Reduced-order Model Research Articles

Quasinormal mode expansion in thin elastic plates

Elastic wave manipulation using large arrays of resonators is driving the need for advanced simulation and optimization methods. To address this we introduce and explore a robust framework for wave control: quasinormal modes (QNMs). Specifically, we consider the problem for thin elastic plates, where the Green's function formalism is well known and readily exploited to solve multiple scattering problems. By studying the associated nonlinear eigenvalue problem we derive a dispersive QNM expansion, providing a reduced-order model for efficient forced response computations which reveals physical insight into the resonant mode excitation. Furthermore, we derive eigenvalue sensitivities with respect to resonator parameters and apply a gradient-based optimization to design quasi-bound states in the continuum and position eigenfrequencies precisely in the complex plane. Scattering simulations validate our approach in structures such as graded line arrays and quasicrystals. Drawing on QNM concepts from electromagnetism we demonstrate significant advances in elastic metamaterials, highlighting their potential for tailored wave manipulation. Published by the American Physical Society 2024

Physics-informed two-tier neural network for non-linear model order reduction

AbstractIn recent years, machine learning (ML) has had a great impact in the area of non-intrusive, non-linear model order reduction (MOR). However, the offline training phase often still entails high computational costs since it requires numerous, expensive, full-order solutions as the training data. Furthermore, in state-of-the-art methods, neural networks trained by a small amount of training data cannot be expected to generalize sufficiently well, and the training phase often ignores the underlying physical information when it is applied with MOR. Moreover, state-of-the-art MOR techniques that ensure an efficient online stage, such as hyper reduction techniques, are either intrusive or entail high offline computational costs. To resolve these challenges, inspired by recent developments in physics-informed and physics-reinforced neural networks, we propose a non-intrusive, physics-informed, two-tier deep network (TTDN) method. The proposed network, in which the first tier achieves the regression of the unknown quantity of interest and the second tier rebuilds the physical constitutive law between the unknown quantities of interest and derived quantities, is trained using pretraining and semi-supervised learning strategies. To illustrate the efficiency of the proposed approach, we perform numerical experiments on challenging non-linear and non-affine problems, including multi-scale mechanics problems.

Self-assembly of lamellae by elastocapillarity in the presence of gravity: experiments and modeling

Abstract Elastocapillarity describes the self-organization of elastic structures under liquid surface forces, for applications to micro- and nano scale self-assembly. It has recently become evident that precise prediction of elastocapillary self-assembly is challenged by the various phenomena arising due to fluid flow. In this study, we present experiments and a corresponding reduced-order model for the dynamic coalescence of flexible lamellae arrays. Vertical arrays of elastomeric lamellae attached to a substrate at the bottom are fabricated using 3D printed molds. The lamellae have sub-mm thickness and a few millimeters height, a size scale that cannot be achieved by traditional microfabrication. The gaps between the elastic lamellae are filled with liquid while the tips of the lamellae, which are at the top with respect to gravity, are initially fixed, then they are released which leads to their spontaneous bending and coalescence. We study the cluster size as a function of system variables both experimentally and numerically. The solid lamellae are modeled as discrete rigid elements connected by torsional springs to the substrate, while the fluid flow uses lubrication theory and capillarity. Importantly, the model incorporates the gravitational forces to effectively capture the systems' behavior ranging from the microscale with negligible gravity effect to the macroscale where gravity significantly influences the dynamics. We further derive a linearized continuum model from our nonlinear formulation, providing an analytical scaling law for the lamellae cluster size. This work is relevant for new polymorphic fins and lamellae devices, capillary self-folding and origami.

Reduced Order Model Based Nonlinear Waveform Inversion for the 1D Helmholtz Equation

Reduced Order Model Based Nonlinear Waveform Inversion for the 1D Helmholtz Equation

Assessing the Effect of Coriolis Acceleration on the Coherent Structures in the Wake of a Wind Turbine using DMD

Abstract The present work aims to study the effect of veer, namely, the height-dependent lateral deflection of wind velocity due to Coriolis acceleration, on the coherent structures in the wake of the NREL 5-MW reference wind turbine using the sparsity-promoting dynamic mode decomposition (SP-DMD) method for the detection of dynamically-relevant flow structures. Large eddy simulation of the flow impacting the wind turbine is carried out by solving the filtered Navier-Stokes equations for incompressible flows. The effects of both tower and nacelle are included in the simulation using the immersed boundary method. Simulations are performed at Re ≈ 108, generating the inlet velocity profile by a precursor simulation of the atmospheric boundary layer subjected to Coriolis acceleration. The analysis of the DMD results allows to study the effects of the Coriolis acceleration on the most relevant dynamic modes in the turbine wake and to understand the basic mechanisms by which the wind veer significantly affects the wake recovery rate. Moreover, as a result of the SP-DMD methodology, the most relevant modes are extracted from the wake and a limited subset of relevant flow features that optimally approximates the original data sequence is identified. This small number of modes represent the kernel that can be employed for developing an accurate reduced order model of the wake.

A non-intrusive reduced-order model for finite element analysis of implant positioning in total hip replacements.

Sophisticated high-fidelity simulations can predict bone mass density (BMD) changes around a hip implant after implantation. However, these models currently have high computational demands, rendering them impractical for clinical settings. Model order reduction techniques offer a remedy by enabling fast evaluations. In this work, a non-intrusive reduced-order model, combining proper orthogonal decomposition with radial basis function interpolation (POD-RBF), is established to predict BMD distributions for varying implant positions. A parameterised finite element mesh is morphed using Laplace's equation, which eliminates tedious remeshing and projection of the BMD results on a common mesh in the offline stage. In the online stage, the surrogate model can predict BMD distributions for new implant positions and the results are visualised on the parameterised reference mesh. The computational time for evaluating the final BMD distribution around a new implant position is reduced from minutes to milliseconds by the surrogate model compared to the high-fidelity model. The snapshot data, the surrogate model parameters and the accuracy of the surrogate model are analysed. The presented non-intrusive surrogate model paves the way for on-the-fly evaluations in clinical practice, offering a promising tool for planning and monitoring of total hip replacements.

Reduced order modelling method for electromagnetic analysis of electrical machines

AbstractThis paper presents a workflow to create a reduced order model for the electromagnetic analysis of electrical machines over its operating range so that the stator forces can be exported for mechanical analysis. A permanent magnet synchronous motor developed within the EDISON (Electric Drivetrain Integration by Simulation and OptimizatioN) project is used as a model to validate the workflow. The response surface model approximation method is used to generate the reduced order model, which is then validated by comparing with the results from finite element analyses.

A brief review of reduced order models using intrusive and non‐intrusive techniques

AbstractReduced Order Models (ROMs) have gained a great attention by the scientific community in the last years thanks to their capabilities of significantly reducing the computational cost of the numerical simulations, which is a crucial objective in applications like real time control and shape optimization. This contribution aims to provide a brief overview about such a topic. We discuss both a classic intrusive framework based on a Galerkin projection technique and hybrid/non‐intrusive approaches, including Physics Informed Neural Networks (PINN), purely Data‐Driven Neural Networks (NN), Radial Basis Functions (RBF), Dynamic Mode Decomposition (DMD) and Gaussian Process Regression (GPR). We also briefly mention geometrical parametrization and dimensionality reduction methods like Active Subspaces (ASs). Then we test the performance of such approaches in terms of efficiency and accuracy against three academic test cases, the lid driven cavity, the flow past a cylinder and the geometrically parametrized Stanford Bunny. Moreover, we also briefly present some preliminary results related to a more complex case involving an industrial application.

Predictive Model and Optimization of Micromixers Geometry using Gaussian Process with Uncertainty Quantification and Genetic Algorithm

Abstract Microfluidic devices are gaining attention for their small size and ability to handle tiny fluid volumes. Mixing fluids efficiently at this scale, known as micromixing, is crucial. This article builds upon previous research by introducing a novel optimization approach in microfluidics, combining Computational Fluid Dynamics (CFD) with Machine Learning (ML) techniques. Our research focuses on enhancing global optimization techniques while minimizing computational costs. It draws inspiration from a Y-type micromixer, initially featuring cylindrical grooves on the main channel's surface and internal obstructions. Simulations, conducted using OpenFOAM software, evaluate the impact of circular obstructions on mixing percentage and pressure drop, considering variations in obstruction diameter and offset. A Gaussian Process (GP) was utilized to model the data, providing model uncertainty. This study optimizes geometries by using genetic algorithm (GA) based on the reduced order model provided by GP. Results align with previous research, showing that medium-sized obstructions (137 mm diameter, 10 mm offset) near the channel wall are optimal. This approach not only provides efficient microfluidic optimization with uncertainty quantification but also highlights the effectiveness of combining CFD and ML techniques in achieving desired outcomes.

Dynamic characteristic of transonic aeroelasticity affected by fluid mode in pre-buffet flow

Transonic aeroelasticity remains a significant challenge in aerospace. The coupling mechanism of aeroelastic problems involving the coexistence of fluid modes and multiple structural modes still needs further investigation. For this purpose, we analysed the dynamic characteristic of a two-degree-of-freedom (2DOF) NACA0012 airfoil in pre-buffet flow. First, we constructed an aeroelastic reduced-order model, which can represent near-unstable transonic flow using the dominant fluid mode. Then, the flutter mechanism was investigated by studying the main eigenvalues of the model that vary with the natural pitching frequency. The results revealed that the existence of the fluid mode transitions the transonic flutter type from coupled-mode flutter to single-DOF (SDOF) flutter, which leads to a reduction in the flutter boundary. Under the effect of the fluid mode, the system produces six aeroelastic phenomena at different structural natural frequencies, including SDOF heaving/pitching flutter, heaving/pitching instability within coupled-mode flutter, forced vibration and stable state. Moreover, we identified two types of SDOF flutter in the 2DOF system. The first type corresponds to the traditional SDOF flutter, where the coupling of other modes has a small impact on the system's stability in most cases. However, within specific ranges of natural frequencies, this type of SDOF flutter may disappear due to coupling with other modes. The second type of SDOF flutter is characterized by strong coupling dominated by the unstable mode. It arises from the interaction among the flow, heaving and pitching modes, and does not manifest in the absence of any of these modes.

Updating reduced-order dynamic model for flexible-joint arm to design adaptive feedback linearization control

This study deals with the construction of an enhanced dynamic model for a lever-arm with flexible-joint to design a feedback linearization controller. The uncertainties of reduced-order model with nominal parameters are compensated via a complementary term to upgrade it to the actual system. The proposed method uses data fusion of encoder/gyroscope to remove the measurement errors. Mathematical analyses are provided to show the stochastic stability of the proposed scheme. Accordingly, a fuzzy adaptive solution is presented to effectively schedule the free parameters of the algorithm. The enhanced reduced-order model is used to design a feedback linearization law for control of lever-arm position. The controller adapts itself to the actual system, and is reliable and cost-effective because of using less sensors. The simulation studies indicate a higher accuracy of the proposed system compared with the controller designed by the full-order model in the presence of parametric uncertainties and disturbances. Also, experimental results conducted on a fabricated platform demonstrate the effectiveness of the suggested system to control the arm position in different scenarios. The comparative results with prevalent controllers indicate higher accuracy with smaller control input for the proposed controller to reject the disturbances and other uncertainties.

Evaluation of reduced-order models for the rapid aerodynamic analysis of supersonic and hypersonic bodies

Abstract This paper presents a comprehensive evaluation of reduced-order models (ROMs) for the determination of pressure coefficient distributions on supersonic and hypersonic bodies. The study investigates the limitations, aerodynamic precision and computational performance associated with various methodologies, ranging from simplistic Newtonian theory-based approaches to more advanced first and second-order shock-expansion theories. Validation is performed by comparing computed results with experimental and computational data for pressure distributions, drag and lift coefficients and centres of pressure for fundamental geometries and authentic vehicle design over a wide range of freestream conditions. The study also includes a comprehensive computational complexity analysis, demonstrating the superiority of finite-element ROM approaches over traditional finite-volume computational fluid dynamics (CFD) simulations. The primary objective of this paper is to scrutinise the extension of these methodological classes to the low supersonic regime. Hence, thermo-chemical reactions within the flow are disregarded, and the ideal gas law is adopted. A value of $\gamma = 1.4$ is chosen for consistency and comparability across the analyses. The proposed ROMs show remarkable potential for reducing high-speed simulation execution times by four orders of magnitude, maintaining accuracy within 20 per cent and as low as 1 per cent. The study unveils three key findings: first, the accuracy degradation of Newtonian-based theories for inclined elements, particularly around 45 degrees, and their reduced dependency on Mach number at large inclination. Secondly, the study presents novel insights into the impact of shock-wave-Mach-wave interactions on pressure distribution calculations, emphasising the Mach number as a crucial metric governing recompression effects. Lastly, the study demonstrates the exceptional accuracy of DeJarnette’s method, providing ${C_P}$ results within 2 per cent for a wide range of conditions, offering an attractive alternative to the Taylor-Maccoll equation.

Prediction of Temperature Distribution on an Aircraft Hot-Air Anti-Icing Surface by ROM and Neural Networks

To address the inefficiencies and time-consuming nature of traditional hot-air anti-icing system designs, reduced-order models (ROMs) and machine learning techniques are introduced to predict anti-icing surface temperature distributions. Two models, AlexNet combined with Proper Orthogonal Decomposition (POD-AlexNet) and multi-CNNs with GRU (MCG), are proposed by comparing several classic neural networks. Design variables of the hot-air anti-icing cavity are used as inputs of the two models, and the corresponding surface temperature distribution data serve as outputs, and then the performance of these models is evaluated on the test set. The POD-AlexNet model achieves a mean prediction accuracy of over 95%, while the MCG model reaches 96.97%. Furthermore, the proposed model demonstrates a prediction time of no more than 5.5 ms for individual temperature samples. The proposed models not only provide faster predictions of anti-icing surface temperature distributions than traditional numerical simulation methods but also ensure acceptable accuracy, which supports the design of aircraft hot-air anti-icing systems based on optimization methods such as genetic algorithms.

Trimming the UCERF3-TD logic tree: Model order reduction for an earthquake rupture forecast considering loss exceedance

The Uniform California Earthquake Rupture Forecast version 3-Time Dependent depicts California’s seismic faults and their activity. Its logic tree has 5760 leaves. Considering 30 more model combinations related to ground motion produces 172,800 distinct models representing so-called epistemic uncertainties. To calculate risk to a portfolio of buildings, one also considers millions of earthquakes and spatially correlated ground-motion variability. We offer a tree-trimming technique that retains the probability distribution of portfolio loss and identifies the leading sources of uncertainty for further study. We applied it to a California statewide building portfolio and various levels of nonexceedance probability between one in 100 and one in 2500. We trimmed the logic tree from 172,800 leaves to as few as 15. The result: a supercomputer that would otherwise run 24 h to estimate the distribution of one-in-250-year loss can calculate it in moments with the reduced-order model. Others can use the reduced-order model to calculate risk to different California portfolios, and scientists can prioritize study to reduce the remaining epistemic uncertainty.

Deep learning based hybrid POD-LSTM framework for laminar natural convection flow in a rectangular enclosure

Abstract Laminar natural convection in side-heated enclosures is characterized by transient phenomena of the working fluid till it reaches steady state. The side heating can done in several ways the most common way being heating one end at constant temperature and cooling the other end. One of the other ways is heating both sides of the enclosure at a constant heat flux. Mathematical modeling of such problems using Computational Fluid Dynamics (CFD) essentially involves considerable amount of computational time and power to predict the flow phenomena observed in actual experimentation. In the last few years, data driven model frameworks have proven to be anefficient way in saving both time and computational cost in several applications. In the present study, a data driven model framework using a combination of unsupervised machine learning (using Proper Orthogonal Decomposition [POD]) and supervised deep learning models (using Long Short Term Memory [LSTM]) has been developed and referred to as POD-LSTM framework. The selection of a few dominant spatial bases and accompanying temporal modes provides us with a reduced order model of the system. The flow is then reconstructed and compared with results of CFD simulations. The Rayleigh number (Ra) chosen for the study is 3.27 × 1010. The estimated time to reach stedy state for this Ra number is 15,000 s. The POD-LSTM framework is trained using data obtained from a validated CFD model for the first 1,000 s. The trained model was then tested to predict temporal dynamics for the entire 15,000 s. The predictions provided by POD-LSTM framework were found upto 98 % accurate compared to the ones predicted by CFD. The computational time and power was however an order of magnitude lower for the POD-LSTM framework than that required for the CFD model.

Modular Modeling of a Half-Vehicle System Using Generalized Receptance Coupling and Frequency-Based Substructuring (GRCFBS)

This paper presents an advanced modular modeling approach for vertical vibration analysis of dynamic systems using the Generalized Receptance Coupling and Frequency-Based Substructuring (GRCFBS) method. The focus is on a four-DoF half-vehicle model comprising three key subsystems: front suspension, rear suspension, and the vehicle’s trimmed body. The proposed technique is designed to predict dynamic responses in reconfigurable systems across various applications, including automotive, robotics, mechanical machinery, and aerospace structures. By coupling the receptance matrices (FRFs) of individual vehicle modules, the overall system receptance matrix is efficiently derived in a disassembled configuration. Two generalized coupling methods, originally developed by Jetmundsen and D.D. Klerk, are employed to determine the complete vehicle’s receptance matrix from its subsystems. Validation is achieved by comparing the results with established methods, such as direct solution and modal analysis, demonstrating high accuracy and reliability for complex dynamic systems. This modular approach allows for the creation of reduced-order models focused on key measurement points without the need for detailed system representation. The method offers significant advantages in early-stage vehicle development, providing critical insights into system vibration behavior.

A low‐cost trans‐scale model for the collaborative analysis of the manufacturing and in‐service process of unidirectional CFRP composites

AbstractDuring the manufacturing of thermoset‐based carbon fiber‐reinforced polymer (CFRP) structures, a curing process involving thermal, chemical, and mechanical interactions occurs. This process gives rise to micro‐scale residual stresses due to differences in fiber and resin properties, leading to decreased mechanical properties compared to nominal values. A trans‐scale analysis method utilizing the reduced order model (ROM) is applied in this study to establish a connection between the manufacturing and in‐service processes for unidirectional CFRP (UD‐CFRP). By employing this method, the evolution of residual stresses at the micro‐scale during UD‐CFRP manufacturing is predicted, and the impact of these residual stresses on structural performance during service is assessed. Specifically, the manufacturing‐induced residual stresses reduce the material strength by a minimum of 19.31%, while also exploring the correlation between macro‐scale and micro‐scale failures. Notably, the computational cost of this method is significantly lower, with a reduction factor of 103 compared to the finite element method. Empirical evidence supports the effectiveness of this method in accurately predicting outcomes throughout both the manufacturing and in‐service processes.Highlights Trans‐scale analysis method links composites manufacturing simulation to in‐service performance. Highly efficient trans‐scale method excels in cost, accuracy, consistency, and convergence. Residual stresses from the curing process significantly impact matrix safety.

Real-Time Modeling for Design and Control of Material Additive Manufacturing Processes

The use of digital twin and shadow concepts for industrial material processes has introduced new approaches to bridge the gap between physical and cyber manufacturing processes. Consequently, many multidisciplinary areas, such as advanced sensor technologies, material science, data analytics, and machine learning algorithms, are employed to create these hybrid systems. Meanwhile, new additive manufacturing (AM) processes for metals and polymers, based on emerging technologies, have shown promise for the manufacturing of sophisticated parts with complex geometries. These processes are undergoing a major transformation with the advent of digital technology, hybrid physical-data-driven modeling, and fast-reduced models. This study presents a fresh perspective on hybrid physical-data-driven and reduced order modeling (ROM) techniques for the digitalization of AM processes within a digital twin concept. The main contribution of this study is to demonstrate the benefits of ROM and machine learning (ML) technologies for process data handling, optimization/control, and their integration into the real-time assessment of AM processes. Therefore, a novel combination of efficient data-solver technology and an architecturally designed neural network (NN) module is developed for transient manufacturing processes with high heating/cooling rates. Furthermore, a real-world case study is presented, showcasing the use of hybrid modeling with ROM and ML schemes for an industrial wire arc AM (WAAM) process.

Method for Constructing a Compact Component Mode Synthesis Model for Analyzing Nonlinear Normal Modes of Structures with Localized Nonlinearities

Abstract Nonlinear dynamics analysis is a crucial topic in mechanical and aerospace engineering. The analysis of nonlinear normal modes (NNMs) provides an effective mathematical tool for interpreting complex nonlinear vibration phenomena. Unlike the invariant normal modes of linear systems, NNMs often exhibit frequency-energy dependence and cannot be computed using the traditional eigen-decomposition method. Calculating NNMs relies on numerical methods that involve expensive iterative computations, especially for systems with numerous degrees-of-freedom. To relieve computational costs, this paper proposes a mode selection method for the component mode synthesis (CMS) technique to enable compact reduced-order modeling of structures with localized nonlinearities. The reduced-order model is then combined with a numerical continuation scheme to establish a low-cost NNM analysis framework. This framework can efficiently predict the NNMs of high-dimensional finite element models. The proposed framework is demonstrated on an I-shaped cantilever beam with localized nonlinear stiffness. The results show that the proposed approach can identify the key modes in the CMS modeling procedure. The NNMs of the nonlinear I-shaped beam structure can then be analyzed with a significant reduction in computational costs.

Multi-timescale modeling and order reduction towards stability analysis of isolated microgrids

Microgrids incorporate a significant proportion of renewable energy sources and power electronic converters in the energy conversion process, creating a sustainable and clean energy infrastructure. However, the multi-timescale dynamics of microgrids are interactively coupled under a nonlinear structure, which makes it difficult to gain insight into the instability mechanisms without a high-fidelity reduced-order model that preserves the main dynamic behaviors of the system. For the isolated AC microgrid dominated by voltage source inverters (VSI), a detailed state-space model of the system, including the inverter, network, and load, is first developed. Based on this model, the eigenvalue analysis is carried out, and a participation factor analysis tool is also utilized to identify the relevant dynamics that have a strong impact on the system's dominant mode. Furthermore, to simplify the system modeling process without losing essential dynamic interactions, a novel multi-timescale coupled reduced-order model is proposed using a transfer function-based order reduction method, which retains the open-loop gain characteristics to preserve the critical couplings between fast inner loop dynamics and slow droop control dynamics. Finally, the accuracy of the reduced-order model is verified by comparing it with the detailed model and the conventional singular perturbation reduced-order model through eigenvalue distribution and time-domain simulation analysis.