Cellular Fourier analysis for geometrically disordered materials

Abstract

Many media are divided into elementary units with irregular shape and size, as exemplified by domains in magnetic materials, bubbles in foams, or cells in biological tissues. Such media are essentially characterized by geometrical disorder of their elementary units, which we term cells. Cells set a reference scale at which parameters and fields reflecting material properties and state are often assessed. In these media, it is difficult to quantify spatial variations of cell-scale fields, because space discretization based on standard coordinate systems is not commensurate with the natural discretization into geometrically disordered cells. Here we consider the spectral analysis of spatially varying fields. We built a method, which we call Cellular Fourier Transform (CFT), to analyze cell-scale fields, which includes both discrete fields defined only at cell level and continuous fields smoothed out from their sub-cell variations. Our approach is based on the construction of a discrete operator suited to the disordered geometry and on the computation of its eigenvectors, which respectively play the same role as the Laplace operator and sine waves in Euclidean coordinate systems. We show that CFT has the expected behavior for sinusoidal fields and for random fields with long-range correlations. Our approach for spectral analysis is suited to any geometrically disordered material, such as biological tissue with complex geometry, opening the way to systematic multiscale analyses of material behavior.