Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency

摘要

A 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.