are looking for a second order accuracy, the coefficients
퐵푖of휅푖,푗expression must vanish in the same time with
the correction term퐶푖,푗. These constraints lead to following
system of equations with respect to variables푎푖:
푎 1 +푎 2 =0,
푎 2 훼퐾−+푎 3 +푎 4 휌+푎 6 훼퐾−휂+푎 8 푐 8 (2)
+푎 10 푐 10 (2)+푎 12 푐 12 (2)+푎 14 푐(2) 14 =퐾−휉,
푎 5 +푎 6 (1 − 훼휒耠耠퐾−)+푎 8 푐 8 (3)−2훼휒耠耠퐾휂−푎 10
+푎 14 푐 14 (3)+휒耠耠(1 − 휌 − 훼휒耠耠퐾−)푎 12 =퐾휂−,
푎 7 +(휌−훼휒耠耠퐾−)푎 8 +훼휒耠耠퐾−푎 10 +3훼(휒耠耠)
2
퐾−푎 14 =퐾−,푎 9 +(1−2훼휒耠耠퐾−)푎 10 +(휌−1+2훼휒耠耠퐾−)푎 8+3휒耠耠(1−휌−2훼휒耠耠퐾−)푎 14 =퐾−,
훼퐾−푎 6 −(2훼퐾−휂−훼휌퐾+휂)푎 8 +2훼퐾휂−푎 10 +푎 11
+(휌+훼휒耠耠퐾−)푎 12 +푎 14 푐 14 (6)=0,
−훼퐾−푎 8 +훼퐾−푎 10 +푎 13 +(휌+3훼휒耠耠퐾−)푎 14 =0,
(17)
where
푐 8 (2)=−(훼퐾휂−퐾휂++훼퐾+퐾−휂휂−퐾−휉+휌퐾+휉)/퐾+
−(1+훼퐾−휉−휌)휒耠耠+훼퐾−(휒耠耠)
2
,푐 10 (2)=휒耠耠(1 − 휌) + 훼퐾휂휂−+훼휒耠耠(퐾−휉−휒耠耠퐾−),
푐 12 (2)=
(퐾휂−−휌퐾휂+)
퐾+
+훼휒耠耠퐾휂−,
푐 14 (2)=
(−2퐾휂−퐾+휂+2휌(퐾+휂)
2
)(퐾+)^2+
(훼휒耠耠퐾휂−퐾휂++퐾휂휂−−휌퐾휂휂+)
퐾+
+3휒耠耠(훼퐾휂휂−+휒耠耠+훼휒耠耠퐾−휉−휌휒耠耠(1 + 훼휒耠耠퐾+)),
푐 8 (3)=
(퐾휂−−퐾+휂)
퐾+
+훼휒耠耠(2퐾−휂+휌퐾+휂),
푐 14 (3)=
−휒耠耠(3퐾−휂−퐾휂+−2휌퐾+휂)
퐾+
−(휒耠耠)
2
(6훼퐾휂−+훼휌퐾휂+),푐 14 (6)=
2(퐾−휂−휌퐾+휂)
퐾+
+훼휒耠耠(6퐾−휂+휌퐾휂+).
(18)
Since all coefficients푎푖are defined as functions of훾푚(see
( 13 )), one could write the system ( 17 )inacompactformas[A]⋅훾=푏 (19)with the vector훾=[훾 1 ,훾 2 ,...,훾푛푠]푡and푏=[0,퐾−휉,퐾−휂,퐾−,
퐾−,0,0]푡.
Thus, the central point to obtain the solution of the
problem ( 2 ) by using IIM is to construct and resolve the
system ( 19 ). Note that, at irregular points, the set of ( 19 )
could be obtained using at least six points stencil as proposed
in the original IIM method [ 21 ]. However, as discussed in
some recent works [ 16 , 17 , 19 ], nine-point stencil (푛푠 = 9)is
preferred because the resulting linear system of equations is
in this case a block tridiagonal one.
However, using of a nine points stencil (푛푠 = 9)leads
to an underdetermined system ( 19 ) whose solution could be
obtained by using themaximum principle[ 16 , 17 , 19 ]andan
optimization approach. For that, the following constrained
quadratic optimization problem is considered:min훾 {1
2
儩儩
儩儩훾−푔
儩儩
儩儩
2
2 }, (20)with훾푚≥0 if(푖푚,푗푚)≠(0,0),훾푚<0 if(푖푚,푗푚) =(0,0).(21)
In this equation, the vector푔 =[푔 1 ,푔 2 ,...,푔푛푠]푡has the
following components:푔푚=
퐾푖+푖푚/2,푗+푗푚/2
ℎ^2
,
if(푖푚,푗푚) ∈{(−1,0),(1,0),(0,−1),(0,1)},푔푚=−
퐾푖,푗−1/2+퐾푖−1/2,푗+퐾푖+1/2,푗+퐾푖,푗+1/2
ℎ^2
,
if(푖푚,푗푚) =(0,0),
푔푚=0otherwise.(22)
This vector푔represents in effect the coefficients of the
standard five points stencil to whom the solution of the
optimization problem has to coincide in case of continuous
properties of constituent phases (i.e., when[퐾] = 0).
The existence of the solution of the optimization problem
( 20 )aswellastheconvergenceofthesubstitutionmethodin
that case has been demonstrated in [ 16 , 17 ] and will not be
discussed here. We just mention that, as in a previous work
of these authors [ 18 ], this optimization problem is solved by
using the optimization toolbox available inMATLAB.2.3. Computing Solution, Its Derivatives at Interface, and
Effective Properties.As previously outlined, the non linear
transfer problem is treated here by using an iterative proce-
dure consisting for each iteration to solve a linearized partial
differential equation. The resolution of this later by using
the IIM method needs to determine the set of weighting