Reduced-order Model Research Articles

An enhanced sparrow search algorithm for model order reduction of proton exchange membrane fuel cell system

An enhanced sparrow search algorithm for model order reduction of proton exchange membrane fuel cell system

Nonlinear unsteady aerodynamic forces prediction and aeroelastic analysis of wind-induced bridge response at multiple wind speeds: A deep learning-based reduced-order model

Machine learning-based aerodynamic reduced-order models (ROMs) combine high accuracy with extremely low computational costs, making them highly effective in predicting nonlinear and unsteady bridge aerodynamic forces. Although several machine learning-based nonlinear aerodynamic models have been developed, the majority are built on a single wind speed parameter. However, in nonlinear aerodynamic prediction and aeroelastic analysis of bridges, the variability in incoming wind speed significantly influences the computed results. A ROM relying solely on a single wind speed lacks the ability to accurately forecast the intricate dynamic behaviors arising from changes in wind speed. When the incoming wind speed changes, the model's prediction accuracy significantly decreases. Usually, it is necessary to establish a new database and train a new model, which not only increases time and cost but also greatly reduces the convenience of the ROM. Addressing this challenge, this study proposes a multiple-wind-speed (MWS) nonlinear unsteady bridge aerodynamic model based on the LSTM deep neural network. Taking the Taohuayu Yellow River Bridge in the Henan Province of China as an example, the modeling process of the proposed MWS-ROM is demonstrated, along with non-linear aerodynamic predictions of the deck under various conditions and aerodynamic-elastic analysis of the deck under different wind speeds. The research results show that the trained LSTM network can accurately predict the nonlinear aerodynamic forces of bridges under single and double degrees of freedom vibration conditions. The MWS-ROM performed well in predicting convergent vibrations at low wind speeds and limits cycle oscillations at high wind speeds, aligning closely with results from the CFD full-order model. Compared to CFD, the aerodynamic ROM based on the LSTM network significantly enhances computational efficiency, consequently boosting the convenience and efficiency of bridge flutter analysis. Additionally, the methodology proposed herein can be extended for wind-induced vibration control and response prediction in other types of deck sections.

Application of a non-intrusive reduced order modeling approach to magnetohydrodynamics

Magnetohydrodynamics (MHD) investigates the intricate relationship between electromagnetism and fluid dynamics, offering a complete insight into the behavior of conducting fluids under the influence of magnetic fields. This theory plays a pivotal role in the framework of magnetic confinement fusion, where it can be applied to describe both thermonuclear plasmas confined inside the vacuum vessel and operating fluids, such as liquid metals and molten salts, flowing within the blanket of future tokamaks. Currently, the state-of-the-art numerical modeling of MHD scenarios employs a multi-physics framework to examine the interplay between magnetic fields and thermal hydraulics; however, due to the complexity of the involved physics, detailed models are required, resulting in a significant computational burden. In this regard, reduced order modeling (ROM) techniques may represent a promising solution, as they enable approximating complex systems with lower-dimensional models. Indeed, ROM methodologies can significantly reduce the required computational time while maintaining accuracy in capturing the convoluted physics involved in fusion reactors, especially in the contexts of sensitivity analysis, uncertainty quantification, and control. Despite their potential, ROM methods are relatively under-explored within the MHD framework; this study applies ROM techniques to MHD scenarios, focusing on their capabilities and possible limitations. To this aim, the backward-facing step, which is well suited for exploring the effects of different magnetic fields on turbulent dynamics, is adopted as case study. In particular, this work evaluates the potentialities of the ROM approach in enhancing computational efficiency within the MHD domain. Each of the methods evaluated was effective in precisely reconstructing flow dynamics at any given time and across the full range of magnetic field values tested while significantly reducing computational costs compared to full-order simulations. Practically, this study demonstrates the feasibility to create simplified models that accurately represent the magnetohydrodynamic flows of fluids within the blanket.

Nonlinear Reduced Order Modeling of Heated Structures with Temperature-Dependent Properties

This paper focuses on extending coupled structural–thermal reduced order modeling approaches for the prediction of the nonlinear geometric response of heated structures to efficiently account for temperature-dependent structural and thermal properties. Structurally, a linear dependence of the elasticity tensor and thermal expansion coefficient is assumed consistently with typical material behavior. The resulting structural reduced order model (ROM) equations exhibit polynomials of the temperature generalized coordinates, the coefficients of which are readily identified. A different strategy is proposed for conductance and capacitance properties which typically depend nonlinearly on temperature. It relies on splitting the temperature generalized coordinates into a small set of dominant ones having a nonlinear effect on the ROM conductances and capacitances and a much larger set affecting them linearly. The inclusion of uncertainty in the structural ROM is also addressed extending earlier work limited to temperature independent properties. These strategies are exemplified on a representative hypersonic vehicle panel undergoing a full mission profile and with aero-thermal–structural coupling. The ROM predictions are observed to be very close to the full finite element ones for the mean model. Moreover, an uncertainty analysis demonstrates the strong sensitivity of the response to the structural model in the strongly nonlinear regions of the trajectory.

Accuracy and Computational Cost Comparison of Numerical Flight Dynamics Reduced-Order Models

This paper compares several approaches to model the unsteady and nonlinear longitudinal aerodynamics of a generic unmanned combat air vehicle configuration based on computational fluid dynamics (CFD). Three different reduced-order models (ROMs) are implemented through unsteady simulations to predict the evolution of forces and moments during prescribed trajectories carried out at M=0.15. The first model investigates a linear quasi-steady model based on a first-order Taylor series expansion of the aerodynamic coefficients. The second one picks up the mathematical formulation of the first model but adds a dynamic coefficient in order to take into account the unsteady effect of rotation rate variations. Finally, the third model is built on indicial functions computed from positive step changes in the angle of attack or pitch rate. The unknown parameters of the first two models and the unknown indicial functions of the third one are obtained, respectively, from the study of forced oscillations and step motions. These prescribed trajectories are performed using unsteady Reynolds-averaged Navier–Stokes simulations coupled with a grid displacement tool. Two different methods are tested and compared to carry out the CFD mesh motions: an overset method and a full moving grid (FMG) method. The indicial functions obtained using the FMG method give results equivalent to those found in the literature in half the time of the overset method. To reduce the computational costs to set up the ROMs, the FMG method is preferred to the overset method. ROMs are then applied, and their predictions are compared with CFD simulations for prescribed trajectories with different angular rates and angles of attack. A study of instantaneous and overall accuracy associated with the implementation cost of the ROMs has allowed to show their strengths and their weaknesses. Similar results are observed at low angles of attack for the ROMs studied, while a major modeling advantage of the indicial method has been identified for higher angles of attack in the nonlinear domain.

Snap-through and indirect reduced-order modelling

Snap-through is a nonlinear jump phenomenon that is increasingly being designed into engineering structures. The rich and varied dynamics of snap-through present a significant bottleneck in the design process as costly dynamic simulations are required to ensure predictable behaviour. The purpose of this research is to investigate the use of a recent projection-based, reduced-order modelling technique, namely, implicit condensation and expansion with inertial compensation, to reproduce the snap-through behaviours found in bistable finite element systems in analogous models with minimal degrees of freedom. In this work, the free response of a simple mass-spring oscillator is used to differentiate different types of snap-through. This provides a framework for discussing the dynamics of a family of buckled beams. The ability of the reduced-order models to faithfully reproduce these dynamics is investigated by testing the existence and stability of predicted periodic orbits. It is shown that the minimum projection basis required to capture snap-through accurately can be found through considering the minimum energy required for different types of snap-through. For bistable beams, this can be further simplified to just requiring an eigenvalue analysis at the most unstable equilibrium.

An adaptive learning strategy for surrogate modeling of high-dimensional functions - Application to unsteady hypersonic flows in chemical nonequilibrium

Many engineering applications rely on the evaluation of expensive, non-linear high-dimensional functions. In this paper, we propose the RONAALP algorithm (Reduced Order Nonlinear Approximation with Adaptive Learning Procedure) to incrementally learn, on-the-fly as the application progresses, a fast and accurate reduced-order surrogate model of a target function. First, a combination of nonlinear auto-encoder, community clustering, and radial basis function networks allows us to learn an efficient and compact surrogate model with limited training data. Secondly, an active learning procedure overcomes any extrapolation issues during the online stage by adapting the surrogate model with high-fidelity evaluations that fall outside its current validity range. This approach results in generalizable, fast, and accurate reduced-order models of high-dimensional functions. The method is demonstrated on three direct numerical simulations of hypersonic flows in chemical nonequilibrium. Accurate simulations of these flows rely on detailed thermochemical gas models that dramatically increase the cost of such calculations. Using RONAALP to learn a reduced-order thermodynamic model surrogate on-the-fly, the cost of such simulations was reduced by up to 75% while maintaining an error of less than 10% on relevant quantities of interest.

Nonlinear parametric models of viscoelastic fluid flows.

Reduced-order models (ROMs) have been widely adopted in fluid mechanics, particularly in the context of Newtonian fluid flows. These models offer the ability to predict complex dynamics, such as instabilities and oscillations, at a considerably reduced computational cost. In contrast, the reduced-order modelling of non-Newtonian viscoelastic fluid flows remains relatively unexplored. This work leverages the sparse identification of nonlinear dynamics (SINDy) algorithm to develop interpretable ROMs for viscoelastic flows. In particular, we explore a benchmark oscillatory viscoelastic flow on the four-roll mill geometry using the classical Oldroyd-B fluid. This flow exemplifies many canonical challenges associated with non-Newtonian flows, including transitions, asymmetries, instabilities, and bifurcations arising from the interplay of viscous and elastic forces, all of which require expensive computations in order to resolve the fast timescales and long transients characteristic of such flows. First, we demonstrate the effectiveness of our data-driven surrogate model to predict the transient evolution and accurately reconstruct the spatial flow field for fixed flow parameters. We then develop a fully parametric, nonlinear model capable of capturing the dynamic variations as a function of the Weissenberg number. While the training data are predominantly concentrated on a limit cycle regime for moderate , we show that the parametrized model can be used to extrapolate, accurately predicting the dominant dynamics in the case of high Weissenberg numbers. The proposed methodology represents an initial step in applying machine learning and reduced-order modelling techniques to viscoelastic flows.

Aerostatic Stability and Bifurcation for Long-Span Bridges Based on Reduced Order Modeling via Singular Value Decomposition

Aerostatic Stability and Bifurcation for Long-Span Bridges Based on Reduced Order Modeling via Singular Value Decomposition

Dimension reduction based on time-limited cross Gramians for bilinear systems

The cross Gramian is a useful tool in model order reduction but only applicable to square dynamical systems. Throughout this paper, time-limited cross Gramians is firstly extended to square bilinear systems that satisfies a generalized Sylvester equation, and then concepts from decentralized control are used to approximate a cross Gramian for non-square bilinear systems. In order to illustrate these cross Gramians, they are calculated efficiently based on shifted Legendre polynomials and applied to dimension reduction, which leads to a lower dimensional model by truncating the states that are associated with smaller approximate generalized Hankel singular values. In combination of the dominant subspace projection method, our reduction procedure is modified to produce a bounded-input bounded-output stable-preserved reduced model under some certain conditions. At last, the performance of numerical experiments indicates the validity of our reduction methods.

Reduced-order condensed-phase kinetic models for polyethylene, polypropylene and polystyrene thermochemical recycling

Thermochemical recycling of plastic waste (PW) into chemicals and energy vectors requires coupling particle and reactor-scale simulations to accurate condensed phase pyrolysis mechanisms for each constituent. This work proposes a methodology to derive reduced-order condensed-phase kinetic models from validated semi-detailed kinetic mechanisms. Two types of kinetic models are obtained for polyethylene (PE), polypropylene (PP) and polystyrene (PS): reduced semi-detailed models and multi-step fully lumped ones. These families offer different compromises between accuracy and computational cost. The former employ 50–100 gas + liquid species and describe both the radical degradation and the detailed carbon distribution of the products. Conversely, the latter involves 5–10 species per polymer tracking only the main petroleum cuts. The kinetic mechanisms are complemented by the definition of thermochemical properties of gas, liquid, and solid-phase species, accounting for phase-transitions through pseudo-chemical reactions. Model validations are performed by comparison with experimental data and the original semi-detailed mechanisms in terms of mass loss, heat fluxes and product distribution profiles. The resulting CHEMKIN-like condensed-phase models are attached as Supplementary Material and as a GitHub repository. Extending the proposed approach to other polymers and coupling it with existing subsets in the CRECK kinetic framework (e.g., biomass, PVC, PET) offers a powerful tool to model thermochemical recycling of PW and biomass/PW mixtures.

Physics-informed machine learning approach for reduced-order modeling of integrally bladed rotors: Theory and application

Integrally bladed rotors are commonly used in aircraft and rocket turbomachinery and known to exhibit complex dynamics when subject to operational loading conditions. Though nominally cyclic symmetric structures, in practice, cyclic symmetry is destroyed due to mistuning caused by random sector-to-sector imperfections in material properties and geometry. Simulating mistuned blisk dynamics using high-fidelity models can be computationally expensive, thus, a variety of physics-based reduced-order models have been previously developed. However, these models cannot easily incorporate experimental data nor leverage potential benefits of data-driven and machine-learning-based approaches. Here, we present a novel first-of-its-kind physics-informed machine learning modeling approach that incorporates physical laws directly into a novel network architecture while maintaining a sector-level viewpoint. The approach is combined with an assembly procedure resulting in a significantly smaller linear system based on blade-alone response data, and can directly incorporate physical response data like that measured with blade tip timing and/or traveling-wave excitation. Validation is shown using a large-scale finite-element model, with multiple traveling-wave forced-response predictions and response selection cases considered. Using only as little as a single degree of freedom per sector from the blade tip, this approach shows high accuracy relative to high-fidelity simulations.

Polytopic autoencoders with smooth clustering for reduced-order modeling of flows

With the advancement of neural networks, there has been a notable increase, both in terms of quantity and variety, in research publications concerning the application of autoencoders to reduced-order models. We propose a polytopic autoencoder architecture that includes a lightweight nonlinear encoder, a convex combination decoder, and a smooth clustering network. Supported by several proofs, the model architecture ensures that all reconstructed states lie within a polytope, accompanied by a metric indicating the quality of the constructed polytopes, referred to as polytope error. Additionally, it offers a minimal number of convex coordinates for polytopic linear-parameter varying systems while achieving acceptable reconstruction errors compared to proper orthogonal decomposition (POD). To validate our proposed model, we conduct simulations involving two flow scenarios with the incompressible Navier-Stokes equation. Numerical results demonstrate the guaranteed properties of the model, low reconstruction errors compared to POD, and the improvement in error using a clustering network.

Investigating ocean circulation dynamics through data assimilation: A mathematical study using the Stommel box model with rapid oscillatory forcings.

We investigate ocean circulation changes through the lens of data assimilation using a reduced-order model. Our primary interest lies in the Stommel box model, which reveals itself to be one of the most practicable models that has the ability of reproducing the meridional overturning circulation. The Stommel box model has at most two regimes: TH (temperature driven circulation with sinking near the north pole) and SA (salinity driven with sinking near the equator). Currently, the meridional overturning is in the TH regime. Using box-averaged Met Office EN4 ocean temperature and salinity data, our goal is to provide a probability that a future regime change occurs and establish how this probability depends on the uncertainties in initial conditions, parameters, and forcings. We will achieve this using data assimilation tools and DAPPER within the Stommel box model with fast oscillatory regimes.

Too hot to print, too slow to handle; finding optimal path characteristics for WAAM

Wire + Arc Additive Manufacturing (WAAM) is recognized as a highly capable metal additive process. However, the quality of manufactured parts is often compromised by defects stemming from process-induced temperature fields and transient weld-bead shapes, largely due to suboptimal path planning. Recent efforts have predominantly focused on optimizing either the thermal aspects or the productivity of WAAM, but the interplay between these factors and the final geometric accuracy of the parts has not been thoroughly investigated. This study introduces an automated framework for path planning optimization in WAAM, effectively bridging the gap between temperature control, geometric accuracy, and productivity. The approach utilizes a reinforcement learning agent to generate paths that minimize deposition time. These paths are subsequently segmented and optimized for their temperature response using a Monte Carlo tree search algorithm, with the thermal effects efficiently approximated through a reduced-order model. The paper presents a comparative analysis of various deposition strategies, offering insights and recommendations to enhance both productivity and part quality in WAAM path planning.

One techno-economic analysis to rule them all: Instant prediction of hydrothermal liquefaction economic performance with a machine learned analytic equation

Hydrothermal liquefaction (HTL) has remarkable potential for efficient conversion of abundant, decentralized organic wastes into renewable fuels. Because waste is a highly distributed resource with context-dependent economic viability, selection of optimal deployment sites is slowed by the need to develop detailed techno-economic analyses (TEA) for the thousands of potential deployment locations, each with their own unique combinates of scale, proximity to infrastructure/markets, and feedstock properties. An economic modeling framework that requires only easily obtainable inputs for assessing economic performance would therefore allow multiplexed analysis of many thousands of cases, whereas traditional TEA would not be possible for more than a handful of cases. Within such a context, the present study uses machine learning to guide development of a TEA and modeling framework which provides accurate cost predictions using three key inputs – feedstock cost, biocrude yield, and process scale – to estimate the minimum fuel selling price (MFSP) that an HTL process can achieve. The structure of the proposed framework is informed and based on empirical observations of cost projections made by a detailed TEA over a wide range of feedstock costs, biocrude yields, and process scales. A machine learning guided process was used to identify, train, and test a series of models using auto-generated data for training and independently reported data for testing. The most accurate model consists of three terms and requires 6 adjustable parameters to predict independently published values of MFSP (N = 28) to within an average value of ± 20.4%. It is demonstrated that the reduced-order model’s predictions fall within 40% of the corresponding published values 95% of the time, and in the worst case, the associated discrepancy is 45.9%, suggesting that the accuracy of the machine learned model is indeed comparable to the TEAs that were used to build it. Moreover, the terms in the model are physically interpretable, conferring greater reliability to the use of its predictions. The model can be used to predict the dependence of MSFP on biocrude yield, scale, and feedstock cost; interestingly, MFSP is insensitive to biocrude yield and/or scale under many situations of interest and identifying the critical value for a given application is crucial to optimizing economic performance. The proposed model can be also extended to evaluate economic performance of newly developed HTL-based processes, including catalytic HTL, and the methodological framework used in this study is deemed appropriate for the development of machine learned TEA models in cases of other similar waste-to-energy technologies.

A reduced-order hybrid model for photobioreactor performance and biomass prediction

This paper introduces a hybrid approach for photobioreactor modeling tailored to microalgae cultivation, combining data-driven and mechanistic concepts to improve modeling efficiency and practicality for industrial scale-up applications. Most growth models for microalgae are nonlinear and require experimental measurement of several parameters. The aim of this work is to develop linear practical models for monitoring purposes. A model based on linear coefficients and polynomial features is proposed, balancing interpretability with non-linear representation focusing on model transparency. To simplify the growth model, Taylor series expansion is applied to the Monod and logistic population models. Two scale-specific models are developed and evaluated, offering practical solutions for monitoring microalgae growth in photobioreactors. Therefore, this reduced order representation allows the biomass growth rate to be dependent directly on the biomass concentration. These models do not require exhaustive data collection of substrate concentration over time, making them cost-effective and efficient for industrial applications. This work provides a step forward in photobioreactor modeling, contributing to the sustainable production of microalgae.

Sliding mode control design using a generalized reduced-order fractional model for chemical processes

This article presents a comprehensive study that evaluates the potential advantages of adopting a unified, systematic, and organized general fractional model within the Sliding Mode Control (SMC) design framework. The exploration extends to the model's applicability in a variety of disciplines. Although prior research has utilized SMC and fractional-order systems independently, no previous work has established a mathematical framework that integrates a generic fractional-order SMC model. The study addresses this gap and highlights potential implications for control design and applications in a variety of fields, particularly for chemical processes.

Data-driven reduced order surrogate modeling for coronary in-stent restenosis

Background:The intricate process of coronary in-stent restenosis (ISR) involves the interplay between different mediators, including platelet-derived growth factor, transforming growth factor-β, extracellular matrix, smooth muscle cells, endothelial cells, and drug elution from the stent. Modeling such complex multiphysics phenomena demands extensive computational resources and time. Methods:This paper proposes a novel non-intrusive data-driven reduced order modeling approach for the underlying multiphysics time-dependent parametrized problem. In the offline phase, a 3D convolutional autoencoder, comprising an encoder and decoder, is trained to achieve dimensionality reduction. The encoder condenses the full-order solution into a lower-dimensional latent space, while the decoder facilitates the reconstruction of the full solution from the latent space. To deal with the 5D input datasets (3D geometry + time series + multiple output channels), two ingredients are explored. The first approach incorporates time as an additional parameter and applies 3D convolution on individual time steps, encoding a distinct latent variable for each parameter instance within each time step. The second approach reshapes the 3D geometry into a 2D plane along a less interactive axis and stacks all time steps in the third direction for each parameter instance. This rearrangement generates a larger and complete dataset for one parameter instance, resulting in a singular latent variable across the entire discrete time-series. In both approaches, the multiple outputs are considered automatically in the convolutions. Moreover, Gaussian process regression is applied to establish correlations between the latent variable and the input parameter. Results:The constitutive model reveals a significant acceleration in neointimal growth between 30−60 days post percutaneous coronary intervention (PCI). The surrogate models applying both approaches exhibit high accuracy in pointwise error, with the first approach showcasing smaller errors across the entire evaluation period for all outputs. The parameter study on drug dosage against ISR rates provides noteworthy insights of neointimal growth, where the nonlinear dependence of ISR rates on the peak drug flux exhibits intriguing periodic patterns. Applying the trained model, the rate of ISR is effectively evaluated, and the optimal parameter range for drug dosage is identified. Conclusion:The demonstrated non-intrusive reduced order surrogate model proves to be a powerful tool for predicting ISR outcomes. Moreover, the proposed method lays the foundation for real-time simulations and optimization of PCI parameters.

Kalman filter-driven state observer for thermal error compensation in machine tool digital twins

Sustainable reduction of thermal errors during production is the key challenge in modern high-precision manufacturing. Numerical compensation models provide an energy-efficient solution, but in the case of data-driven models, high-quality experimental data must be time-consuming and expensive to produce, negatively impacting overall productivity. Furthermore, robustness concerns arise in the case of new operating conditions, which were not contained in the training data. This paper presents a novel use of a Kalman filter together with model order reduced finite element models to observe the entire thermal state, which allows the subsequent solution of the mechanical model and computation of the thermal errors in real-time without requiring any training data but instead purely based on the physical system model. The effectiveness of this approach is evaluated using experiments on a thermal test bench with 16 out of 40 temperature sensors employed for observation and demonstrated on a 5-axis machine tool (MT) with 13 out of 25 temperature sensors used. Due to the combination of the reduced order model and Kalman filter these 13 temperature sensors are sufficient to represent a MT mesh of more than 350’000 elements. The entire temperature profile of the thermal test bench is reconstructed to achieve a root mean square error (RMSE) of the unobserved temperature sensors of only 2.7 °C, which accounts for more than 83% of all temperature variations and 1.3 °C for the 5-axis MT. For the thermal error of the thermal test bench, the RMSE could be reduced from 67.4μm to 33.3μm, corresponding to a reduction of 52.7 %. This could be achieved without the need for experimental data for model calibration, in a real-time capable physics-based model.