Accuracy Of Reduced-order Model Research Articles

Data-driven closure of projection-based reduced order models for unsteady compressible flows

A data-driven closure modeling based on proper orthogonal decomposition (POD) temporal modes is used to obtain stable and accurate reduced order models (ROMs) of unsteady compressible flows. Model reduction is obtained via Galerkin and Petrov–Galerkin projection of the non-conservative compressible Navier–Stokes equations. The latter approach is implemented using the least-squares Petrov–Galerkin (LSPG) technique and the present methodology allows pre-computation (i.e., not requiring hyper-reduction) of both Galerkin and LSPG coefficients. Closure is performed by adding linear and non-linear coefficients to the original ROMs and minimizing the error with respect to the POD temporal modes. In order to further reduce the computational cost of the ROMs, an accelerated greedy missing point estimation (MPE) hyper-reduction method is employed. A canonical compressible cylinder flow is first analyzed and serves as a benchmark. The second problem studied consists of the turbulent flow over a plunging airfoil undergoing deep dynamic stall. For both cases, regularization is required and an iterative Tikhonov methodology is proposed. For the first case, linear and non-linear closure coefficients are both low in intrusiveness, capable of providing results in excellent agreement with the full order model. Regularization of calibrated models is also straightforward for this case. On the other hand, the dynamic stall flow is significantly more challenging, specially when only linear coefficients are used. Results show that non-linear calibration coefficients outperform their linear counterparts when a POD basis with fewer modes is used in the reconstruction. However, determining a correct level of regularization is more complicated with non-linear coefficients. Hyper-reduced models show good results when combined with non-linear calibration and an appropriate sized POD basis.

Direct computation of nonlinear mapping via normal form for reduced-order models of finite element nonlinear structures

The direct computation of the third-order normal form for a geometrically nonlinear structure discretised with the finite element (FE) method, is detailed. The procedure allows to define a nonlinear mapping in order to derive accurate reduced-order models (ROM) relying on invariant manifold theory. The proposed reduction strategy is direct and simulation free, in the sense that it allows to pass from physical coordinates (FE nodes) to normal coordinates, describing the dynamics in an invariant-based span of the phase space. The number of master modes for the ROM is not a priori limited since a complete change of coordinate is proposed. The underlying theory ensures the quality of the predictions thanks to the invariance property of the reduced subspace, together with their curvatures in phase space that accounts for the non-resonant nonlinear couplings. The method is applied to a beam discretised with 3D elements and shows its ability in recovering internal resonance at high energy. Then a fan blade model is investigated and the correct prediction given by the ROMs are assessed and discussed. A method is proposed to approximate an aggregate value for the damping, that takes into account the damping coefficients of all the slave modes, and also using the Rayleigh damping model as input. Frequency–response curves for the beam and the blades are then exhibited, showing the accuracy of the proposed method.

Detecting internal resonances during model reduction

Model order reduction of geometrically nonlinear dynamic structures is often achieved via a static condensation procedure, whereby high-frequency modes are assumed to be quasi-statically coupled to a small set of lower frequency modes, which form the reduction basis. This approach is mathematically justifiable for structures characterized by slow/fast dynamics, such as thin plates and slender beams, and has been shown to provide highly accurate results. Nevertheless, selecting the reduction basis without a priori knowledge of the full-order dynamics is a challenging task; retaining redundant modes will lead to computationally suboptimal reduced-order models (ROMs), while omitting dynamically significant modes will lead to inaccurate results, and important features such as internal resonances may not be captured. In this study, we demonstrate how the error associated with static condensation can be efficiently approximated during model reduction. This approximate error can then be used as the basis of a method for predicting when dynamic modal interactions will occur, which will guide the reduction basis selection process. Equivalently, this may serve as a tool for verifying the accuracy of ROMs without the need for full-order simulations. The proposed method is demonstrated using a simple oscillator and a finite element model of a clamped–clamped beam.

DYNAMIC MODE DECOMPOSITION FOR CONSTRUCTION OF REDUCED-ORDER MODELS OF HYPERBOLIC PROBLEMS WITH SHOCKS

Construction of reduced-order models (ROMs) for hyperbolic conservation laws is notoriously challenging mainly due to the translational property and nonlinearity of the governing equations. While the Lagrangian framework for ROM construction resolves the translational issue, it is valid only before a shock forms. Once that occurs, characteristic lines cross each other and projection from a high-fidelity model space onto a ROM space distorts a moving grid, resulting in numerical instabilities. We address this grid distortion issue by developing a physics-aware dynamic mode decomposition (DMD) method based on hodograph transformation. The latter provides a map between the original nonlinear system and its linear counterpart, which coincides with the Koopman operator. This strategy is consistent with the spirit of physics-aware DMDs in that it retains information about shock dynamics. Several numerical examples are presented to validate the proposed physics-aware DMD approach to constructing accurate ROMs.

Accurate Reduced-Order Models for Heterogeneous Coherent Generators

We introduce a novel framework to approximate the aggregate frequency dynamics of coherent generators. By leveraging recent results on dynamics concentration of tightly connected networks, and frequency weighted balanced truncation, a hierarchy of reduced-order models is developed. This hierarchy provides increasing accuracy in the approximation of the aggregate system response, outperforming existing aggregation techniques.

L1-based calibration of POD-Galerkin models of two-dimensional unsteady flows

This paper discusses a physics-informed methodology aimed at reconstructing efficiently the fluid state of a system. Herein, the generation of an accurate reduced order model of two-dimensional unsteady flows from data leverages on sparsity-promoting statistical learning techniques. The cornerstone of the approach is l1 regularised regression, resulting in sparsely-connected models where only the important quadratic interactions between modes are retained. The original dynamical behaviour is reproduced at low computational costs, as few quadratic interactions need to be evaluated. The approach has two key features. First, interactions are selected systematically as a solution of a convex optimisation problem and no a priori assumptions on the physics of the flow are required. Second, the presence of a regularisation term improves the predictive performance of the original model, generally affected by noise and poor data quality. Test cases are for two-dimensional lid-driven cavity flows, at three values of the Reynolds number for which the motion is chaotic and energy interactions are scattered across the spectrum. It is found that: (A) the sparsification generates models maintaining the original accuracy level but with a lower number of active coefficients; this becomes more pronounced for increasing Reynolds numbers suggesting that extension of these techniques to real-life flow configurations is possible; (B) sparse models maintain a good temporal stability for predictions. The methodology is ready for more complex applications without modifications of the underlying theory, and the integration into a cyber-physical model is feasible.

Lagrangian Reduced Order Modeling Using Finite Time Lyapunov Exponents

There are two main strategies for improving the projection-based reduced order model (ROM) accuracy—(i) improving the ROM, that is, adding new terms to the standard ROM; and (ii) improving the ROM basis, that is, constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose two new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct two new Lagrangian ROMs, which we denote α-ROM and λ-ROM. We show that both Lagrangian ROMs are more accurate than the standard Eulerian ROMs, that is, ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). In particular, the α-ROM can be orders of magnitude more accurate than the standard Eulerian ROMs. We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs’ accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis.

Reduced-Order Models for Ranking Damage Initiation in Dual-Phase Composites Using Bayesian Neural Networks

The design and development of materials with increased damage resilience is often impeded by the difficulty in establishing the precise linkages, with quantified uncertainty, between the complex details of the internal structure of materials and their potential for damage initiation. We present herein a novel machine-learning-based approach for establishing reduced-order models (ROMs) that relate the microstructure of a material to its susceptibility to damage initiation. This is accomplished by combining the recently established materials knowledge system framework with toolsets such as feedforward neural networks and variational Bayesian inference. The overall approach is found to be versatile for training scalable and accurate ROMs with quantified prediction uncertainty for the propensity to damage initiation for a variety of microstructures. The approach is applicable to a large class of challenges encountered in multiscale materials design efforts.

Data-driven variational multiscale reduced order models

We propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS–ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS–ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS–ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS–ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient ν=10−3; (ii) a 2D flow past a circular cylinder at Reynolds numbers Re=100, Re=500, and Re=1000; (iii) the quasi-geostrophic equations at Reynolds number Re=450 and Rossby number Ro=0.0036; and (iv) a 2D flow over a backward facing step at Reynolds number Re=1000. The numerical results show that the data-driven VMS–ROM is significantly more accurate than standard ROMs.

Reduced-Order Modeling of Nuclear Reactor Kinetics Using Proper Generalized Decomposition

In this work, we attempt to overcome the “curse of dimensionality” inherent to neutron diffusion kinetics problems by employing a novel reduced-order modeling technique known as proper generalized decomposition (PGD). The novelty of this work is that it represents the first attempt at applying PGD reduced-order modeling to time-dependent multigroup neutron diffusion kinetics. The performance of PGD reduced-order models (ROMs) will be quantified by comparing PGD ROMs to reference high-fidelity solutions using Rattlesnake for the TWIGL problem, a standard reactor kinetics benchmark. We show that for problems that exhibit sufficient spatial regularity, our proposed PGD algorithm computes accurate ROMs in less time than the reference high-fidelity calculation. By considering a variation of the TWIGL benchmark that maintains an analogous delayed supercritical behavior but has a smooth spatial solution, we compute PGD ROMs with a maximum relative difference in total power of less than 2.2% using 103 fewer degrees of freedom and a speedup of nearly 13× when compared to reference solutions. However, when introducing the stronger spatial irregularities of the reference benchmark, the accuracy and timing of the proposed PGD algorithm diminishes. We show that by using continuous finite elements, PGD ROMs are subject to undesirable numerical oscillations. In this paper, we motivate the use of PGD in neutron diffusion kinetics, discuss the adopted mathematical framework, and using our results, discuss the challenges and unique aspects of our implementation.

Integration of moment equations in a reduced-order modeling strategy for Monte Carlo simulations of groundwater flow

We illustrate and test an approach grounded on embedding moment equations (MEs) of groundwater flow within a Monte Carlo based modeling strategy to yield a Reduced-Order Model (ROM) that enables the efficient and accurate evaluation of probability distributions of hydraulic heads in randomly heterogeneous transmissivity fields. The projection space determining the accuracy of the ROM solution is typically computed through the principal component analysis of a selected number of full system model solutions (the so-called snapshots). Computationally expensive sensitivity analyses are then required to assess the independence of the ROM from the snapshots. Here, we propose to compute the projection vectors upon relying on the hydraulic head covariance evaluated from the solution of corresponding MEs of groundwater flow.Our workflow to compute hydraulic head distributions is organized according to the following steps: (i) approximation of mean hydraulic head and head covariance matrix through (second-order accurate) solutions of MEs; (ii) computation of the leading eigenvectors of the head covariance matrix to form the basis set for the ROM projection space; and (iii) construction of the ROM. Sample probability density functions of hydraulic heads are then efficiently obtained via Monte Carlo simulations relying on the developed ROM. The proposed methodology is compared against snapshot-based ROMs and the full system model in a two- and a three-dimensional steady-state groundwater flow setting where pumping from a point source is superimposed to a mean uniform flow. Our results show that the projection space computed by relying on MEs provides a more accurate ROM solution than the one resulting from reliance on snapshots.

An investigation into the electrostatic force representation for electrically actuated microelectromechanical systems

The ever-increasing demand for microelectromechanical systems (MEMS) in modern electronics has reinforced the need for extremely accurate analytical and reduced-order models to aid in the design of MEMS devices. Many MEMS designs consist of cantilever beams with a tip mass attached at the free end to act as a courter electrode for electrical actuation. One critical modeling aspect of electrically actuated MEMS is the electrostatic force that drives these systems. The two most used representations in the literature approximate the electrostatic force between two electrodes as a point force. In this work, the effects of the representation of the electrostatic force for electrically actuated microelectromechanical systems are investigated. The system under investigation is composed of a beam with an electrode attached to its end. The distributed force, rigid body, and point mass electrostatic force representations are modeled, studied, and their output results are compared qualitatively. Static and frequency analyses are carried out to investigate the influences of the electrostatic force representation on the static pull-in, fundamental natural frequency, and mode shape of the system. A nonlinear distributed-parameter model is then developed in order to determine and characterize the response of electrically actuated systems when considering various representation of the electrostatic forces. The results show that the size of the electrode may strongly affect the natural frequencies and static pull-in when the point mass, rigid body, and plate representations are considered. From nonlinear analysis, it is also proven that the representation may affect the hardening behavior of the system and its dynamic pull-in. This modeling and analysis give guidelines about the usefulness of the electrostatic force representations and possible erroneous assumptions that can be made which may result in inaccurate design and optimal performance detection for electrostatically actuated systems.

Deep-learning-based surrogate model for reservoir simulation with time-varying well controls

A new deep-learning-based reduced-order modeling (ROM) framework is proposed for application in subsurface flow simulation. The reduced-order model is based on an existing embed-to-control (E2C) framework and includes an auto-encoder, which projects the system to a low-dimensional subspace, and a linear transition model, which approximates the evolution of the system states in low dimension. In addition to the loss function for data mismatch considered in the original E2C framework, we introduce a physics-based loss function that penalizes predictions that are inconsistent with the governing flow equations. The loss function is also modified to emphasize accuracy in key well quantities of interest (e.g., fluid production rates). The E2C ROM is shown to be analogous to an existing ROM, POD-TPWL, which has been extensively developed for subsurface flow simulation. The new ROM is applied to oil-water flow in a heterogeneous reservoir, with flow driven by nine wells operating under time-varying control specifications. A total of 300 high-fidelity training simulations are performed in the offline stage, and the network training requires 10-12 minutes on a Tesla V100 GPU node. Online (runtime) computations achieve speedups of over a factor of 1000 relative to full-order simulations. Extensive test case results, with well controls varied over large ranges, are presented. Accurate ROM predictions are achieved for global saturation and pressure fields at particular times, and for injection and production well responses as a function of time. Error is shown to increase when 100 or 200 (rather than 300) training runs are used to construct the E2C ROM. Overall the E2C ROM is shown to provide reliable predictions with levels of perturbations in the well controls that are much larger than those used with existing POD-TPWL treatments. The current model is however limited to 2D systems, and the required number of training simulations is much larger than that for POD-based ROMs.

An [formula omitted]-error bound for time-limited balanced truncation

Model order reduction (MOR) is often applied to spatially-discretized partial differential equations to reduce their order and hence decrease computational complexity. A reduced system can be obtained, e.g., by time-limited balanced truncation, a method that aims to construct an accurate reduced order model on a given finite time interval [0,T]. This particular balancing related MOR technique is studied in this paper. An LT2-error bound based on the truncated time-limited singular values is proved and is the main result of this paper. The derived error bound converges (as T→∞) to the well-known H∞-error bound of unrestricted balanced truncation, a scheme that is used to construct a good reduced system on the entire time line. The techniques within the proofs of this paper can also be applied to unrestricted balanced truncation so that a relatively short time domain proof of the H∞-error bound is found here.

Lagrangian dynamic mode decomposition for construction of reduced-order models of advection-dominated phenomena

Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are two complementary singular-value decomposition (SVD) techniques that are widely used to construct reduced-order models (ROMs) in a variety of fields of science and engineering. Despite their popularity, both DMD and POD struggle to formulate accurate ROMs for advection-dominated problems because of the nature of SVD-based methods. We investigate this shortcoming of conventional POD and DMD methods formulated within the Eulerian framework. Then we propose a Lagrangian-based DMD method to overcome this so-called translational problem. Our approach is consistent with the spirit of physics-aware DMD since it accounts for the evolution of characteristic lines. Several numerical tests are presented to demonstrate the accuracy and efficiency of the proposed Lagrangian DMD method.

Memory embedded non-intrusive reduced order modeling of non-ergodic flows

Generating a digital twin of any complex system requires modeling and computational approaches that are efficient, accurate, and modular. Traditional reduced order modeling techniques are targeted at only the first two, but the novel nonintrusive approach presented in this study is an attempt at taking all three into account effectively compared to their traditional counterparts. Based on dimensionality reduction using proper orthogonal decomposition (POD), we introduce a long short-term memory neural network architecture together with a principal interval decomposition (PID) framework as an enabler to account for localized modal deformation. As an effective partitioning tool for breaking the Kolmogorov barrier, our PID framework, therefore, can be considered a key element in the accurate reduced order modeling of convective flows. Our applications for convection-dominated systems governed by Burgers, Navier-Stokes, and Boussinesq equations demonstrate that the proposed approach yields significantly more accurate predictions than the POD-Galerkin method and could be a key enabler toward near real-time predictions of unsteady flows.

Evaluating reduced order models of curved beams for random response prediction using static equilibrium paths

Thin curved structural components are susceptible to dynamic snap-through when subjected to the high amplitude aerodynamic loading that is present in extreme environments. In order to estimate the life of the structures, time simulations must extend over many seconds so that the statistical response due to the random loading environment can be adequately characterized. In order to compute the geometrically nonlinear response of these structures in a reasonable time, Reduced Order Models (ROMs) have been developed that reduce the computational burden dramatically. However, the accuracy of a ROM can vary drastically depending on how it is created, specifically depending on which modes are included within the basis set and the magnitude of the static loads used to estimate the nonlinear stiffness coefficients in the ROM. This work presents a procedure to check that a ROM is accurate in dynamic snap through prior to computing the dynamic response. The Riks method is used to compute the static equilibrium path, or force-displacement behavior, for both the FE model and the ROM, and by comparing these one gains insight into the character of the ROM. Two curved beams are studied to validate the procedure and we identify load levels and mode sets that produce accurate ROMs for these structures. The ROMs are further validated by computing the dynamic response, and the results show that, for these two structures, if the ROM accurately predicts the static snap through behavior of the model to a uniformly distributed static load, then it also correctly reproduces the dynamic response to a uniform pressure that varies randomly in time.

Reduced order models with local property dependent transfer coefficients for real time simulations of monolith reactors

We present a novel and accurate reduced order model with local property dependent internal and external mass transfer coefficients for real time simulations of monolith reactors. We compare the accuracy of the present approach with other reduced order models in the literature and with exact (numerical) solution of the detailed washcoat diffusion model. We illustrate the application of the reduced order models with three examples: a four reaction model of a three-way catalytic converter (TWC) to determine the steady-state as well as transient behavior in real time; a simplified TWC model with oxygen storage and release, and the case of three reversible reactions with linear kinetics. We show that the use of local property dependent transfer coefficients in terms of the Jacobian of the reaction rate vector leads to the best accuracy followed by its diagonal approximation, followed by the asymptotic (slow reaction) approximation for the internal Sherwood number. It is shown that the proposed method has sufficient accuracy for most practical applications, especially in the washcoat diffusion controlled regime, while speeding up the calculations by orders of magnitude compared to detailed washcoat diffusion models.

A nonlinear data-driven reduced order model for computational homogenization with physics/pattern-guided sampling

Developing an accurate nonlinear reduced order model from simulation data has been an outstanding research topic for many years. For many physical systems, data collection is very expensive and the optimal data distribution is not known in advance. Thus, maximizing the information gain remains a grand challenge. In a recent paper, Bhattacharjee and Matouš (2016) proposed a manifold-based nonlinear reduced order model for multiscale problems in mechanics of materials. Expanding this work here, we develop a novel sampling strategy based on the physics/pattern-guided data distribution. Our adaptive sampling strategy relies on enrichment of sub-manifolds based on the principal stretches and rotational sensitivity analysis. This novel sampling strategy substantially decreases the number of snapshots needed for accurate reduced order model construction (i.e., ∼5× reduction of snapshots over Bhattacharjee and Matouš (2016)). Moreover, we build the nonlinear manifold using the displacement rather than deformation gradient data. We provide rigorous verification and error assessment. Finally, we demonstrate both localization and homogenization of the multiscale solution on a large particulate composite unit cell.

Development of Geometrically Accurate Continuum-Based Tire Models for Virtual Testing

AbstractIn this paper, an approach based on the integration of computer-aided design and analysis (I-CAD-A) is used to develop new continuum-based finite element (FE) tire models for the small and large deformation analyses. Based on given tire specifications, the mechanics-based geometry/analysis absolute nodal coordinate formulation (ANCF) is used to define the tire geometry with the same degree of accuracy as B-splines and nonuniform rational B-spline (NURBS), widely used in the computer-aided design (CAD) systems. In the case of large deformations, the ANCF geometry can be used directly as the analysis mesh without the need for conversion or adjustments. In order to define the material parameters that characterize the ANCF tire composite structure, a virtual test rig is developed, and the tire calibration process is performed according to the standards defined by the Society of Automotive Engineers (SAE). In order to develop small-deformation models that can be used in the prediction of the tire frequencies and mode shapes, the ANCF position vector gradients are consistently written in terms of rotation parameters, leading to geometrically accurate floating frame of reference (FFR) finite elements, referred to as ANCF/FFR elements. Using this mechanics-based geometry/analysis approach, new geometrically accurate reduced-order tire models are systematically developed and used to define vibration equations for the prediction of the tire frequencies, which are verified using a commercial FE software. The element stiffness matrix is calculated using the general continuum mechanics approach (GCM), and the effectiveness of the strain split method (SSM) for locking alleviation is tested. The results obtained in this investigation show that the I-CAD-A tire modeling approach can be used to develop geometrically accurate tire models suited for the large-deformation multibody system (MBS) problems as well as for the prediction of the tire frequencies and mode shapes.