The effective conductivity of arrays of squares: large random unit cells and extreme contrast ratios
This work provides a fast and accurate computational method for materials scientists studying composites with extreme property contrasts, though it is an incremental extension of prior work on multi-component checkerboards.
The paper presents an integral equation scheme for computing effective conductivities of two-component checkerboard-like composites with large random unit cells and extreme contrast ratios, achieving at least nine-digit accuracy for systems with over a million squares at contrast ratio 10^6.
An integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two-component checkerboard-like composites with complicated unit cells at very high contrast ratios. The scheme extends recent work on multi-component checkerboards at medium contrast ratios. General improvement include the simplification of a long-range preconditioner, the use of a banded solver, and a more efficient placement of quadrature points. This, together with a reduction in the number of unknowns, allows for a substantial increase in achievable accuracy as well as in tractable system size. Results, accurate to at least nine digits, are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 10^6. Furthermore, the scheme is flexible enough to handle complex valued conductivities and, using a homotopy method, purely negative contrast ratios. Examples of the accurate computation of resonant spectra are given.