Waves in composites and metamaterials/Effective tensors using Hilbert space formalism

The content of these notes is based on the lectures by Prof. Graeme W. Milton (University of Utah) given in a course on metamaterials in Spring 2007.

Recap edit

In the previous lecture we introduced the Hilbert space   of of square-integrable,  -periodic, complex vector fields with the inner product


We then decomposed   into three orthogonal subspaces.

  1. The subspace   of uniform fields, i.e.,   is independent of  , or in Fourier space,   unless  .
  2. The subspace   of zero divergence, zero average value fields, i.e.,   and  , or in Fourier space,   and  .
  3. The subspace   of zero curl, zero average value fields, i.e.,   and  , or in Fourier space,   and  .

To determine the effective permittivity, we introduced the operator   with the properties


if and only if


We also found that in Fourier space,




Deriving a formula for the effective permittivity edit

Let us now derive a formula for the effective tensor. Recall that the polarization is defined as


where the permittivity tensor is being thought of as an operator that operates on  .

Also notice that


From (3) we have


From the definition of   (equations (1) and (2)) and using (4) and (5), we can show that


Let   act on both sides of (6). Then we get




Inverting (7) gives


Averaging (8) gives


Averaging (3) leads to the relation


Comparing (9) and (10) shows us that


Recall from the previous lecture that, in equation (11), the operator   is local in real space while the operator   is local in Fourier space. It is therefore not obvious how one can invert  .

Let us define


Then (8) can be written as


Assuming that  , we can expand the first operator in terms of an infinite series, i.e.,


Then we have


Also, from the definition of  , we have










Let us now define


Then we can write


These recurrence relations may be used to compute these fields inductively. An algorithm that may be used is outlined below:

  • Set  . Then
  • Compute   in real space using the relation
  • Take a fast Fourier transform to find  .
  • From (14) we get
  • Compute   in Fourier space.
  • Take an inverse fast Fourier transform to find   in real space.
  • Increment   by 1 and repeat.

This is the method of Moulinec and Suquet (Mouli94). The method also extends to nonlinear problems (Mouli98). However, there are other iterative methods that have faster convergence.

Convergence of expansions edit

For simplicity, let us assume that   is isotropic, i.e.,  . Then,


where   is the projection from   onto  .

Define the norm of a field   as


Also, define the norm of a linear operator   acting on   as








In addition, we have the triangle inequality


So from (12) and (13), we have


But   since it is a projection onto  . Hence,


Therefore the series converges provided


In that case




To get a better understanding of the norm  , let us consider a  -phase composite with isotropic phases, i.e.,




In this case,






Since,   are weights, it makes sense to put the weights where   is maximum. Hence, we can write


For   to be less than 1, we therefore require that, for all  ,


A geometrical representation of this situation is shown in Figure 1.

Figure 1. Allowable values of   for convergence of series expansion.

If the value of   is sufficiently large, then we get convergence if all the   s lie in one half of the complex plane (shown by the green line in the figure).

Similarly, we can expand


in the form


where   is a projection onto  , i.e.,


In this case, we find that the series converges provided


Note that each term in (15) is an analytic function of   (in fact a polynomial). So, if we truncate the series, we have an analytic function of  .

Since a sequence of analytic functions which is uniformly convergent in a domain converges to a function which is analytic in that domain (see, for example, Rudin76), we deduce that   is an analytic function of   in the   disk (see Figure~1) with   provided   for  .

Similarly, the effective tensor is an analytic function of  ,  , etc.

Since   is independent of  , by taking the union of all such regions of analyticity, we conclude that   is an analytic function of   provided all these   s lie inside a half-plane (see Figure~1). This means that there exists a   such that


Corollary: edit

A corollary of the above observations is the following. If each   is an analytic function of   for   (which is what one expects with   as the frequency) and   for all   with  , then   will be analytic in  .

Another interesting property: edit

Now, if  , we have




This means that


Therefore,   is homogeneous of degree one and


For a two-phase composite, if we take


we get


Therefore, it suffices to study the analytic function  . For further details see Milton02.

References edit

  • [Milton02]     G. W. Milton. Theory of Composites. Cambridge University Press, New York, 2002.
  • [Mouli94]     H. Moulinec and P. Suquet. A fast numerical method for computing the linear and nonlinear mechanical properties of composites. Comptes rendus de l'Académie des sciences II, 318(11):1417--1423, 1994.
  • [Mouli98]     H. Moulinec and P. Suquet. A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput. Methods Appl. Mech. Engrg., 157:69--94, 1998.
  • [Rudin76]     W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, New York, 1976.