Order Reduction Method Research Articles

Consensus Control and Optimization of Time-Delayed Multiagent Systems: Analysis on Different Order-Reduction Methods

Consensus Control and Optimization of Time-Delayed Multiagent Systems: Analysis on Different Order-Reduction Methods

Synchronization of fractional-order neural networks with inertia terms via cumulative reduced-order method

Synchronization of fractional-order neural networks with inertia terms via cumulative reduced-order method

Ablation study of a non-intrusive data-driven surrogate model for the Rayleigh-Bénard convection problem

Non-intrusive data-driven models are promising class of methodologies to build surrogate models for challenging parametric engineering problems. POD-DL [1,2], a method from this class uses deep neural networks as the main data-driven ingredient. Following a first linear dimensionality reduction, based on a POD basis, it applies a second nonlinear dimension reduction, in this case a deep autoencoder, and a nonlinear regression using a forward neural network to account for the temporal and parametric coefficients. This setting generates a large number of hyperparameters, such as the number of modes of the POD basis, the number of layers and amount of neurons per layer for each network (the autoencoder one and the regression one), besides all typical hyperparameters of neural networks, as learning rate, batch size, etc. This all impacts the amount of time for the training process and the surrogate model accuracy.Based on our previous experience with this methodology for coupled transport-fluid problems, we decided to study the Rayleigh-Bénard convection problem [3] with a fixed Prandtl number, a well documented heat transfer problem coupled with fluid flow. Besides the temporal character of the problem, the Rayleigh number serves as a parameter, i.e., each different number generates a different dynamic that is learned by the surrogate model. Then, the regression neural network can predict the dynamics for an unseen Rayleigh number. In order to understand the contribution of each component of the method, we perform an ablation study in this controlled setup. [1] - M. Cracco et al., “Deep learning-based reduced-order methods for fast transient dynamics”, Arxiv Preprint 2212.07737, 2022.[2] - S. Fresca and A. Manzoni, “POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition”, Comput. Methods Appl. Mech. Engrg., 2022.[3] - R. Kessler, “Nonlinear transition in three-dimensional convection”, Journal of Fluid Mechanics, 1987.

Improving accuracy of parametric surrogate model for turbidity currents using diffusion-based super-resolution

Turbidity currents are a sort of density-driven flow carrying particles that are generated between fluids with small density differences. They also are a mechanism responsible for the deposition of sediments on a seabed. A deep understanding of this phenomenon may help geologists on strategic knowledge in oil exploration. We simulate such currents using a stabilized finite element formulation in a Eulerian-Eulerian framework.This brings the challenge of a high computational cost for the evaluation of the high-fidelity model. We thus use a surrogate model composed of one linear (Proper Orthogonal Decomposition) and one non-linear (Autoencoder) reduction [1,2]. Once the surrogate is trained for a set of parameters with data generated by the high-fidelity model, one can use such a surrogate model for predicting the dynamics for unseen values of the parameter.Denoising Diffusion Probabilistic Models (DDPM) [3] is the method behind SOTA generative AI algorithms with very good performance in tasks like super-resolution. Based on such usage of diffusion for generative AI, the authors in [4] developed a framework for flow field reconstruction using a super-resolution task. Such a diffusion model is trained only using high-resolution data, without a pairing between low and high-resolution data, common in other techniques in the area.The present work aims to use the dataset of high-fidelity parametric simulations of turbidity currents for training both the surrogate and the diffusion model for the super-resolution task. The goal is: using the trained super-resolution model, improve the fields predicted by the reduced model for unseen points of the parametric space.[1] - M. Cracco et al., “Deep learning-based reduced-order methods for fast transient dynamics”, Arxiv Preprint 2212.07737, 2022.[2] - S. Fresca and A. Manzoni, “POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition”, Comput. Methods Appl. Mech. Engrg., 2022.[3] - J. Ho, A. Jain and P. Abbeel, “Denoising Diffusion Probabilistic Models”, arXiv: 2006.11239, 2020.[4] - D. Shu, Z. Li, A. Barati Farimani, “A physics-informed diffusion model for high-fidelity flow field reconstruction”, Journal of Computational Physics, Volume 478, 2023

Analysis of the influence of clamp installation position on vibration stress for spatial pipeline

In aero-engines, small changes in the clamp position can sometimes significantly alter the vibration stress of the pipeline system, so there is an urgent need to study the influence of clamp position on pipeline vibration stress. Firstly, a dynamic modeling method of spatial pipeline system is proposed based on the finite element method, the modeling method can more accurately simulate the complex boundary constraints caused by multiple clamps and pipe fittings. Then, an improved dynamic substructure method (reduced-order method) is developed by combining the dynamic substructure method with the clamp position parametric model to improve the efficiency of the subsequent clamp position parameters influence analysis. Further, the rationality and efficiency of the reduced-order modeling approach are verified by numerical and experimental studies. Finally, the influence of the clamp position and the key parameters of different components (pipe body, clamps, fittings) on the pipeline vibration stress is investigated. The results show that when the main vibration region of the pipeline system coincides with the installation region of the clamp, the vibration stress can be significantly reduced by installing the clamp in the maximum modal displacement region of the corresponding pipe segment. The related modeling methods and conclusions can provide valuable references for the dynamics design of pipeline system in engineering practice.

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.

Two-grid reduced-order method based on POD for a nonlinear poroelasticity model

In this paper, we consider a two-field nonlinear poroelasticity model, in which the unknown variables are the solid displacement and the pore pressure, and the permeability of the elastic material depends on the dilatation. A standard Galerkin method based on Picard iteration is presented. To reduce the computational costs, we develop the two-grid method, the reduced-order finite element (ROFE) method based on proper orthogonal decomposition (POD) technique, and a combination of the first two methods, i.e., the two-grid reduced-order finite element (TGROFE) method. The associated numerical error has four components due to the coarse triangulation discretization, fine triangulation discretization, time discretization, and eigenvalue truncation by POD. Numerical experiments show the accuracy and efficiency of these methods. To further illustrate the application of the POD technique to the quasi-static poroelasticity problem, we investigate two reduced-order mixed finite element methods, which avoid pressure oscillations and/or have locking-free property.

Multiscale preconditioning of Stokes flow in complex porous geometries

Fluid flow through porous media is central to many subsurface (e.g., CO2 storage) and industrial (e.g., fuel cell) applications. The optimization of design and operational protocols, and quantifying the associated uncertainties, requires fluid-dynamics simulations inside the microscale void space of porous samples. This often results in large and ill-conditioned linear(ized) systems that require iterative solvers, for which preconditioning is key to ensure rapid convergence. We present robust and efficient preconditioners for the accelerated solution of saddle-point systems arising from the discretization of the Stokes equation on geometrically complex porous microstructures. They are based on the recently proposed pore-level multiscale method (PLMM) and the more established reduced-order method called the pore network model (PNM). The four preconditioners presented are the monolithic PLMM, monolithic PNM, block PLMM, and block PNM. Compared to existing block preconditioners, accelerated by the algebraic multigrid method, we show our preconditioners are far more robust and efficient. The monolithic PLMM is an algebraic reformulation of the original PLMM, which renders it portable and amenable to non-intrusive implementation in existing software. Similarly, the monolithic PNM is an algebraization of PNM, allowing it to be used as an accelerator of direct numerical simulations (DNS). This bestows PNM with the, heretofore absent, ability to estimate and control prediction errors. The monolithic PLMM/PNM can also be used as approximate solvers that yield globally flux-conservative solutions, usable in many practical settings. We systematically test all preconditioners on 2D/3D geometries and show the monolithic PLMM outperforms all others. All preconditioners can be built and applied on parallel machines.

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.

Reduced-order method for nuclear reactor primary circuit calculation

Reduced-order method for nuclear reactor primary circuit calculation

Stabilized weighted reduced order methods for parametrized advection-dominated optimal control problems governed by partial differential equations with random inputs

Abstract In this work, we analyze Parametrized Advection-Dominated distributed Optimal Control Problems with random inputs in a Reduced Order Model (ROM) context. All the simulations are initially based on a finite element method (FEM) discretization; moreover, a space-time approach is considered when dealing with unsteady cases. To overcome numerical instabilities that can occur in the optimality system for high values of the Péclet number, we consider a Streamline Upwind Petrov–Galerkin technique applied in an optimize-then-discretize approach. We combine this method with the ROM framework in order to consider two possibilities of stabilization: Offline-Only stabilization and Offline-Online stabilization. Moreover we consider random parameters and we use a weighted Proper Orthogonal Decomposition algorithm in a partitioned approach to deal with the issue of uncertainty quantification. Several quadrature techniques are used to derive weighted ROMs: tensor rules, isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo methods. We compare all the approaches analyzing relative errors between the FEM and ROM solutions and the computational efficiency based on the speedup-index.

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

A high-frequency reduced-order model for parameters optimization of AC/DC distribution systems considering random disturbances

Accompanying the rapid development of AC/DC distribution systems, along with the random disturbances introduced by a significant number of distributed generations and loads, the difficulty of system stability analysis has increased. The phenomenon of DC bus voltage instability has become severe, affecting the safe and stable operation of power systems. This paper focuses on the research of multi-terminal AC/DC distribution systems. A reduced-order method for multi-terminal distribution systems is proposed. Through parameter equivalent transformation, the system model is simplified while maintaining its dynamic characteristics. Parameter sensitivity analysis is utilized to identify key parameters influencing the system’s dynamic behavior, achieving an analytical expression of the system’s dynamic characteristics. And stability domain analysis is applied to delineate the variations in system dynamic behavior and responses to random disturbances under different parameter settings. Then a multi-parameter optimization method is designed to enhance system stability while achieving fast system response. Finally, theoretical analysis is validated through software simulation and hardware-in-the-loop experiments.

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.

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.

L1‐type finite element method for time‐fractional diffusion‐wave equations on nonuniform grids

AbstractAs is well known, one of the main challenges in dealing with the time‐fractional diffusion‐wave equation is that the solutions have weak regularity at the initial values. This necessitates the use of nonuniform grids for error analysis. However, the theoretical analysis of nonuniform grids is relatively complex and lacks efficient and stable numerical methods. The main research problem addressed in this article is the time‐fractional diffusion‐wave equation on nonuniform grids. We use the order reduction method and discrete complementary convolution kernel to discretize the Caputo derivative of the equation and obtain the L1‐type finite element method on nonuniform grids. The properties related to discrete complementary convolutional kernels are often used for algorithm convergence analysis, which simplifies the process of finite element theory analysis and is efficient and innovative. This article demonstrates the solvability, stability, and convergence of the L1‐type finite element schemes on nonuniform grids. Finally, the consistency between the proposed finite element scheme and the convergence order results obtained from theoretical analysis was verified through experiments.

A new mixed order reduction method using bonobo optimizer and stability equation

A new mixed order reduction method using bonobo optimizer and stability equation

Research on designing dynamic model of TITAN-VIII quadruped robot based on the robot structure

Quadruped robots have many advantages over wheeled robots in that they have high mobility and can move on any terrain. However, the control problem of four-legged robots will be more complicated because the robot's kinematic model has a high order and the dynamic process is complicated, so it is necessary to implement order reduction methods when constructing kinematic equations for four-legged robots. The article proposes a simplified method to determine the kinematic equations for four-legged robots called TITAN-VIII. The Denavit-Hartenberg (D-H) rule is used to analyze the forward kinematics. The inverse kinematic equations are determined on the basis of mathematics and the structure of the robot's legs. The research results are simulated on MATLAB Simulink software.

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.