundary which hasintersection with global boundary Γ over which the Neumann boundary conditionsare applied. The local sub-domains could be of any geometric shape and size. Forsimplicity they are taken to be B(y, σ) ∩Ω. It is clear that for every internal nodey, Γyσ = ∅ and Lyσ = ∂Ωyσ. In MLBIE method the solution of the governing equation(3.8) in two dimensional space can be found in a second weak form via a modifiedfundamental solution (companion solution [17])
u∗(x, y) =1
2πlnσ
r
corresponds to the Poisson equation
∆u∗(x, y) + δ(x, y) = 0,
where δ(x, y) is the Dirac delta function, r is the distance of the field and sourcepoints, i.e., r = ‖x− y‖2 and σ is the radius of sub-domains.
By the use of the Gauss divergence theorem to the weak form of Equation (3.8)and then applying the Neumann boundary condition (3.10), we obtain an integral
8 DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
representation
E
∫Ωyσ
1αuT (x)u∗(x, y)dΩ
+G
[c(y)uT (y) +
∫Lyσ
uT (x)∂u∗
∂n(x, y)dΓ +−
∫Γyσ
uT (x)∂u∗
∂n(x, y)dΓ
]−G
∫Ωyσ
1κ
[∇κ · ∇u(x)
]Tu∗(x, y)dΩ
=∫
Ωyσ
[ 1κfT (x) +
1αu0(x)
]u∗(x, y)dΩ +GE−1
∫Γyσ
1κqT (x)u∗(x, y)dΓ,
(3.11)
which is held for every y ∈ int(Ω) ∪ Γq, providing E is an invertible matrix. Notethat the value c(y) depends on the location of y in the domain Ω or its boundary.If y ∈ int(Ω) then c(y) = 1. For y ∈ Γ it depends on whether the source point ylies on the smooth boundary 1/2 or at corners β(y)/(2π) where β(y) is the interiorangle at point y. The symbol −
∫denotes the Cauchy Principle Value (CPV) integral.
The right hand side of Equation (3.11) is a known vector of size F × 1, say r(y).Note that neither Lagrange multiplier nor penalty parameters are introduced in
the local weak form, because the Dirichlet boundary conditions are imposed directlyusing the MLS approximation. Hence, for y ∈ Γu, Equation (3.9) is repeatedreplacing x by y.
Now consider a quasi-uniform set X = x1, x2, ..., xN in Ω with the fill distancehX,Ω. The set X should also satisfy the unisolvency condition to have a well-definedMLS approximation [22]. For simplicity suppose that xj ∈ Γu for 1 ≤ j ≤ N1 andxj ∈ Γq ∪ int(Ω) for N1 +1 ≤ j ≤ N . Thanks to the MLS approximations (2.3) and(2.4), the vectors u and its derivatives can be written in terms of values of u atcenters X. To make difference between the MLS approximation in time and space,we use the notation φj(x) instead of ϕj(x) for shape functions in (2.3) and (2.4).Thus equation (3.9) can be approximated by
E( N∑j=1
φj(y)uT (xj))
= uT (y), y ∈ Γu. (3.12)
and equation (3.11) by
E( N∑j=1
[ ∫Ωyσ
1αφj(x)u∗(x, y)dΩ
]uT (xj)
)
+G( N∑j=1
[c(y)φj(y) +
∫Lyσ
φj(x)∂u∗
∂n(x, y)dΓ +−
∫Γyσ
φj(x)∂u∗
∂n(x, y)dΓ
−∫
Ωyσ
1κ∇κ · ∇φj(x)u∗(x, y)dΩ
]uT (xj)
)= r(y).
(3.13)
12 9
Imposing (3.12) at y = xi, 1 ≤ i ≤ N1 and (3.13) at y = xi, N1 + 1 ≤ i ≤ N andsetting
H1(i, j) = φj(xi), 1 ≤ i ≤ N1, 1 ≤ j ≤ N,
H2(i, j) =∫
Ωxiσ
1α(x)
φj(x)u∗(x, xi)dΩ, N1 + 1 ≤ i ≤ N, 1 ≤ j ≤ N,
K2(i, j) = c(xi)φj(xi) +∫Lxiσ
φj(x)∂u∗
∂n(x, xi)dΓ +−
∫Γxiσ
φj(x)∂u∗
∂n(x, xi)dΓ
−∫
Ωxiσ
1κ∇κ · ∇φj(x)u∗(x, xi)dΩ, N1 + 1 ≤ i ≤ N, 1 ≤ j ≤ N,
R1(i, :) = u(xi), N1 + 1 ≤ i ≤ N,
R2(i, :) = r(xi), N1 + 1 ≤ i ≤ N,
U(j, :) = u(xj), 1 ≤ j ≤ N,
give E(H1U)T = RT1
E(H2U)T +G(K2U)T = RT2.
Let H = [HT1 HT
2 ]N×N , K = [0 KT2 ]N×N and R = [RT1 RT2 ]F×N , thus we have
EUTH +GUTK = R, (3.14)
which is a generalized Sylvester equation. This equation should be solved by somelinear algebra techniques that we will discuss in the following. Note that U(j, `) =u(xj , t`) for 1 ≤ j ≤ N and 1 ≤ ` ≤ F . When U is derived from equation (3.14),the solution at every point (x, t) ∈ Ω× [0, tF ] can be approximated by
u(x, t) ≈ u(x, t) =N∑j=1
φj(x)u(xj , t)
=N∑j=1
F∑`=1
φj(x)ϕ`(t)u(xj , t`).
If we set φ(x) = [φ1(x), ..., φN (x)] and ϕ(t) = [ϕ1(t), ..., ϕF (t)] then
u(x, t) ≈ φ(x)UϕT (t). (3.15)
Note that, E and G are time and H and K are space matrices. The method providesthe time and space matrices separately. Therefore other meshless methods in spacecan be easily replaced by the MLBIE, presented in this paper.
The advantages of this method over the finite difference discretization in timecan be expressed as bellow. In time discretization method we should use a differenceequation of order O(∆t) like [u(t+ ∆t)− u(t)]/∆t instead of ∂u/∂t. Thus a smalltime step ∆t requires to get accurate results. In the method of this paper the
10 DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
convergence rate in the time domain is of order hm+1T,[0,tF ]. Therefore, using not so
many small time fill distance hT,[0,tF ], accurate results are obtained.Laplace transform method is another technique to treat the time dependent
problems. Several linear system of equations should be solved to get the solutionat a sample time t. Unfortunately, the accuracy of the space method (here ML-BIE) will finally lose due to the poor accuracy of Laplace inversion techniques. Inaddition Laplace transforms of the body force and the boundary conditions shouldbe provided.
However, the method presented in this paper needs a linear algebra technique tosolve the generalized Sylvester equation (3.14). There are several strategies.
This equation can be converted to
TUT + UTS = C, (3.16)
with the time matrix T = G−1E, the space matrix S = KH−1 and the right-handside C = G−1RH−1, providing G and H are invertible matrices. Equation (3.16)is a standard Sylvester equation with unknown U . In Matlab the function
X = lyap(T,S,-C)
gives the solution U = XT . Numerical results show, matrix H is invertible and if0 is not included in t1, ..., tF , matrix G is also invertible. Note that there is noforce to include zero.
Equation (3.14) can be also solved directly using the Bartels-Stewart (BS) orHessenberg-Schur (HS) algorithms presented in [25] based on QZ factorization,without any assumption on invertibility. Gardiner et.al [25] derived: if N ≥ F (ifis not just change N and F ), the computational costs of BS and HS algorithms are
BS : 33N3 + 33F 3 + 3N2F + 3F 2N,
HS : 8.8N3 + 33F 3 + (5 + 5.5p)N2F + 3F 2N,
where 0 ≤ p ≤ 0.5. In this method, F is the number of meshless points in one-dimensional time domain, and in many cases can be ignored in contrast to thenumber of meshless points in multi-dimensional space domain, i.e. N .
These costs are roughly the same as the costs of solving many linear system ofequations which is required in time discretization technique. Note that, solving asingle generalized sylvester equation (3.14) gives the results at all time and spacepoints, at once. This means that solving a generalized Sylvester equation is not anoticeable problem in sense of computational costs. Anyway, since in time differencemethod, LU decomposition and backward and forward substitutions can be usedin some cases, the method of this paper is slightly more expensive.
12 11
4. Numerical results
In numerical results, the basis
B =
(x− z)β
h|β|
0≤|β|≤m
is used where h can be qX , hX,Ω or an average of them. And z is a fixed evalu-ation point such as a test point or a Gaussian point for integration in weak-formbased techniques. Here β = (β1, β2) ∈ N2
0 is a multi-index and |β| = β1 + β2.If x = (x1, x2) then xβ = xβ1
1 xβ22 . This choice of basis function, instead of
B = xβ0≤|β|≤m, leads to a well-conditioned matrix A = PWPT in the MLSapproximation (Equation (2.2)) and A−1 is computed accurately, especially whenthe mesh-size is reduced. The effect of this variation on the conditioning of theMLS algorithm has been analytically investigated in [26].
We take m = 2 to span the trial spaces by the MLS approximation in bothtime and space domains. The local sub-domain’s size is chosen as σ = 0.002.Numerical results in [27] show that it is a good choice. For integration over thelocal sub-domains in MLBIE (space) we use the Gauss integration formulas [12].A fifteen points Gauss formula is employed for regular integrals and a seven pointsspecial Gauss formula for integrations with singular kernel ln r in [0, 1]. For the timeintegration (to generate the matrix G), if the time domain is large it is better to usesome more accurate quadratures. Here we use a fifty points Gauss formula. If thetime domain is small we can use some fewer points quadrature formulas. Note thatthe integration in the time domain is based on calling the one-dimensional MLSshape functions which has extremely low cost in comparison to the integration inmulti-dimensional space domain. Therefore, the cost of integration in time can bereally ignored.
4.1. Problem 1. Consider a finite strip with a unidirectional variation of the ther-mal conductivity and diffusivity. The same exponential spatial variation for bothparameters is taken
κ(x) = κ0 exp(γx1),
α(x) = α0 exp(γx1),(4.1)
with α0 = 0.17 × 10−4 m2s−1 and κ0 = 17 Wm−1C−1. Three different exponen-tial parameters γ = 0.2, 0.5 and 1.0 cm−1 (20, 50 and 100 m−1) are selected innumerical calculation. On both opposite sides parallel to the x2-axis two differenttemperatures are prescribed. One side is kept to zero temperature and the otherhas the Heaviside step time variation i.e., u = TH(t) with T = 1 C. On the lateralsides of the strip the heat flux vanishes. See Fig. 1. Such problem has been consid-ered in [5] using MLBIE method with Laplace transform in time, and in [9] usingan RBF based meshless collocation method with time difference approximation.
12 DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
In numerical calculations, a square with a side-length a = 0.04 m and a 11× 11regular node distribution is used for the MLS approximation in the space domain(Fig. 1). Also a regular node distribution in the time domain t ∈ [0, 60] with filldistance (mesh-size) hT,[0,tF ] = 2 sec. is used. The values of unknown function u
at arbitrary point (x, t) ∈ Ω × [0, tF ] is approximated using equation (3.15), aftersolving U from (3.14).
The special case with an exponential parameter γ = 0 corresponds to a homo-geneous material. In such a case an analytical solution is available
u(x, t) =Tx1
a+
2π
∞∑n=1
T cosnπn
sinnπx1
a× exp
(−α0n
2π2t
a2
),
which can be used to check the accuracy of the present numerical method. Numer-ical results are computed at three locations along the x1-axis with x1/a = 0.25,0.5 and 0.75. Results are depicted in Fig. 2. An excellent agreement betweennumerical and analytical results is obtained.
It is known that the numerical results are rather inaccurate at very early timeinstants and at points close to the application of thermal shocks. Therefore inFig. 3 we have compared the numerical and analytical solutions at very early timeinstants. Numerical results are obtained for x1/a = 0.5, tF = 2 and hT,[0,tF ] = 0.02and then given for time-lengths 4× 10−3 (using Equation (3.15)) by full-rectanglesin this figure. The absolute error is approximately bounded by 2 × 10−4. Theresults are more accurate for points closer to the starting time. Besides, in Fig.4 the numerical and analytical solutions at points very close to the applicationof thermal shocks are given and compared for t = 10 sec. The maximum absoluteerror is 1.45×10−3. Consequently, the method gives accurate results in these cases.
The discussion above concerns heat conduction in homogeneous materials onlysince analytical solutions can be used for verification. To illustrate the applicationof the proposed algorithm, consider now the cases γ = 20, and 50 m−1, respectively.The variation of temperature with time for three γ-values and at position x1 = 0.01and 0.02 m are presented in Figs. 5 and 6, respectively.
Fig. 6 is in good agreement with Figure 11 presented in [9], but some differencesare evident between Fig. 5 of this paper and Figure 4 of [5]. This disagreement hasalso been reported in Table 1 of [9]. Note that in this case the results of this paperare in agreement with Table 1 of [9].
For final steady state an analytical solution can be obtained as
u(x, t→∞) = Texp(−γx1)− 1exp(−γa)− 1
,(u→ T
x1
a, as γ → 0
).
Analytical and numerical results computed at time t = 60 sec. corresponding tostationary or static loading conditions are presented in Fig. 7. Numerical resultsare in good agreement with analytical ones for the steady state temperatures.
12 13
As expected, it is found from Figs. 5, 6 and 7 that the temperature increasesalong with an increase in γ-values (or equivalently in thermal conductivity), andthe temperature approaches a steady state when t > 20 sec.
4.2. Problem 2. In this example, an infinitely long and functionally graded thick-walled hollow cylinder is considered, where the following radii R1 = 8 × 10−2 mand R2 = 10× 10−2 m are selected. This problem has been considered in [5] usingMLBIE method, where Laplace transform employed to eliminate the time variable.
Heaviside step boundary condition is prescribed on the external surface of thehollow cylinder for the time-dependent thermal loading as a thermal shock withT = 1 C. The inner surface is kept at zero temperature. Due to the symmetry ingeometry and boundary conditions it is sufficient to analyze only a quarter of thecross section of the hollow cylinder. The boundary conditions and the total nodeslying in the domain and on the global boundary are depicted in Fig. 8.
The thermal conductivity and diffusivity functions are chosen such as (4.1) withsame constants α0 and κ0. For comparison purpose, a homogeneous material (γ =0) is first considered. In this homogeneous case, an analytical solution is known as:
u(r, t) = Tln(r/R1)
ln(R2/R1)− π
∞∑n=1
TJ2
0 (R1βn)U0(rβn)J2
0 (R1βn)− J20 (R2βn)
× exp(−α0β
2nt),
whereU0(rβn) = J0(rβn)Y0(R2βn)− J0(R2βn)Y0(rβn),
and βn are the roots of the following transcendental equation
J0(r)Y0(rR2/R1)− Y0(r)J0(rR2/R1) = 0,
with J0(r) and Y0(r) being Bessel functions of first and second kinds and zerothorder. In numerical results we use time step hT,[0,tF ] = 0.5 sec. in the time domain[0, 16].
Fig. 9 depicts the numerical and exact solutions for various times at r = 0.09m. The results show an excellent agreement with the analytical solution.
Fig. 10 shows the variation of the temperature with radial coordinate at fourdifferent time instants.
Finally, we consider the functionally graded hollow cylinder with the thermaldiffusivity and conductivity being graded in the radial direction. Numerical resultsfor the time variation of the temperature are shown in Fig. 11. Similar to theresults of a finite strip in the first example, the temperature level at interior pointsin the steady state increases with increasing γ-value. Here, again there are somedifferences between the results of this paper and the results presented in Figure 9 of[5] using MLBIE method and Laplace transform. It can be seen that, before gettingthe steady state solution, the results of this paper are larger than those obtainedby MLBIE and Laplace transform. This disagreement also observed in the previousexample. For γ = 50 m−1 and 100 m−1 the temperature grows rapidly and gets
14 DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
the steady-state in lower times. To deal with this argument we use very fine timesteps hT,[0,tF ] = 0.1 and hT,[0,tF ] = 0.005, respectively. The results are presentedin Fig. 12 and 13. As we can see, in the fist case for t > 1 sec., and in the secondcase for t > 0.05, we get the steady state solutions.
Finally Fig. 14 illustrates the variation of the temperature with the normalizedradial coordinate r/R1 for different choices of γ at t = 2 sec.
5. Conclusion
Heat conduction problem in functionally graded materials (FGMs) is investigatedin this paper using the truly meshless local boundary integral equation (MLBIE)method. This method is based on the moving least squares (MLS) approximationin trial space. We avoid using Laplace transform and time discretization methodsto eliminate the time variable. We employ again MLS as an approximation methodin the time domain. Up to the best of our knowledge, it is a new technique to treatthe time variable (at least) in the field of meshless methods. Numerical resultswhich are provided for a finite strip and a hallow cylinder, show the accuracy andefficiency of this technique. Although the process has been presented for the MLBIEmethod, it can be extend to other MLS based meshless methods, similarly.
Acknowledgements
The first author was partially supported by the Center of Excellence for Math-ematics, University of Isfahan.
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12 17
0 1 2 3 40
1
2
3
4
u = T.H(t)
q = 0
q = 0
x1 (cm)
x2 (cm)
u = 0
Figure 1. Boundary conditions and node distribution for a finite strip.
0 10 20 30 40 50 60-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time t (sec.)
Tem
pera
ture
u (
C )
x1/a = 0.25, Numerical " Analytical x1/a = 0.50, Numerical " Analyticalx1/a = 0.75, Numerical " Analytical
Figure 2. Time variation of the temperature in a finite strip at three
positions with γ = 0.
18 DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4-5
-2.5
0
2.5
5x 10
-4
Time t (sec.)
Tem
pera
ture
u (
C )
Analytical
Numerical
Figure 3. Accuracy of method for early time instants at position
x1/a = 0.5 of the functionally graded finite strip.
0.9 0.92 0.94 0.96 0.98 10.8
0.85
0.9
0.95
1
x1 /a (m)
Tem
prat
ure
u (C
)
Analytical
Numerical
Figure 4. Accuracy of method for points close to the thermal shock
at time t = 10 sec. of the functionally graded finite strip.
12 19
0 10 20 30 40 50 60-0.1
0
0.1
0.2
0.3
0.4
Time t (sec.)
Tem
prat
ure
u (C
)
= 0 m-1
= 20 m-1
= 50 m-1
Figure 5. Time variation of the temperature at position x1/a = 0.25
of the functionally graded finite strip.
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time t (sec.)
Tem
pera
ture
u (
C )
= 0 m-1
= 20 m-1
= 50 m-1
Figure 6. Time variation of the temperature at position x1/a = 0.5
of the functionally graded finite strip.
20 DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
0 0.2 0.4 0.6 0.8 1-0.2
0
0.2
0.4
0.6
0.8
1
x1 /a (m)
Tem
pera
ture
u (
C )
= 0 m-1, Num. " Anal.
= 20 m-1, Num.
" Anal.
= 50 m-1, Num. " Anal.
= 100 m-1, Num.
" Anal.
Figure 7. Distribution of temperature along x1-axis for a functionally
graded finite square strip under steady-state loading conditions.
0 2 4 6 8 100
2
4
6
8
10
x1 (cm)
x 2 (cm
)
u = 0
u = T.H(t)
q = 0
q = 0
R2
R1
Figure 8. Boundary conditions and node distribution for a hollow cylinder.
12 21
0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
0.4
0.5
Time t (sec.)
Tem
pera
ture
u (
C )
Analytical solution
Numerical solution
Figure 9. Exact and numerical solution at r = 0.09 m in hallow
cylinder with homogeneous material properties (γ = 0).
0.08 0.085 0.09 0.095 0.10
0.2
0.4
0.6
0.8
1
r (m)
Tem
pera
ture
u (
C )
t = 2 sec.t = 4 sec.t = 6 sec.t = 10 sec.
Figure 10. Temperature variation with radial coordinate r in hallow
cylinder at different time levels with homogeneous material properties
(γ = 0).
22 DAVOUD MIRZAEI∗,†,‡, MEHDI DEHGHAN‡
0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
0.4
0.5
0.6
Time t (sec.)
Tem
pera
ture
u (
C )
= 0 m-1
= 10 m-1
= 20 m-1
Figure 11. Time variation of the temperature at r = 0.09 m in a
functionally graded hollow cylinder.
0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time t (sec.)
Tem
pera
ture
u (
C )
= 50 m-1
Figure 12. Time variation of the temperature at r = 0.09 m for
γ = 50 m−1 for a functionally graded hollow cylinder.
12 23
0 0.02 0.04 0.06 0.08 0.10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time t (sec.)
Tem
pera
ture
u (
C )
= 100 m-1
Figure 13. Time variation of the temperature at r = 0.09 m for
γ = 100 m−1 for a functionally graded hollow cylinder.
0.08 0.085 0.09 0.095 0.1-0.2
0
0.2
0.4
0.6
0.8
1
1.2
r (m)
Tem
pera
ture
u (
C )
= 0 m-1
= 20 m-1
= 50 m-1
= 100 m-1
Figure 14. Temperature variation with radial coordinate r at t = 2
sec. for a functionally graded hollow cylinder
