Numerical material representation using proper orthogonal decomposition and diffuse approximation

From numerical point of view, analysis and optimization in computational material engineering require efficient approaches for microstructure representation. This paper develops an approach to establish an image-based interpolation model in order to efficiently parameterize microstructures of a representative volume element (RVE), based on proper orthogonal decomposition (POD) reduction of density maps (snapshots). When the parameters of the RVE snapshot are known a priori, the geometry and topology of individual phases of a parameterized snapshot is given by a series of response surfaces of the projection coefficients in terms of these parameters. Otherwise, a set of pseudo parameters corresponding to the detected dimensionality of the data set are taken from learning the manifolds of the projection coefficients. We showcase the approach and its potential applications by considering a set of two-phase composite snapshots. The choice of the number of retained modes is made after considering both the image reconstruction errors as well as the convergence of the effective material constitutive behavior obtained by numerical homogenization.