On the characterization of effective properties in 
highly heterogeneous media

ANDERS HOLMBOM, JEANETTE SILFVER
Department of Technology, Physics and Mathematics
Mid Sweden University
SE-831 25 Östersund
SWEDEN

Abstract: We discuss methods related to homogenization theory and G-convergence for the computation of effective properties in heterogeneous materials. Especially we investigate a method to detect deviations from the arithmetic mean in heterogeneous materials that are not necessarily periodic.

Key-Words: Heterogeneous materials, G-convergence, homogenization, multi-scale, effective properties.

1 Introduction
Let us consider a sequence of pieces of heterogeneous materials, with the corresponding heat conduction numbers {Ah}, that converge to a limit A in some well defined meaning. All these objects are assumed to have the same size and shape as a certain open bounded set W.
    For the simple case of stationary heat conduction, with a heat source f defined on W, and temperature zero at the boundary, we achieve a sequence {uh} of temperature distributions as the solutions to the equations
          (1)    
If the stabilization of {Ah} near A is distinct enough, e.g.,
                              (2)
(strong convergence in L2(W)), we will find the seemingly very natural result that the limit u is the solution to 
                     (3)      
This is, however, in general not the case if the convergence of {Ah} is of somewhat less strength.

We discuss this phenomenon in the context of 
periodic homogenization and G-convergence, in the next section.

2 Homogenization
In periodic homogenization (see [1], [2]) we study (1) for
                   ,
where A is periodic with respect to the unit cube Y, and when . It holds that
         ,	          (4)
but the matrix representing the effective properties does not coincide with the limit in (4). The convergence takes place in the sense of certain weighted averages (weak convergence in L2(W)), and this is not strong enough.
    Convergence similar to that in (1) and (3) is well studied under the name of G-convergence (see [CiDo]). It is proven, under not to rigorous assumptions, that there exists a matrix B(x) such 

that the solutions uh  to (1) converge to the solution u to
                    (5)
    However, like we can see in (4), it is not obvious how to decide B from the limit behaviour of Ah, unless the convergence is quite strong.
Still, there are cases where B can be computed, even though it does not coincide with any obvious limit for {Ah}. The most well known example is the case of periodic homogenization, represented by (4). Here, the effective matrix can be computed as        
          
where  zi is periodic with respect to Y, and solves 
           in Y.
Obviously, the correction of the limit in (4) is    contained in the term   
                .
    In fact, there are cases where this correction vanishes. In [JKO] it is proven that this happens in periodic homogenization when            
                  .
In our next and final section we will identify a similar criterion, for a situation without periodicity assumptions.

3 A criterion for non-periodic media
The discussion in the preceding section raises the question if there are strengths of convergence between those in (2) and (4), where the matrix B ceases to be a limit to {Ah} in any traditional sense. In other words, how much can the convergence of {Ah} be destabilized until the need for a correcting term appears? Below, we provide an example, where we can identify a precise criterion for how strong oscillations there can be allowed in {Ah}, until this takes place. 
   Let us first introduce a sequence of matrices        
               ,           (6) 

where A is smooth and
                weakly in L2(Y). 
   (This means that 

for all v that are square integrable over Y, i.e. 
for all v in L2(Y)).
    The operators in (6) are so called Hilbert- 
Schmidt operators and for this kind of functions 
they turn weak convergence into strong, i.e.,
.
The solutions to (1) for Ah as in (6) will then converge to the solution to (5) with 
                .                             (7)
    If we instead study (1) for 
,
the situation is less obvious. We cannot be sure that {Ah} converges strongly, because wh does now depend on the variable x as well. It turns out that the result hinges on the behaviour of the sequence
. 
The distinction between when (6) holds, and when a correction appears, is exactly whether
       weakly in L2(W)
or not.
Detailed studies of this, and closely related questions, are found in [3], [4] and [6]. The methods introduced in these papers mean further developments of the two-scale convergence method by Nguetseng (see [5]), and concern generalization to non-periodic cases. The relation between these more general forms of two-scale convergence and traditional concepts, such as 
G-convergence and compensated compactness (see [2]), are investigated.
The key to these investigations is the observation that for a sequence of operators 
               ,

where , that obey certain uniform bounds, it holds for any bounded sequence in  that, up to a subsequence,                

for some  and all . 
W and A are arbitrary open bounded sets 
in RN  .

References
[1] A. Bensoussan,J.L.Lions,G. Papanicoloau,
Asymptotic Analysis for periodic structures, North-Holland, (1978).
[2] D. Cioranescu and P. Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, Oxford Univ. Press, New York, 1999.
[3] A. Holmbom, J. Silfver, N. Svanstedt, 
N. Wellander, On two-scale convergence and related sequential compactness topics. Submitted.
[4] A. Holmbom, J. Silfver, N. Svanstedt, 
N. Wellander, On the relationship between some weak compactnesses with different numbers of scales. Chalmers finite element centre preprints 2003-25.
[5] D. Lukkassen, G. Nguetseng, P. Wall, Two-scale convergence, Int. J. Pure and Appl. Math., Vol.2 No.1 (2002), pp. 35-86.
[6] J. Silfver, Sequential Convergence for Functions and Operators, Licentiate Thesis 10, Mid Sweden University, Östersund 2004.