Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X
Boundary integral formulation for non-linear
heat conduction in anisotropic heterogeneous
media
Eduardo Divo & Alain Kassab
Institute for Computational Engineering
Department of Mechanical, Materials and Aerospace
Or/aWo,
Abstract
The recently developed BEM formulation of the authors for heat conduction in
media with spatially dependent thermophysical properties is herein implemented
to approximate the solution of problems in which the temperature dependency of
the material properties results into a non-linear governing equation. The thermal
conductivity of the medium is also considered to be anisotropic. An iterative
approach is used to lag the thermal conductivity as a spatially varying function
Numerical examples of the present application are provided in regular and
irregular geometries. BEM solutions are found to be in agreement with
analytical solutions. Convergence of the iterative scheme is very fast satisfying
convergence criteria in a small number of updates.
1 Introduction
Non-linear heat conduction has been the focus of much research in the
BEM community. Integral transformations of the dependent variable to
linearize the governing equation, such as the Kirchoff transform, have been
the traditional way to approach this type of problem. However, when the
thermal conductivity additionally depends on space, no mathematical
transformation has been found to generally linearize the problem. On the
other hand, purely space dependent thermal conductivity problems have
Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X
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been addressed by a wide variety of techniques, including Dual Reciprocity
BEM, perturbation methods, and new modified Green's function for
linearly and one-dimensional varying thermal conductivity. The
combination of the temperature dependence with the explicit space
dependence of the thermal conductivity poses a great challenge to BEM
Recently, the authors proposed a general formulation for steady-state and
transient heat conduction in media whose thermal conductivity varies
arbitrarily with respect to space, and further more, extended this technique
to directional dependent conductivity. This particular formulation relies on
the aid of a new non-symmetric forcing function, D, tailored with the
thermal conductivity variation, which is used in place of the Dirac Delta
function to perturb the adjoint equation. This allows the derivation of
locally radially symmetric fundamental solutions, E, which replace Green's
free space solutions in the resulting BIE. The advantage of this formulation
is that permits general multi-dimensional variations of the thermal
conductivity with space while keeping the boundary-only feature of the
BEM technique.
An iterative approach is developed to address the temperature dependency
of the thermal conductivity by transforming it to a lagged space
dependency. At each step the generalized fundamental solution is found for
the space variation at hand and introduced in the generalized BIE. Iteration
is repeated until convergence. An efficient scheme is developed to speed up
computation. The [G] and [H] influence coefficient matrices are expressed
as a linear combination of sub-matrices [G]*> and [#]**, as the p^ term in a
Taylor series expansion of the generalized fundamental solution and its
normal derivative. The iterative scheme is reduced to a simple update of the
coefficients for the linear combination of the influence coefficient matrices.
2 Generalized BEM
for Anisotropic
Heterogeneous Media
The steady-state heat conduction governing equation in anisotropic
heterogeneous media is[l],
]=0
(1)
Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X
Boundary Elements
661
where ^ (x) is the spatially varying thermal conductivity symmetric tensor,
and T(x) is the temperature. This variable coefficient partial differential
equation is converted to an integral equation by introducing a function
E(x, £) referred to as a generalized fundamental solution [2-4], Integrating
the product of the governing equation and the function E(x, £) over the
domain Q and using Green'sfirstidentity twice, the differential equation is
converted to a boundary integral equation with a domain term containing
the adjoint equation for the fundamental solution E(x,£). This adjoint
equation is perturbed by a singular non-symmetric forcing function called
D(x, £) defined by its operational properties in Divo and Kassab[4] and
shown to be comprised of a Dirac Delta function and a dipole-like nonsymmetric singular function as D(x, $ = D,(x, f) + 8(x, £). The integral
equation is then expressed in terms of boundary only integrals as,
n
(2)
- T(x) [ J (x)-VE(x, OJ • n }dT(x)
The new terms that arise from this derivation are the amplification factor
4(£) and the sifting deviation e(£), which are both evaluated in terms of
boundary information through derivations detailed in [4]
The 2-D fundamental solution is defined in terms of a local polar
coordinate system centered about the source point £ = (x,,y,) which is
defined by the inverse transformation from local polar to global Cartesian
coordinates as, x = x* + rcosQ, and y = y, + rsinO. Seeking a locally
symmetric solution to the adjoint equation, that is E = £(r,o:.-,y,.), and
using the properties of the non-symmetric forcing function D, the
generalized fundamental solution is,
dO
It is noted that the variable coefficients k^, k^ and *„„, of the thermal
conductivity tensor, h , need to be transformed into the shifted polar
coordinates system to express them in terms of (r,0,z,-,y,-), before
carrying out the integration process. Following a similar approach and
using a local spherical coordinate system with backward transformation as
x = Xi + r sin 9cos<p, y = y^ + r sin 6 sin 0, and z = z, + r cosd, the
three-dimensional fundamental solution can be derived as,
Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X
662
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dr
527%
(4)
The krr component of the conductivity tensor in terms of the Cartesian
components is,
, 6, 0, KIJ Hi, Z{) — sir? 9 cos^(f> k^x 4- 2 sir?Q coscj) sin<f) k^y +
2 sin9 cos(f) cosBk^z + sir?6 sir?^> kyy 42 sinO sin<f) cosG kyz + cos^G k^z
Standard BEM techniques are then followed to solve the problem once the
fundamental solution is explicitly known. It is important to point out that
not all thermal conductivity spatial dependencies are integrable under the
transformed definite integral present on the fundamental solution form.
However, any multi-dimensional polynomial variation of k is generally
integrable. This expansion allows any thermal conductivity variation to be
fitted using polynomials, leading to an explicit expression for E(x, £).
3 Non-Linear Formulation and Iterative
Approach
The non-linear heat conduction governing equation can be written as [1],
V.[fcCT)VT(z)]=0
(6)
This equation may be linearized by implementing the well-known Kirchoff
transform of the dependent variable T as U(T) — WTM/T;^(^)^^- The
resulting Laplace equation in the variable U can be easily solved by
standard BEM
techniques. This transformation is not as useful in
simplifying the governing equation when the thermal conductivity also
varies spatially, e.g. k(T,x). However, in the limited case where the
conductivity is limited to the form fc(T, x) = kr(T}ks(x], then the
Kirchoff transform leads to a linear but heterogeneous governing equation,
V.f ks(x}VU(x}] = 0
(7)
The generalized boundary integral formulation for heat conduction in
heterogeneous media detailed in the previous section can be directly
implemented to solve the above equation. However, requiring that
(5)
Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X
Boundary Elements
663
k(T,x] = kr(T)ks(x) places the major restriction that the temperature
variation of thermal conductivity at any location in space is the same.
A more general approach to the non-linear heterogeneous problem
including anisotropy consists in an iterative scheme that transforms the
temperature dependence on the thermal conductivity into an explicit spatial
dependence provided an initial guess of the temperature field an its
correspondents updates. Therefore, the thermal conductivity tensor function
h (T,x) = £ *(x), where p corresponds to each iteration step, reducing
the governing equation to,
V.[^(z).VT(z)] =0
(8)
which can be solved using the method described in the previous section. An
efficient way of transforming the temperature dependence of the thermal
conductivity into an explicit space dependence for every iteration step is a
least-square fit, which in addition offers the opportunity to obtain a direct
polynomial approximation of the variation of the thermal conductivity
allowing for a closed form expression of the fundamental solution E?(x, £)
at every iteration step p. If we, for example, choose to expand the values of
the vv component of the thermal conductivity tensor in a two-dimensional
bi-quadratic polynomial fit with the form,
Wz,
2/) = Q_ + C2_z + C3_?y + c^z^ 4- c.^zi/ 4- c^z/
(9)
the fundamental solution has the form,
and the gradient of the fundamental solution,
X—
where the terms,
+ W^,, W
+ 2c. + 3
(12)
4-
with similar expressions in three dimensions. The dependence of the
fundamental solution and its gradient upon the coefficients of the thermal
conductivity is explicit but non-linear, therefore, the influence coefficient
Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X
664
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matrices must be evaluated for every iteration step. This process is
computationally intensive and requires a collocation process and its
correspondent integration algorithm over all boundary elements for every
iteration step. In order to relieve the computational burden, Taylor series
expansions of the fundamental solution and its gradient are used to
transform the explicit dependence upon the thermal conductivity
coefficients into a linear dependence as,
, ,
and the gradient,
allowing to generate sub-sets of the influence coefficient matrices which
relate together through the thermal conductivity coefficients to form the
total influence coefficient matrices as,
7r/q L %+ n2
and,
i
(16)
*
n=l
where, in the case of constant elements and neglecting the constant
(17)
The terms h^ are the integrals of the Taylor expansion function n over the
element j at the collocation point i for the gradient component k. The
thermal conductivity tensor J^ has been assumed to be constant over each
boundary element. The advantage of this approach is that the sub-sets of
influence coefficient matrices are computed just once due to their
geometry-only dependence and is just a matter of updating the thermal
conductivity coefficients to form the total matrices [G] and [H] for every
iteration step. This process speeds up the solution because the integration
routines, which are computationally expensive, are performed just once.
Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X
Boundary Elements
665
4 Numerical Implementation
The numerical solution of the boundary integral equation, Eq. (2), follows
standard BEM, Brebbia etal.[5]. The domain boundary, T, is discretized
using TV - boundary nodes, and boundary elements are used to model the
temperature and its flux on the boundary. Equation (2) is discretized as,
N
j= l
where p is the current iteration step. Applying Eq. (18) at all surface nodes
leads to a matrix set of equations in the standard BEM
form,
[H]P{T}P+* = [G]*{q}r+\ where the modified matrix [H]P includes the
effect of the sifting deviation e(&), see Divo and Kassab[4]. The above
discretized boundary integral equation is then used at every iteration step p
to update the temperature field {T}^ until a convergence criteria, 6, is
satisfied as,
2=1
<e
(19)
The convergence value selected for the numerical implementation of the
method is e = 10~*. It will be shown in the example section that the above
convergence criteria is satisfied in a few iteration steps, providing a fast
and accurate temperature field solution for the non-linear heat conduction
problem in anisotropic heterogeneous media.
5 Examples
A 1x1 square region is used as the geometry for thefirstexample in which
a non-linear thermal conductivity is imposed having the following form,
(20)
Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X
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666
The top and bottom surfaces of the square are imposed with adiabatic
conditions while the left surface is maintained at zero and the right surface
is held at one degree. The exact solution for this problem is
T(x) = 2* — 1. Sixteen equally spaced constant elements are used to
discretize the boundary of the problem, and six terms are used for the
Taylor expansion of the fundamental solution. In all cases, the initial guess
is taken as the mean value of imposed first kind conditions or the mean
value of the ambient temperature in the case of imposed convective
condition. The convergence criteria is satisfied after only three updates.
The BEM solution is found to be in very close agreement with the exact
solution giving a maximum relative percent error of 0.41% for the
temperature field and 1.66% for the normal heat flux. Figure 1 is a plot the
one dimensional temperature distribution for 0 < x < 1 showing high
accuracy of the BEM approximation.
A
%
J,
Figure 1. Exact and BEM computed temperature (Max error = 0.41%).
The same geometry is used now for the second example refining the
discretization to forty equally spaced constant boundary elements. The
temperature is held at zero degrees at the left, right, and bottom surfaces
and a unit heat flux is imposed into the body at the top surface. The nonlinear thermal conductivity distribution used in this case is taken as,
k(T) = T + 1
A series expansion solution for the two-dimensional
distribution can be found to have the following form,
(21)
temperature
-1
n=l
(22)
Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X
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667
The BEM computed isotherms are plotted in figure 2 along with the
relative deviation with respect to the exact solution. The maximum relative
deviation is 1.72%. The iterative solution converges in four steps.
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
Figure 2. BEM computed isotherms and relative % error (Max = 1.72%).
For the third example, a two-dimensional airfoil is discretized with 115
constant boundary elements. A non-linear, space dependent, orthotropic
thermal conductivity is considered, with the components having the form,
10
(25-
10
(20
(23)
It can be shown that with the above orthotropic components of the thermal
conductivity the following temperature distribution
339
(300 + 25z + 20y -
5xy -,,21
- y'
(24)
satisfies the governing equation. The above is used to imposed first kind
boundary conditions along the pressure side of the airfoil and second kind
along the suction side. Figure 3 shows the exact and BEM computed
isotherms and the relative percent error where a maximum of 1.8% is found
after only four iterations.
6 Concluding Remarks
The method recently proposed by the authors to generate a BIE for heat
conduction in media with spatially dependent properties is applied in an
iterative fashion to solve heat conduction in media whose conductivity also
varies with temperature. The spatial variation of k is lagged and
Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X
Boundary Elements
668
approximated with a polynomial least-squares fit at each step. Numerical
examples reveal the iteration converges quickly requiring few updates.
4.5-1
4.5-q
4-
353-
252-
1.51-
0.5-
0 0.5
I ' 'T >I ' ' < 'I>
0 0.5 1
15
2
0- I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I
0 0.5 1 1.5 2
Figure 3. Exact and BEM computed isotherms and relative % error
(Max =1.8%)
7 References
[1] Ozisik, N.M., Heat Conduction, John Wiley and Sons, New York,
1993.
[2] Kassab, A J and Divo, E., 'A General Boundary Integral Equation for
Isotropic Heat Conduction Problems in Bodies with Space Dependent
Properties,' Engineering Analysis, Vol. 18, No. 4, pp. 273-286, 1996.
[3] Divo, E. and Kassab, A.J., 'A Boundary Integral Equation for Steady
Heat Conduction in Anisotropic and Heterogeneous Media,' Numerical
#caf 7)w%^r, Vol. 32, No. 1, pp. 37-61, 1997.
[4] Divo, E. and Kassab, A J , 'A New BEM for Heat Conduction in
Media With Spatially Varying Thermal Conductivity,' Chapter in
Advances in Boundary Elements: Numerical and Mathematical
Aspects, Golberg, M.A. (ed.), Computational Mechanics, Boston,
1998 (in press).
[5] Brebbia, CA, Telles, J.C.F., and Wrobel, L, Boundary Element
Techniques in Engineering: theory and application in engineering,
Springer-Verlag, New York, 1985