Model Order Reduction Method Research Articles

Model Order Reduction Methods for Rotating Electrical Machines: A Review

Due to the rise of e-mobility applications, there is an increased demand to create more accurate control methods, which can reduce the loss in an e-drive system. The accurate modeling of the rotating machines needs to resolve a partial differential equation system that describes the thermal and mechanical behavior of the different parts in addition to the electromagnetic design. Due to these models’ limited resources and high computation demand, they cannot be used directly for real-time control. Model order reduction methods have been of growing interest in the past decades and offer solutions for this problem. According to the processed literature, many model order reduction-based methods are used for a wide range of problems. However, a paper has not been published that discusses a model order reduction-based real-time control model that is actually used in the industry. This paper aims to summarize and systematically review the model order reduction methods developed for rotating electrical machines in the last two decades and examine the possible usage of these methods for a real-time control problem.

Order Reduction of Real Time Electromechanical Systems by Using a New Model Order Reduction Method and Controller Design

Order Reduction of Real Time Electromechanical Systems by Using a New Model Order Reduction Method and Controller Design

Model order reduction for the input–output behavior of a geothermal energy storage

In this article, we consider a geothermal energy storage system in which the spatio-temporal temperature distribution is modeled by a heat equation with a time-dependent convection term. Such storage systems are often embedded in residential heating systems. The control and management of such systems requires knowledge of aggregated characteristics of the temperature distribution in the storage. These describe the input–output behavior of the storage, the associated energy flows, and their response to charging and discharging processes. Our aim is to derive an efficient, approximate description of these characteristics by using low-dimensional systems of ordinary differential equations (ODEs). This leads to a model order reduction problem for a large-scale linear system of ODEs resulting from the semidiscretization of the heat equation combined with a linear algebraic output equation. In a first step, we approximated the nonautonomous system of ODEs by a linear time-invariant system. Then, we applied Lyapunov balanced truncation model order reduction to approximate the output by a reduced-order system that has only a few state equations but almost the same input–output behavior. The results of our extensive numerical experiments show the efficiency of the applied model order reduction method. We found that only a few suitably chosen ODEs are sufficient to achieve good approximations of the input–output behavior of the storage.

Reduced-order model-inspired experimental identification of damped nonlinear structures

In this work, we address Nonlinear System Identification (NSI) of geometrically nonlinear structures using experimental response data. Specifically we consider nonlinear structures with large inertia effects. A laboratory scale cantilever-type beam structure is considered, which is chosen for its large inertial effects. Free decay data is gathered from the experimental cantilever-type beam system using an image-based measurement technique. A general mathematical model is derived for nonlinear systems with large inertia, by taking inspiration from model reduction methods. Specifically, a reduced-order modelling method, which accounts for the kinetic energy of the modes not included in the reduction basis, is utilised for experimental NSI. Experimental system identification using the ROM-inspired model shows superior accuracy compared to standard stiffness nonlinear models typically used for modelling such systems. We also identify the damping of the structure by projecting the effect of unmodelled modes onto non-conservative dissipative forces in the ROM-inspired model. We show that this addresses issues of poor fit associated with using a linear damping model. This results in a nonlinear damping force in the ROM-inspired model which accounts for the damping effect of the unmodelled modes. These nonconservative effects of unmodelled modes are often neglected, which results in an incorrect damping estimation. A crude Finite-Element (FE) model of the cantilever-type beam is used to generate the nonlinear mapping of nonlinear damping force. The free decay response of the identified nonconservative ROM-inspired model closely matches the measurement decay response. This further validates the accuracy of the ROM-inspired model in NSI.

Real-time control of a soft manipulator based on reduced order extended position-based dynamics

Soft robots are high-dimensional nonlinear systems coupled with both geometric and material nonlinearity. Control of such a system is complex and time-consuming. In this study, a real-time trajectory tracking control framework is established based on the reduced order extended position-based dynamics. In contrast to the common nonlinear model order reduction methods that require to collect a large number of data to create the motion subspace, this article's motion subspace is constructed based on the model configuration and material properties. The linear modes of the model and the related modal derivatives provide the reduced order matrix, which streamlines and increases the efficiency of model construction. Then, coupled with the instantaneous optimal control, a real-time reduced order model-based control framework of soft robots can be constructed. Experiments on trajectory tracking of a soft manipulator are conducted to verify the accuracy and efficiency of the proposed controller. The average error of all experiments is within 1 cm; and the single-step calculation time of the controller is about 0.057 s, which is less than the sampling period 0.1 s.

Model order reduction of time-domain acoustic finite element simulations with perfectly matched layers

This paper presents a stability-preserving model reduction method for an acoustic finite element model with perfectly matched layers (PMLs). PMLs are often introduced into an unbounded domain to simulate the Sommerfeld radiation condition. These layers act as anisotropic damping materials to absorb the scattered field, of which the material properties are frequency- and coordinate-dependent. The corresponding time-domain model size is often very large due to this frequency-dependent property and the number of elements needed per wavelength. Therefore, to enable efficient transient simulations, this paper proposes a two-step method to generate stable reduced order models (ROMs) of such systems. Firstly, the modified and stable version of PMLs is projected by a one-sided split basis, which gives a stable intermediate ROM. Secondly, the intermediate ROM is modified to satisfy the stability-preserving condition by applying the modal transformation. Applying any one-sided model order reduction method on this modified model leads to a stable and small ROM. This two-step method is further extended to account for the locally-conformal PML model by reformulating it in curvilinear coordinates, which works for arbitrary convex truncated domains. The proposed method is successfully verified by several numerical simulations.

Multifidelity Methodology for Reduced-Order Models with High-Dimensional Inputs

In the early stages of aerospace design, reduced-order models (ROMs) are crucial for minimizing computational costs associated with using physics-rich field information in many-query scenarios requiring multiple evaluations. The intricacy of aerospace design demands the use of high-dimensional design spaces to capture detailed features and design variability accurately. However, these spaces introduce significant challenges, including the curse of dimensionality, which stems from both high-dimensional inputs and outputs necessitating substantial training data and computational effort. To address these complexities, this study introduces a novel multifidelity, parametric, and nonintrusive ROM framework designed for high-dimensional contexts. It integrates machine learning techniques for manifold alignment and dimension reduction—employing proper orthogonal decomposition and model-based active subspace—with multifidelity regression for ROM construction. Our approach is validated through two test cases: the 2D RAE 2822 airfoil and the 3D NASA CRM wing, assessing various fidelity levels, training data ratios, and sample sizes. Compared to the single-fidelity principal component–active subspace (PCAS) method, our multifidelity solution offers improved cost-accuracy benefits and achieves better predictive accuracy with reduced computational demands. Moreover, our methodology outperforms the manifold-aligned ROM method by 50% in handling scenarios with large input dimensions, underscoring its efficacy in addressing the complex challenges of aerospace design.

A novel model order reduction technique for solving horizontal refraction equations in the modeling of three-dimensional underwater acoustic propagation

Modeling three-dimensional (3D) underwater acoustic propagation is vital for underwater detection, localization, and navigation. This article introduces a novel model order reduction (MOR) technique to solve horizontal refraction equations (HREs) in the modeling of 3D underwater acoustic propagation. This approach relies on an adiabatic approximation of the 3D sound field, representing the field as a combination of local vertical modes with their modal coefficients governed by a system of two-dimensional (2D) HREs. Inspired by normal mode theory, the coefficients in the expansion over vertical modes are determined by projecting them onto a lower-dimensional, orthogonal space defined by their transverse eigenfunctions. By artificially truncating the horizontal domain in the transverse directions using two perfectly matched layers (PMLs), the eigenproblem associated with the transverse eigenfunctions of the modal coefficients is closed and thus can be solved through a modal projection method. The modal projection method enables fast computation of modal coefficients in a longitudinally invariant environment within seconds, offering a naturally wide-angle solution covering ±90°. The MOR method is extended to encompass fully 3D cases by introducing an admittance matrix, a memory-saving strategy that prevents numerical overflow when the longitudinal range is large. Moreover, the fact that the outer boundaries of the PMLs are range-independent allows the proposed MOR technique to perform well on a coarse grid when employing the Magnus scheme, significantly saving the numerical cost for the fully 3D simulation. Numerical simulations are provided for both longitudinally invariant and fully 3D scenarios, demonstrating the high accuracy and efficiency of the proposed MOR technique in solving HREs.

On the benefits of a multiscale domain decomposition method to model-order reduction for frictional contact problems

In this paper, the efficiency of a multiscale strategy based on a domain decomposition method (DDM) for model-order reduction of time-dependent frictional contact problems is presented. The proposed strategy relies on the LArge Time INcrement (LATIN) nonlinear solver combined with model reduction based on the Proper Generalized Decomposition (PGD). The LATIN presents a robust treatment of contact conditions, sharing similarities with augmented Lagrangian approaches, and naturally leads to a mixed DDM. In addition, the global space–time formulation of the method allows PGD-based model reduction to be used during computations, creating and enriching on-the-fly reduced bases per substructure to better track sliding fronts and propagative phenomena. The introduction of a multiscale strategy in the LATIN framework is consistent with the physics of contact problems, in which phenomena with different wavelengths interact: local solutions at contact interfaces presents high gradient effects with a short wavelength compared to the characteristic length of the structure. By taking advantage of this, the coarse scale problem of the strategy enables to capture efficiently the behavior of the problem at the structural level, focusing then on capturing the local contact variations at the contact interfaces. The most important features of the approach are shown comprehensively on a simple one-dimensional frictional contact problem. Then, its robustness and effectiveness are tested on a two-dimensional multibody frictional contact problem with more complex loadings. Guidelines are also given for choosing the parameters of the strategy, in particular those driving the construction of the reduced basis.

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

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

Special structural Gramian approximation methods for model order reduction of time‐delay systems

AbstractModel order reduction methods via low‐rank approximation of Gramians for time‐delay systems are developed in this paper. The main contribution is to achieve the balancing and truncation of the system by utilizing low‐rank decomposition of the Gramians combined with the low‐rank square root framework. Here, based on Laguerre expansion technique, the low‐rank factorization of the system Gramians is realized via a linear system with special structure, thus enabling an efficient implementation of the reduction process. Furthermore, the issue of stability preservation is briefly described. We employ the dominant subspaces projection model reduction method to mitigate the effects which may accidentally produce unstable reduced models. Finally, numerical results verify the performance of the approximation‐Gramian methods.

A new hybrid reduced order modeling for parametrized Navier–Stokes equations in stream-vorticity formulation

In the context of traditional reduced order modeling methods (ROMs), time and parameter extrapolation tasks remain a formidable challenge. To this end, we propose a hybrid projection/data-driven framework that leverages two subspaces to improve the prediction accuracy of traditional ROMs. We first obtain inaccurate mode coefficients from traditional ROMs in the reduced order subspace. Then, in the prior dimensionality reduced subspace, we correct the inaccurate mode coefficients and restore the discarded mode coefficients through neural network. Finally, we approximate the solutions with these mode coefficients in the prior dimensionality reduced subspace. To reduce the computational cost during the offline training stage, we propose a training data sampling strategy based on dynamic mode decomposition (DMD). The effectiveness of the proposed method is investigated with the parameterized Navier–Stokes equations in stream-vorticity formulation. In addition, two additional time extrapolation methods based on DMD are also proposed and compared.

Efficient strategy for topology optimization of stochastic viscoelastic damping structures

In this study, the dynamic topology optimization (TO) of stochastic viscoelastic damping structures (VDSs) is performed for the first time. However, the expensive computation cost seriously hinders the TO process. To address this problem, an efficient strategy is proposed. On the one hand, in the stochastic structural response analysis, a fully adaptive method based on direct probability integral method is presented to determine the number and locations of samples. Meanwhile, to improve the computational efficiency of structural response for each sample, a piecewise model-order reduction method based on Krylov subspace is adopted to generate the orthonormal basis and project the original large-scale system onto a low-order system. On the other hand, to overcome the optimization challenge arising from large number of design variables in the density-based topology method, a material-field series-expansion method is employed to describe the topology layout and significantly reduce the number of design variables. Moreover, the sensitivity of the optimization model is derived by the adjoint method and the method of moving asymptotes (MMA) is used to efficiently update the design variables. Some numerical examples comprehensively demonstrate the effectiveness and efficiency of the proposed strategy. The results indicate that the uncertainty of structural parameters greatly affect both the topology layout of VDS and vibration damping capacity.

Broadband topology optimization of three-dimensional structural-acoustic interaction with reduced order isogeometric FEM/BEM

This paper presents a model order reduction method to accelerate broadband topology optimization of structural-acoustic interaction systems by coupling Finite Element Methods and Boundary Element Methods. The finite element method is used for simulating thin-shell vibration and the boundary element method for exterior acoustic fields. Moreover, the finite element and boundary element methods are implemented in the context of isogeometric analysis, whereby the geometric accuracy and high order continuity of Kirchhoff-Love shells can be guaranteed and meantime no meshing is necessary. The topology optimization method takes continuous material interpolation functions in the density and bulk modulus, and adopts adjoint variable methods for sensitivity analysis. The reduced order model is constructed based on second-order Arnoldi algorithm combined with Taylor's expansions which eliminate the frequency dependence of the system matrices. Numerical results show that the proposed algorithm can significantly improve the efficiency of broadband topology optimization analysis.

Reduced-order modeling and solution method for nonlinear frequency response analysis of large space truss structures

Reduced-order modeling and solution method for nonlinear frequency response analysis of large space truss structures

Structure-preserving reduced order model for cross-diffusion systems

In this work, we construct a structure-preserving Galerkin reduced-order model for the resolution of parametric cross-diffusion systems. Cross-diffusion systems are often used to model the evolution of the concentrations or volumic fractions of mixtures composed of different species, and can also be used in population dynamics (as for instance in the SKT system). These systems often read as nonlinear degenerated parabolic partial differential equations, the numerical resolutions of which are highly ex- pensive from a computational point of view. We are interested here in cross-diffusion systems which exhibit a so-called entropic structure, in the sense that they can be formally written as gradient flows of a certain entropy functional which is actually a Lyapunov functional of the system. In this work, we propose a new reduced-order modelling method, based on a reduced basis paradigm, for the resolution of parameter-dependent cross-diffusion systems. Our method preserves, at the level of the reduced-order model, the main mathematical properties of the continuous solution, namely mass conservation, non- negativeness, preservation of the volume-filling property and entropy-entropy dissipation relationship. The theoretical advantages of our approach are illustrated by several numerical experiments.

A Parametrized Model Order Reduction Method Based on Integral Equation for Array Structures

A Parametrized Model Order Reduction Method Based on Integral Equation for Array Structures

Model order reduction, a novel method using krylov sub-spaces and genetic algorithm

Model Order Reduction (MOR) of complex and large systems in Electrical engineering, continuous to be an attractive field for Engineers and Scientists over the last few decades, this complexity of models makes the control designs and simulation using Computer Aided Design (CAD) more and more difficult and consuming a lot of time. There for, accurate, robust and fast algorithms for simulation are needed. The goal of MOR is to replace the original system by an appropriate reduced system which preserves the main properties of the original one such that stability and passivity. Several analytical MOR techniques have been proposed in the literature over the past few decades, to approximate high order linear dynamic systems like Krylov sub-space techniques and SVD (Singular Value Decomposition) techniques. However, most of these techniques lead to computationally demanding, time consuming, iterative procedures that usually result in non-robustly stable models with poor frequency response resemblance to the original high order model in some frequency ranges. Recently a set of new techniques based on Artificial Intelligence (AI) were proposed in [1] for MOR. This article considers the problem of model order reduction of Linear Time In varying (LTI) systems. It is described by first and second order ordinary differential equations model. A tow steps method for model order reduction of LTI systems is proposed here, which combined features of an analytic technique (Krylov approach) and an AI technique (Genetic Algorithm). In the first step, the size of the original model is reduced to an intermediate order, using an analytical technique based on Krylov sub-spaces. In the final step of the reduction process, an AI approach based on Genetic Algorithm (GA) is applied to obtain an optimized nominal model.

Low-rank balanced truncation of discrete time-delay systems based on Laguerre expansions

This paper introduces a novel model order reduction method based on low-rank Gramian approximations for discrete time-delay systems. Firstly, an efficient algorithm based on Laguerre functions to compute the low-rank decomposition factors of the controllability and observability Gramians for discrete time-delay systems is given, in which the low-rank factors satisfy the iterative recursive formulas of the expansion coefficients of the Laguerre functions. It effectively avoids the direct solutions of Gramians. Then, the reduced-order model of the discrete time-delay system is obtained by combining the low-rank square root method. Additionally, a modified algorithm that combines the dominant subspace projection method is introduced, which alleviates certain drawbacks of the above technique and enhances the stability in certain cases. Finally, two numerical examples are given to verify the accuracy and efficiency of our proposed algorithm.

High-order Krylov subspace model order reduction methods for bilinear time-delay systems

Model order reduction methods via high-order Krylov subspace for bilinear time-delay systems are developed in this paper. The proposed methods are based on the expansion of the Taylor series or Laguerre series. The obtained reduced systems can not only match certain expansion coefficients but also preserve the structure of the original system. We also briefly discuss the two-sided projection reduction method. To address the implementation of our approach, we utilize the high-order block Arnoldi algorithm to generate projection matrices and employ the genetic algorithm to optimize parameter selection during the reduction process. Finally, we validate the performance of the proposed reduction methods through numerical results.