Accelerated mesh sampling for the hyper reduction of nonlinear computational models

Abstract

Summary 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 © 2016 John Wiley & Sons, Ltd.

Hyper-Reduction Over Nonlinear Manifolds for Large Nonlinear Mechanical Systems

“…We have included its pseudo-code in Algorithm 1, where ζ E and G E denote, respectively, the restriction of ζ ∈ R ne and column-wise restriction of G to the elements in the active subset E. The set Z is the disjoint inactive subset which contains the zero entry-indices of ξ and ζ. More recent work [24] proposes and discusses different alternatives to accelerate the sampling procedure during hyper-reduction, including its parallel implementation to reduce offline costs.…”

Reduced-Order Modelling and Homogenisation in Magneto-Mechanics: A Numerical Comparison of Established Hyper-Reduction Methods

“…for snapshots of the stress field P j with m u and m y being the number of elements in the reduced meshes. Different algorithms for the minimisation of (36) are discussed in [46]. Since this minimisation is numerically expensive, collateral bases [29]…”

A multilevel projection‐based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics

“…In the above expressions, the superscript e designates the restriction of a global vector or matrix to element e, |V ′ k | ≪ |V k |, and V ′ k can be efficiently computed using one of three different active set algorithms that are described and contrasted in [44]. Furthermore, the ECSW method distinguishes itself from alternative hyper reduction methods through its preservation of the Lagrangian structure associated with Hamilton's principle.…”

“…k , (i) k , and (i) k as the prescribed displacements, history variables, and material properties associated with the snapshot u (i) k , respectively. The vector of ECSW weights, k ∈ R |V k | , can be defined as the approximate solution of the following non-negative least squares (NNLS) problem [44] minimize…”

“…Typically, iterates are generated using the Lawson and Hanson algorithm [45], and the training tolerance is chosen as ∼ 0.01. An element e is included in V ′ k if the corresponding weight e k is non-zero, that is, V ′ k = {e ∈ V k | e k > 0} (note that while the sparsity of k is not strictly enforced in (33), it is promoted by the non-negativity constraint [44]). Once the reduced mesh and its corresponding weights are constructed, the reducedorder internal force is approximated as…”