Model Reduction Method Research Articles

Analysis of the Flow Field at the Tip of an Axial Flow Compressor during Rotating Stall Process Based on the POD Method

Rotating stalls are one of the most dangerous phenomena to be avoided in the designing and operating of axial flow compressors. An understanding of the evolution of the flow characteristics around the rotor tip region is important to study the process of stall development. In this paper, some critical characteristics of the stall-related structures, which could not be observed by the traditional analysis methods, are understood by using the proper orthogonal decomposition (POD) method. The detailed unsteady flow fields in a typical transonic axial flow compressor during the stall process are obtained by the validated numerical simulation. Thereafter, the proper orthogonal decomposition method, as a method for model reduction and decomposition, is adopted to extract the flow characteristics from the numerical results of the stall process. It is found that the flow characteristics during the stall process can be well decomposed by the POD method. The pre-stall POD results show that the important flow features can be extracted and revealed in the low-order modes, which are not obvious in the original flow field. When the stall cells are formed and developed, the flow characteristics are gradually determined by the modes, which are related to the features of the stall cells. When the compressor is operated under stable stall conditions, the low-order POD modes are composed of a series of harmonic modes, which are sinusoid-like in space and time with the frequency of the stall cell rotation.

A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity

We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained optimization statement that penalizes the jump at the components’ interfaces subject to the approximate satisfaction of the PDE in each local subdomain. Furthermore, the approach relies on the decomposition of the local states into a port component – associated with the solution on interior boundaries – and a bubble component that vanishes at ports: since the bubble components are uniquely determined by the solution value at the corresponding port, we can recast the constrained optimization statement into an unconstrained statement, which reads as a nonlinear least-squares problem and can be solved using the Gauss–Newton method. We present thorough numerical investigations for a two-dimensional neo-Hookean nonlinear mechanics problem to validate our method; we further discuss the well-posedness of the mathematical formulation and the a priori error analysis for linear coercive problems.

A Light Weight Multisensory Fusion Model for Induction Motor Fault Diagnosis

Fault diagnosis keeps an essential tool to ensure the safety and reliability of a motor system. Based on deep learning, fault diagnosis models constructed by mining historical fault data of equipment have received extensive attention. However, the high computational cost constrains the application of deep learning models for fault diagnosis, especially when coping with multisource data. Inspired by the model reduction and neural network structure automatic search method, this article proposed a light weight multisensory fusion model for induction motor data fusion and diagnosis. Inverted residual block and network architecture search technology are introduced to accelerate the training speed and prediction speed of the diagnostic model, so that the diagnostic accuracy is maintained at an acceptable level. The effectiveness of the proposed model is demonstrated through motor fault diagnosis experiments. Compared with other popular neural networks, the proposed method is capable of judging fault patterns accurately with shorter prediction time.

Interpolatory Methods for Model Reduction [Bookshelf

The surge of interest in machine learning has led to increased emphasis on the value of accurate, efficient surrogate models. The book is divided into four parts. Part 1, consisting of Chapters 1 and 2, introduces the reader to the basic ideas underlying general model reduction and provides the relevant system-theoretic background needed for the remainder of the book. Part 2, consisting of Chapters 3–5, covers core concepts of interpolatory model reduction where wellknown results and state-of-the-art implementations are presented in a compact yet deep way. Part 3, consisting of Chapters 6 and 7, provides the first comprehensive textbook exposition of recent developments in parameter dependent dynamical systems and nonlinear interpolatory model reduction. This book is a timely reminder that before we reach for black-box approximation functions (which often require prohibitive amounts of data to train), there is a long and rigorous history (and many recent advances) in system-theoretic model-reduction methods that can deliver structured, reduced-order models in a broad range of scientific and engineering settings.

Three-Dimensional Effect of Feedback Controlled Winglets in Flutter Suppression of Suspension Bridge

To account for the influence of winglet’s arrangement length and installing position along the longitudinal direction of bridge deck, as well as actuators quantities on flutter control effect, a three-dimensional model of deck-winglet system was been derived in state space domain, and a model reduction method was also discussed to enhance control efficiency. Then, a suspension bridge with a main span of 3[Formula: see text]000[Formula: see text]m was used to investigate the control effect of active winglets, and to explore the influence of winglet length and actuator quantity on flutter control effect. Finally, the flutter and buffeting performance of winglet itself was investigated. The simulation results shows that the modes involved in flutter analysis should be determined according to mode-energy participation ratio. When installing a pair of winglets whose width is 10% of bridge deck, the critical wind speed of bridge deck may be twice of a non-controlled condition. In addition, the critical wind speed increases by increasing the length of winglets and actuator quantity. However, the critical wind speed may remain constant once the winglet length and actuator quantity exceed a critical value. By ensuring the critical wind speed of winglet itself is higher than bridge deck, winglet’s width should be less than 10% of bridge deck’s width, and the independent length winglet is proposed to be shorter than 15% of the length of center span of bridge deck.

Fast data-driven model reduction for nonlinear dynamical systems

We present a fast method for nonlinear data-driven model reduction of dynamical systems onto their slowest nonresonant spectral submanifolds (SSMs). While the recently proposed reduced-order modeling method SSMLearn uses implicit optimization to fit a spectral submanifold to data and reduce the dynamics to a normal form, here, we reformulate these tasks as explicit problems under certain simplifying assumptions. In addition, we provide a novel method for timelag selection when delay-embedding signals from multimodal systems. We show that our alternative approach to data-driven SSM construction yields accurate and sparse rigorous models for essentially nonlinear (or non-linearizable) dynamics on both numerical and experimental datasets. Aside from a major reduction in complexity, our new method allows an increase in the training data dimensionality by several orders of magnitude. This promises to extend data-driven, SSM-based modeling to problems with hundreds of thousands of degrees of freedom.

A Review of the Accuracy of Primal Assembly Model Order Reduction Techniques

Model order reduction (MOR) techniques have been used in structural dynamics for over sixty years as a quick and effective way to examine the dynamic behavior of complex finite element (FE) models. Various macroscopic phenomena arise from the physics and mechanics of the underlying microstructure, and MOR approaches can be employed in conjunction with the multiscale computational homogenization approach, which is deriving local macroscopic constitutive responses from underlying microstructure; to predict the dynamic behavior of heterogeneous macrostructures. The accuracy and efficiency of the most representative approaches from dynamic condensation, component mode synthesis (CMS), and general model reduction (GMR) methods are compared in this paper. The techniques considered are Guyan and the improved reduced system (IRS) from classical dynamic condensation, Craig-Bampton (CB) and enhanced Craig-Bampton (ECB) from CMS, IRS-based substructuring, GMR, and GMRPlus. All the substructuring methods employ primal assembly at the boundary interface of the substructures. The accuracy and computing efficiency of the techniques are compared in-depth using real-world engineering problems, and the effect of the number of substructures on the accuracy and computational cost is also examined.

Predicting solar wind streams from the inner-heliosphere to Earth via shifted operator inference

Solar wind conditions are predominantly predicted via three-dimensional numerical magnetohydrodynamic (MHD) models. Despite their ability to produce highly accurate predictions, MHD models require computationally intensive high-dimensional simulations. This renders them inadequate for making time-sensitive predictions and for large-ensemble analysis required in uncertainty quantification. This paper presents a new data-driven reduced-order model (ROM) capability for forecasting heliospheric solar wind speeds. Traditional model reduction methods based on Galerkin projection have difficulties with advection-dominated systems—such as solar winds—since they require a large number of basis functions and can become unstable. A core contribution of this work addresses this challenge by extending the non-intrusive operator inference ROM framework to exploit the translational symmetries present in the solar wind caused by the Sun's rotation. The numerical results show that our method can adequately emulate the MHD simulations and is more accurate than a reduced-physics surrogate model, the Heliospheric Upwind Extrapolation model.

Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods

Vibration and dissipation in vibro-acoustic systems can be assessed using frequency response analysis. Evaluating a frequency sweep on a full-order model can be very costly, so model order reduction methods are employed to compute cheap-to-evaluate surrogates. This work compares structure-preserving model reduction methods based on rational interpolation and balanced truncation with a specific focus on their applicability to vibro-acoustic systems. Such models typically exhibit a second-order structure and their material properties as well as their excitation may be depending on the driving frequency. We demonstrate the effectiveness of all considered methods in terms of their accuracy and computational cost by applying them to numerical models of vibro-acoustic systems depicting structural vibration, sound transmission, acoustic scattering, and poroelastic problems. The results of the experiments are extensively discussed to derive guidelines for the choice of model reduction methods based on the problem setting.

Symplectic model reduction methods for the Vlasov equation

AbstractParticle‐based simulations of the Vlasov equation typically require a large number of particles, which leads to a high‐dimensional system of ordinary differential equations. Solving such systems is computationally very expensive, especially when simulations for many different values of input parameters are desired. In this work, we compare several model reduction techniques and demonstrate their applicability to numerical simulations of the Vlasov equation. The necessity of symplectic model reduction algorithms is illustrated with a simple numerical experiment.

Assessing HD-EEG functional connectivity states using a human brain computational model

Objective. Electro/Magnetoencephalography (EEG/MEG) source-space network analysis is increasingly recognized as a powerful tool for tracking fast electrophysiological brain dynamics. However, an objective and quantitative evaluation of pipeline steps is challenging due to the lack of realistic ‘controlled’ data. Here, our aim is two-folded: (a) provide a quantitative assessment of the advantages and limitations of the analyzed techniques and (b) introduce (and share) a complete framework that can be used to optimize the entire pipeline of EEG/MEG source connectivity. Approach. We used a human brain computational model containing both physiologically based cellular GABAergic and Glutamatergic circuits coupled through Diffusion Tensor Imaging, to generate high-density EEG recordings. We designed a scenario of successive gamma-band oscillations in distinct cortical areas to emulate a virtual picture-naming task. We identified fast time-varying network states and quantified the performance of the key steps involved in the pipeline: (a) inverse models to reconstruct cortical-level sources, (b) functional connectivity measures to compute statistical interdependency between regional signals, and (c) dimensionality reduction methods to derive dominant brain network states (BNS). Main results. Using a systematic evaluation of the different decomposition techniques, results show significant variability among tested algorithms in terms of spatial and temporal accuracy. We outlined the spatial precision, the temporal sensitivity, and the global accuracy of the extracted BNS relative to each method. Our findings suggest a good performance of weighted minimum norm estimate/ Phase Locking Value combination to elucidate the appropriate functional networks and ICA techniques to derive relevant dynamic BNS. Significance. We suggest using such brain models to go further in the evaluation of the different steps and parameters involved in the EEG/MEG source-space network analysis. This can reduce the empirical selection of inverse model, connectivity measure, and dimensionality reduction method as some of the methods can have a considerable impact on the results and interpretation.

A new reduced model of damped moment frame for rapid optimization of viscous damper placement

AbstractA new model reduction method is presented for moment‐resisting frames with viscous dampers. The proposed method considers (1) the interactive effect of added damping between the horizontal‐vertical directions and (2) the direct effect with regard to the vertical direction different from the conventional static condensation method. When brace‐type dampers are treated, these effects cannot be neglected originally. The proposed method gives a better correspondence of the fundamental eigenmode with that of the non‐reduced moment frame than the static condensation method. In addition, the proposed reduced model can accurately evaluate the maximum interstory drifts of the non‐reduced moment frame. The proposed model reduction method is formulated in the frequency domain. The frequency dependency of the stiffness matrix and the damping matrix of the reduced model is investigated. The influences of the added damping matrix by viscous dampers on the natural circular frequencies, the damping ratios and the participation vectors are also investigated. Finally, it is demonstrated through a numerical example that the displacement‐based optimization of the damper allocation can be accurately and rapidly conducted by using the reduced models throughout the optimization process.

A highly efficient method for structural model reduction

AbstractModel reduction is a commonly used method for quickly computing the lower‐order eigenvalues and eigenvectors of a structure. This article proposed a direct model reduction method without any iterations. Compared with the existing methods, the proposed technique has two main advantages. The first one is that the explicit expression of the transformation matrix can be directly obtained using a recurrence formula. The second one is that the calculation efficiency of the proposed method is very high because there is no need for iterative calculation. Three types of structure are used as examples to verify the proposed method. It is found that the lower‐frequency eigenpairs calculated by the reduced model are very close to the exact ones obtained by the full model. Moreover, the calculation time of the proposed method is much less than that of the existing methods. It has been shown that the proposed method is reliable and effective.

Enhanced model reduction method via combined supervised and unsupervised learning for real-time solution of nonlinear structural dynamics

Enhanced model reduction method via combined supervised and unsupervised learning for real-time solution of nonlinear structural dynamics

Online Multiscale Model Reduction for Nonlinear Stochastic PDEs with Multiplicative Noise

In this paper, an online multiscale model reduction method is presented for stochastic partial differential equations (SPDEs) with multiplicative noise, where the diffusion coefficient is spatially multiscale and the noise perturbation nonlinearly depends on the diffusion dynamics. It is necessary to efficiently compute all possible trajectories of the stochastic dynamics for quantifying a model’s uncertainty and statistic moments. The multiscale diffusion and nonlinearity may cause the computation to become very intractable. To overcome the multiscale difficulty, a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) is used to localize the computation and obtain an effective coarse model. However, the nonlinear terms are still defined on a fine scale space after the Galerkin projection of CEM-GMsFEM is applied to the nonlinear SPDEs. This significantly impacts the simulation efficiency by CEM-GMsFEM. To this end, a stochastic online discrete empirical interpolation method (DEIM) is proposed to treat the stochastic nonlinearity. The stochastic online DEIM incorporates offline snapshots and online snapshots. The offline snapshots consist of the nonlinear terms at the approximate mean of the stochastic dynamics and are used to construct an offline reduced model. The online snapshots contain some information of the current new trajectory and are used to correct the offline reduced model in an incremental manner. The stochastic online DEIM substantially reduces the dimension of the nonlinear dynamics and enhances the prediction accuracy for the reduced model. Thus, the online multiscale model reduction is constructed by using CEM-GMsFEM and the stochastic online DEIM. A priori error analysis is carried out for the nonlinear SPDEs. We present a few numerical examples with diffusion in heterogeneous porous media and show the effectiveness of the proposed model reduction.

Event-triggered disturbance attenuation for multi-agent systems with input delay

This paper designs event-triggered delay compensation and disturbance attenuation control algorithms for multi-agent systems. First, through predictor-based model reduction method, an unknown large input delay can be compensated and delay-free dynamics is obtained. Based on the transformed systems, predictive descriptor observers are proposed to obtain estimated system states and sensor faults. Then, an event-triggered fault-tolerant algorithm is designed to save energy resources for communication among the agents while attenuating the disturbances. Moreover, it is validated that the systems do not exhibit Zeno behaviour using the designed event-triggering sequence. Numerical simulations are conducted to demonstrate the effectiveness of designed algorithms.

Efficient hyperreduction of high-order discontinuous Galerkin methods: Element-wise and point-wise reduced quadrature formulations

We develop and assess projection-based model reduction methods for high-order discontinuous Galerkin (DG) discretizations of parametrized nonlinear partial differential equations with applications in aerodynamics. Our emphasis is on the choice of hyperreduction methods. We analyze computational complexity and use numerical examples to show that typical hyperreduction methods based on element-wise or index-wise sampling of the residual vector cannot effectively reduce high-order DG discretizations, with a large number of degrees of freedom and quadrature points per element and conversely a small number of elements. To overcome this limitation, we devise a hyperreduction method for high-order DG methods that samples individual quadrature points, instead of elements, to provide a finer decomposition of the residual. We compare the formulation, implementation, and computational complexity of the element-wise and point-wise formulations. We then numerically assess the two formulations using Euler flow over a shape-parametrized airfoil and laminar and turbulent Navier-Stokes flow over a three-dimensional aerodynamic body. In each case, we observe that the point-wise hyperreduced model reduces the computational time by several orders of magnitude relative to the high-order DG method and a few orders of magnitude relative to the element-wise hyperreduced model. In addition, the computational performance of the point-wise formulation does not deteriorate with the DG polynomial degree, unlike for the element-wise formulation.

Non-intrusive balancing transformation of highly stiff systems with lightly damped impulse response

Balanced truncation (BT) is a model reduction method that uses a coordinate transformation to retain eigen-directions that are highly observable and reachable. To address realizability and scalability of BT applied to highly stiff and lightly damped systems, a non-intrusive data-driven method is developed for balancing discrete-time linear systems via the eigensystem realization algorithm (ERA). The advantage of ERA for balancing transformation makes full-state outputs tractable. Further, ERA enables balancing despite stiffness, by eliminating computation of balancing modes and adjoint simulations. As a demonstrative example, we create balanced reduced-order models (ROMs) for a one-dimensional reactive flow with pressure forcing, where the stiffness introduced by the chemical source term is extreme (condition number [Formula: see text]), preventing analytical implementation of BT. We investigate the performance of ROMs in prediction of dynamics with unseen forcing inputs and demonstrate stability and accuracy of balanced ROMs in truly predictive scenarios, whereas without ERA, proper orthogonal decomposition-Galerkin and least-squares Petrov-Galerkin projections fail to represent the true dynamics. We show that after the initial transients under unit impulse forcing, the system undergoes lightly damped oscillations, which magnifies the influence of sampling properties on predictive performance of the balanced ROMs. We propose an output domain decomposition approach and couple it with tangential interpolation to resolve sharp gradients at reduced computational costs. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

3D anisotropic TEM modeling with loop source using model reduction method

Abstract For model reduction techniques, there have been relatively few studies performed regarding the forward modeling of anisotropic media in comparison to transient electromagnetic (TEM) forward modeling of isotropic media. The transient electromagnetic method (TEM) responses after the current has been turned off can be represented as a homogeneous ordinary differential equation (ODE) with an initial value, and the ODE can be solved using a matrix exponential function. However, the order of the matrix exponential function is large and solving it directly is challenging, thus this study employs the Shift-and-Invert (SAI-Krylov) subspace algorithm. The SAI-Krylov subspace technique is classified as a single-pole approach compared to the multi-pole rational Krylov subspace approach. It only takes one LU factorization of the coefficient matrix, along with hundreds of backward substitutions. The research in this paper shows that the anisotropic medium has little effect on the optimal shift ${\gamma _{opt}}$ and subspace order m. Furthermore, as compared to the mimetic finite volume method (SAI-MFV) of the SAI-Krylov subspace technique, the method proposed in this paper (SAI-FEM) can further improve the computing efficiency by roughly 13%. In contrast to the standard implicit time step iterative technique, the SAI-FEM method does not require discretization in time, and the TEM response at any moment within the off-time period can be easily computed. Next, the accuracy of the SAI-FEM algorithm was verified by 1D solutions for an anisotropic layer model and a 3D anisotropic model. Finally, the electromagnetic characteristics of the anisotropic anomalous body of the center loop device and separated device of the airborne transient electromagnetic method were analyzed, and it was found that horizontal conductivity has a considerable influence on the TEM response of the anisotropic medium.

Assessing the wall energy efficiency design under climate change using POD reduced order model

Within the environmental context, numerical modeling is a promising approach to assess the energy efficiency of building. Resilient buildings need to be designed, capable of adapting to future extreme heat. Simulations are required assuming a one-dimensional heat transfer problem through walls and a simulation horizon of several years (nearly 30). The computational cost associated with such modeling is quite significant and model reduction methods are worth investigating. The objective is to propose a reliable reduced-order model for such long-term simulations. For this, an alternative model reduction approach is investigated, assuming a known Proper Orthogonal Decomposition reduced basis for time, and not for space as usually. The model enables computing parametric solutions using basis interpolation on the tangent space of the Grassmann manifold. Three study cases are considered to verify the efficiency of the reduced-order model. Results highlight that the model has a satisfying accuracy of 10-3 compared to reference solutions. The last case study focuses on the wall energy efficiency design under climate change according to a four-dimensional parameter space. The latter is composed of the load material emissivity, heat capacity, thermal conductivity and thickness insulation layer. Simulations are carried over 30 years considering climate change. The solution minimizing the wall work rate is determined with a computational ratio of 0.1% compared to standard approaches.