Reduced Order Modeling of Turbulent Flows Using Statistical Coarse-graining

Computer Methods in Applied Mechanics and Engineering

2020

We formulate a new projection-based reduced-ordered modeling technique for non-linear dynamical systems. The proposed technique, which we refer to as the Adjoint Petrov-Galerkin (APG) method, is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. A Markovian finite memory assumption within the Mori-Zwanzig formalism is then used to develop a reduced-order representation of the coarse-scales. This procedure leads to a closed reduced-order model that displays commonalities with the adjoint stabilization method used in finite elements. The formulation is shown to be equivalent to a Petrov-Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the Least-Squares Petrov-Galerkin method. Theoretical analysis examining a priori error bounds and computational cost is presented. Numerical experiments on the compressible Navier-Stokes equations demonstrate that the proposed method can lead to improvements in numerical accuracy, robustness, and computational efficiency over the Galerkin method on problems of practical interest. Improvements in numerical accuracy and computational efficiency over the Least-Squares Petrov-Galerkin method are observed in most cases.(G ROM) has been used successfully in a variety of problems. When applied to general non-self-adjoint and non-linear problems, however, theoretical analysis and numerical experiments have shown that Galerkin ROM lacks a priori guarantees of stability, accuracy, and convergence [9]. This last issue is particularly challenging as it demonstrates that enriching a ROM basis does not necessarily improve the solution [10]. The development of stable and accurate reduced-order modeling techniques for complex non-linear systems is the motivation for the current work.A significant body of research aimed at producing accurate and stable ROMs for complex non-linear problems exists in the literature. These efforts include, but are not limited to, "energy-based" inner products [9,11], symmetry transformations [12], basis adaptation [13,14], L 1 -norm minimization [15], projection subspace rotations [16], and least-squares residual minimization approaches [17,18,19,20,21,22,23,24]. The Least-Squares Petrov-Galerkin (LSPG) [22] method comprises a particularly popular leastsquares residual minimization approach and has been proven to be an effective tool for non-linear model reduction. Defined at the fully-discrete level (i.e., after spatial and temporal discretization), LSPG relies on least-squares minimization of the FOM residual at each time-step. While the method lacks a priori stability guarantees for general non-linear systems, it has been shown to be effective for complex problems of interest [24,23,25]. Additionally, as it is formulated as a minimization problem, physical constraints such as conservation can be naturally incorporated into the ROM formulation [26]. At the fully-discrete level, LSPG is sensitive to both the time integration scheme as ...

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confidence: 99%

“…This manuscript leverages work that the authors have performed on the use of the MZ formalism to develop closure models of partial differential equations [50,51,52,53,54]. In addition to focusing on the development and analysis of MZ models, the authors have examined the formulation of the MZ formalism within the context of the VMS method [54].…”

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confidence: 99%

The Adjoint Petrov–Galerkin method for non-linear model reduction

Parish

Wentland

Computer Methods in Applied Mechanics and Engineering

2020

“…The initial conditions are generated following a specific energy spectrum [23,34] shown in eq. (32).…”

Section: Training With Multiple Trajectories With Random Initializationmentioning

confidence: 99%

Long‐Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks

Pan

2018

Complexity

We study the use of feedforward neural networks (FNN) to develop models of nonlinear dynamical systems from data. Emphasis is placed on predictions at long times, with limited data availability. Inspired by global stability analysis, and the observation of strong correlation between the local error and the maximal singular value of the Jacobian of the ANN, we introduce Jacobian regularization in the loss function. This regularization suppresses the sensitivity of the prediction to the local error and is shown to improve accuracy and robustness. Comparison between the proposed approach and sparse polynomial regression is presented in numerical examples ranging from simple ODE systems to nonlinear PDE systems including vortex shedding behind a cylinder, and instability-driven buoyant mixing flow. Furthermore, limitations of feedforward neural networks are highlighted, especially when the training data does not include a low dimensional attractor. Strategies of data augmentation are presented as remedies to address these issues to a certain extent.Keywords data-driven modeling · artificial neural networks · dynamical system modeling · machine learning IntroductionThe need to model dynamical behavior from data is pervasive across science and engineering. Applications are found in diverse fields such as in control systems [48], time series modeling [42], and describing the evolution of coherent structures [9]. While data-driven modeling of dynamical systems can be broadly classified as a special case of system identification [26], it is important to note certain distinguishing qualities: the learning process may be performed off-line; physical systems may involve very high dimensions; and the goal may involve the prediction of long-time behaviour from limited training data.Artificial neural networks (ANN) have attracted considerable attention in recent years in domains such as image recognition in computer vision [39,19] and in control applications [12]. The success of ANNs arises from their ability to effectively learn low-dimensional representations

“…It can be shown that, when the original dynamical system is linear, that solutions to the orthogonal dynamics equation can be obtained by solving a corresponding set of auxiliary ordinary differential equations. This methodology was suggested in [22] as a procedure to approximate the orthogonal dynamics in non-linear systems and was further developed and investigated in detail in [23]. To gain insight into the orthogonal dynamics, we consider a simple linear system.…”

Section: A the Orthogonal Dynamics Equation And The Memory Kernelmentioning

confidence: 99%

Non-Markovian closure models for large eddy simulations using the Mori-Zwanzig formalism

Parish

2017

Phys. Rev. Fluids

This work uses the Mori-Zwanzig (M-Z) formalism, a concept originating from non-equilibrium statistical mechanics, as a basis for the development of coarse-grained models of turbulence. The mechanics of the generalized Langevin equation (GLE) are considered and insight gained from the orthogonal dynamics equation is used as a starting point for model development. A class of sub-grid models is considered which represent non-local behavior via a finite memory approximation (Stinis, P., "Mori-Zwanzig reduced models for uncertainty quantification I: Parametric uncertainty," arXiv:1211.4285, 2012.), the length of which is determined using a heuristic that is related to the spectral radius of the Jacobian of the resolved variables. The resulting models are intimately tied to the underlying numerical resolution and are capable of approximating non-Markovian effects. Numerical experiments on the Burgers equation demonstrate that the M-Z-based models can accurately predict the temporal evolution of the total kinetic energy and the total dissipation rate at varying mesh resolutions. The trajectory of each resolved mode in phase-space is accurately predicted for cases where the coarse-graining is moderate. LES of homogeneous isotropic turbulence and the Taylor Green Vortex show that the M-Z-based models are able to provide excellent predictions, accurately capturing the sub-grid contribution to energy transfer. Lastly, LES of fully developed channel flow demonstrate the applicability of M-Z-based models to non-decaying problems. It is notable that the form of the closure is not imposed by the modeler, but is rather derived from the mathematics of the coarse-graining, highlighting the potential of M-Z-based techniques to define LES closures.