Summary: A reduction/hyper reduction framework is presented for dramatically accelerating the solution of nonlinear dynamic multiscale problems in structural and solid mechanics. At each scale, the dimensionality of the governing equations is reduced using the method of snapshots for proper orthogonal decomposition, and computational efficiency is achieved for the evaluation of the nonlinear reduced-order terms using a carefully designed configuration of the energy conserving sampling and weighting method. Periodic boundary conditions at the microscales are treated as linear multipoint constraints and reduced via projection onto the span of a basis formed from the singular value decomposition of Lagrange multiplier snapshots. Most importantly, information is efficiently transmitted between the scales without incurring high-dimensional operations. In this proposed proper orthogonal decomposition-energy conserving sampling and weighting nonlinear model reduction framework, training is performed in two steps. First, a microscale hyper reduced-order model is constructed in situ, or using a mesh coarsening strategy, in order to achieve significant speedups even in non-parametric settings. Next, a classical offline-online training approach is performed to build a parametric hyper reduced-order macroscale model, which completes the construction of a fully hyper reduced-order parametric multiscale model capable of fast and accurate multiscale simulations. A notable feature of this computational framework is the minimization, at the macroscale level, of the cost of the offline training using the in situ or coarsely trained hyper reduced-order microscale model to accelerate snapshot acquisition. The effectiveness of the proposed hyper reduction framework at accelerating the solution of nonlinear dynamic multiscale problems is demonstrated for two problems in structural and solid mechanics. Speedup factors as high as five orders of magnitude are shown to be achievable.
{Copyright © 2017 John Wiley & Sons, Ltd.}