Structure‐preserving, stability, and accuracy properties of the energy‐conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models

Abstract

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.

Generalization of quadratic manifolds for reduced order modeling of nonlinear structural dynamics

“…The second aspect deals with the complexity reduction in the computation of certain nonlinear terms in the governing equations, which need to be updated for every evaluation in both, static and dynamic problems. In this aspect, usually referred to as Hyper Reduction, huge strides have been made recently in the context of Finite Elements [3,13].…”

Uncertainty Quantification

“…The knowledge about the HDM response is obtained during a training procedure: the design parameter vector μ is sampled at a few points using an effective sampling strategy [5,79,160]. The projection of the μparametric nonlinear HDM of dimension m onto a subspace of dimension N << m is done by using an (m×N ) reduced-order basis [V ] independent of μ, which yields the μ-parametric nonlinear ROM (note that we have used the notation m instead of m DOF contrarily to the notation used in this chapter, because we need to simplify the reading of the mathematical symbols that we use hereinafter).…”

“…Despite its low dimension, the resulting μ-parametric nonlinear ROM does not necessarily guarantee computational feasibility because the construction of the nonlinear ROM does not scale only with its size N , but also with that of the underlying μ-parametric nonlinear HDM, m >> N. A remedy consists in equipping the nonlinear ROM with a procedure for approximating the resulting reduced operators whose computational complexity scales only with the small size N of the ROM hyper-reduction method) [78,79,173,91,45]. Despite its low dimension, the resulting μ-parametric nonlinear ROM does not necessarily guarantee computational feasibility because the construction of the nonlinear ROM does not scale only with its size N , but also with that of the underlying μ-parametric nonlinear HDM, m >> N. A remedy consists in equipping the nonlinear ROM with a procedure for approximating the resulting reduced operators whose computational complexity scales only with the small size N of the ROM hyper-reduction method) [78,79,173,91,45].…”

Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm

“…Nevertheless, for optimization with largescale nonlinear computational models, the introduction of parametric reduced-order models are often necessary (see for instance [3,24]). Concerning the algorithms for solving optimization problems under uncertainties, many methods have been proposed in the literature such as the gradientbased learning that is adapted to convex problems [48,89], the global search algorithms such as the stochastic algorithms, the genetic algorithm, and the evolutionary algorithms [14,46].…”