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

• Published in: International journal for numerical methods in engineering Vol. 102; no. 5; pp. 1077 - 1110
• Main Authors: Farhat, Charbel, Chapman, Todd, Avery, Philip
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 04.05.2015; Wiley Subscription Services, Inc
• Subjects: Accuracy; Approximation; Computational efficiency; ECSW; Finite element method; Galerkin projection; Hamilton's principle; hyper reduction; Mathematical analysis; Mathematical models; mesh sampling; model reduction; nonlinear dynamics; Nonlinearity; Projection; proper orthogonal decomposition; Reduction; structure preserving
• DOI: 10.1002/nme.4820
• ISSN: 0029-5981

SummaryThe 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. Copyright © 2015 John Wiley & Sons, Ltd.