Projection-based and neural-net reduced order model for nonlinear Navier–Stokes equations

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Highlights

  • Explicit projection of the pressure Poisson equation and the predictor-corrector scheme for solving the pressure equation.

  • Dirichlet and Neumann boundary conditions are considered for general fluid dynamics problems.

  • OpenFoam platform used to build the ROMs for problems with complex geometries and boundary conditions.

  • Neural network (Nonlinear autoregressive eXogenous model) approach is explored for mapping among variables.

  • ROMs are tested for true predictions, i.e. modelling new cases that have not seen in the model construction.

Abstract

This paper presents complete steps for the construction and solution algorithm for a projection-based reduced order model coupled with an artificial neural net model. The reduced model is constructed based on the Galerkin projection of the equations governing the physical processes on reduced subspaces obtained by the Proper Orthogonal Decomposition method. The projection is done using numerical discretisation schemes implemented in the commonly used OpenFOAM platform. The neural net model is trained using the Nonlinear AutoRegressive eXogenous model network (NARX) architecture. The reduced order models are demonstrated for true predictions of incompressible flows at low Reynolds numbers driven by various boundary conditions. Compared with the full CFD simulations, this model shows excellent agreement while only requiring a fraction of the computational resources.

Introduction

Numerical modelling has been used extensively in engineering and manufacturing processes, especially in product design and testing. This trend is particularly fuelled by the continuous growth in computational power and recent advances in numerical algorithms and modelling techniques. Numerical studies are generally cheap (although still time consuming) and data descriptions obtained via such numerical simulations are complete and typically free from physical restrictions. However, using high-fidelity numerical models requires enormous computational resources and thus, is not practical for achieving real-time simulations. Conversely, the use of surrogate models can be more advantageous in providing near-real-time simulations, albeit with a potential loss in accuracy.

There are several surrogate modelling techniques that can be used, depending on the structure of the problem and modelling purposes [1]. Data-fit models are response surface methods that use interpolation or regression of numerical or experimental data to fit a model for the system output as a function of input parameters, with examples including neural net models such as the typical multilayer perceptron. However, the underlying physics is often ignored in such data-fit models, and their accuracy can depend heavily on the completeness and availability of data. Hierarchical models are physics-based models derived from higher-fidelity models by simplifying physics assumptions or using coarser grids. Important physics might be lost due to these simplifications, but they can typically run much faster. Currently, building a robust yet reliable and accurate surrogate model remains a challenging task, and an active area of research.

Projection-based reduced order models (PB-ROM) are projections of results from high-fidelity numerical models onto a reduced subspace which can be derived from a combination of high-fidelity full numerical simulations and experimental data. Important physical structures such as dynamical evolutions and spatial gradients can be retained in these reduced subspaces for further modelling. The projection-based reduced modelling approach is widely used for modelling fluid dynamical systems. Readers can refer to [2,3] for comprehensive reviews of methodologies and applications related to this approach.

This paper considers a PB-ROM approach for the surrogate model as applied to the classical fluid dynamical system of flow past a stationary cylinder at low Re. ROMs have been constructed for similar problems in recent studies [3,4]. Critically, much of the prior literature tested these ROMs only for reconstruction of flow dynamics, i.e. modelling the same transient cases that were used to construct the models, but with further extensions in time. In this work, the ROMs are not only tested for reconstruction but are extended to true predictions, i.e. modelling new cases with boundary conditions that have not been used for the model construction.

Full model simulations that constitute the data for this paper were run on the commonly-used open source platform OpenFOAM® [5]. The PB-ROM is constructed based on the Galerkin projection of the equations governing the flow on reduced subspaces obtained by Proper Orthogonal Decomposition (POD) [4,[6], [7], [8], [9] of the generated full model solutions. This lower-order representation of flow solutions has been generalised to take into account Dirichlet and Neumann boundary conditions. In addition, the numerical discretisation schemes in OpenFOAM are employed for the projection of the mathematical operators of the governing equations on the subspaces. The development of ROMs within the framework of OpenFOAM ensures consistency in the ROM's numerical discretisation with the full model and facilitates the integration of PB-ROM into the OpenFOAM platform.

While the Galerkin projection can accurately reproduce a numerical operator in the governing equations, one major drawback is that this projection can be impractical for nonlinear operators. Methods such as Discrete Empirical Interpolation Method (DEIM) [10] have been proposed to handle the nonlinear functions, but may still not work well for the projection of complex functions such as the pressure Poisson equation that is typically solved when modelling incompressible flows [11]. Although the pressure Poisson equation can be handled by the Galerkin projection, Machine Learning (ML) data fit or empirical models can be used as an alternative way to compute the pressure [4,11]. Machine Learning techniques have also been extended to areas such as turbulence modelling and flow prediction [12,13]. ML-based models have advantages over sparse polynomial regression in terms of adaptability, but face difficulties in extrapolation and large training cost [14]. In the majority of the results, ML-based models are only able to predict a few time steps from the training set [15,16].

In this study, we explore the possibility of using a ML model to approximate the pressure Poisson equation in a PB-ROM approximation of Navier–Stokes equations. The use of a ML data fit approach for mapping nonlinear relationships among pressure and velocity variables is to alleviate the difficulty in projecting nonlinear operators in the governing equations while the PB-ROM model provides the core underlying physics of the system. The Nonlinear Autoregressive Exogenous (NARX) model [17], [18], [19] is employed here as it has been shown in prior work to model time-varying complex systems well. The NARX model is a powerful method for modelling time series data as it has an important property which is the ‘memory’ of the dynamical system. In this case, it is the relationship of the variables in previous calculation.

The paper is structured as follows. Section 2 presents the construction of a ROM on OpenFOAM platform via the PB-ROM algorithm and its integration with a NARX model. The full model for incompressible flows and a brief introduction to OpenFOAM are first outlined in Section 2.1. The POD and NARX methods are briefly presented in Sections 2.2 and 2.4. The construction and solution algorithms for the reduced order models are described in detail in Sections 2.3 and 0. Simulation results and discussions are presented in Section 3. The conclusions will be in Section 4.

Section snippets

Full model of incompressible flows

Incompressible flows are governed by the Navier–Stokes (NS) equations which, in Cartesian coordinates x = (x,  y, z), have the formut=u·u+νΔu1ρp·u=0where the velocity u = (u, v, w) and the pressure p are spatio-temporal functions in a computational domain Ωf; ρ and ν are the density and kinematic viscosity of the fluid.

In Computational Fluid Dynamics (CFD), the above NS equations can be solved by the “pisoFoam” solver in OpenFOAM. The solver uses the Finite Volume Method (FVM) for spatial

Simulation results and discussions

In this section, the PB-ROM and PB-NN-ROM are demonstrated for an incompressible flow past a stationary circular cylinder. The full CFD simulations are done using OpenFOAM with 2nd-order discretisation schemes. The ensemble means, basis vectors and modal coefficients for velocity components and pressure are computed using POD method detailed in Section 2.2. The projections of the differential operators on the basis vectors are done in OpenFOAM. The NARXNET model, the PB-ROM and PB-NN-ROM

Conclusions

Development of two surrogate models, the PB-ROM which is based purely on Galerkin projection and the PB-NN-ROM that combines Galerkin projection and a neural network are presented. The models are demonstrated for classical low-Re flows past a stationary cylinder.

The PB-ROM results match very well with the full CFD solutions, not only for the reconstruction cases but also for the true prediction cases. Using the explicit pressure equation in the PB-ROM allows the flow fields to transiently

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