University of Huddersfield Repository

University of Huddersfield Repository
University of Huddersfield Repository
Ge, Q., Yap, Y.F., Zhang, M. and Chai, John
Modeling anisotropic diffusion using a departure from isotropy approach
Original Citation
Ge, Q., Yap, Y.F., Zhang, M. and Chai, John (2013) Modeling anisotropic diffusion using a
departure from isotropy approach. Computers & Fluids, 86. pp. 298-309. ISSN 00457930
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Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.
Modeling Anisotropic Diffusion Using A Departure From Isotropy
Approach
Q. Ge1, Y.F. Yap1*, M. Zhang2 and J.C. Chai1
1
2
Depart. of Mechanical Eng., Petroleum Institute, Abu Dhabi, UAE.
School of Energy & Power Eng., Nanjing University of Science & Technology, Nanjing, P.R.
China.
*Corresponding author:
Tel: +971 2 607 5175
Fax: +971 2 607 5200
Email address: [email protected]
Address:
The Petroleum Institute
Department of Mechanical Engineering
P.O. Box 2533, Abu Dhabi, UAE.
Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.
Abstract
There are a large number of finite volume solvers available for solution of isotropic diffusion
equation. This article presents an approach of adapting these solvers to solve anisotropic diffusion
equations. The formulation works by decomposing the diffusive flux into a component associated
with isotropic diffusion and another component associated with departure from isotropic diffusion.
This results in an isotropic diffusion equation with additional terms to account for the anisotropic
effect. These additional terms are treated using a deferred correction approach and coupled via an
iterative procedure. The presented approach is validated against various diffusion problems in
anisotropic media with known analytical or numerical solutions. Although demonstrated for twodimensional problems, extension of the present approach to three-dimensional problems is straight
forward. Other than the finite volume method, this approach can be applied to any discretization
method.
Keywords: Anisotropic diffusion, finite volume method
1 Introduction
Isotropic diffusion equation governs a very wide range of physical processes occurring in isotropic
media, including heat, mass and momentum transfers. Most media encountered in physical and
engineering applications are however anisotropic in nature. For these media, the directional
dependence of their diffusion coefficients must be accounted for. This equation needs to be further
generalized by introducing the generalized Fick’s law [1-2] with an anisotropic diffusion coefficient,
and thus forming the anisotropic diffusion equation with additional mixed derivative terms. These
mixed derivative terms characterize the more complicated interactions in the physical process
originated from the anisotropy of the media under investigation. Isotropic diffusion equation is
therefore a very special limiting case of an anisotropic diffusion equation.
Anisotropic diffusion equation arises in very diverse physical processes. Diffusion of water vapours,
organic vapours and gases in soil, diffusion of nutrients away from fertilizer granules towards plant
roots in soil and diffusion of contaminants within subsurface geological formations are examples of
solutal diffusion transport in porous media [3-5]. The structure of these naturally occurring porous
media is highly irregular in terms of the pore distribution with respect to both size and shape. Given
the anisotropy (and the heterogeneity) of the media, such diffusion processes can be appropriately
modelled with an anisotropic diffusion equation. Besides, the generalized Darcy’s law coupled with
the continuity equation for modeling fluid flow in anisotropic heterogeneous porous media, e.g.
subsurface geological formations, gives rise to a similar anisotropic diffusion equation in terms of
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Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.
the fluid pressure [6-7]. Heat transfer in structural materials, e.g. wood and laminated metal sheets,
and crystals is another flourishing field where anisotropic diffusion equation is generally applied [810]. Interestingly, in the recent years, anisotropic diffusion equation finds its application in the field
of imaging, e.g. diffusion-tensor magnetic resonance imaging [11] and more generally PDE-based
anisotropic diffusion filters [12-14].
From a historical point of view, solutions of isotropic diffusion equation were attempted much
earlier than that of anisotropic diffusion equation. Isotropic diffusion equation has a lucidly simpler
mathematical structure and therefore is more amenable to both analytical and numerical approaches.
Some of these developed numerical approaches, e.g. based on the finite difference (FD), finite
volume (FV), finite element (FE), boundary element (BE) methods and fast Poisson solver [15-19],
are now well established and implemented routinely as standard solvers, at least for simple
geometrical configurations. For more complicated geometrical configurations, numerical solution
implemented on unstructured mesh is still being actively pursued for example in the recent work of
[20]. Driven by the pressing needs of the above mentioned practical applications involving
anisotropic media, these methods are then generalized to anisotropic diffusion equation. Such
generalizations require careful consideration of the discretization procedure that gives proper
discretization of the diffusion terms.
The applications of FD method for various anisotropic diffusion problems were made in [7, 21-22].
In [21], the coordinate system is realigned with the principal direction of the anisotropic diffusion
coefficient so that the cross derivative terms vanish. This approach is however difficult to be
generalized for heterogeneous media where the principal direction changes spatially. Of particular
interest is the improvement made by the introduction of mimetic approach [23-24]. Mimetic
approach incorporates the essential property of conservation during the discretization procedure and
gives locally conservative discretization equations. The FV method was also extended and
employed in the works of [25-27]. For FV method, an accurate approximation of the flux at the
control volume face remains one of the challenges. Flux-continuity across the control volume faces
has been given extra attention to produce locally conservative schemes [27-29]. Matrix- [30] and
flux-splitting [31] approaches were formulated for the FV method on structured and unstructured
mesh. These approaches employ a deferred correction approach so that the coefficient matrix and
the flux vector retain similar forms as those resulted from simple diffusion problems. The FE
method was employed in [32-33] with proper modifications in the treatment of the additional mixed
derivative terms. One notable effort that increase the method’s accuracy is incorporation of the
adaptive mesh approach into the framework of a FE method where the underlying mesh adapts
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Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.
dynamically during the solution process was developed [34]. This adaptive mesh approach although
costly gives excellent results with much lesser numerical smearing for diffusion in highly
anisotropic media. Extension and applications of the BE method in various problems involving
conduction heat transfer, fluid flow in porous media and structural problem of an elliptical bar
under torsion have been demonstrated in [35-36]. It should be mentioned that some of these
extensions require intricate discretization procedure and therefore not straight forward to implement
numerically.
Here in this article, an alternative approach that adapts the existing solvers for isotropic diffusion
equation to solve anisotropic diffusion equation is presented. In this approach, the diffusive flux is
decomposed into a component associated with isotropic diffusion and another component
associated with departure from isotropic diffusion. This decomposition transforms an anisotropic
diffusion equation into the form of an isotropic diffusion equation with additional terms to account
for the anisotropic effect. These additional terms are treated using a deferred correction approach
and coupled via an iterative procedure. The advantage of the decomposition approach proposed here
is that it allows existing solvers for isotropic problems to be extended easily to anisotropic diffusion
problems (at least for orthogonal coordinate systems). The main contribution of this proposed
approach is the simplicity it offers in the implementation of such an extension. It just requires an
additional subroutine be written to evaluate the departure from isotropic term and called from the
original solver. No other modification on the original code of the solver is required.
It should be noted that different deferred correction approaches have been proposed for the solution
of anisotropic diffusion problems. In the flux-splitting approach at the flux level [30-31], the flux is
split into the form of a leading two-point flux and additional cross-diffusion terms. The leading twopoint flux term is approximated implicitly but the remainder flux term is treated explicitly and
coupled iteratively. With this, the standard five-point (seven-point) stencil is preserved for twodimensional (three-dimensional) problems. For flux-splitting at the matrix level [30-31], the
coefficient matrix in the system of linear equations is effectively decomposed into a penta-diagonal
matrix and a residual matrix. The penta-diagonal matrix is in the similar form that would be
obtained by discretizing a simple diffusion equation. In the work of [25], only the cross diffusion
terms are approximated explicitly via a deferred correction approach and coupled iteratively.
Adaptation of these approaches into existing solvers for isotropic diffusion problems requires more
modifications on the original code for the case where the diagonal components of the diffusion
coefficient are different.
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Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.
The remaining of the article is separated into five sections. A description of the problem is given in
Section 2. Section 3 is the core of the article where the reformulation of the anisotropic diffusion
equation is presented. The numerical solution procedure, including discretization, implementation
of boundary conditions and convergence criterion, is discussed in Section 4. Validations of the
present approach against seven different problems are given in Section 5. Finally, the article
concludes with a few remarks in Section 6.
2 Problem Description
The domain of interest  , shown in Fig. 1, consists of an anisotropic medium. The steady-state

diffusion transport of a quantity  ( x) within  is governed by the conservation equation
 


(1)
   q ( x)  S( x)  0 , x  
 

where q ( x) and S( x) are the diffusion flux and the volumetric source/sink respectively. When
 
q ( x) is described by the generalized Fick’s law, the anisotropic diffusion flux is as
 


q ( x )    ( x )   ( x )
(2)

where ( x ) is the anisotropic diffusion coefficient, a second order tensor characterizing the

anisotropy of the medium. Constrained by the Second Law of Thermodynamics, ( x ) is positive
definite [42]. Substitution of Eq. (2) into Eq. (1) gives a second order elliptic anisotropic diffusion


equation of the form




  ( x )   ( x )  S ( x )  0 , x  
(3)
Equation (3) is subjected to the following mixed boundary condition, i.e. a combination of a


prescribed  P ( x ) and a prescribed flux qnP ( x ) .
 ( x)   P ( x), x  



   


q ( x )  n ( x )  qnP ( x ), x   q
(4a)
(4b)
where

  q   ,   q   and   . n is the unit normal vector pointing out of  .
Dirichlet boundary condition is the limiting case of Eq. (4) when  q  . This article focuses on
adapting solvers for isotropic diffusion equation to solve anisotropic diffusion equation (Eq. 3)
subjected to the boundary condition of Eq. (4).
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Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.


n̂
Figure 1: Domain of interest for an anisotropic diffusion problem.
3 Mathematical Formulation
Equation (3) can be reformulated into an isotropic diffusion equation. For this purpose, the
anisotropic diffusion coefficient is decomposed as



( x )  max ( x ) I  D ( x )
(5)
where




max ( x)  max xx ( x ), yy ( x), zz ( x )




 xx ( x )  max ( x )

xy ( x )
xz ( x )
 





D ( x )  
yx ( x )
yy ( x )  max ( x )
yz ( x )



 

zx ( x )
zy ( x )
zz ( x )  max ( x )

(6a)
(6b)
The operator max a , b, c returns the largest of a , b and c . It was suggested by one of the
reviewers of this article that for a symmetric  , max can also be set to the maximum of the
 
eigenvalue so that the decomposition would be frame-independent. With this, q ( x) can be written
as
  
  
q ( x)  qmax ( x)  qD ( x)
where




qmax ( x)  max ( x) ( x)
 


q D ( x )   D ( x )   ( x )
Upon substitution of Eqs. (7) and (8) into to Eq. (1), Eq. (1) can now be expressed as





  max ( x) ( x)  SD ( x)  S( x)  0 , x  
(7)
(8a)
(8b)
(9)
where




S D ( x)     D ( x)   ( x)  , x  


(10)
Equation (9) is the familiar isotropic diffusion equation. The first term on the left-side (LS) is
referred to as the isotropic term as it captures diffusion in an isotropic medium of diffusion
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Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.


coefficient max ( x) . The additional source term SD ( x) accounts for the effects due to the

anisotropy of the medium. SD ( x) is therefore referred as departure from isotropic term. The

treatment of SD ( x) will be further discussed in the next section. Essentially, the anisotropic
diffusion equation (Eq. 3) is now in the form of an isotropic diffusion equation (Eq. 9). By

incorporating the additional source term SD ( x) , a standard solver for isotropic diffusion equation
can now be employed to solve Eq. (9).
4 Solution Procedure
4.1 Discretization
The finite volume method [17, 37] is employed to numerically solve Eq. (9) in two dimensions on a
Cartesian mesh. Extension of the presented approach to three dimensions is straight forward. The
physical domain is first partitioned into a number of non-overlapping control volumes (CVs) as
shown in Fig. 2. A node is located at the centre of every CV. There are two types of CVs, i.e.
internal CVs (e.g. CV P) and boundary CVs (e.g. CV PB). For the internal CV P, the neighbouring
nodes are labeled as W, E, N and S. It has four boundaries, denoted by e, w, n and s (with area of
Ae , Aw , An and As respectively). The four corners of the CV P are denoted ne, nw, se and sw. The
volume of the CV P is denoted as VP .  , max and all the components of  are stored at the
nodes. For ease of reference and without the loss of generality, the dependence of the variables on

x is implied implicitly and will therefore be dropped henceforth. Integration of Eq. (9) over the CV
P gives
VP   max  dV  VP SD dV  VP SdV  0
(11)
Employing the Gauss’ divergence theorem, the volume integrations can be converted into surface
integrations as
AP
max    dA  A


      dA 
D
VP SdV  0

P 
(12)
The conversion of the 2nd term on the LS of Eq. (11) into a surface integration is important to
ensure flux consistency in the discrezation equation. The terms in Eq. (12) can be approximated
AP
max    dA
numerically as



 
 
 
 








  max
A
A
A
As
e
max
w
max
n
max




y  n
x  w
y  s
x  e



m
m
m
m
(13a)
7
A D    dA
Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.

m1

 

 xx  max  
  xy
x  e
y

P


m1

 

 yy  max
 yx
y n
x

VP SdV  Save VP
m1

 

Ae   xy
 xx  max  
x  w
y

m1


Aw
m1


 
An  yx
 yy  max
x
y  s

As
(13b)
(13c)
The superscript m refers to quantities of the m iteration, i.e. the current iteration. Note that Eq.
(13b) is calculated using the existing values of  from the known m  1 iteration, i.e. previous
iteration. By doing so, it becomes a ‘known’ source term. With this treatment, the discretization
equation for Eq. (9) becomes that of an isotropic diffusion equation with ‘known’ source terms
accounting for the anisotropy of the medium. A standard solver for isotropic diffusion equation can
therefore be modified easily to solve Eq. (9). In Eqs. (13a) and (13b),




,
,
and
x e x w y n
y s
are approximated with central differences, assuming a linear variation of  between two adjacent
CVs. max and all the components of D at the interface e, w, n and s are calculated via a
harmonic mean for its superiority of maintaining the normal flux continuity. With these
approximations, Eq. (12) becomes
m
m
m1
m1
a P Pm  a E Em  a W W
 a N N
 a SSm  Save
VP  Sdep
where
aE 
aW 
aN 
aS 
max, e Ae
(14)
xPE
(15a)
xPW
(15b)
yPN
(15c)
yPS
(15d)
max, w Aw
max, n An
max, s As
a P  a E  aW  a N  a S
(15e)
8
Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.

m1
  xy,e Ae ne
Sdep
m1
 se
 m1  Pm1
 xx  max e Ae E
y
xPE
m1
m1
m1
 m1  W
 m1  sw
 xy, w Aw nw
 xx  max w Aw P
xPE
y
m1
m1
 Nm1  Pm1
ne
 nw
 yx,n An
 yy  max  An
n
x
yPN
(15f)
m1
m1
se
Pm1  Sm1
 sw
 yx, s As
 yy  max  As
s
x
yPS
In the evaluation of Eq. (15f), the values of ne , nw , se , sw are interpolated linearly from  of
the neighbouring nodes.
x
n̂
N
P
An
N
n
PB
B
W
nw
E
qnP
S
P
w
W
ne
e
Aw
sw
E
y
Ae
s
As
S
se
VP
Figure 2: Partition of the domain of interest into non-overlapping control volumes.
4.2 Implementation of Boundary Conditions
The implementation of both a prescribed value  P and a prescribed flux qnP at the boundary are
discussed here. For a prescribed  P , the imposition of Eq. (4a) is straight forward.  at the
boundary nodes, i.e. the hollow nodes in Fig. 2, is set to the prescribed  P . Of particular important
is the imposition of a prescribed flux qnP , i.e. Eq. (4b). To implement this condition, the prescribed
diffusion flux out of the domain at the boundary qnP is brought into the adjacent boundary CVs as a
volumetric sink and the diffusion flux at the corresponding boundary is set to zero. Note that for
flux of  diffuses out of the domain, qnP  0 . For example, an additional sink term given by
qnP AB
Sextra  
VPB
(16)
is added into the boundary CV PB shown in Fig. 2. With this sink term added to account for the
amount of  diffuses out of the domain, the flux at the boundary can now be set zero, i.e.
9
Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.
 
 
q ( xB )  n ( xB )  0
(17)
This can be achieved numerically by setting the associated diffusion coefficients in the
discretization equation (Eqs. 13a and 13b) to zero, i.e.
xx e  xy  max e  0
e
(18)
4.3 Solution Algorithm
The overall solution procedure can be summarized as follows:
(1) Specify max and D in Eqs. (6).

(2) Calculate the volumetric source/sink S( x) in Eq. (9).
(3) Calculate SD from Eq. (10).
(4) Solve Eq. (9) for  .
(5) Repeat steps (2) to (4) until the solution converges.
4.4 Convergence Criterion and Error Calculations
Application of the discretization equation (Eq. 14) to every CV gives a system of linear equation
with penta-diagonal coefficient matrix. This system of linear equation is solved iteratively using the
Thomas algorithm sweeping alternatively in the x  and y  direction. No special initial guess is
employed, i.e. i , j  0 . For convergence, the relative changes in  between to successive iterations
is monitored. The solution is assumed converged when e   10 8 where
1 m
e   max 1  im
, j / i , j
i 1,.., M
(19)
j 1,..., N
For the cases considered in this article, a uniform mesh is employed. Mesh refinement is performed
with a refinement ratio of h 
x
where x0 is the coarsest mesh. Error of the present solution
x0
will be reported in terms of maximum and root mean square norms discretely defined respectively
as
 inum
E   max iexact
,j
,j
i 1,.., M
(20)
j 1,..., N
E 2 2

M ,N
i 1, j 1
 inum
Vi , j
iexact
,j
,j
2
(21)
10
Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.
The order of accuracy is then estimated from the slope of the graph log E 2 vs. log( h ) fitted using
the least square method.
5 Results and Discussions
The present approach is validated against known solutions for diffusion in various media subjected
to either the limiting case of Dirichlet boundary condition or the more general mixed boundary
condition. The first three validation exercises concern diffusion in heterogeneous anisotropic media
with various forms of diffusion coefficients. The exact solutions for these cases were setup using
the method of manufactured solution [38]. Two additional cases of diffusion in anisotropic media
with discontinuous diffusion coefficient are then considered. This is followed finally by two
applications of the presented approach for heat conduction in anisotropic medium with internal heat
generation and flow in multi-layered anisotropic porous medium.
5.1 Diffusion in a heterogeneous anisotropic medium with a linear variable diffusion
coefficient
The validation exercise starts with a medium with a simple linear diffusion coefficient. The domain
of interest is a unit square of   0,12 . The diffusion coefficient varies linearly in the domain as
1  x  y 0.5  y 


 0.5  y 1  2 x  y
(22)
Within the domain of interest, there is a non-zero variable source term given by
S  3  x  y e x  cos y  1  2 x  y sin y
(23)
With these, it can be verified easily that the exact solution is given by
  e x  sin y
(24)
Two tests of which the enforced boundary conditions are different will be considered. In the first
test, the domain is subjected to the following Dirichlet boundary condition.
 P  x,0  e x , 0  x  1
 P  x,1  e x  sin 1, 0  x  1
 P 0, y   1  sin y, 0  y  1
 P 1, y   e  sin y, 0  y  1
(25a)
(25b)
(25c)
(25d)
The second test, proposed in [39], subjects the domain to a mixed boundary condition of Eq. (26).
While a prescribed  P is applied at the lower and upper boundaries, a prescribed flux qnP is
enforced at both the left and right boundaries of the domain.
11
 P  x,0  e x , 0  x  1
Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.
(26a)
 P  x,1  e x  sin 1, 0  x  1
(26b)
qnP 0, y   1  y  0.5  y cos y, 0  y  1
(26c)
qnP 1, y   2  y e  0.5  y cos y, 0  y  1
(26d)
Given the smooth nature of the solution, it is expected that most, if not all, of the essential features
of the solutions can be well captured by a relatively coarse mesh. Mesh independent solutions were
indeed obtained on the coarsest mesh of 25 25 CVs using the present approach for both boundary
conditions. These solutions are presented together with the exact solution in Fig. 3. The present
solutions agree well with the exact solution. The errors are plotted in Fig. 4. Both E  and E 2
decreases as the mesh is refined. Estimated from the slope of the graphs, the order of accuracy for
the case of Dirichlet and mixed boundary conditions are respectively 1.7442 and 2.0816.
(a)
(b)
Figure 3: Solutions for diffusion in a medium with a linear variable diffusion coefficient subjected
to (a) Dirichlet boundary condition and (b) mixed boundary condition.
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Figure 4: Errors for the case of diffusion in a medium with a linear variable diffusion coefficient.
5.2 Diffusion in a heterogeneous anisotropic medium with a non-linear variable diffusion
coefficient
This exercise demonstrates the capability of the present approach in handling a more general form
of the diffusion coefficient. Diffusion in again a unit square is considered for a non-linear variable
diffusion coefficient of the form
 x 2  y 2  1   xy

2
2 
 1   xy x  y 
(27)
where  characterizes the degree of anisotropy in the medium [40]. For   1 , the diffusion
coefficient reduces to that of an isotropic medium. There is a non-zero variable source term in 
given by
S  15  7 x 2 y  8  4 x 2  24  4  y 2  2 y 3
(28)
This problem has an exact solution in the form of
  1  x2 y  x2  3 y2
(29)
Note that the exact solution is independent of  . The following mixed boundary condition of Eq.
(27) is imposed on the boundary of the domain.
qnP  x,0  x 4 , 0  x  1
 P  x,1  2 x 2  3, 0  x  1
qnP 0, y   0, 0  y  1


(30a)
(30b)


qnP 1, y      y 2 2 y  2   1    y  6 y 2 , 0  y  1
(30c)
(30d)
For the case of   0.5 , mesh independent solution obtained on a coarsest mesh of 25 25 CVs is
shown in Fig. 5a. For comparison purpose, the exact solution is superimposed. The present solution
agrees well with the exact solution. Although not shown here, similar agreement is achieved where
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the boundary is subjected to a Dirichlet boundary condition. The order of accuracy estimated from
Fig. 6 is 1.7567.
The degree of anisotropy in the medium increases when  is decreasing. For   1 , the medium is
highly anisotropic. Either a very fine mesh or a multi-scale approach for example in [32] is then
required to resolve the fine anisotropic features of the medium. For demonstration, the case of
  0.01 is now considered using the present approach. The solution on a mesh of 400 400 CVs,
much finer than the case of   0.5 , is plotted in Fig. 5b. Agreement with the superimposed exact
solution is good. The order of accuracy is estimated to be 1.6962 surprisingly close to than that for
the case of   0.5 .
(a)
(b)
Figure 5: Solutions for diffusion in a medium with a non-linear variable diffusion coefficient with
(a)   0.50 and (b)   0.01.
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Figure 6: Errors for the case of diffusion in a medium with a non-linear variable diffusion
coefficient.
5.3 Diffusion in a heterogeneous anisotropic medium with an asymmetric non-linear variable
diffusion coefficient
Asymmetric diffusion coefficient can occur physically, e.g. the electron heat conductivity tensor
used in plasma physics [41-42]. The ability to handle asymmetric diffusion tensors is therefore of
interest. This exercise concerns diffusion in a unit square with an asymmetric non-linear diffusion
coefficient of the form
1  x  y
 x2

1  2x 
 y


y
S  161  x  ycos 4 x  91  2 x  ysin 3 y  3cos 3 y  8 sin 4 x  6 x cos 3 y
(31)
The medium has a non-zero variable source term of the form
(32)
At the boundary of the domain, the following mixed boundary condition is enforced.
qnP  x,0  31  2 x , 0  x  1
qnP  x,1  61  x  cos 3  4 sin 4 x, 0  x  1
 P 0, y   1  sin 3 y, 0  y  1
 P 1, y   cos 4  sin 3 y, 0  y  1
(33a)
(33b)
(33c)
(33d)
The exact solution for this diffusion process is given by
  cos 4 x  sin 3 y
(34)
Figure 7 shows the mesh independent solution computed on a mesh of 50 50 CVs and the exact
solution. A finer mesh is required in this case to sufficiently resolve most features of the solution.
The present solution agrees reasonably well with the exact solution. Show in Fig. 8 is the plots of
E  and E 2 against h . This approach retains a comfortable order of accuracy of 1.4966 even in
the event of an asymmetric diffusion coefficient in this case. Figure 9 shows the reduction of e 
and E  during the iterative solution process for both the present and the original formulation on a
mesh of 50 50 CVs. The original formulation entails directly solving Eq. (3) via the same iterative
procedure as discussed in Section 4.4. Of course without introducing the deferred correction, the
original formulation converges within fewer number of iterations. This is generally true for other
cases considered here in this article. Therefore, having relatively more iteration to achieve
convergence is the price to pay of using the present formulation which allow easy adaption of
existing solvers for isotropic diffusion problem to anisotropic diffusion problem by adding a
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departure from isotropic term implemented in a deferred correction approach. An additional
computation is also made where the Dirichlet boundary condition is applied. Visually
indistinguishable solution is obtained and therefore is not shown here.
Figure 7: Solution for diffusion in a medium with an asymmetric non-linear variable diffusion
coefficient.
Figure 8: Errors for the case of diffusion in a medium with an asymmetric non-linear variable
diffusion coefficient.
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Figure 9: Reduction of e  and E  during the iterative solution process for both the present and
the original formulation.
5.4 Diffusion in an anisotropic medium with discontinuous diffusion coefficient
The present approach is now applied to an anisotropic medium in which the diffusion coefficient is
discontinuous across an irregular internal interface. Here, irregular interface refers to an interface
that cannot be represented exactly on a computational mesh, e.g. a Cartesian mesh in the current
problem. Figure 10 depicts a domain consisting of two different homogeneous sub-domains defined
respectively by 0  rx  sy   and   rx  sy  1 . The diffusion coefficient in each sub-domain is
constant. It is however discontinuous across the interface between the two sub-domains defined by
rx  sy   . The diffusion coefficient is given by
 1 1 / 10
,0  r x  sy  

1 
1 / 10

 1 1 / 10  ,   r x  sy  1
1 / 10
1 
(35)
where 0  r , s,   1 and r  s  1. The diffusion coefficient in the sub-domain of   rx  sy  1 is
 times larger than that of the sub-domain of 0  rx  sy   . Note that the directions of both the
major and minor principle diffusion coefficients are identical in the two sub-domains. The
anisotropy in each sub-domain is not high, e.g. the sub-domain of 0  rx  sy   has a principal
1.1 0 
diffusion coefficient of  *  
 . The case of enforcing given flux on the top and bottom
 0 0.9
boundaries and given  on the left and right boundaries is considered in [43-44]. Under such
  r x  sy 
,0  r x  sy  
1     1

r x  sy      1 ,   r x  sy  1

 1     1
boundary conditions, the exact solution is given by
(36)
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As for boundary conditions, a mixed boundary condition is enforced. Prescribed  P 0, y  and
 P 1, y  are enforced respectively at the left and right boundaries. These are evaluated from Eq.
(36). Prescribed flux of qnP  x,0  and qnP  x,1 are imposed at the lower and upper boundaries
respectively. qnP  x,0  and qnP  x,1 can be constructed from Eq. (36) using Eq. (4b). The case of
r  2 / 3 , s  1 / 3 ,   1 / 3 and   10 is considered here. Mesh independent solution, obtained on a
mesh of 50 50 CVs, is shown in Fig. 11 together with the exact solution. The present solution is in
good agreement with the exact solution. The effect of the discontinuity in the diffusion coefficient is
clearly evident in the solution with a much stepper gradient of  in the sub-domain of
0  rx  sy   where the diffusion coefficient is much smaller. Errors as the mesh is refined are
plotted in Fig. 12. The Cartesian mesh employed cannot represent the discontinuity of the diffusion
coefficient exactly. More accurate solution can only be obtained by refining the mesh. Therefore it
is not surprising that the order of accuracy decreases substantially to 1.1157.
0  rx  sy  
y
rx  sy   (interface)
  rx  sy  1
1
x
1
Figure 10: Schematic of an anisotropic medium with discontinuous diffusion coefficient.
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Figure 11: Solution for diffusion in a medium with discontinuous diffusion coefficient.
Figure 12: Errors for the case of diffusion in a medium with discontinuous diffusion coefficient.
5.5 Diffusion in an orthotropic medium with discontinuous diffusion coefficient
Figure 13 shows an orthotropic physical domain with a smaller computational domain 
embedded. The physical domain consists of two sub-domains labelled respectively as l (left,
x*  0.5 ) and r (right, x*  0.5 ). The diffusion coefficient is given by
 50

 0
 *  
 1
 0
0
, x*  0.5
1
0
, x*  0.5
10
(37)
There is a discontinuity in the diffusion coefficient across x*  0.5 . The major principle diffusion
coefficients in these sub-domains are oriented 90o to each other. Within the domain, there is no
source/sink term, i.e. S  0 . For such a system, the exact solution can be expressed as [44-45]
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
, x*  0.5
cl x *2  d l y *2
2
2

a r  br x *  c r x *  d r y * , x*  0.5
 x*, y *  
(38a)
where

*yy,l
4a r
,  * , f 
*
  2  1
yy,r
xx,l
*
xx
,r
a r  1 , br    1 f , cr  f , d r  c r
cl  c r , d l  d r
(38b)
*
xx
,r
*yy,r
(38c)
(38d)
Solution is sought for the computational domain of   0,12 oriented with an angle  from the
physical domain. With such an orientation, the discontinuity in the diffusion coefficient generally
does not align with the boundary of the Cartesian mesh in  . The coordinate system and the
diffusion coefficient in  are related to those of the physical domain via


x  Rx *


( x )  R T  *( x*)R
(39)
(40)
where the rotational matrix R is given by
cos
R
 sin 
 sin  
cos 
(41)
At the boundary of the computational domain, prescribed  P derived from Eq. (38) is enforced.
Figure 14 shows the present mesh independent solutions obtained on a mesh of 100100 CVs for
  0 and 30o. For the case of   0o , the discontinuity in the diffusion coefficient is captured
exactly by the boundary of the Cartesian CVs. This is no longer the case for   30o . The effect of
the discontinuity in the diffusion coefficient across the interface is captured reasonable well by the
present solution although a much finer mesh is required. These solutions are in good agreement
with the exact solution of Eq. (38). From Fig. 15, it can be estimated that the order of accuracy for
the case of   0o and   30o are respectively 1.1825 and 0.8989.
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l
y*
physical
domain
r
0.5
computational
domain
y

x

interface
x*
Figure 13: Schematic of an orthotropic medium with discontinuous diffusion coefficient.
(a)
(b)
Figure 14: Solutions for diffusion in an orthotropic medium with (a)   0o and (b)   30o .
Figure 15: Errors for the case of diffusion in an orthotropic medium with discontinuous diffusion
coefficient.
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5.6 Heat Conduction in an Anisotropic Medium
Heat conduction in the anisotropic medium with an internal heat source is considered. Figure 16
depicts the domain of interest. Within the hatched region, there is a volumetric heat source of
Q gen  10 5 cosh 0.84851 xc exp 0.20579 xc  yc 
(42a)
xc  x  0.2 / 0.8
(42b)
where
yc   y  0.4 / 0.8
(42c)
The domain is characterized by the principal conductivity coefficients of k *xx  6.5 and k *yy  11 .3 .
The principal axis of k*xx is orientated 60o to the x -axis. Based on these, the tensorial conductivity
coefficient in the x, y coordinate system can be calculated using Eq. (40). As for boundary
conditions, the temperatures at the bottom and top of the domain are fixed respectively at T  0 oC
and T  50 oC. The two side walls are insulated. Solutions were obtained on two different meshes,
i.e. 16  32 CVs and 32  64 CVs. The isotherms of T / 50 for these solutions are shown in Fig. 17.
Note that the plot is rotated 90o for a better presentation given large aspect ratio of the domain. A
mesh of 16  32 CVs is sufficient to resolve the essential features of the solution. As expected, the
highest temperature occurs at the center of the region with heat source. Due to the anisotropy of the
medium, the solution is not symmetric along x  0.2 and the isotherms are not perpendicular to the
physical domain. The temperature along the two insulated side walls at x  0 and x  0.4 can be
evaluated using Eq. (2) in a post-processing calculation. These are shown in Fig. 18. Comparison is
made against those of the boundary element method and the commercial software ANSYS of which
are both presented in [46]. The present solution agrees well with these solutions.
y
0.15
0.1
0.15
T=50oC
0.35
Qgen
k *yy
0.10
k *xx
60o
0.35
o
T=0 C
x
Figure 16: Schematic of an anisotropic medium with an internal heat source.
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Figure 17: Normalized isotherms for heat conduction in an isotropic medium.
Figure 18: Normalized temperature along the insulated side walls.
5.7 Flow in a multi-layered anisotropic porous medium
Fluid flow through a multi-layered anisotropic porous medium shown in Fig. 19 is considered. The

flow is governed by the Darcy’s law in the form of Eq. (2) where q ,  and  are respectively the
velocity vector, permeability tensor and pressure. Coupling this with the continuity equation, an
anisotropic diffusion equation of the form of Eq. (1) can be derived for the pressure. The multilayered porous medium consists of a total of three different homogeneous anisotropic layers
labelled as A, B and C. The principal permeability tensors for the layer A, B and C are respectively
* 6 0
o
A

 , A  60
0
1


(43a)
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4 0
B*  
, B  60o

 0 1
(43b)
2 0
C*  
,C  60o

0 1
(43c)
The domain is subjected to the following mixed boundary conditions:
 P  x,0  1, 0  x  1
(44a)
 P  x,2   0, 0  x  1
qnP 0, y   qnP 1, y   0, 0  y  1
(44b)
(44c)
Note that the right and left boundaries are impermeable. A unit pressure difference is applied across
the medium in the y -direction. The computed pressure is presented in the first plot of Fig. 20.
Comparison was made against that of Lorinczi et al. [39] where good agreement is attained. The
velocity field is shown in the second plot of Fig. 20. Generally, due to the anisotropic nature of the
porous medium, the flow direction is strongly dictated by the permeability tensor characterising the
medium.
y
layer C
3 x  y  0.75  0
3 x  y  0.25  0
2
layer B

layer A
x
1
Figure 19: Schematic of a multi-layered anisotropic porous medium.
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Figure 20: Pressure and velocity field for flow through a multi-layered anisotropic porous medium.
6 Concluding Remarks
This article presents an approach of adapting solvers for isotropic diffusion equations to solve
anisotropic diffusion equations. The formulation works by decomposing the diffusive flux into a
component associated with isotropic diffusion and another component associated with departure
from isotropic diffusion. This results in an isotropic diffusion equation with additional terms to
account for the anisotropic effect. These additional terms are treated using a deferred correction
approach and coupled via an iterative procedure. Validations of the present approach were
performed for diffusion in various anisotropic media. The approach is applied to investigate heat
conduction in anisotropic medium with internal heat generation and flow in a multi-layered
anisotropic porous medium. Although demonstrated for two-dimensional problems, extension of the
present approach to three-dimensional problems is straight forward. In this article, the approach is
demonstrated using the finite volume method. It can however be applied equally well for any
discretization method including the finite difference and finite element methods.
Acknowledgement
Fruitful discussions with I.G. Economou is gratefully acknowledged. This work is supported by the
Petroleum Institute under Grant: RAGS-11017.
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Nomenclature
A
surface area of a control volume
E2
root mean square norm error
E
maximum norm of error
h
refinement ratio
I
identity matrix
k

q

qD

q max
thermal conductivity
diffusive flux
departure from isotropic component of the diffusive flux
isotropic component of the diffusive flux
Q gen volumetric heat generation
R
rotational matrix
S
source term
SD
source term due to departure from isotropic
T

x
temperature
position vector
x
coordinate axis
y
coordinate axis
Greek letters
V
A
volume

degree of anisotropic

surface area
transported quantity

diffusion coefficient
D
departure from isotropic component of the diffusion coefficient
max isotropic component of the diffusion coefficient

 q


domain of interest
boundary with prescribed flux
boundary with prescribed 
orientation of the principal directions
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Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.
Superscript
m
current iteration
P
prescribed value
*
principal directions
Subscript
ave
average
B
boundary control volume
n
normal
P
control volume P
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Computers & Fluids, 2013, 86, pp. 298–309, ISSN 00457930.
List of Figures
Figure 1: Domain of interest for an anisotropic diffusion problem.
Figure 2: Partition of the domain of interest into non-overlapping control volumes.
Figure 3: Solutions for diffusion in a medium with a linear variable diffusion coefficient subjected
to (a) Dirichlet boundary condition and (b) mixed boundary condition.
Figure 4: Errors for the case of diffusion in a medium with a linear variable diffusion coefficient.
Figure 5: Solutions for diffusion in a medium with a non-linear variable diffusion coefficient with
(a)   0.50 and (b)   0.01.
Figure 6: Errors for the case of diffusion in a medium with a non-linear variable diffusion
coefficient.
Figure 7: Solution for diffusion in a medium with an asymmetric non-linear variable diffusion
coefficient.
Figure 8: Errors for the case of diffusion in a medium with an asymmetric non-linear variable
diffusion coefficient.
Figure 9: Reduction of e  and E  during the iterative solution process for both the present and
the original formulation.
Figure 10: Schematic of an anisotropic medium with discontinuous diffusion coefficient.
Figure 11: Solution for diffusion in a medium with discontinuous diffusion coefficient.
Figure 12: Errors for the case of diffusion in a medium with discontinuous diffusion coefficient.
Figure 13: Schematic of an orthotropic medium with discontinuous diffusion coefficient.
Figure 14: Solutions for diffusion in an orthotropic medium with (a)   0o and (b)   30o .
Figure 15: Errors for the case of diffusion in an orthotropic medium with discontinuous diffusion
coefficient.
Figure 16: Schematic of an anisotropic medium with an internal heat source.
Figure 17: Normalized isotherms for heat conduction in an isotropic medium.
Figure 18: Normalized temperature along the insulated side walls.
Figure 19: Schematic of a multi-layered anisotropic porous medium.
Figure 20: Pressure and velocity field for flow through a multi-layered anisotropic porous medium.
31
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