A simple and flexible model order reduction method for FFT-based homogenization problems using a sparse sampling technique

This work is concerned with the development of a novel model order reduction technique for FFT solvers. The underlying concept is a compressed sensing technique which allows the reconstruction of highly incomplete data using non-linear recovery algorithms based on convex optimization, provided the data is sparse or has a sparse representation in a transformed basis. In the context of FFT solvers, this concept is utilized to identify a reduced set of frequencies on a sampling pattern in the frequency domain based on which the Lippmann–Schwinger equation is discretized. Classical fixed-point iterations are performed to solve the local problem. Compared to the unreduced solution, a significant speed-up in CPU times at a negligibly small loss of accuracy in the overall constitutive response is observed. The generation of highly resolved local fields is easily possible in a post-processing step using reconstruction algorithms which are available as open source routines. The developed reduction technique does not require any time-consuming offline computations, e.g. for the generation of snapshots, is not restricted to any kinematic or constitutive assumptions and its implementation is straightforward. Composites consisting of elastic inclusions embedded in an i) elastic and ii) elastoplastic matrix are investigated as representative simulation examples.