Statistically compatible hyper-reduction for computational homogenization

Abstract

The computational bottleneck of reduced order models (ROMs) in nonlinear homogenization is usually given by the local material laws, which need to be evaluated in a large number of microscopic integration points. Hyper-reduction methods use only a small subset of the integration points and reach tremendous speed-ups at high accuracy. However, the underintegration breaks the overall compatibility of the microscopic strain field and is in this sense disrespecting the microscopic boundary value problem. Here, a new type of generalized integration points is introduced in strain space in order to remedy this shortcoming. Being inspired by results from nonlinear homogenization theory, the concept of statistical compatibility is developed and forms the theoretical basis for the new integration points, which respect the compatibility of the microscopic strain field in a statistical sense. The statistically compatible integration points can be derived offline and replace the conventional ones in a Galerkin-projection based setting with global modes identified via proper orthogonal decomposition (POD). The method is tested for various reinforced composites, indicating that 10-20 integration points are often sufficient to reach errors smaller than 3%, with CPU-times in the μs-range (per time step). A possible extension of the method for problems with higher nonlinearity and stronger field fluctuations is discussed within the context of a porous microstructure.