Accelerated mesh sampling for the hyper reduction of nonlinear computational models

In nonlinear model order reduction, hyper reduction designates the process of approximating a projection-based reduced-order operator on a reduced mesh, using a numerical algorithm whose computational complexity scales with the small size of the projection-based reduced-order model. Usually, the reduced mesh is constructed by sampling the large-scale mesh associated with the high-dimensional model underlying the projection-based reduced-order model. The sampling process itself is governed by the minimization of the size of the reduced mesh for which the hyper reduction method of interest delivers the desired accuracy for a chosen set of training reduced-order quantities. Because such a construction procedure is combinatorially hard, its key objective function is conveniently substituted with a convex approximation. Nevertheless, for large-scale meshes, the resulting mesh sampling procedure remains computationally intensive. In this paper, three different convex approximations that promote sparsity in the solution are considered for constructing reduced meshes that are suitable for hyper reduction and paired with appropriate active set algorithms for solving the resulting minimization problems. These algorithms are equipped with carefully designed parallel computational kernels in order to accelerate the overall process of mesh sampling for hyper reduction, and therefore achieve practicality for realistic, large-scale, nonlinear structural dynamics problems. Conclusions are also offered as to what algorithm is most suitable for constructing a reduced mesh for the purpose of hyper reduction. Copyright (c) 2016 John Wiley & Sons, Ltd.