Reduced-order Model Research Articles

Adaptive reduced basis trust region methods for parameter identification problems

In this contribution, we are concerned with model order reduction in the context of iterative regularization methods for the solution of inverse problems arising from parameter identification in elliptic partial differential equations. Such methods typically require a large number of forward solutions, which makes the use of the reduced basis method attractive to reduce computational complexity. However, the considered inverse problems are typically ill-posed due to their infinite-dimensional parameter space. Moreover, the infinite-dimensional parameter space makes it impossible to build and certify classical reduced-order models efficiently in a so-called “offline phase”. We thus propose a new algorithm that adaptively builds a reduced parameter space in the online phase. The enrichment of the reduced parameter space is naturally inherited from the Tikhonov regularization within an iteratively regularized Gauß-Newton method. Finally, the adaptive parameter space reduction is combined with a certified reduced basis state space reduction within an adaptive error-aware trust region framework. Numerical experiments are presented to show the efficiency of the combined parameter and state space reduction for inverse parameter identification problems with distributed reaction or diffusion coefficients.

Dynamics of small solid particles on substrates of arbitrary topography

We study the dynamics of a small solid particle arising from the dewetting of a thin film on a curved substrate driven by capillarity, where mass transport is controlled by surface diffusion. We consider the case when the size of the deposited particle is much smaller than the local radius of curvature of the substrate surface. The application of the Onsager variational principle leads to a reduced-order model for the dynamic behavior of particles on arbitrarily curved substrates. We demonstrate that particles move towards region of the substrate surface with lower mean curvature with a determined velocity. In particular, the velocity is proportional to the substrate curvature gradient and inversely proportional to the size of the particle, with a coefficient that depends on material properties that include the surface energy, surface diffusivity, density, and Young’s (wetting) angle. The reduced model is validated by comparing with numerical results for the full, sharp-interface model in both two and three dimensions.

Higher-order sensitivity analyses to understand the role of FE input parameters on the simulation of composites in progressive fracture tests

With the rise of data-driven engineering methods, such as machine learning, a comprehensive understanding of the underlying data fed into these algorithms is required for effectively applying these methods. This study presents two highly efficient Finite Element (FE) models that operate at different length scales, based on Continuum Damage Mechanics (CDM). The combination with Random Sampling-High Dimensional Model Representation (RS-HDMR) enables the determination of influential FE input parameters and their correlations to simulate Fibre Reinforced Polymer (FRP) composites subjected to a variety of progressive fracture tests in tension and compression. The results indicate that laminate-based models utilise the FE input parameters efficiently where all parameters are found to be influential. On the other hand, only fibre-related properties are relevant in ply-based FE models, with all input parameters related to transverse properties deemed to be non-influential. The findings of this study aid in identifying suitable fracture tests for a meaningful calibration and validation of CDM-based FE models. Moreover, the study lays the foundation for understanding the role of input parameters within data-driven methods, and for developing efficient reduced-order models or surrogates for uncertainty quantification on the damage tolerance of FRP composites.

Nonlinear dynamics of wing-like structures using a momentum subspace-based Koiter-Newton reduction

Cantilevers find a wide range of applications in the design of scientific equipment and large-scale engineering structures such as aircraft wings. Analysis techniques based on linearization approximations are unable to capture the large amplitude oscillation behaviour of such structures and thus, necessitates development of dedicated nonlinear methods. In this work, the recent developments in the Koiter-Newton model reduction method are utilized to obtain nonlinear reduced order models (ROMs) from full finite element structural models in order to simulate large amplitude dynamics of cantilevers. The method describes a nonlinear system of governing equations comprising quadratic and cubic terms which are obtained as higher order derivatives of the in-plane strain energy. To ensure that the large rotations in cantilevers and the resultant foreshortening effect is also accounted for, a ROM updating algorithm is adopted where the ROM parameters are varied with the structural deflections. Linear eigenmodes of the structure are utilized to form the reduction subspace. To validate the methodology, the ROM solution is compared against experimental results and a convergence study is conducted to identify the number of modes needed to replicate the nonlinear response. Finally, a composite wingbox structure is considered for which time domain simulations are conducted and frequency response curves, obtained through a frequency sweep, are presented.

An Optimized Power-Angle and Excitation Dual Loop Virtual Power System Stabilizer for Enhanced MMC-VSG Control and Low-Frequency Oscillation Suppression

Modular Multilevel Converter Virtual Synchronous Generator (MMC-VSG) technology is gaining widespread attention for its ability to enhance the inertia and frequency stability of the power grid integrated with converter-interfaced renewable energy sources. However, the excitation voltage regulation in the MMC-VSG can generate equivalent negative damping torque and cause low-frequency oscillation problems similar to those in synchronous machines. This article aims to improve the system’s damping torque and minimize low-frequency oscillations by introducing a Virtual Power System Stabilizer (VPSS) into the power control loop. Building on the study of dynamic interactions between various control links of the MMC, this research establishes a reduced-order model (ROM) and a Phillips–Heffron state equation for the MMC-VSG single machine infinite bus system, using a hybrid modeling approach and a zero-pole truncation method. It also analyzes the mechanism of low-frequency oscillations in the MMC-VSG system through the damping torque method. The analysis reveals that the negative damping torque produced during the excitation voltage regulation process causes changes in the virtual power angle, which in turn increases the risk of low-frequency oscillation in the MMC-VSG. To address this issue, the article proposes an optimized control method for the MMC-VSG dual power loop architecture (power-angle/excitation) VPSS. This strategy compensates for the inadequate damping torque of a single loop VPSS and effectively suppresses low-frequency oscillations in the system.

Effective lateral dispersion of momentum, heat and mass in bubbling fluidized beds

The lateral dispersion of bed material in a bubbling fluidized bed is a key parameter in the prediction of the effective in-bed heat transfer and transport of heterogenous reactants, properties important for the successful design and scale-up of thermal and/or chemical processes. Computational fluid dynamics simulations offer means to investigate such beds in silico and derive effective parameters for reduced-order models. In this work, we use the Eulerian-Eulerian two-fluid model with the kinetic theory of granular flow to perform numerical simulations of solids mixing and heat transfer in bubbling fluidized beds. We extract the lateral solids dispersion coefficient using four different methods: by fitting the transient response of the bed to (1) an ideal heat or (2) mass transfer problem, (3) by extracting the time-averaged heat transfer behavior and (4) through a momentum transfer approach in an analogy with single-phase turbulence. The method (2) fitting against a mass transfer problem is found to produce robust results at a reasonable computational cost when assessed against experiments. Furthermore, the gas inlet boundary condition is shown to have a significant effect on the prediction, indicating a need to account for nozzle characteristics when simulating industrial cases.

Data-Driven Modeling of Tire-Soil Interaction with POD-Based Model Order Reduction

Abstract A data-driven model capable of predicting time-domain solutions of a high-fidelity tire-soil interaction model is developed to enable quick prediction of mobility capabilities on deformable terrain. The adaptive model order reduction based on the proper orthogonal decomposition (POD), for which the high-dimensional equations are projected onto the reduced subspace, is utilized as the basis for predicting the time-domain tire-soil interaction behavior. The projection-based model order reduction, however, requires many online matrix operations due to the successive updates of the nonlinear functions and Jacobians at every time step, thereby hindering the computational improvement. Therefore, a data-driven approach using a Long Short Term Memory (LSTM) neural network is introduced to predict the reduced order coordinates without the projection and time integration processes for computational speedup. With this model, a hybrid data-driven/physics-based off-road mobility model is proposed, where four separate LSTM-POD data-driven tire-soil interaction models are integrated into the physics-based multibody dynamics (MBD) vehicle model through a force-displacement coupling algorithm. By doing so, the individual data-driven tire-soil interaction model can be constructed efficiently, and the MBD and LSTM models are assembled as a single off-road mobility model and analyzed with existing off-road mobility solvers. The predictive ability and computational benefit of the proposed data-driven tire-soil interaction model with the POD-based model order reduction are examined with several numerical examples.

Proper orthogonal decomposition reduced-order model of the global oceans

AbstractA reduced-order model (ROM) of the global oceans is developed by projecting the hydrostatic Boussinesq equations of motion onto a proper orthogonal decomposition (POD) basis. Three-dimensional POD modes are calculated from the ocean fields of an ensemble climate reanalysis dataset. The coefficients in the POD ROM are calculated using a regression approach. The performance of various POD ROM configurations are assessed. Each configuration is derived from an alternate sea-water equation of state, linking the density and temperature fields. POD ROM variants incorporating an equation of state in which density is a quadratic function of temperature, are able to reproduce the statistics of the large-scale structures at a fraction of the computational cost required to numerically simulate this flow. Due to the speed and efficiency of calculation, such reduced-order models of the global geophysical system will enable researchers and policy makers to assess the physical risk for a broader range of potential future climate scenarios.

Neural downscaling for complex systems: from large-scale to small-scale by neural operator

Researchers have long been working on interpreting and predicting the dynamics of complex systems in various fields. Conventional methods including full-scale simulations and reduced-order models are used to solve related problems, but often yield unsatisfactory results in terms of precision and computational efficiency. This is primarily due to the challenges posed by simulating small-scale dynamics. An alternative consideration is to leverage well-represented and readily accessible large-scale dynamics to assist small-scale modelling. This paper proposes a novel methodology, named as Neural Downscaling (ND), that effectively captures small-scale dynamics within a complementary subspace from corresponding large-scale dynamics well-represented in a low-dimensional space. The framework of ND integrates neural operator techniques with the principles of inertial manifold and nonlinear Galerkin theory. The effectiveness and efficiency of ND are demonstrated in the complex systems governed by Kuramoto–Sivashinsky and Navier–Stokes equations. Being an unprecedentedly deterministic model targeted at general small-scale dynamics, ND sheds light on the intricate spatiotemporal nonlinear dynamics of complex systems and reveals how small-scale dynamics are interacting with large-scale dynamics.

Research on Model Reduction of AUV Underwater Support Platform Based on Digital Twin

Digital twin technology, as a data-driven and model-driven innovation means, plays a crucial role in the process of digital transformation and intelligent upgrading of the marine industry, helping the industry to move towards a new stage of more intelligent and efficient development. In order to solve the defects of the Autonomous Underwater Vehicle (AUV) underwater support platform structure deformation field, digital twin technology and model reduction technology are applied to an AUV underwater support platform, and a five-dimensional digital twin model of the AUV underwater support platform is studied, including five dimensions: physical world, digital world, twin data center, service application, and data connection. The digital twin of the subsea support platform is established by using the digital twin modeling technology. The POD method is used to calculate the deformation field matrix of the support structure of the subsea support platform under the 0–5 sea state, and the corresponding eigenvalues and eigenvectors are obtained. By intercepting the eigenvectors corresponding to the eigenvalues of the high energy proportion, the low-order equation is constructed, and the reduced-order model under each sea state can be quickly solved. The experimental results show that the model reduction technology can greatly shorten the model solving time, and the calculated results are highly consistent with the simulation results of the finite element full-order model, which can realize the rapid analysis of the deformation response of the subsea support platform structure, and provide a theoretical basis and technical support for the subsequent simulation, state evaluation, visual monitoring, and predictive maintenance.

Mass and Heat Exchange in Rotating Compressor Cavities With Variable Cob Separation

Abstract Next generation aeroengines will operate at ever-increasing pressure ratios with smaller cores, where the control of blade-tip clearances across the flight cycle is an emerging design challenge. Such clearances are affected by the thermal expansion of the compressor disks that hold the blades, where acute thermal stresses govern operating life. The cavities formed by corotating disks feature a heated shroud at high radius and cooler cobs at low radius. A three-dimensional, unsteady and unstable flow structure is induced by destabilizing buoyancy forces. The radial distribution of disk temperature is driven by a conjugate heat transfer at Grashof numbers of order 1013. Such flows are further influenced by the heat and mass exchange with an axial throughflow of cooling air at low radius, where the interaction depends on the Rossby number and separation of the disk cobs. This paper is the first to study the effect of cob separation ratio on mass and heat exchange for compressor cavities. A model is developed to predict the cavity-throughflow interaction, and disk and fluid-core temperatures. The judicious use of a physics-based methodology provides reliable, reduced-order solutions to the complex conjugate problem, thereby making it appropriate for practical engine thermo-mechanical design. The model is validated by detailed experimental measurements using the Bath Compressor Cavity Rig, where variable disk cob spacings were investigated over a range of engine-representative conditions. The unsteady pressure measurements collected in the frame of reference of the rotating disks reveal new insight into the fundamentally aperiodic nature of the flow structure. This new understanding of heat transfer informs an expedient reduced-order model and enables more efficient design of future high pressure-ratio aeroengines.

Reduced order modeling for real-time monitoring of structural displacements due to electromagnetic forces in large scale tokamaks

Abstract The real-time monitoring of the structural displacement of the vacuum vessel of thermonuclear fusion devices caused by electromagnetic loads is of great interest. In this paper, model order reduction is applied to the integral equation methods and the finite elements method to develop electromagnetic and structural reduced order models (ROMs) compatible with real-time execution which allows for the real-time monitoring of strain and displacement in critical positions of Tokamaks machines. Low-rank compression techniques based on hierarchical matrices are applied to reduce the computational cost during the offline stage when the ROMs are constructed. Numerical results show the accuracy of the approach and demonstrate the compatibility with real-time execution in standard hardware.

Least-squares pressure recovery in reduced order methods for incompressible flows

In this work, we introduce a method to recover the reduced pressure for Reduced Order Models (ROMs) of incompressible flows. The pressure is obtained as the least-squares minimum of the residual of the reduced velocity with respect to a dual norm. We prove that this procedure provides a unique solution whenever the full-order pair of velocity-pressure spaces is inf-sup stable. We also prove that the proposed method is equivalent to solving the reduced mixed problem with reduced velocity basis enriched with the supremizers of the reduced pressure gradients. Optimal error estimates for the reduced pressure are obtained for general incompressible flow equations and specifically, for the transient Navier-Stokes equations. We also perform some numerical tests for the flow past a cylinder and the lid-driven cavity flow which confirm the theoretical expectations, and show an improved convergence with respect to other pressure recovery methods.

Dynamic mode decomposition based on expectation–maximization algorithm for simultaneous system identification and denoising

The present study proposes a novel dynamic mode decomposition (DMD) that can simultaneously estimate the reduced-order model, the original signal, and the system/observation noise model only from the noisy data. An expectation–maximization (EM)-algorithm DMD (EMDMD) combines DMD and the parameter adjustment of the linear dynamical system (LDS) based on the EM algorithm. The initial parameters based on the linearity of the reduced-order data are set by using DMD. Subsequently, the log-likelihood of the complete data is maximized by adjusting the LDS parameters while separating the noise. The proposed algorithm is applied to the benchmark data of the short-fat and tall-skinny data matrices with different noise and the time-series velocity fields of the flow around a circular cylinder and the separated flow around an airfoil. The performance of EMDMD in terms of system identification and noise separation from the noisy data is evaluated, and the EMDMD shows the highest system identification and noise separation performance in all data.

Optimizing indoor air models through k-means clustering of nanoparticle size distribution data

Sectional physics-based aerosol models imply a computational effort that hinders their use in building digital twins, real-time predictive control, and computer-based iterative optimization versus black-box approaches. The innovation of this paper lies in the proposal of a novel systematic methodology to optimize the number of size bins in sectional reduced-order models for particle concentration simulations. This allows its application in indoor air quality management and overcomes generalizability and data-dependency issues of black-box models. This method, based on k-means clustering, aims to ensure precision when tackling relatively fast fine and ultrafine particle simulations targeting the reduction of time and resource consumption from experimentally-determined aerosol size distribution time series. Consequently, three tests were carried out using combustion aerosols inside a custom-designed emission chamber to simulate emission hotspots in non-commercial and occupational settings. 2-, 3-, 4-, and 5-cluster classifications were evaluated for data coming from 13 particle size bins through silhouette analysis and the study of their temporal profiles. Results show that the 4-cluster classification summarizes the behavior of data in the 10–420 nm range, ensuing up to a 77 % improvement in the model's computational demand. Moreover, this method allows an accurate definition of the necessary size ranges to calculate nanoparticle concentrations inside the chamber and facilitates the interpretation of aerosol behavior and processes through the resulting clusters' temporal profiles.

A new proper orthogonal decomposition method with second difference quotients for the wave equation

Recently, researchers have investigated the relationship between proper orthogonal decomposition (POD), difference quotients (DQs), and pointwise in time error bounds for POD reduced order models of partial differential equations. In a recent work (Eskew and Singler, Adv. Comput. Math., 49, 2023, no. 2, Paper No. 13), a new approach to POD with DQs was developed that is more computationally efficient than the standard DQ POD approach and it also retains the guaranteed pointwise in time error bounds of the standard method. In this work, we extend this new DQ POD approach to the case of second difference quotients (DDQs). Specifically, a new POD method utilizing DDQs and only one snapshot and one DQ is developed and used to prove ROM error bounds for the damped wave equation. This new approach eliminates data redundancy in the standard DDQ POD approach that uses all of the snapshots, DQs, and DDQs. We show that this new DDQ approach also has pointwise in time data error bounds similar to DQ POD and use it to prove pointwise and energy ROM error bounds. We provide numerical results for the POD ROM errors to demonstrate the theoretical results. We also explore an application of POD to simulate ROMs past the training interval for collecting the snapshot data for the standard POD approach and the DDQ POD method.

A Power-RPM Reduced-Order Model and Power Control Strategy of the Dual Three-Phase Permanent Magnet Synchronous Motor in a V/f Framework for Oscillation Suppression

A Power-RPM Reduced-Order Model and Power Control Strategy of the Dual Three-Phase Permanent Magnet Synchronous Motor in a V/f Framework for Oscillation Suppression

Improved a posteriori error bounds for reduced port-Hamiltonian systems

Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem, to the case of port-Hamiltonian systems. The approaches rely on a secondary approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are adapted to port-Hamiltonian systems. The mathematical relationship between the two methods is discussed both theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid–structure interaction.

Selective actuation of higher-order modes of an electromagnetically driven micro drum

In this paper, we experimentally investigate selective actuation of the first few higher-order modes of a micromachined silicon nitride circular membrane (micro drum), which is actuated by electromagnetic forces. A novel design of electrodes is proposed to enable selective actuation for portions of the membrane with the Lorentz force, which excites efficiently the higher-order modes. With different electrode configurations, we show that the (0, 1) mode, the degenerate (1, 1) modes, and the (2, 1) mode can be selectively activated or deactivated using the orthogonality between the actuation force and the mode shapes. A reduced-order model for the circular membrane is established and used to explain the results.

Small signal analysis of DC voltage control based on a virtual resistance of DC/DC converter integrated in a multiterminal DC grid

AbstractThe future multi‐terminal direct‐current (MTDC) grid will require the interconnection of point‐to‐point high‐voltage (HV) DC links with different specifications such as DC voltage level, system grounding configuration and HVDC technology. To adapt these differences, it is obligatory for DC/DC converters to interconnect HVDC links. Additionally, they are capable of providing supplementary functionalities as they are highly controllable devices. In this article, a primary virtual resistance DC voltage controller associated with DC/DC converter is proposed for managing DC grid voltages of the interconnected HVDC grids, increasing the reliability of the system. The commonly known topology, Front‐to‐Front Modular Multilevel Converter (F2F‐MMC) is adopted for DC/DC converter. Time‐domain simulations are performed using EMTP software for validating the controller behaviour under power disturbances and large events of loss of one converter in a MMC‐based MTDC system. The converters are modelled using reduced order modelling (ROM) methodology. Apart from this, dynamic studies have been carried out using a linear state space model for small‐signal stability analysis of a HVDC system integrating DC/DC converter with a virtual resistance DC voltage controller. The results are examined through parametric sensitivity analysis.