Subspace collections with orthogonal subspaces obviously have a lot of relevance to physical problems. Here we give an example to show that subspace collections with nonorthogonal subspaces are an important tool in analysis, reaching beyond their connection with rational functions of several complex variables. This chapter is mostly self-contained. While it uses nonorthogonal subspace collections, it is not necessary for the reader to have read Chapter 7 of . However, it is recommended for the reader to look at Chapters 1 and 2 in that book.
The specific problem we consider is a two-component conducting medium with conductivities and , phase being the inclusion phase. The dimension is or . Without loss of generality we can take . Assume for simplicity that the composite has square or cubic symmetry so that the effective conductivity tensor is isotropic, taking the form .  made the pioneering observation that the function is an analytic function of and has all its singularities on the negative real axis. He made some assumptions which were not valid . These assumptions could be circumvented by approximating the composite by a large resistor network (). Subsequently Golden and Papanicolaou (, ) gave a rigorous proof of the analytic properties. Here we make the additional assumption (only true for some geometries) that the function has all its singularities confined to an interval on the negative real axis, and (see 1). This information results in tighter bounds on the conductivity and complex conductivity, or equivalently the complex dielectric constant, (; ; ), and these bounds may be inverted to yield information about and from experimental measurements (). By contrast, here our goal is to utilize the information to improve the speed of convergence of Fast Fourier Transform methods for computing the fields in composites and their associated effective tensors. These methods were first introduced by Moulinec and Suquet (, ): see  for a recent review. One important application has been to viscoplastic polycrystals (, ). For materials with a linear response, previous significant advances in the acceleration of these schemes were made by  (see also the generalization in section 14.9 of , and in particular equation (14.38)), and by  and .
The singularities in the interval could be poles or could be a branch cut (if the inclusion has sharp corners, or if there is some randomness in the geometry). Note it is not only the effective conductivity which has this analytic form but also the electric field and current field at fixed , and fixed applied field, with varying. If we didn’t know anything about and there could potentially be singularities anywhere along the negative real axis. Let’s first look at the case , .
for the effective conductivity , and electric field , with being the applied (average) electric field. [To obtain (1.1) from (2.35) in  we use the fact that .] Their key and beautiful idea was that since the action of is readily evaluated in Fourier space, while the action of is readily evaluated in real space, both and can be readily calculated from these series by going back and forth between real and Fourier space, using Fast Fourier Transforms to do so.
In a two-phase medium with isotropic conductivities and , and with the choice of used by Moulinec and Suquet, takes the value in phase 1 and the value in phase 2. So it is clear that this scheme gives an expansion of the form
The transformation maps the right half of the complex plane to the unit disk (see 2).
So with the series will converge if , i.e., in the entire right half of the complex plane, . (Recall that it is the distance from the origin to the nearest singularity in the plane which determines the radius of convergence in the plane, and hence the region of convergence in the plane.)
The transformation where maps the complex plane minus the slit along the negative real axis to the unit disk (see 3).
So with the series will converge if , i.e., in the entire complex plane minus the slit along the negative real axis. These arguments show that when the accelerated scheme should have a larger region of convergence, and by the same line of reasoning a faster rate of convergence when both schemes converge. (Note, however, that if and the scheme of Moulinec and Suquet could outperform that of Eyre and Milton for small and very large values of , since the scheme of Moulinec and Suquet should then converge for sufficiently small or sufficiently large negative real values of .)
If we know and (or bounds for them) then it makes sense to use a transformation which maps the complex plane minus the slit to the unit circle . Since the transformation
maps the interval to and takes to , it is clear that the transformation
maps the complex plane minus the slit to the unit disk (see 4).
So we want to get a Fast Fourier Transform (FFT) method associated with an expansion
It looks rather formidable but really it is just a matter of substituting
in the  scheme associated with
But of course one wants to do this substitution at the level of the underlying Hilbert space, not just in the conductivity function.
2 Substitution at the level of the Hilbert space
Substitution at the level of the Hilbert space requires the substitution of subspace collections where the subspaces do not satisfy the orthogonality property. At a discrete level it is easy to get an idea of how this can be done. If we consider the composite as a resistor network, in phase with resistors having resistance , while in phase with resistors having resistance , so e.g., the composite is replaced by a network (see 5(a)).
Then at this level we could replace each resistor by the compound resistor (see 5(b)) with real constants. That is the resistance gets replaced by
when , and so gets replaced by
which is a fractional linear transformation of . In other words, in phase the number of fields is multiplied by (replacing one resistor by ). Note that with real positive values of , and the transformation (2.2) maps the nonnegative real -axis onto an interval on the positive real axis, and the negative real -axis onto its complement, which cannot be the desired interval . Essentially, instead of making a transformation which has the effect of lengthening the branch cut as desired in the map at the top of 4, we have a transformation which presumably (in some suitably defined metric) shortens the branch cut. This is why it is necessary to substitute nonorthogonal subspace collections, rather than orthogonal ones [see also the discussion below (4.17)], since, as we will see, they only act to lengthen the branch cut.
How to do this in general in the continuum case is described in section 29.1 of , bottom of page 621 and page 622, for orthogonal subspace collections, and in the second half of Section 7.8 of , without assuming orthogonality. That is a bit abstract so let us go through it for the case in question.
3 The original subspace collection
Our starting point is the Hilbert space of fields (real or complex
dimensional vector fields), that are cell-periodic and square-integrable in the unit cell. We denoteas the projection onto all fields which are zero in phase , as the space of constant vector fields, as the space of gradients which derive from periodic potentials (i.e., if and ) and as the space of divergence free fields with zero average value (i.e., if and ). In our Hilbert space the inner product is taken to be
where the overline denotes complex conjugation. With respect to this inner product the spaces and are orthogonal. The projection onto we denote by . In Fourier space
The field equations are solved once we have found electric fields and current fields such that
The effective tensor is then defined via
where is the projection onto . We assume the composite is isotropic so that is a scalar, although the method is easily generalized to anisotropic materials where the effective conductivity is a second order tensor.
4 The vector subspace collection we substitute into the original subspace collection
Now consider a dimensional subspace collection consisting of component vectors with inner product
where the overline denotes complex conjugation. The projection projects onto the one dimensional space of fields proportional to the unit vector where and are given constants such that . The ’s could be complex but we do not mean . Thus is a projection but not an orthogonal projection when the ’s are complex, as then is not Hermitian. We take the following:
The field equations become
where the constant will be chosen so the associated “effective modulus” is . That is
where is the projection onto , so that
Without loss of generality we can choose so the field equations become
From the middle equation we get
which with gives
So we have
which with is satisfied with
Note that (respectively ) is obtained by substituting (respectively ) in (4.16). Given real we need to choose and so that these equations are satisfied. This will necessitate complex solutions for and since otherwise will be negative. Note that with and being complex, is no longer Hermitian, even though it is a (non self-adjoint) projection, and so we have a subspace collection which is not orthogonal: is not orthogonal to . Also from the field equations with we get
5 The subspace collection after the substitution
Now consider the Hilbert space consisting of all periodic fields of the form
Fields in take the form
Fields in take the form
where . Fields in take the form
where . The space consists of all vectors of the form
and consists of all fields of the form
Also consists of all vectors of the form
and consists of all fields of the form
The inner product on is defined to be
where the overline denotes taking a complex conjugate. With this inner product, the subspaces and are mutually orthogonal. We define , i.e.,
where is the identity matrix which defines even if and are complex. Now the field equations are
These are easy to solve given periodic solutions and to the equations in the Hilbert space , i.e.,
We take (with )
Note that we have and . Also, with , we have
Finally if is the projection onto we have
and since we deduce that
That is, is still the effective tensor. Now the idea is to apply, either the basic Fast Fourier Transform scheme of Moulinec and Suquet (, ) to the Hilbert space that is associated with the expansion
The operator is easily evaluated in real space. The operator which projects onto is easily evaluated in Fourier space since
where in Fourier space
Hence the accelerated Fast Fourier Transform method of Eyre and Milton can be directly applied in the Hilbert space .
6 Proof of acceleration
Let us suppose is fixed such that . The rate of convergence will be determined by the magnitude of
Since , when , the convergence will be quicker the smaller is. Now as increases,
decreases monotonically from the value at , to the value at . So the larger the value of , the faster the rate of convergence. Similarly as decreases, decreases monotonically from the value at to the value of at . So the smaller the value of , the faster the rate of convergence. More generally when is complex, to see the rate of convergence one should plot the contours in the complex plane. It might be instructive to do this for particular values of and , and compare the contours with .
7 The numerical example of Moulinec and Suquet
This numerical example is due to Hervé Moulinec and Pierre Suquet (to be published) and compares in a model example the speeds of convergence of results in the Hilbert spaces and for the Fast Fourier Transform scheme first proposed by Moulinec and Suquet (, ). Then the speeds of convergence of results are compared in the Hilbert spaces and for the accelerated scheme of  [see also the generalization in section 14.9 of ]. In both cases the Fast Fourier Transform schemes converge substantially faster in the Hilbert space .
It is to be emphasized that while these numerical results are only for the effective tensor, the real interest in the new algorithm is for obtaining results for the fields inside the body: accelerated rates of convergence for the effective conductivity alone could easily be obtained by using Padé approximants () or (almost equivalently) the associated bounds which use series expansion coefficients up to a given order (; ; : see also Chapter 27 in  and Chapter 10 in , and references therein). Using the new method I expect there will be a similar acceleration of the convergence rates for the fields. Indeed, bounds on the norm of the difference between the actual field and the field obtained by truncating the series expansion show these improved convergence rates. However this does not guarantee pointwise convergence, and in any case it needs to be numerically explored. Padé approximants methods could also be used for the fields, but this approach has the disadvantage that one needs to simultaneously store a lot of information: not just the fields, but also the fields that appear up to a given order in the perturbation expansion for a nearly homogeneous medium.
The model example is a regular array of squares of conductivity occupying a volume fraction of 25% in a matrix of conductivity , as illustrated in 6.
An exact formula for the effective conductivity of this array (and for the interior fields) was discovered by ,
.] Taking a wider estimate for the branch cut (withand ), Moulinec and Suquet used the algorithm described in this chapter and found that the acceleration provided by the new method was generally substantially improved as shown in their 7 and 8.
8 Estimating the parameters and
 has derived rigorous lower bounds on and upper bounds on , for suspensions of separated spheres in a medium. Actually we need only estimates on and . For instance if an estimate on is slightly too large, then the transformations will look like those depicted in 9. The series will still converge but only for , where will be close to (but less than ). However, the support of the measure will be highly dependent on small details of the microstructure: a tiny sharp cusp on the surface of an otherwise smooth inclusion will in general dramatically change the support of the measure: see , page 378 of , and equation (15) of . As remarked on page 378 of , this is related to the fact that these sharp corners behave as sinks of energy in the mathematically equivalent dielectric problem with the conductivities and being replaced by electrical permittivities and , where is real and positive, while is almost real and negative with a tiny positive imaginary part. Related observations and analysis include those of , , , and . Also there is a close connection with the essential spectrum of the Neumann–Poincaré operator on planar domains with corners (; ).
9 Bounds on the support of the measure using -convex functions
For many other equations in periodic composite materials robust things can be said about the measure which restrict its support. We assume we have a Hilbert space of square integrable periodic fields with an inner product
where the angular brackets denote an average over the unit cell of periodicity. We also assume has a decomposition into three orthogonal subspaces
We are interested in the equations
and where the local tensor field takes the form
is the orthogonal matrix (satisfying) associated with a rotation acting on elements in the tensor space, while is a field of rotation matrices giving the local orientation of each phase, and is the indicator function that is in phase and zero elsewhere.
First, following section 14.8 of  let us show that there are series expansions for the effective tensor that converge even when the local tensor is not everywhere positive definite. To obtain the series expansion we introduce a constant reference tensor and an associated operator : we say
Explicitly, is given by the operator
where the inverse is to be taken on the subspace . Introducing the polarization field
we see it satisfies , and hence we have the identity
Averaging both sides yields a formula for the effective tensor, and associated series expansion:
where the angular brackets denote a volume average.
A sufficient condition for convergence is that the operator has norm less than . We take a reference tensor of the form
where the tensor is assumed to be bounded and self-adjoint, but need not be positive definite. We are interested in seeing whether the expansion for the effective tensor converges when is very large. Expanding the operator in powers of gives
with a remainder term
When this remainder term has norm satisfying the bound