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

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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 | edit source]

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,

where

Deriving a formula for the effective permittivity[edit | edit source]

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

Therefore,

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

Hence,

Now,

Therefore,

or,

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 | edit source]

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

Therefore,

Hence,

So

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

where

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

where

In this case,

where

Hence,

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 | edit source]

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 | edit source]

Now, if , we have

Therefore,

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 | edit source]

  • [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.