Randomized nonlinear model order reduction methods for large dynamical problems with explicit time integration

Abstract

Projection-based nonlinear model order reduction (MOR) often involves the computation of a truncated singular value decomposition (SVD) of a snapshot matrix A ∈ IR m×n , m ≥ n, computed from training simulations, where only the first k basis vectors are retained. A can however become very large in case of detailed models with a large number of degrees of freedom (DoFs), or when many snapshots are present. This is often the case for explicit FEM simulations of industrial problems. Computing the SVD can then become problematic, as most widely used SVD algorithms for MOR have an asymptotic complexity of O(mn 2 ) [1]. Randomized algorithms can alternatively be used to compute only the k most significant basis vectors. Such methods only scale with O(mnk) or even O(mn log(k)) [2]. We present a modified hybrid randomized incremental algorithm for efficient reduced basis (RB) computation with low memory requirements, and apply the method to a large-scale example problem. The results show that the hybrid algorithm is capable of efficiently computing a RB even for very large problems at a manageable computational effort. Projection based nonlinear model order reductionConsider the second-order nonlinear equation Mẍ = f (x,ẋ, t) describing the behavior of a mechanical system, where M ∈ IR m×m is the mass matrix, x ∈ IR m are the nodal displacements of a FEM discretization, and f is the vector of nonlinear nodal forces, depending on x, its first time derivativeẋ, and the time t. The principal assumption of projection based nonlinear MOR methods is that the solutions of physical problems described by partial differential equations (PDEs) typically reside on a low-dimensional manifold of dimension k, with k m [3]. The displacements x can thus be approximated bywhere x 0 is a constant (e.g. the initial conditions), V is the reduced basis, andx is the vector of reduced global DoFs. In practice, V is often computed using the method of snapshots [3], more precisely as the truncated left-singular vectors of an SVD of the resulting snapshot matrix A ∈ IR m×n . We refer to [3][4][5] for further details, including additional clustering and hyper-reduction methods. Singular value decomposition algorithmsThere are several methods for computing the (truncated) SVD of a given m × n snapshot matrix A. Perhaps the simplest one is to compute the full SVD following a bidiagonalization of A [1], and subsequently truncate it to a desired rank k or up to a specified tolerance. Methods based on first computing an eigenvalue decomposition of A T A can be more efficient; however both approaches have an asymptotic time complexity of O(mn 2 ). By contrast, efficient iterative (e.g. Arnoldi or Lanczos) and incremental algorithms [6] can compute a rank-k truncated SVD at an asymptotic complexity as low as O(mnk). Over the past decade, randomized low-rank approximations have emerged as an interesting alternative to conventional approaches. A randomized rank-k SVD can be computed at an asymptotic complexity of O(mn log(k)) using structured random matrice...

Dimensional reduction for parametric projection‐based reduced‐order models in crash

“…To summarize, the speedup in crash models can be achieved by basically two mechanisms: hyperreduction and a probably larger timestep in the explicit time integration [11, 13]. So far, pROMs in crash were mainly applied to reproductive examples [14]; however, the key to use them in the previously mentioned multiquery analyses is the variation of the input parameters.…”