Substitution of subspace collections with nonorthogonal subspaces to accelerate Fast Fourier Transform methods applied to conducting composites

We show the power of the algebra of subspace collections developed in Chapter 7 of the book "Extending the Theory of Composites to Other Areas of Science (edited by Milton, 2016). Specifically we accelerate the Fast Fourier Transform schemes of Moulinec and Suquet and Eyre and Milton (1994, 1998) for computing the fields and effective tensor in a conducting periodic medium by substituting a subspace collection with nonorthogonal subspaces inside one with orthogonal subspaces. This can be done when the effective conductivity as a function of the conductivity ${\sigma }_1$ of the inclusion phase (with the matrix phase conductivity set to $1$) has its singularities confined to an interval $[-\beta ,-\alpha ]$ of the negative real ${\sigma }_1$ axis. Numerical results of Moulinec and Suquet show accelerated convergence for the model example of a square array of squares at $25\mathrm{\%}$ volume fraction. For other problems we show how $Q_C^{\ast }$-convex functions can be used to restrict the region where singularities of the effective tensor as a function of the component tensors might be found.