Reduced-order Model Research Articles

Reduced-Order Modeling of Steady and Unsteady Flows with Deep Neural Networks

Large-eddy and direct numerical simulations generate vast data sets that are challenging to interpret, even for simple geometries at low Reynolds numbers. This has increased the importance of automatic methods for extracting significant features to understand physical phenomena. Traditional techniques like the proper orthogonal decomposition (POD) have been widely used for this purpose. However, recent advancements in computational power have allowed for the development of data-driven modal reduction approaches. This paper discusses four applications of deep neural networks for aerodynamic applications, including a convolutional neural network autoencoder, to analyze unsteady flow fields around a circular cylinder at Re = 100 and a supersonic boundary layer with Tollmien–Schlichting waves. The autoencoder results are comparable to those obtained with POD and spectral POD. Additionally, it is demonstrated that the autoencoder can compress steady hypersonic boundary-layer profiles into a low-dimensional vector space that is spanned by the pressure gradient and wall-temperature ratio. This paper also proposes a convolutional neural network model to estimate velocity and temperature profiles across different hypersonic flow conditions.

Reduced-order modeling of modular, position-dependent systems with translating interfaces

Many complex mechatronic systems consist of multiple interconnected dynamical subsystems, which are designed, developed, analyzed, and manufactured by multiple independent teams. To support such a design approach, a modular model framework is needed to reduce computational complexity and, at the same time, enable multiple teams to develop and analyze the subsystems in parallel. In such a modular framework, the subsystem models are typically interconnected by means of a static interconnection structure. However, many complex dynamical systems exhibit position-dependent behavior (e.g., induced by translating interfaces) which cannot be captured by such static interconnection models. In this paper, a modular model framework is proposed, which allows to construct an interconnected system model, which captures the position-dependent behavior of systems with translating interfaces, such as linear guide rails, through a position-dependent interconnection structure. Additionally, this framework allows to apply model reduction on subsystem level, enabling a more effective reduction approach, tailored to the specific requirements of each subsystem. Furthermore, we show the effectiveness of this framework on an industrial wire bonder. Here, we show that including a position-dependent model of the interconnection structure (1) enables to accurately model the dynamics of a system over the operating range of the system and, (2) modular model reduction methods can be used to obtain a computationally efficient interconnected system model with guaranteed accuracy specifications.

Projection-based reduced-order modelling of time-periodic problems, with application to flow past flapping hydrofoils

Simulating forced time-periodic flows in industrial applications presents significant computational challenges, partly due to the need to overcome costly transients before achieving time-periodicity. Reduced-order modelling emerges as a promising method to speed-up computations. We extend upon the work of Lotz et al. (2024) where a time-periodic space–time model is introduced. We present a time-periodic reduced-order model that directly finds the time-periodic solution without requiring extensive time integration. The reduced-order model gives a reduction in variables in both space and time. Our approach involves a POD-Galerkin reduced-order model based on a time-periodic full-order model that employs isogeometric analysis, residual-based variational multiscale turbulence modelling and weak boundary conditions. The projection-based reduced-order model inherits these features. We evaluate the reduced-order model with numerical experiments on moving hydrofoils. The motion is known a priori and we restrict ourselves to two spatial dimensions. In these experiments we vary the Strouhal and Reynolds numbers, and the motion profile respectively. Reduced-order model solutions agree well with those of the full-order model. The errors over the entire time period of thrust and lift forces are less than 0.2%. This includes complex scenarios such as the transition from drag to thrust production with increasing Strouhal number. Our time-periodic reduced-order model offers speed-ups ranging from O(102) to O(103) compared to the full-order model, depending upon the basis size. This makes it an appealing solution for prescribed time-periodic problems, with potential for additional speedup through nonlinear reduction techniques such as hyper-reduction.

Model order reduction in parallel of discrete-time linear systems based on Meixner and Krawtchouk polynomials

Abstract In this paper, based on the partition technique, we use Meixner and Krawtchouk polynomials to present an input-independent model order reduction method. Our main contributions are twofold. First, the explicit difference relations of Meixner polynomials and Krawtchouk polynomials are expressed in an unified form. The parallel computation is carried out on the partitioned subsystems using the Krylov subspaces by which one can generate reduced systems independent of the expansion coefficients of input and can save the computation time. Second, a parallel adaptive enrichment strategy is used to choose the reduced order of reduced systems. Theoretical analysis shows that the proposed method characterizes the property of invariable coefficients. Finally, two numerical examples demonstrate that the proposed method achieves good reduction results in terms of accuracy and reduced CPU time.

Non-intrusive reduced order models for partitioned fluid–structure interactions

The main goal of this work is to develop a data-driven Reduced Order Model (ROM) strategy from high-fidelity simulation result data of a Full Order Model (FOM). The goal is to predict at lower computational cost the time evolution of solutions of Fluid–Structure Interaction (FSI) problems. For some FSI applications, the elastic solid FOM (often chosen as quasi-static) can take far more computational time than the fluid one. In this context, for the sake of performance one could only derive a ROM for the structure and try to achieve a partitioned FOM fluid solver coupled with a ROM solid one. In this paper, we present a data-driven partitioned ROM on two study cases: (i) a simplified 1D-1D FSI problem representing an axisymmetric elastic model of an arterial vessel, coupled with an incompressible fluid flow; (ii) an incompressible 2D wake flow over a cylinder facing an elastic solid with two flaps. We evaluate the accuracy and performance of the proposed ROM-FOM strategy on these cases while investigating the effects of the model’s hyperparameters. We demonstrate a high prediction accuracy and significant speedup achievements using this strategy.

Mode selection for component mode synthesis with guaranteed assembly accuracy

In this work, a modular approach is introduced to select the most important eigenmodes for each component of a composed structural dynamics system to obtain the required accuracy of the reduced-order assembly model. To enable the use of models of complex (structural) dynamical systems in engineering practice, e.g., in a design, optimization and/or control context, the complexity of the models needs to be reduced. When the model consist of an assembly of multiple interconnected structural components, component mode synthesis is often the preferred model reduction method. The standard approach to component mode synthesis for such system is to select the eigenmodes of a component that are most important to accurately model the dynamic behaviour of this component in a certain frequency range of interest. However, often, a more relevant goal is to obtain, in this frequency range, an accurate model of the assembly. In the proposed approach, accuracy requirements on the level of the assembly are translated to accuracy requirements on component level, by employing techniques from the field of systems and control. With these component-level requirements, the eigenmodes that are most important to accurately model the dynamic behaviour of the assembly can be selected in a modular fashion. We demonstrate with two structural dynamics benchmark systems that this method based on assembly accuracy allows for a computationally efficient selection of eigenmodes that (1) guarantees satisfaction of the assembly accuracy requirements and (2) results in most cases in reduced-order models of significantly lower order with respect to the industrial standard approach in which component eigenmodes are selected using a frequency criterion.

Feasible model-order reduction approach for analysis of composite rotor blades based on geometrically exact beam formulation

This paper presents a novel and practical reduced-order model (ROM) approach designed specifically for analyzing composite rotor blades, leveraging the geometrically exact beam formulation. The slender nature of rotor blades necessitates aero-structure coupled analysis to precisely predict their response, which involved numerous iterative computations. By capturing large displacements and rotations, this approach enhances the accuracy of blade behavior predictions under operational conditions. To address the computational demands of large-scale analyses, a ROM technique that effectively reduces system dimensionality while preserving critical structural features is introduced. The proposed method is then applied to the aero-structure interaction analysis of a hingeless hovering rotor trim, incorporating geometrical nonlinearity and centrifugal effects. The analysis results demonstrate that the ROM accurately captures the dynamics of these complex aeroelastic systems while significantly reducing computational costs. This approach exhibits significant potential for various aerospace applications, particularly in the design and analysis of structures undergoing geometrical nonlinear behavior and rotation-induced loads.

Bayesian reduced-order deep learning surrogate model for dynamic systems described by partial differential equations

We propose a reduced-order deep-learning surrogate model for dynamic systems described by time-dependent partial differential equations. This method employs space–time Karhunen–Loève expansions (KLEs) of the state variables and space-dependent KLEs of space-varying parameters to identify the reduced (latent) dimensions. Subsequently, a deep neural network (DNN) is used to map the parameter latent space to the state variable latent space.An approximate Bayesian method is developed for uncertainty quantification (UQ) in the proposed KL-DNN surrogate model. The KL-DNN method is tested for the linear advection–diffusion and nonlinear diffusion equations, and the Bayesian approach for UQ is compared with the deep ensembling (DE) approach, commonly used for quantifying uncertainty in DNN models. It was found that the approximate Bayesian method provides a more informative distribution of the PDE solutions in terms of the coverage of the reference PDE solutions (the percentage of nodes where the reference solution is within the confidence interval predicted by the UQ methods) and log predictive probability. The DE method is found to underestimate uncertainty and introduce bias.For the nonlinear diffusion equation, we compare the KL-DNN method with the Fourier Neural Operator (FNO) method and find that KL-DNN is 10% more accurate and needs less training time than the FNO method.

Local reduced subspaces of subchannel‐inspired subdomains

AbstractIn this article, we study an update of the traditional subchannel approximation utilizing local reduced order bases. Through employing the symmetries and periodicity of a 7‐pin bundle, the global domain is decomposed into numerous repeating subdomains following several dividing strategies. We locally study the reduced basis generated by proper orthogonal decomposition. We analyze the similarities, assessing the truncation error and the distance between the linear subspaces spanned by the reduced bases. We focus on the first stage of building a reduced order model, the generation of the reduced subspace, which is usually not regarded in detail in our application problem. Our assessment related to flow blockage in liquid metal‐cooled nuclear reactors, a postulated high‐risk accident that results in potential fuel damage.

MIRACLE—a microphone array impulse response dataset for acoustic learning

This work introduces a large dataset comprising impulse responses of spatially distributed sources within a plane parallel to a planar microphone array. The dataset, named MIRACLE, encompasses 856,128 single-channel impulse responses and includes four different measurement scenarios. Three measurement scenarios were conducted under anechoic conditions. The fourth scenario includes an additional specular reflection from a reflective panel. The source positions were obtained by uniformly discretizing a rectangular source plane parallel to the microphone for each scenario. The dataset contains three scenarios with a spatial resolution of 23mm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$23\\,\ extrm{mm}$$\\end{document} at two different source-plane-to-array distances, as well as a scenario with a resolution of 5mm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5\\,\ extrm{mm}$$\\end{document} for the shorter distance. In contrast to existing room impulse response datasets, the accuracy of the provided source location labels is assessed and additional metadata, such as the directivity of the loudspeaker used for excitation, is provided. The MIRACLE dataset can be used as a benchmark for data-driven modelling and interpolation methods as well as for various acoustic machine learning tasks, such as source separation, localization, and characterization. Two timely applications of the dataset are presented in this work: the generation of microphone array data for data-driven source localization and characterization tasks and data-driven model order reduction.

Automated construction of effective potential via algorithmic implicit bias

We introduce a novel approach for decomposing and learning every scale of a given multiscale objective function in Rd, where d⩾1. This approach leverages a recently demonstrated implicit bias of the optimization method of gradient descent [44], which enables the automatic generation of data that nearly follow Gibbs distribution with an effective potential at any desired scale. One application of this automated effective potential modeling is to construct reduced-order models. For instance, a deterministic surrogate Hamiltonian model can be developed to substantially soften the stiffness that bottlenecks the simulation, while maintaining the accuracy of phase portraits at the scale of interest. Similarly, a stochastic surrogate model can be constructed at a desired scale, such that both its equilibrium and out-of-equilibrium behaviors (characterized by auto-correlation function and mean path) align with those of a damped mechanical system with the original multiscale function being its potential. The robustness and efficiency of our proposed approach in multi-dimensional scenarios have been demonstrated through a series of numerical experiments. A by-product of our development is a method for anisotropic noise estimation and calibration. More precisely, Langevin model of stochastic mechanical systems may not have isotropic noise in practice, and we provide a systematic algorithm to quantify its covariance matrix without directly measuring the noise. In this case, the system may not admit closed form expression of its invariant distribution either, but with this tool, we can design friction matrix appropriately to calibrate the system so that its invariant distribution has a closed form expression of Gibbs.

Accelerating particle-in-cell kinetic plasma simulations via reduced-order modeling of space-charge dynamics using dynamic mode decomposition.

We present a data-driven reduced-order modeling of the space-charge dynamics for electromagnetic particle-in-cell (EMPIC) plasma simulations based on dynamic mode decomposition (DMD). The dynamics of the charged particles in kinetic plasma simulations such as EMPIC is manifested through the plasma current density defined along the edges of the spatial mesh. We showcase the efficacy of DMD in modeling the time evolution of current density through a low-dimensional feature space. Not only do such DMD-based predictive reduced-order models help accelerate EMPIC simulations, they also have the potential to facilitate investigative analysis and control applications. We demonstrate the proposed DMD-EMPIC scheme for reduced-order modeling of current density and speedup in EMPIC simulations involving electron beam under the influence of magnetic field, virtual cathode oscillations, and backward wave oscillator.

Dynamic Mode Decomposition With Gaussian Process Regression for Control of High-Dimensional Nonlinear Systems

Abstract In this work, we consider the problem of learning a reduced-order model of a high-dimensional stochastic nonlinear system with control inputs from noisy data. In particular, we develop a hybrid parametric/nonparametric model that learns the “average” linear dynamics in the data using dynamic mode decomposition with control (DMDc) and the nonlinearities and model uncertainties using Gaussian process (GP) regression and compare it with total least-squares dynamic mode decomposition (tlsDMD), extended here to systems with control inputs (tlsDMDc). The proposed approach is also compared with existing methods, such as DMDc-only and GP-only models, in two tasks: controlling the stochastic nonlinear Stuart–Landau equation and predicting the flowfield induced by a jet-like body force field in a turbulent boundary layer using data from large-scale numerical simulations.

Theoretical and experimental study of unsteady behavior of marine diesel engine under fluctuating back pressure condition

In order to meet the increasingly urgent emission reduction requirements, underwater exhaust has emerged as a promising technology for marine diesel engines and has received widespread attention in recent years. However, under fluctuating high back pressure conditions, the diesel engine as a whole operates in an unsteady state, with significant fluctuations in output power and exhaust temperature, posing a serious threat to their stability and safety. This paper focuses on the unsteady operating characteristics of diesel engines and its impact on the engine output power, efficiency and safety under fluctuating high back pressure conditions. Firstly, based on model order reduction and the first Lyapunov approximation theorem, the theoretical derivation reveals that the filling and emptying effects in the intake and exhaust pipes and the shaft moment of inertia of the turbocharger are the main causes of the unsteady of the engine under fluctuating high back pressure conditions. The factors influencing the strength of engine unsteadiness, including wave period, wave amplitude, mean back pressure and engine properties, are identified. Then, diesel engine experiments under different fluctuating high back pressure conditions are performed and the conclusions derived from the theoretical analysis are validated through experiments. Based on the research results mentioned above, this paper proposes a criterion to determine the unsteadiness of diesel engines. Through comparison with experimental data, it is found that this criterion effectively reflects the strength of engine unsteadiness, with a correlation coefficient of 0.9169 with the unsteady parameter.

Optimization of air distribution and coal blending in pulverized coal boilers for high-temperature corrosion prevention based on POD reduced-order modeling

High-temperature corrosion in coal-fired boilers poses a significant threat to safe operation. However, there is currently a lack of effective online monitoring and optimization methods for high-temperature corrosion. Therefore, this study proposes a novel approach to rapidly predict the distribution of chemical species and evaluate high-temperature corrosion degrees, along with optimizing operating conditions to prevent high-temperature corrosion. Firstly, a total of 564 Computational Fluid Dynamics (CFD) simulations are performed on a 330 MW tangentially fired boiler, covering various operating conditions, including coal blending, air distribution, boiler load, etc., to obtain a database of O2, CO, and H2S distributions within the boiler. Then a method based on Proper Orthogonal Decomposition (POD) and Support Vector Regression (SVR) is used to process the database to realize real-time prediction of boiler chemical species distribution, which is next utilized as inputs of a high-temperature corrosion evaluation to obtain in-situ corrosion degree distribution. Finally, Particle Swarm Optimization (PSO) is used to optimize coal blending and air distribution schemes, effectively reducing severe corrosion ratio from 36.34 % to 10.04 % in a typical case by improving the atmosphere near the water walls. This study thus provides a new perspective for online boiler diagnostics and digital twin construction, particularly by achieving online monitoring of high-temperature corrosion and optimizing operating conditions to prevent corrosion.

Semi-decentralized temperature control in district heating systems

The supply temperature in district heating systems has traditionally been controlled using feedforward – a robust and well-validated approach for district heating networks with few producers and relatively high supply temperatures. The transition towards lower temperature district heating networks allows for efficient reuse of excess heat from, e.g., industrial processes and data centers. Excess heat is often intermittent, cannot always be assumed to be possible to control with a centralized controller, and can cause temperature disturbances in the grid. Closing the loop using PID control is challenging due to the process’s time-varying nature and long time delays. Model Predictive Control (MPC) suffers from a higher complexity, long computational times, and the need for a well-validated and maintained centralized model. The paper suggests a semi-decentralized approach using the Smith predictor with an event-driven assignment of active controllers and sensors. A reduced order model based on a more comprehensive state space model is derived and used for gain scheduling and input–output pairing using the normalized relative gain array. The focus is on temperature disturbance rejection, and appropriate tuning rules and controller structures are suggested. Simulation results show that the proposed control structure can handle various types of temperature disturbances, even in the presence of model estimation errors.

High-resolution and reduced-order multi-physics simulation on thermal–hydraulic and thermoelectric characteristics of the thermionic space reactor core

The simulation of a thermionic space reactor core necessitates a comprehensive consideration of the intricate interplay between fluid flow, heat transfer, and thermionic energy conversion, constituting a multi-physics coupling challenge. Given the absence of a thermionic conversion model in contemporary CFD (Computational Fluid Dynamics) codes, accurately simulating the thermal–hydraulic and thermo-electric behavior of thermionic reactors in existing CFD software is particularly challenging. To analyze the multi-physics coupling phenomena in thermionic reactors, this paper revises the conjugate heat transfer solver in the proprietary 3D (three-dimensional) finite-volume CFD code, mFVM (modular Finite-Volume Method Code), to enable bidirectional coupling with the thermionic conversion model. This enhancement aims to predict the high-resolution three-dimensional temperature field and thermoelectric characteristics of the thermionic reactor core. Furthermore, a reduced-order multi-physics model of the thermionic reactor core is established using RESYS (Reactor System Simulation Code). The numerical simulation outcomes for both the high-resolution and reduced-order multi-physics models, pertaining to electrically heated and fission-heated TFEs (Thermionic Fuel Elements), are validated against experimental data. The results indicate that the emitter electron cooling effects caused by thermionic emission reduced the maximum temperature of emitter by about 200 K for both electrically heated and fission-heated TFE. Therefore, a bidirectional coupling between the thermal–hydraulic model and the thermionic conversion model is essential for precisely calculating the temperature field within the TFE and the thermionic reactor core. Thereafter, both the high-resolution model and the reduced-order model are utilized to investigate the thermal–hydraulic and thermoelectric characteristics of the TOPAZ-II reactor core. The calculated electrical power of the TOPAZ-II reactor is 5.39 kWe using the mFVM model and 5.47 kWe using the RESYS model. Finally, an analysis of the thermoelectric conversion characteristics based on the simulation results of the reduced-order model reveals that the TOPAZ-II reactor system is capable of load-following only when the load resistance exceeds 0.06 Ω under nominal full power operation conditions.

Nonlinear Reduced Order Modelling for Korteweg-de Vries Equation

Efficient computational techniques that maintain the accuracy and invariant preservation property of the Korteweg-de Vries (KdV) equations have been studied by a wide range of researchers. In this paper, we introduce a reduced order model technique utilizing kernel principle component analysis (KPCA), a nonlinear version of the classical principle component analysis, in a non-intrusive way. The KPCA is applied to the data matrix, which is formed by the discrete solution vectors of KdV equation. In order to obtain the discrete solutions, the finite differences are used for spatial discretization, and linearly implicit Kahan's method for the temporal one. The back-mapping from the reduced dimensional space, is handled by a non-iterative formula based on the idea of multidimensional scaling (MDS) method. Through KPCA, we illustrate that the reduced order approximations conserve the invariants, i.e., Hamiltonian, momentum and mass structure of the KdV equation. The accuracy of reduced solutions, conservation of invariants, and computational speed enhancements facilitated by classical (linear) PCA and KPCA are exemplified through one-dimensional KdV equation.

A novel data-driven reduced-order model for the fast prediction of gas-solid heat transfer in fluidized beds

Computational fluid dynamics − discrete element method (CFD-DEM) emerges as a promising tool to model dense gas–solid two-phase flow in fluidized beds, yet the numerical efficiency is still far from being satisfactory. Accordingly, this work develops a proper orthogonal decomposition (POD) reduced-order model (ROM) based on the CFD-DEM method for the fast prediction of gas–solid flow and heat transfer in a bubbling fluidized bed (BFB). The POD method is employed to decompose particle coordinates and temperature into spatial modes and time evolution coefficients. The radial basis function neural network (RBFNN) and temporal convolutional neural network (TCN) are utilized to predict the time evolution coefficients. Regarding the POD modes, the particle temperature exhibits an obvious monotonic or periodic characteristic compared to the particle coordinates, indicating the potential for long-term prediction. The RBFNN-ROM and TCN-ROM reconstruct the flow field effectively, achieving acceleration ratios of 2000 and 3000, respectively. The ROM developed in this study holds the potential to enable real-time prediction of industrial processes, thus paving the way for the realization of digital twins.

Solid–fluid force modeling: Insights from comparing a reduced order model for a pair of particles with resolved CFD-DEM

Solid–fluid force models are essential to efficiently model multiple industrial apparatuses such as fluidized beds, spouted beds, and slurry transport. They are generally built using strong hypotheses (e.g. fully developed flow and no relative motion between particles) that affect their accuracy. We study the effect of these hypotheses on particle dynamics using the sedimentation of a pair of particles. We develop new induced drag, lift and torque models for pairs of particles based on an artificial neural network (ANN) regression. The fluid force model covers a range of Reynolds numbers of 0.1 to 100 and particle centroid distance of up to 9 particle diameters. The ANN model uses 3475 computational fluid dynamics (CFD) simulation results as the training data set. Using this fluid force model, we develop a reduced-order model (ROM), which includes the virtual mass force, the Meshchersky force, the history force, the lubrication force, and the Magnus force. Using the results of a resolved computational fluid dynamics coupled with a discrete element method (CFD-DEM) model as a reference, we analyze the discrepancies between the ROM and CFD-DEM results for a series of sedimentation cases that cover particle Archimedes number from 20 to 2930 and particle to fluid density ratio of 1.5 to 1000. The errors primarily stem from particle history interactions that are not accounted for by the fully developed flow hypothesis. The importance of this effect on the dynamic of two particles is isolated and it is shown that it is more pronounced in cases with a lower particle-to-fluid density ratio (such as solid–liquid cases). This work underscores the need for more research on these effects to increase the precision of solid–fluid force models for small particle-to-fluid density ratios (1.5).