Projection-based nonlinear model order reduction (MOR) methods typically make use of a reduced basis V ∈ R m×k to approximate high-dimensional quantities. However, the most popular methods for computing V, eg, through a singular value decomposition of an m × n snapshot matrix, have asymptotic time complexities of (min(mn 2 , m 2 n)) and do not scale well as m and n increase. This is problematic for large dynamical problems with many snapshots, eg, in case of explicit integration. In this work, we propose the use of randomized methods for reduced basis computation and nonlinear MOR, which have an asymptotic complexity of only (mnk) or (mn log(k)). We evaluate the suitability of randomized algorithms for nonlinear MOR and compare them to other strategies that have been proposed to mitigate the demanding computing times incurred by large nonlinear models. We analyze the computational complexities of traditional, iterative, incremental, and randomized algorithms and compare the computing times and accuracies for numerical examples. The results indicate that randomized methods exhibit an extremely high level of accuracy in practice, while generally being faster than any other analyzed approach. We conclude that randomized methods are highly suitable for the reduction of large nonlinear problems. KEYWORDSexplicit FEM, low-rank approximation, nonlinear dynamics, nonlinear model order reduction, randomized numerical linear algebra, randomized SVD Int J Numer Methods Eng. 2019;118:209-241.wileyonlinelibrary.com/journal/nme

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Summary Many model order reduction (MOR) methods employ a reduced basis boldV∈Rm×k to approximate the state variables. For nonlinear models, V is often computed using the snapshot method. The associated low‐rank approximation of the snapshot matrix boldA∈Rm×n can become very costly as m,n grow larger. Widely used conventional singular value decomposition methods have an asymptotic time complexity of scriptOfalse(minfalse(mn2,m2nfalse)false), which often makes them impractical for the reduction of large models with many snapshots. Different methods have been suggested to mitigate this problem, including iterative and incremental approaches. More recently, the use of fast and accurate randomized methods was proposed. However, most work so far has focused on fixed‐rank approximations, where rank k is assumed to be known a priori. In case of nonlinear MOR, stating a bound on the precision is usually more appropriate. We extend existing research on randomized fixed‐precision algorithms and propose a new heuristic for accelerating reduced basis computation by predicting the rank. Theoretical analysis and numerical results show a good performance of the new algorithms, which can be used for computing a reduced basis from large snapshot matrices, up to a given precision ε.

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...

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Because of the drawbacks of standard lumped mass–spring models discussed at the beginning of this paper, a new approach for simplified modeling of frontal impacts appropriate for early phase crashworthiness design is proposed. It is based on a first step, the Geometry Space Model, representing the real location of the structural components with deformable, non-deformable, and gap parts. This is then transformed by a new algorithm into the Deformation Space Model which considers only the available free deformation lengths and can be used to assess the correct deformation modes of complex structural systems. These developments are embedded in a wider research field, already published, where Solution Spaces are established for set-based design of the force–displacement curves for all springs. Together with this Solution Space technology, the proposed new simplified modeling approach for frontal impacts will make early phase development more efficient in the future.

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