Reduced-Order, Trajectory Piecewise-Linear Models for Nonlinear Computational Fluid Dynamics

Abstract

Computational fluid dynamics (CFD) is now widely used throughout the fluid dynamics community and yields accurate models for problems of interest. However, due to its high computational cost, CFD is limited for some applications. Therefore, model reduction has been used to derive low-order models that replicate CFD behavior over a restricted range of inputs, and various frameworks have been developed. Unfortunately, the majority of those methods are limited to linear cases and do not properly handle reduction of nonlinear systems.In order to overcome restrictions of weak nonlinearity and the costly representation of the system's nonlinearity found in other nonlinear reduction approaches, a trajectory piecewise-linear (TPWL) scheme is developed for a CFD model of the twodimensional Euler equations. The approach uses a weighted combination of linearized models to represent the nonlinear CFD system. Using a set of training trajectories obtained via a simulation of the nonlinear CFD model, algorithms are presented for linearization point selection and weighting of the models. Using the same training trajectories to provide a snapshot ensemble, the proper orthogonal decomposition (POD) is used to create a reduced-space basis, onto which the TPWL model is projected. This projection yields an efficient reduced-order model of the nonlinear system, which does not require the evaluation of any full-order system residuals, while capturing a large portion of the nonlinear space.The method is applied to the case of flow through an actively controlled supersonic diffuser. Convergence of the TPWL approach is presented for both full-order and reduced-order cases. The TPWL approach and the POD combine naturally to form an efficient reduction procedure and the methodology is found to yield accurate results, including cases with significant shock motion. Reduced-order PWL models are shown to be three orders of magnitude more efficient than the nonlinear CFD for simulation of a representative test case. AcknowledgmentsA number of peoples contributed in various ways to the last two years of my life here, in Boston, and made them more than enjoyable and memorable.Foremost on my list is my advisor, Karen, who helped me more than anything else in getting through MIT. Even though a communication problem initially restricted the flow of information between us, and despite the fact that I thought for a long time that she was English, she has always shown me a great deal of patience. Also, the availability, guidance and responsiveness she provided during all that time went beyond any expectations I possibly could have had for a research advisor. I will never thank her enough for the opportunities to go to Rotterdam and Singapore.Thanks to Jim Paduano and all the member of the supersonic project for their initial interest toward me and allowing me such an easy switch to model order reduction. Thanks to Mark Drela who provided me with precious hints in debugging mtflow. I must also acknowledge Michal Rewienski and Alexandre Megretsk...

Model-Order Reduction of Moving Nonlinear Electromagnetic Devices

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Nonlinear Model Reduction via Piecewise Balance Truncation

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“…Gu and Roychowdhury [7] proposed a similar but distinguishably different technique and demonstrated its advantages over the TPWL. Gratton and Wilcox [8] combined the TPWL and POD methods to derive a reduced model to study a flow control scheme for supersonic diffuser. Jesmani et al [9], taking advantage of linear characteristics of flow in streamline coordinates, applied the TPWL for prediction of pressure and water saturation in oil reservoirs.…”

Linearized reduced-order models for subsurface flow simulation

“…Here, by contrast, we apply POD in conjunction with the linearized representation. We note that many previous TPWL implementations [14,[17][18][19][20] used Krylov techniques to construct the reduced-order basis, though POD was used by [16,21].…”