Efficient Reduced-order Model Research Articles

Localized Model Reduction in Porous Media Flow

This paper introduces a new localized approach to construct an efficient reduced order model for fluid flow simulation and optimization in porous media flow. For nonlinear systems, one of the most common methodology used is the proper orthogonal decomposition (POD) combined with discrete empirical interpolation method (DEIM) due to its computational efficiency and good approximation. Whereas regular POD-DEIM approximates the fine scale model with just one single reduced subspace, the localized POD (LPOD) and localized DEIM (LDEIM) that are introduced in this work compute several local subspaces. Each subspace characterize a region of solutions and all together they not only can approximate the high fidelity model better, but also reduced the computational cost of simulation. LPOD and LDEIM use classification approach to find these regions in the offline computational phase. After obtaining each class, POD and DEIM is applied to construct the basis of the reduced space. In the online phase, at each time step, the reduced states and functions will be used to find the most representative basis for POD and DEIM without requiring fine scale information. The advantages of LPOD and LDEIM are shown in a numerical example of two phase flow in porous media.

Computational design of patterned interfaces using reduced order models.

Patterning is a familiar approach for imparting novel functionalities to free surfaces. We extend the patterning paradigm to interfaces between crystalline solids. Many interfaces have non-uniform internal structures comprised of misfit dislocations, which in turn govern interface properties. We develop and validate a computational strategy for designing interfaces with controlled misfit dislocation patterns by tailoring interface crystallography and composition. Our approach relies on a novel method for predicting the internal structure of interfaces: rather than obtaining it from resource-intensive atomistic simulations, we compute it using an efficient reduced order model based on anisotropic elasticity theory. Moreover, our strategy incorporates interface synthesis as a constraint on the design process. As an illustration, we apply our approach to the design of interfaces with rapid, 1-D point defect diffusion. Patterned interfaces may be integrated into the microstructure of composite materials, markedly improving performance.

Efficient Surrogate Modelling of Nonlinear Aerodynamics in Aerostructural Coupling Schemes

This paper considers the substitution of the computational-fluid-dynamics solver within an aerostructural coupling scheme by an efficient reduced-order surrogate model to realize high-fidelity nonlinear aeroelastic analyses within a fraction of the time compared to the corresponding coupled computational-fluid-dynamics analysis. The presented surrogate model approach is a combination of parameter reduction via proper orthogonal decomposition and system identification methods designed to cover nonlinear aerodynamic effects such as viscosity as well as transient effects. The investigated test case is the thick NLR7301 airfoil that shows significant nonlinear aerodynamic behavior along with nonnegligible viscous effects. The model is identified from a set of transient forced motion computational-fluid-dynamics analyses. After identification, the model is used to predict the discrete surface force distribution of the NLR7301 airfoil in static as well as transient coupled analyses. It is shown that the static aeroelastic equilibrium and the limit cycle oscillation case is predicted by the surrogate model with sufficient accuracy.

Interpolation-based reduced-order modelling for steady transonic flows via manifold learning

This paper presents a parametric reduced-order model (ROM) based on manifold learning (ML) for use in steady transonic aerodynamic applications. The main objective of this work is to derive an efficient ROM that exploits the low-dimensional nonlinear solution manifold to ensure an improved treatment of the nonlinearities involved in varying the inflow conditions to obtain an accurate prediction of shocks. The reduced-order representation of the data is derived using the Isomap ML method, which is applied to a set of sampled computational fluid dynamics (CFD) data. In order to develop a ROM that has the ability to predict approximate CFD solutions at untried parameter combinations, Isomap is coupled with an interpolation method to capture the variations in parameters like the angle of attack or the Mach number. Furthermore, an approximate local inverse mapping from the reduced-order representation to the full CFD solution space is introduced. The proposed ROM, called Isomap+I, is applied to the two-dimensional NACA 64A010 airfoil and to the 3D LANN wing. The results are compared to those obtained by proper orthogonal decomposition plus interpolation (POD+I) and to the full-order CFD model.

Real-Time Ensemble Control with Reduced-Order Modeling

The control of spatially distributed systems is often complicated by significant uncertainty about system inputs, both time-varying exogenous inputs and time-invariant parameters. Spatial variations of uncertain parameters can be particularly problematic in geoscience applications, making it difficult to forecast the impact of proposed controls. One of the most effective ways to deal with uncertainties in control problems is to incorporate periodic measurements of the system's states into the control process. Stochastic control provides a convenient way to do this, by integrating uncertainty, monitoring, forecasting, and control in a consistent analytical framework. This paper describes an ensemble-based approach to closed-loop stochastic control that relies on a computationally efficient reduced-order model. The use of ensembles of uncertain parameters and states makes it possible to consider a range of probabilistic performance objectives and to derive real-time controls that explicitly account for uncertainty. The process divides naturally into forecast/update and forecast/control steps carried out recursively and initialized with a prior ensemble that describes parameter uncertainty. The performance of the ensemble controller is investigated here with a numerical experiment based on a solute transport control problem. This experiment evaluates the performance of open- and closed-loop controllers with full and reduced-order models as well as the performance obtained with a controller based on perfect knowledge of the system and the nominal performance obtained with no control. The experimental results show that a closed-loop controller that relies on measurements consistently performs better than an open-loop controller that does not. They also show that a reduced-order forecasting model based on offline simulations gives nearly the same performance as a significantly more computationally demanding full-order model. Taken together, these results confirm that reduced-order ensemble closed-loop control is a flexible and efficient option for uncertain spatially distributed systems.

Efficient Predictions of Unsteady Viscous Flows Around Bluff Bodies by Aerodynamic Reduced Order Models

This paper describes an efficient reduced order model (ROM) applied in the aerodynamic analysis of bluff bodies. The proposed method, which is based on eigensystem realization algorithm (ERA), uses the impulse response of the system obtained by computational fluid dynamics (CFD) analysis to construct a ROM that can accurately predict the response of the system to any arbitrary input. In order to study the performance of the proposed technique, three different geometries including elliptical and rectangular sections as well as the deck cross section of Great Belt Bridge (GBB) were considered. The aerodynamic coefficients of the impulse responses of the three sections are used to construct the corresponding ROM for each section. Then, the aerodynamic coefficients from an arbitrary sinusoidal input obtained by CFD are compared with the predicted one using the ROM. The results presented illustrate the ability of the proposed technique to predict responses of the systems to arbitrary sinusoidal and other generic inputs, with significant savings in terms of CPU time when compared with most CFD codes. The methodology described in this paper has wide application in many offshore engineering problems where flexible structures interact with unsteady fluid flow, and should be useful in preliminary design, in design optimization, and in control algorithm development.

A reduced-order model for groundwater flow equation with random hydraulic conductivity: Application to Monte Carlo methods

[1] We present a model-order reduction technique that overcomes the computational burden associated with the application of Monte Carlo methods to the solution of the groundwater flow equation with random hydraulic conductivity. The method is based on the Galerkin projection of the high-dimensional model equations onto a subspace, approximated by a small number of pseudo-optimally chosen basis functions (principal components). To obtain an efficient reduced-order model, we develop an offline algorithm for the computation of the parameter-independent principal components. Our algorithm combines a greedy algorithm for the snapshot selection in the parameter space and an optimal distribution of the snapshots in time. Moreover, we introduce a residual-based estimation of the error associated with the reduced model. This estimation allows a considerable reduction of the number of full system model solutions required for the computation of principal components. We demonstrate the robustness of our methodology by way of numerical examples, comparing the empirical statistics of the ensemble of the numerical solutions obtained using the traditional Monte Carlo method and our reduced model. The numerical results show that our methodology significantly reduces the computational requirements (CPU time and storage) for the solution of the Monte Carlo simulation, ensuring a good approximation of the mean and variance of the head. The analysis of the empirical probability density functions at the observation wells suggests that our reduced model produces good results and is most accurate in the regions with large drawdown.

Position-Dependent Multibody Dynamic Modeling of Machine Tools Based on Improved Reduced Order Models

Dynamic response of a machine tool structure varies along the tool path depending on the changes in its structural configurations. The productivity of the machine tool varies as a function of its frequency response function (FRF) which determines its chatter stability and productivity. This paper presents a computationally efficient reduced order model to obtain the FRF at the tool center point of a machine tool at any desired position within its work volume. The machine tool is represented by its position invariant substructures. These substructures are assembled at the contacting interfaces by using novel adaptations of constraint formulations. As the tool moves to a new position, these constraint equations are updated to predict the FRFs efficiently without having to use computationally costly full order finite element or modal models. To facilitate dynamic substructuring, an improved variant of standard component mode synthesis method is developed which automates reduced order determination by retaining only the important modes of the subsystems. Position-dependent dynamic behavior and chatter stability charts are successfully simulated for a virtual three axis milling machine, using the substructurally synthesized reduced order model. Stability lobes obtained using the reduced order model agree well with the corresponding full-order system.

Proper orthogonal decomposition for model reduction of a vibroimpact system

The application that inspires this work is the percussion drilling. This problem has impacts and presents uncertainties. In this first analysis the focus is on the construction of an efficient reduced-order model to deal with the nonlinear dynamics due to the impacts. It is important to have an efficient reduced-order model to perform the stochastic analysis. The simplified full model is constructed using the finite element method, and three different bases are used to construct the reduced-order models: LIN-basis (composed by the normal modes of the associated linear problems), PODdir-basis (obtained through proper orthogonal decomposition -direct method) and PODsnap-basis (obtained through proper orthogonal decomposition -snapshot method). The shapes of the elements of LIN-basis, PODdir-basis, and PODsnap-basis are compared. One important conclusion is that the information necessary to represent the details of a vibroimpact dynamics, measured by the proper orthogonal values, is more than the usual 99% recommended.

An Efficient Reduced-Order Model to Investigate the Behavior of an Imperfect Microbeam Under Axial Load and Electric Excitation

In this study an efficient reduced-order model for a MEMS device is developed and investigations of the nonlinear static and the dynamic behavior are performed. The device is constituted of an imperfect microbeam under an axial load and an electric excitation. The imperfections, typically due to microfabrication processes, are simulated assuming a shallow arched initial shape. The axial load is deliberately added with an elevated value. The structure has a bistable static configuration of double potential well with possibility of escape. We derive a single-mode reduced-order model via the Ritz technique and the Padé approximation. This model, while simple, is able to combine both a sufficient accuracy, which enables to detect the main qualitative features of the device response up to elevated values of electrodynamic excitation, and a remarkable computational efficiency, which is essential for systematic global nonlinear dynamic simulations. We illustrate the nonlinear phenomena arising in the device, such as the coexistence of various competing in-well and cross-well attractors, which leads to a considerable versatility of behavior. We discuss their physical meaning and their practical relevance for the engineering design of the microstructure, since this is an uncommon and very attractive aspect in applications.

Advanced Computational Dissipative Structural Acoustics and Fluid-Structure Interaction in Low-and Medium-Frequency Domains. Reduced-Order Models and Uncertainty Quantification

This paper presents an advanced computational method for the prediction of the responses in the frequency domain of general linear dissipative structural-acoustic and fluid-structure systems, in the low-and medium-frequency domains and this includes uncertainty quantification. The system under consideration is constituted of a deformable dissipative structure that is coupled with an internal dissipative acoustic fluid. This includes wall acoustic impedances and it is surrounded by an infinite acoustic fluid. The system is submitted to given internal and external acoustic sources and to the prescribed mechanical forces. An efficient reduced-order computational model is constructed by using a finite element discretization for the structure and an internal acoustic fluid. The external acoustic fluid is treated by using an appropriate boundary element method in the frequency domain. All the required modeling aspects for the analysis of the medium-frequency domain have been introduced namely, a viscoelastic behavior for the structure, an appropriate dissipative model for the internal acoustic fluid that includes wall acoustic impedance and a model of uncertainty in particular for the modeling errors. This advanced computational formulation, corresponding to new extensions and complements with respect to the state-of-the-art are well adapted for the development of a new generation of software, in particular for parallel computers.

Low-order modelling of shallow water equations for sensitivity analysis using proper orthogonal decomposition

This article presents a reduced-order model (ROM) of the shallow water equations (SWEs) for use in sensitivity analyses and Monte-Carlo type applications. Since, in the real world, some of the physical parameters and initial conditions embedded in free-surface flow problems are difficult to calibrate accurately in practice, the results from numerical hydraulic models are almost always corrupted with uncertainties. The main objective of this work is to derive a ROM that ensures appreciable accuracy and a considerable acceleration in the calculations so that it can be used as a surrogate model for stochastic and sensitivity analyses in real free-surface flow problems. The ROM is derived using the proper orthogonal decomposition (POD) method coupled with Galerkin projections of the SWEs, which are discretised through a finite-volume method. The main difficulty of deriving an efficient ROM is the treatment of the nonlinearities involved in SWEs. Suitable approximations that provide rapid online computations of the nonlinear terms are proposed. The proposed ROM is applied to the simulation of hypothetical flood flows in the Bordeaux breakwater, a portion of the ‘Rivière des Prairies' located near Laval (a suburb of Montreal, Quebec). A series of sensitivity analyses are performed by varying the Manning roughness coefficient and the inflow discharge. The results are satisfactorily compared to those obtained by the full-order finite volume model.

Theoretical and experimental investigation of the nonlinear response of an electrically actuated imperfect microbeam

In this study a theoretical and experimental investigation of the nonlinear response of an electrically actuated microbeam is performed. A mechanical model is proposed, which accounts for two common imperfections of microbeams, due to microfabrications, which are the compliant support conditions and the initially deformed profile. A computationally efficient single-mode reduced-order model is derived by combining the Ritz technique and the Pade approximation. Numerical simulations of the harmonic response of the device near primary resonance are shown illustrating nonlinear phenomena arising in the device response. Experimental investigation is conducted on a polysilicon imperfect microbeam confirming the simulation results. The concurrence between the theoretical results and the experimental data reveals that this model, while simple, is capable of properly capturing the response both at low and, especially, at higher electrodynamic voltages.

Aerodynamic System Modeling based on Proper Orthogonal Decomposition

The main goal of present paper is to construct an efficient reduced order model (ROM) for aerodynamic system modeling. Proper Orthogonal Decomposition (POD) is presented to address the problem. First, the snapshots are collected to form the POD kernel, and then Singular Values Decomposition (SVD) is used to obtain POD modes, finally PODROM can be constructed by projecting full order aerodynamic system to POD modes subspace. Two problems are addressed: (1) aerodynamic data inverse design; (2) aeroelastic structure active control. For the second problem, POD method with balanced modification is introduced to improve the robustness of original POD method. Results in problem (1) suggest POD method works efficiently not only for interpolation inverse design but also for extrapolation problems. The results in problem (2) demonstrate POD method with balanced modification is efficient and accurate enough for aeroelastic system analysis.

A computationally efficient reduced order model to generate multi-parameter fluid-thermal databases

A reduced order model (ROM) is proposed to generate multi-parameter databases of some fluid-thermal problems, using a combination of proper orthogonal decomposition, a gradient-like method, and a continuation method. The resulting ROM greatly reduces the CPU time required by slower methods based on genetic algorithm formulations. As a byproduct, the number of required snapshots is also reduced, which yields an additional improvement of the computational efficiency. The work presented in this article aims to facilitate the use of ROMs in industrial environments, in which time is a very important asset. The methodology is illustrated with the non-isothermal flow past a backward-facing step in the laminar regime, which is a representative problem, related to the engineering design of micro-heat sinks.

Aero-Elastic Active Control Modeling via Improved POD Method

The main goal of present paper is to construct an efficient reduced order model (ROM) for aerodynamic system modeling. Proper Orthogonal Decomposition (POD) is presented to address the problem. First, the snapshots are collected to form the POD kernel, and then Singular Values Decomposition (SVD) is used to obtain POD modes, finally POD-ROM can be constructed by projecting full order aerodynamic system to POD modes subspace. However, the robustness of ROM constructed via conventional POD method is not guaranteed. To improve the robustness of conventional POD method, balanced truncation modification was introduced. Aero-elastic active control case was chosen for this new method validation. The results demonstrate POD method with balanced modification is efficient and accurate enough for aeroelastic system analysis.

An efficient reduced-order model for low Mach number reacting flows using auxiliary variables

In the nonisothermal gaseous flows through open vessels with chemical reaction and combustion processes, the flow can be approximated with compressible Navier–Stokes equation where density depends on temperature but is independent of velocity and pressure fields, which is called the low mach number approximation. To implement advanced estimation and control schemes such as Kalman filtering and optimal feedback control to low Mach number flows, it is necessary to find an accurate and reliable reduced-order model. One of the best methods available for this purpose is the Karhunen–Loève Galerkin (KLG) procedure. The application of the KLG procedure to low Mach number flows, however, is never straightforward because some variables depend on temperature nonlinearly or exponentially and the velocity field is not solenoidal which prohibits the elimination of pressure in the Galerkin procedure. In the present investigation, we have overcome these difficulties by introducing appropriate auxiliary variables and have successfully derived a reduced-order model for low Mach number flows with radiative heat transfer and chemical reactions. The resulting reduced-order model is found to yield accurate results efficiently. The present work sets up the first stage for the real-time estimation and control of various gaseous nonisothermal reactors and combustors.

The accuracy and efficiency of a reduced-order model for transient flow analysis in gas pipelines

A reduced-order modeling approach for natural gas transient flow in pipelines is proposed. The Euler equations are considered as the governing equations and solved numerically using the implicit Steger–Warming flux vector splitting method (FSM). Next, the linearized form of the equations is derived and the corresponding eigensystem is obtained. Then, a few dominant flow eigenmodes are used to construct an efficient reduced-order model. Two well-known test cases are presented to demonstrate the accuracy and the computational efficiency of the proposed method. The results obtained using the proposed reduced-order model, are in good agreement with those of the direct numerical method. Moreover, it is shown that the proposed reduced-order model is more efficient than the conventional numerical techniques for transient flow analysis of natural gas in pipelines.

The Influence of Higher-Order Mode Shapes for Reduced-Order Models of Electrostatically Actuated Microbeams

Reduced-order models for micro-electromechanical structures possess several attractive features when compared with computational approaches using, e.g., finite-element packages. However, also within the business of reduced-order modeling, there are different approaches that yield different results. The efficiency of such approaches has to be judged according to, first, the purposes and aims of the model and, second, according to computational expenses and modeling efforts. This paper deals specifically with the frequently asked question of how many modes have to be considered in the discretization procedure to ensure an efficient reduced-order model. A consistent nonlinear continuum model is employed to describe a doubly clamped microbeam subject to two cases of electromechanical actuation. The analysis, confined to the static behavior, concentrates on two discretization techniques and addresses the differences between the final reduced-order models, accordingly. The results show significant differences with respect to the number of implemented linear-undamped mode shape functions, which are used as basis functions in the approximation procedure. This is demonstrated for the two mentioned distinct excitation schemes of the doubly clamped microbeam. The purposes of this paper are twofold. First, it draws attention to the differences between reduced-order models, which have been discretized one way or the other according to investigation goals and purposes. Second, it serves as a guideline for future micro- and nano-electromechanical system modeling by elaborating the advantages and disadvantages of both techniques.

A novel model reduction method based on balanced truncation

The main goal of this paper is to construct an efficient reduced-order model (ROM) for unsteady aerodynamic force modeling. Balanced truncation (BT) is presented to address the problem. For conventional BT method, it is necessary to compute exact controllability and observability grammians. Although it is relatively straightforward to compute these matrices in a control setting where the system order is moderate, the technique does not extend easily to high order systems. In response to the challenge, snapshots-BT (S-BT) method is introduced for high order system ROM construction. The outline idea of the S-BT method is that snapshots of primary and dual system approximate the controllability and observability matrices in the frequency domain. The method has been demonstrated for 3 high order systems: (1) unsteady motion of a two-dimensional airfoil in response to gust, (2) AGARD 445.6 wing aeroelastic system, and (3) BACT (benchmark active control technology) standard aeroservoelastic system. All the results indicate that S-BT based ROM is efficient and accurate enough to provide a powerful tool for unsteady aerodynamic force modeling.