Model Order Reduction Algorithm Research Articles

Reduced-Order Model of Coal Seam Gas Extraction Pressure Distribution Based on Deep Neural Networks and Convolutional Autoencoders

There has been extensive research on the partial differential equations governing the theory of gas flow in coal mines. However, the traditional Proper Orthogonal Decomposition–Radial Basis Function (POD-RBF) reduced-order algorithm requires significant computational resources and is inefficient when calculating high-dimensional data for coal mine gas pressure fields. To achieve the rapid computation of gas extraction pressure fields, this paper proposes a model reduction method based on deep neural networks (DNNs) and convolutional autoencoders (CAEs). The CAE is used to compress and reconstruct full-order numerical solutions for coal mine gas extraction, while the DNN is employed to establish the nonlinear mapping between the physical parameters of gas extraction and the latent space parameters of the reduced-order model. The DNN-CAE model is applied to the reduced-order modeling of gas extraction flow–solid coupling mathematical models in coal mines. A full-order model pressure field numerical dataset for gas extraction was constructed, and optimal hyperparameters for the pressure field reconstruction model and latent space parameter prediction model were determined through hyperparameter testing. The performance of the DNN-CAE model order reduction algorithm was compared to the POD-RBF model order reduction algorithm. The results indicate that the DNN-CAE method has certain advantages over the traditional POD-RBF method in terms of pressure field reconstruction accuracy, overall structure retention, extremum capture, and computational efficiency.

An enhanced sparrow search algorithm for model order reduction of proton exchange membrane fuel cell system

An enhanced sparrow search algorithm for model order reduction of proton exchange membrane fuel cell system

A greedy MOR method for the tracking of eigensolutions to parametric elliptic PDEs

In this paper we introduce a Model Order Reduction (MOR) algorithm based on a sparse grid adaptive refinement, for the approximation of the eigensolutions to parametric problems arising from elliptic partial differential equations. In particular, we are interested in detecting the crossing of the hypersurfaces describing the eigenvalues as a function of the parameters.The a priori matching is followed by an a posteriori verification, driven by a suitably defined error indicator. At a given refinement level, a sparse grid approach is adopted for the construction of the grid of the next level, by using the marking given by the a posteriori indicator.Various numerical tests confirm the good performance of the scheme.

Dimension reduction based on approximate gramians via Laguerre polynomials for coupled systems

Abstract In this paper, we focus on the topic of model order reduction (MOR) for coupled systems. At first, an approximation via Laguerre polynomials expansions to controllability and observability gramians for such systems are presented, which provides a low-rank decomposition form whose factors are constructed from a recurrence formula instead of Lyapunov equations. Then, in combination of balanced truncation and dominant subspace projection method, a series of MOR algorithms are proposed that preserve the coupled structures. What’s more, some main properties of reduced-order models, such as stability preservation, are well discussed. Finally, three numerical simulations are provided to illustrate the effectiveness of our algorithms.

Reduced-order model for RBDO of multiple TMDs on eccentric L-shaped buildings subjected to seismic excitations

L-shaped buildings subjected to ground accelerations undergo coupled lateral and torsional vibrations that can result in critical stress concentrations at their re-entrant corners. Even so, a comprehensive literature review indicates that the Reliability-Based Design Optimization (RBDO) of multiple Tuned Mass Dampers (TMDs) in L-shaped buildings has not been studied yet, to the best of the authors’ knowledge. Indeed, the more refined structural model required to capture the wings motion, rather than assuming shear-framed behavior as performed on asymmetric rectangular buildings, becomes critical in an RBDO framework. Thus, integrating a non-modal Model Order Reduction (MOR) strategy allows for a massive reduction in the computational burden. Emerging from control engineering, these techniques do not form their orthogonal subspace based on modal information. Furthermore, they dispense previous snapshots of the system solution while being suitable for non-classically damped systems. Within this context, this paper presents a novel MOR-based RBDO study of multiple TMDs in L-shaped buildings subjected to seismic excitation. The MOR algorithm connects second-order Krylov subspaces to build the reduced-order model at a specific expansion frequency with a windowing strategy for sweeping the complete frequency domain. Then, it is integrated into the RBDO, aiming to determine bi-directional TMD configurations that minimize the building’s annual failure rate for the serviceability limit state. The application case consists of a 4-story L-shaped building with 2880 active degrees of freedom (DOFs). The results show that a reduction in failure probability of up to 65% can be achieved while reaching a speed-up factor (SUF) of more than 35.

Stability analysis of the Robust model order reduction algorithm for parametric linear dynamical systems

AbstractWe study stability of a interpolatory model order reduction (MOR) algorithm for first‐order and second‐order parametric linear dynamical systems, with respect to inexact linear solves. This analysis is easily extendible to other MOR algorithms for such systems. Besides deriving the two conditions for stability, and subsequent experimentation, our most novel contribution here is achieving a backward stable algorithm.

MORpH: Model reduction of linear port-Hamiltonian systems in MATLAB

Abstract We present a novel software toolbox MORpH for the efficient storage, analysis, interconnection and structure-preserving model order reduction (MOR) of linear port-Hamiltonian differential-algebraic equation systems (pH-DAEs). The model class of pH-DAEs enables energy-based modeling and a flexible coupling of models across different physical domains. This makes them particularly suited for the simulation and control of complex technical systems. To promote the use of recent theoretical findings in engineering practice, efficient software solutions are required. In this work, we illustrate how possibly large-scale pH-DAEs can be efficiently stored and interconnected in MATLAB in an object-oriented way. We discuss three structure-preserving MOR strategies that are supported by MORpH and demonstrate the application and performance of selected MOR algorithms by means of two benchmark examples.

Application of dynamic condensation for model order reduction in real-time hybrid simulations

This paper presents an algorithm for model order reduction in real-time hybrid simulations. The bottleneck of hybrid simulations are usually the finite element computations that must be performed in real time. The common approach to deal with this inconvenience is to involve powerful computing hardware. In the following paper opposite approach is presented—a new algorithm is proposed for reducing the model order. This allows to perform hybrid simulations more efficiently while maintaining high accuracy. The algorithm is based on dynamic condensation where the degrees of freedom are divided into masters and slaves, and system matrices are transformed in a way that only masters are kept. The transformation process is complex however, it is performed offline, therefore a relatively high computational effort is made a priori to make real-time calculations more efficient. The dynamic condensation algorithm was adapted to the requirements of hybrid simulations. The proper selection of masters is crucial for accuracy therefore, a novel approach based on evolutionary optimization is implemented. Numerical and experimental examples are provided. The examples prove that by implementing the proposed algorithms the following effects are achieved: (a) the time step of hybrid simulation can be significantly decreased when using implicit integration (increasing the accuracy of the measured dynamic behavior), (b) explicit integration can be sometimes implemented where it was previously not possible in real time, (c) smaller hardware resources can be involved (all computations in real-time hybrid simulations during the experiments were performed on a small microcontroller).

Hybrid data assimilation methods, Part I: Numerical comparison between GEIM and PBDW

Hybrid Data Assimilation (HDA) methods are a class of numerical methods that aim at integrating Model Order Reduction (MOR) techniques into a Data Assimilation (DA) framework, thus combining mathematical models and experimental data. The objective is to reduce the solution time using MOR algorithms whilst keeping the accuracy of the models at the desired level using observations, which serve as an update to the a priori prediction of the model. This two-part work investigated HDA techniques by applying them to two classes of problems: numerical benchmark cases (part 1) and experimental facilities (part 2). In particular, this paper discusses the former, focusing on the numerical formulation of the methodologies and on the effect of noisy data. Indeed, real-world experimental data are always polluted by errors and uncertainties; therefore, it is critical to first assess the performance of these techniques on numerical benchmark cases with the artificial introduction of random noise before applying them to real-world experimental facilities. As such, this paper applies the Generalised Empirical Interpolation Method (GEIM) and the Parameterised-Background Data-Weak (PBDW) formulation to a non-adiabatic airflow over the classical computational fluid-dynamics benchmark of the 3D Backward Facing Step (BFS). Results show how both algorithms are valuable tools to reconstruct the state of the system when measurements are available, whilst assessing the effect of noise on the available data; in particular, the GEIM is a bit better than the PBDW since a lower reconstruction error is achieved with fewer sensors.

Oblique Projection-Based Modal Matching Algorithm for LPV Model Order Reduction of Aeroservoelastic Systems

An aeroservoelastic system can be described as a gridding-based linear parameter-varying (LPV) model, whose dynamic characteristics usually vary with the airspeed. Due to the high order of the system, it is necessary to perform order reduction on LPV models to overcome the control design challenges. However, when directly extending linear time-invariant (LTI) model order reduction technologies to the LPV system, states of the reduced-order LTI models generated separately at different grid points could be inconsistent. In this paper, a novel modal matching algorithm is proposed to solve the problem of state inconsistency by identifying the internal connection between the models at adjacent grid points. An oblique projection-based distance metric is defined to improve the reliability of the modal matching algorithm. The reduced-order LPV model constructed based on this method would have a high fidelity relative to the original model and a smooth interpolation performance between grid points. The proposed algorithm is applied to the X-56A aircraft, and numerical results show its effectiveness.

SOBMOR: Structured Optimization-Based Model Order Reduction

Model order reduction (MOR) methods that are designed to preserve structural features of a given full order model (FOM) often suffer from a lower accuracy when compared to their non-structure-preserving counterparts. In this paper, we present a framework for structure-preserving MOR, which allows us to compute structured reduced order models (ROMs) with a much higher accuracy. The framework is based on parameter optimization, i.e., the elements of the system matrices of the ROM are iteratively varied to minimize an objective functional that measures the difference between the FOM and the ROM. The structural constraints can be encoded in the parametrization of the ROM. The method only depends on frequency response data and can thus be applied to a wide range of dynamical systems. We illustrate the effectiveness of our method on a port-Hamiltonian and on a symmetric second-order system in a comparison with other structure-preserving MOR algorithms.

On frequency- and time-limited [formula omitted]-optimal model order reduction

In this paper, the problems of frequency-limited and time-limited H2-optimal model order reduction of linear time-invariant systems are considered within the oblique projection framework. It is shown that it is inherently not possible to satisfy all the necessary conditions for the local minimizer in the oblique projection framework. The conditions for exact satisfaction of the optimality conditions are also discussed. Further, the equivalence between the tangential interpolation conditions and the gramians-based necessary condition for the local optimum is established. Based on this equivalence, iterative algorithms that nearly satisfy these interpolation-based necessary conditions are proposed. The deviations in satisfaction of the optimality conditions decay as the order of the reduced-model is increased. Moreover, stationary point iteration algorithms that satisfy two out of three necessary conditions for the local minimizer are also proposed. There also, the deviation in satisfaction of the third optimality condition decays as the order of the reduced-model is increased. The efficacy of the proposed algorithms is validated by considering benchmark examples known for testing model order reduction algorithms.

Structure-preserving model order reduction for K-power bilinear systems via Laguerre functions

ABSTRACT This paper presents a series of structure-preserving model order reduction algorithms for K-power bilinear systems via Laguerre functions. The method first aims to rewritten the K-power bilinear system as a general bilinear system and calculate the approximate low-rank factors of the cross Gramian of the bilinear system by combining the idea of Laguerre functions expansion of the matrix exponential function. After that, the approximate balanced system of the K-power bilinear system is constructed by the corresponding projection transformation of each subsystem. In order to achieve the purpose of model order reduction, the states with smaller singular values are then truncated, so as to further obtain the reduced order model. For this approach, there is a disadvantage that unstable systems may be generated although the original one is stable. To alleviate the inadequacies of this approach, we have improved the model reduction procedure, which is based upon the dominant subspace projection method. In addition, we also carried out a correlation analysis on the stability of improved algorithms. Finally, numerical experiments are employed to substantiate the effectiveness of the presented algorithms.

Applying Model Order Reduction Algorithm for Control Design of the Digital Filter

Applying Model Order Reduction Algorithm for Control Design of the Digital Filter

A novel hybrid of Nonlinear Sine Cosine Algorithm and Safe Experimentation Dynamics for model order reduction

ABSTRACT The current study introduces the hybridization of the Nonlinear Sine Cosine Algorithm (NSCA) and Safe Experimentation Dynamics (SED) as a novel optimization method for model order reduction of high-order single-input single-output (SISO) systems. Reciprocated synergism between both meta-heuristic algorithms is achieved by appropriating the nonlinear position-updated mechanism of NSCA for enhanced exploration/exploitation competencies and proficiency of SED in maximizing stagnation avoidance within the local optima. Named the NSCA-SED algorithm, the applicability of the proposed method is assessed by scholastic adoption of a sixth-order numerical transfer function towards two independent high-order systems enclosing Double-Pendulum Overhead Crane and Flexible Manipulator. Experimentation results further suggested NSCA-SED as the superior alternative in terms of execution robustness and consistency excellence against other available optimization-based methods for tackling model order reduction. Exemplified simulations sequentially demonstrated considerable improvements by the employment of NSCA-SED over conventional SCA following respective enhanced proportions of 97.17%, 13.17% and 29.03% for Example 1, Example 2 and Example 3.

A non-intrusive neural network model order reduction algorithm for parameterized parabolic PDEs

Reduced-order modeling based on projection-driven neural network (PDNN) generally needs sufficient data set while physics-informed machine learning (PINN) and physics-reinforced neural network (PRNN) take the reduced order systems into consideration. However, the physics-informed machine learning technique used in these two methods gives rise to expensive time consumption for complex neural network, higher reduced basis and a large amount of residual points. With understanding of PDNN, PINN and PRNN, a model-based neural network (MBNN) is proposed to cope with nonlinear parabolic partial differential equations (PDEs) without source terms. Compared with PINN, the fully discrete scheme of PDEs is adopted to avoid expensive cost for automatic differentiation technique. Moreover, initial conditions at residual points in parameter space are added to loss function with a proper proportion. Since the reduced order equation is taken into account, the proposed MBNN can predict solutions in a larger time range than the time range to which the snapshot belongs. Numerical results show that MBNN achieves better performance than the projection-driven neural network. Generally, the proposed method for lower order reduction can get a consistent L2 error with POD after the convergent tolerance is reached.

System-level modeling and design method of an array of piezoelectric energy harvesters under typical ambient vibration

Vibration-based piezoelectric energy harvesting has received significant attention recently due to its ability to convert vibration energy from the surrounding environment into electric energy to power wireless sensors. In this work, two typical types of vibration sources, that is, engine-induced floor vibration and train-induced bridge vibration, are considered. The array of piezoelectric energy harvesters (PEHs) is adopted to harvest the energy of vibration with different frequencies. The aim of this work is to give a stable system-level modeling method and present a method to select operating point for designing energy harvesters connected with nonlinear energy harvesting circuits under complex excitations. The system-level modeling method is based on Krylov’s subspace model order reduction algorithm. With the help of this method, the harvester can be modeled using conventional finite element software and co-simulated with nonlinear circuits at a system-level. The proposed method is validated using equivalent circuit method, equivalent impedance method and experiment results, and then employed to investigate the performances of harvesters under typical real vibrations. Finally, design suggestions for PEHs under different types of vibration sources are presented.

An MOR Algorithm Based on the Immittance Zero and Pole Eigenvectors for Fast FEM Simulations of Two-Port Microwave Structures

The aim of this article is to present a novel model-order reduction (MOR) algorithm for fast finite-element frequency-domain simulations of microwave two-port structures. The projection basis used to construct the reduced-order model (ROM) comprises two sets: singular vectors and regular vectors. The first set is composed of the eigenvectors associated with the poles of the finite-element method (FEM) state-space system, while the second one is made up from the eigenvectors corresponding to the zeros of the diagonal elements of the matrix-valued immittance transfer function. Importantly, just one LU factorization of the FEM system is required to construct the projection basis during the reduction process, due to the application of a new formulation based on the Schur complement. The sets of eigenvectors that are used in the basis are independent of one another, which makes the new technique better suited for parallel computing compared with previously developed methods, which are sequential in nature. The reliability and accuracy of the proposed scheme are compared with that of the standard MOR technique, namely, the reduced-basis method (RBM), and verified through the analysis of three microwave structures: an eighth-order dual-mode waveguide filter, a dielectric resonator filter, and a folded waveguide filter.

A New Passivity Preserving Model Order Reduction Method: Conic Positive Real Balanced Truncation Method

This article is dedicated to model order reduction of linear time-invariant passive systems, i.e., positive real (PR) systems, based on the balanced truncation (BT) concept. The main feature of this article is that we have used the phase angle of the transfer function, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> , in model order reduction for the first time in the proposed method, which results in more accurate reduced-order models. It is worth noting that this hypothesis has been never made before in any other model order reduction techniques. Furthermore, new gramians are calculated corresponding to their associated new Riccati equations and Lur’e equations, which are in turn achieved by using Kalman–Yakubovic–Popov (KYP) lemma and linear matrix inequalities (LMI). Thereafter, a novel passivity preserving model order reduction algorithm, which takes the phase angle of the transfer function, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> , into account, is presented. It is demonstrated that the proposed method is a generalization of the positive real balanced truncation (PRBT) method, and it is perfectly capable of providing more accurate approximation error compared to PRBT. Finally, numerical examples are included to figure out the effectiveness of the presented method.

Model order reduction of linear and bilinear systems via low-rank Gramian approximation

In this paper, we propose a series of model order reduction algorithms based on low-rank Gramian approximation for linear and bilinear systems. The main idea of the approach for linear systems is to use approximate low-rank factors of the controllability and observability Gramians to generate approximate balanced system for the large-scale system. Then, the reduced-order models are obtained by truncating the states corresponding to the smaller approximate Hankel singular values. The low-rank factors are constructed directly from the Laguerre functions expansion coefficient vectors of the matrix exponential functions by solving a recurrence formula instead of Lyapunov equations. In addition, the reduction procedure is modified with the idea of dominant subspace projection method to produce a stable reduced model under certain conditions. Furthermore, our algorithms are extended to bilinear systems successfully, with a series of corresponding algorithms for bilinear systems derived. Finally, numerical experiments are provided to demonstrate the effectiveness of the proposed methods.