Keywords: multiscale modeling, reduced order modeling, optimal prediction, Duffing oscillator.

A novel multiscale reduced order modeling (ROM) method called optimal spatiotemporal reduced order modeling (OPSTROM), is presented in this paper, which was recently proposed by the authors [1,2,3,4], and can be applied to any nonlinear dynamical system. The governing equations are modified for an under-resolved numerical simulation in space and time with an arbitrary discretization scheme. Basic filtering concepts are used to demonstrate the manner in which subgrid-scale dynamics arise with a coarse computational grid. Models are then developed to account for the underlying spatiotemporal structure using inclusion of statistical information into the governing equations on a multi-point stencil. These subgrid-scale models are designed to provide closure by accounting for the interactions between the spatiotemporal microscales and macroscales as the system evolves. Predictions for the modified system are based upon principles of mean-square error minimization, conditional expectations and stochastic estimation, thus rendering the optimal solution with respect to the chosen resolution.

The OPSTROM approach has practical applications for a wide range of multiscale problems in physics and engineering. When fully resolved simulations are not feasible, the technique can be used to expedite simulations by coarsening the computational grid in space and time while maintaining reliable predictions for the quantities of interest. Three variants of the technique are conceivable: OPTROM (which accounts for the subgrid temporal structure), OPSROM (which accounts for the subgrid spatial structure), and OPSTROM (which accounts for the complete subgrid spatiotemporal structure). The OPTROM variant was originally proposed by the authors [1,2] whereas OPSROM is analogous to the numerical method known as large-eddy simulation for turbulent flows.

The primary objective of this paper is to test the ability of the OPTROM approach to improve the accuracy of Lyapunov exponents for a system in which the response may be regular or chaotic. The Duffing oscillator, has been selected for study, which contains no spatial dependence, as it enables exclusive focus on the effects of a coarse temporal grid. Maps are drawn for the Lyapunov exponents in a large forcing parameter space in which the response may undergo symmetric time-periodicity, asymmetry, period N-tupling and chaos. The exponents are used to discriminate between regular and chaotic response types. In order to expedite simulations, coarse timesteps are used with an implicit time-marching scheme. Substantial discretization errors occur in the form of strong numerical damping, which produces artificial regularity in some of the numerical solutions. In order to improve the accuracy of the under-resolved simulations, optimal subgrid-scale models are developed efficiently by means of direct calculation and estimation [3]. By resolving more time scales with OPTROM, under-resolved predictions for the Lyapunov exponents significantly improve.

1

A. LaBryer, P.J. Attar, P. Vedula, "Optimal temporal reduced order modeling for nonlinear dynamical systems", Journal of Sound and Vibration, under review.

2

A. LaBryer, P.J. Attar, P. Vedula, "An optimal prediction method for under-resolved time-marching and time-spectral schemes", International Journal for Multiscale Computational Engineering, in press.

3

A. LaBryer, P.J. Attar, P. Vedula, "Optimal spatiotemporal reduced order modeling, Part I: Proposed framework", Computational Mechanics, under review.

4

A. LaBryer, P.J. Attar, P. Vedula, "Optimal spatiotemporal reduced order modeling, Part II: Application to a nonlinear beam", Computer Methods in Applied Mechanics and Engineering, under review.

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