SUMMARYA rigorous computational framework for the dimensional reduction of discrete, high-fidelity, nonlinear, finite element structural dynamics models is presented. It is based on the pre-computation of solution snapshots, their compression into a reduced-order basis, and the Galerkin projection of the given discrete high-dimensional model onto this basis. To this effect, this framework distinguishes between vector-valued displacements and manifold-valued finite rotations. To minimize computational complexity, it also differentiates between the cases of constant and configuration-dependent mass matrices. Like most projection-based nonlinear model reduction methods, however, its computational efficiency hinges not only on the ability of the constructed reduced-order basis to capture the dominant features of the solution of interest but also on the ability of this framework to compute fast and accurate approximations of the projection onto a subspace of tangent matrices and/or force vectors. The computation of the latter approximations is often referred to in the literature as hyper reduction. Hence, this paper also presents the energy-conserving sampling and weighting (ECSW) hyper reduction method for discrete (or semi-discrete), nonlinear, finite element structural dynamics models. Based on mesh sampling and the principle of virtual work, ECSW is natural for finite element computations and preserves an important energetic aspect of the high-dimensional finite element model to be reduced. Equipped with this hyper reduction procedure, the aforementioned Galerkin projection framework is first demonstrated for several academic but challenging problems. Then, its potential for the effective solution of real problems is highlighted with the realistic simulation of the transient response of a vehicle to an underbody blast event. For this problem, the proposed nonlinear model reduction framework reduces the CPU time required by a typical high-dimensional model by up to four orders of magnitude while maintaining a good level of accuracy.

The computational efficiency of a typical, projection-based, nonlinear model reduction method hinges on the efficient approximation, for explicit computations, of the scalar projections onto a subspace of a residual vector. For implicit computations, it also hinges on the additional efficient approximation of similar projections of the Jacobian of this residual with respect to the solution. The computation of both approximations is often referred to in the literature as hyper reduction. To this effect, this paper focuses on the analysis and comparative performance study for nonlinear finite element reduced-order models of solids and structures of the recently developed energy-conserving mesh sampling and weighting (ECSW) hyper reduction method. Unlike most alternative approaches, this method approximates the scalar projections of residuals and/or Jacobians directly, instead of approximating first these vectors and matrices then projecting the resulting approximations onto the subspaces of interest. In this paper, it is shown that ECSW distinguishes itself furthermore from other hyper reduction methods through its preservation of the Lagrangian structure associated with Hamilton's principle. For second-order dynamical systems, this enables it to also preserve the numerical stability properties of the discrete system to which it is applied. It is also shown that for a fixed set of parameter values, the approximation error committed online by ECSW is bounded by its counterpart error committed off-line during the training of this method. Therefore, this error can be estimated in this case a priori and controlled. The performance of ECSW is demonstrated first for two academic but nevertheless interesting nonlinear dynamic response problems. For both of them, ECSW is shown to preserve numerical stability and deliver the desired level of accuracy, whereas the discrete empirical interpolation method and its recently introduced unassembled variant are shown to be susceptible to failure because of numerical instability. The potential of ECSW for enabling the effective reduction of nonlinear finite element dynamic models of solids and structures is also highlighted with the realistic simulation of the nonlinear transient dynamic response of a complete car engine to thermal and combustion pressure loads using an implicit scheme. For this simulation, ECSW is reported to enable the reduction of the CPU time required by the high-dimensional nonlinear finite element dynamic analysis by more than four orders of magnitude, while achieving a very good level of accuracy.

A model to predict the macroscopic in-plane residual stress state of semicrystalline thermoplastic composite laminates induced by process cooling is presented. Heat transfer during processing is based upon an incremental transient formulation that consists of a finite difference heat transfer analysis coupled to the crystallization kinetics. Micromechanics is used to evaluate the instantaneous spatial variation of mechanical properties as a function of temperature and degree of crystallinity. Residual stresses are based upon an incremental laminate theory that includes temperature gradients, shrinkage due to crystallization and thermal contraction. Temperature dependent relaxation times are used to model first order viscoelastic effects. A parametric study is conducted to explore the sensitivity of residual stresses to process history. Input parameters varied include surface temperature history (cooling rate), the amount of shrinkage caused by crystallization and the relaxation time at the reference temperature. The model predictions were in good agreement with experimental residual stress measurements for unidirectional graphite (AS4) reinforced polyetheretherketone (PEEK) laminates.

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.