Model Reduction Method Research Articles

Convolution tensor decomposition for efficient high-resolution solutions to the Allen–Cahn equation

This paper presents a convolution tensor decomposition based model reduction method for solving the Allen–Cahn equation. The Allen–Cahn equation is usually used to characterize phase separation or the motion of anti-phase boundaries in materials. Its solution is time-consuming when high-resolution meshes and large time scale integration are involved. To resolve these issues, the convolution tensor decomposition method is developed, in conjunction with a stabilized semi-implicit scheme for time integration. The development enables a powerful computational framework for high-resolution solutions of Allen–Cahn problems, and allows the use of relatively large time increments for time integration without violating the discrete energy law. To further improve the efficiency and robustness of the method, an adaptive algorithm is also proposed. Numerical examples have confirmed the efficiency of the method in both 2D and 3D problems. Orders-of-magnitude speedups were obtained with the method for high-resolution problems, compared to the finite element method. The proposed computational framework opens numerous opportunities for simulating complex microstructure formation in materials on large-volume high-resolution meshes at a deeply reduced computational cost.

Model Reduction Method for Spacecraft Electrical System Based on Singular Perturbation Theory

Model Reduction Method for Spacecraft Electrical System Based on Singular Perturbation Theory

Dynamic substructuring for visco-elastic materials using a POD-based model reduction method: application to laminated panels

Dynamic substructuring is now a well established and powerful tool for engineers used to analyze the dynamic response of mechanical systems. The technique is particularly favored for the simulation of mechanical vibrations allowing different aspects to be studied. It consists in splitting the global structure into several substructures from which reduced models are built. In Finite Element software, these reduced models, also called superelements, are generally built using the concept of Component Mode Synthesis. Difficulties arise when structures made of frequency-dependent materials are considered since classical modal reduction algorithms are inapplicable as they are because of the frequency-dependence of the stiffness matrix. Based on the Balanced Truncation approach, a novel methodology for the construction of a superelement for the dynamic analysis of elastic structures made with viscoelastic materials is presented. The methodology takes advantage of the Golla-Hughes-McTavish rheological model (GHM) before reducing the system via the Balanced Proper Orthogonal Decomposition (BPOD). The coupling of the superelement with a host structure is achieved via the use of Lagrange multipliers. The reduced system takes the form of a constant matrix of very small size compared to the original FE model. The method is applied to laminated panels such as an automobile windscreen.

Impact of Phase Angle Jump on a Doubly Fed Induction Generator under Low-Voltage Ride-Through Based on Transfer Function Decomposition

During the fault period, a phase angle jump may occur at the stator or the point of common coupling, which will deteriorate the low-voltage ride-through (LVRT) characteristics of a doubly fed induction generator (DFIG). The existing LVRT studies focus on the impact of a voltage drop on DFIGs but often ignore that of a phase angle jump. The time-domain simulation is accurate in describing the response of a DFIG during the LVRT process, but it is time-consuming for a DFIG with the full-order model. In this paper, by using the voltage magnitude and phase angle of the stator or the point of common coupling as the inputs, and the state variables as the outputs, the transfer function of a DFIG is derived to analyze its response and find the LVRT measures against the voltage drop and, especially, the phase angle jump. Firstly, the differential-algebraic equations of the DFIG are linearized to propose their transfer function model. Secondly, considering its high-order characteristic, a model reduction method for the transfer function of the DFIG using the Schur decomposition is proposed, and the analytical expression of the output variables of the DFIG with the phase angle jump is derived by the inverse Laplace transformation to judge the necessity of the LVRT measures. Finally, the simulation results of the DFIG are provided to verify the accuracy of the transfer function model and its reduced-order form and validate the feasibility of the LVRT against the phase angle jump with the proposed models.

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

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

Graph-based, dynamics-preserving reduction of (bio)chemical systems.

Complex dynamical systems are often governed by equations containing many unknown parameters whose precise values may or may not be important for the system's dynamics. In particular, for chemical and biochemical systems, there may be some reactions or subsystems that are inessential to understanding the bifurcation structure and consequent behavior of a model, such as oscillations, multistationarity and patterning. Due to the size, complexity and parametric uncertainties of many (bio)chemical models, a dynamics-preserving reduction scheme that is able to isolate the necessary contributors to particular dynamical behaviors would be useful. In this contribution, we describe model reduction methods for mass-action (bio)chemical models based on the preservation of instability-generating subnetworks known as critical fragments. These methods focus on structural conditions for instabilities and so are parameter-independent. We apply these results to an existing model for the control of the synthesis of the NO-detoxifying enzyme Hmp in Escherichia coli that displays bistability.

A new model reduction method based PBC control for grid-connected inverter with LCL-filter

Passivity-based control (PBC) exhibits robustness against parameters shift, providing a substantial level of stability. However, for the LCL-filtered grid-connected inverter (GCI), the conventional PBC (called C-PBC) controller has a narrow control bandwidth due to the control time delay, resulting in poor dynamic performance, especially in the weak grid state. This paper proposes a new PBC (called P-PBC) method to increase the closed-loop bandwidth of the system by setting an appropriate feedback proportional coefficient between the grid-side current and the inverter-side current. By extending the closed-loop bandwidth of the system, the proposed P-PBC method offers improved dynamic performance, particularly in challenging grid conditions. In addition, a state observer is also employed to reduce sensors, which saves costs and enhances the system's reliability. A 110 V/50 Hz/3 kW/3-phase experimental setup has been developed using the dSPACE DS1202 platform to validate the effectiveness of the proposed control method.

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.

Improvement to dynamic condensation of structural systems using an innovative virtual model reduction

Model reduction methods have primarily been used in the field of the structural dynamic analysis to reduce the computational time. However, it has progressively been extending to other engineering practices like the damage detection to extract the structural response by means of limited monitored degrees-of-freedom. This paper aims to improve the iterative dynamic condensation method, as a convenient method of the model reduction, for better assessment of the structural dynamic properties. The improvement consists of: (1) Capturing different modal frequencies by applying an iterative perturbation near the frequency obtained from the static condensation. The perturbation is evaluated in a manner that there is no need to solve eigenvalue problem successively; (2) Developing virtual dynamic condensation (VDC) procedure to generalize unique reduced stiffness and mass matrices, reflecting the influence of all modes of vibration, needed for a complete dynamic analysis. Finally, the accuracy of the proposed VDC method was proved by solving three examples.

A combinatorial model reduction method for the finite element analysis of wind instruments

AbstractA high‐fidelity finite element model is proposed for the complete simulation of the time‐harmonic acoustic propagation in wind instruments. The challenge is to meet the extremely high accuracy required by professional musicians, in a complex domain, for all fingerings and over a wide frequency range, within an affordable computational time. Several modelling assumptions are made to limit the numerical complexity of the problem while preserving all relevant physics. A dedicated high‐performance solution strategy is also proposed, based on partitioning, condensation and model order reduction, exploiting the combinatorial nature of wind instrument fingerings. Finally, the proposed approach is applied to the simulation of an alto saxophone. An order of magnitude reduction in memory and computational cost is achieved.

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.

Structure-preserving model reduction based on multiorder Arnoldi for discrete-time second-order systems

This paper investigates the new structure-preserving model reduction methods based on multiorder Arnoldi for discrete-time second-order systems by using discrete Laguerre polynomials. The expansion coefficients of discrete-time second-order system and its equivalent first-order system satisfy the implicit recursive expressions in space spanned by discrete Laguerre polynomials, and then multiorder Arnoldi algorithm is applied to the implicit recursive expressions to produce the column-orthogonal matrices, where the orthogonal projection matrix for the equivalent first-order system is constructed by partitioning the resulting column-orthogonal matrix appropriately. The reduced-order systems can preserve the special structures of the original systems as well as match a desired number of expansion coefficients of the original outputs. The error estimate of output variables is also given for discrete-time second-order system. Two benchmark examples are simulated to illustrate the effectiveness of the proposed methods.

Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: a study using POD-Galerkin and dynamical low-rank approximation

Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques that allow for efficient and accurate simulations while guaranteeing physical properties like mass conservation. In this paper, we develop the first model reduction for the hyperbolic shallow water moment equations and achieve mass conservation. This is accomplished using a macro-micro decomposition of the model into a macroscopic (conservative) part and a microscopic (non-conservative) part with subsequent model reduction using either POD-Galerkin or dynamical low-rank approximation only on the microscopic (non-conservative) part. Numerical experiments showcase the performance of the new model reduction methods including high accuracy and fast computation times together with guaranteed conservation and consistency properties.

Dynamic Model Reduction for Viscously Damped Structures with Statically Determinate Interfaces

A novel model reduction method for viscously damped structures with statically determinate interfaces, such as spacecraft flexible appendages, is proposed. The paper presents a derivation of the complete complex modal expansion of the interface dynamic stiffness of these structures. Based on the identity relation for all complex modes, which is obtained during the derivation, it is found that the interface acceleration impedance can be expressed as a rational fraction with high accuracy using only low-order complex modes. Using this rational fraction as an approximation model, numerical results of the frequency response can be fitted. The fitted interface acceleration impedance can be applied to real-time control as a reduced model in the form of a transfer function. Furthermore, it can be transformed into the form of system matrices by introducing auxiliary variables, which then participate in the dynamic analysis of the assembly. The reduction process circumvents complex modal analysis and necessitates only the results of frequency responses. Thanks to the powerful ability of conventional finite element software to perform frequency response analysis, this reduction method can be used for large-scale complex models in actual engineering applications.

Signal generator framework-based variation expansion-matching method for polynomial systems

This paper proposes an interpolatory model reduction method rooted in the signal generator framework for polynomial systems. This innovative approach not only surpasses the constraints on input spaces found in moment-matching methods, such as u(t)∈L2([0,+∞]) recommended in H2 optimal interpolation method, but also eliminates redundant reduced bases through variations of solution in the time domain. Our contributions can be delineated into three key aspects. Firstly, we establish the closure under addition, multiplication, and composition for the transformations from all elementary functions and rational functions to a signal generator-driven system. This validates the applicability of the signal generator framework to general inputs. Secondly, we enhance the numerical stability of two-sided reduced bases for polynomial systems through a variation expansion. Thirdly, we employ tensor decomposition techniques to streamline the complexity of the model reduction process. The simulation results have verified the effectiveness and applicability of the proposed method, which requires smaller computational complexity while maintaining the same level of accuracy.

Order reduction of special valve models based on pod method

Abstract The opening size of the disc-type special regulating valve (referred to as the special valve) can be used to regulate the fluid pressure and flow inside the cabin, and is an important equipment in the intake system of the high-altitude simulation test bench (referred to as the high-altitude bench). Due to the high dimensionality and complex solution of the special valve model, the calculation time is relatively long. In this paper, the Proper Orthogonal Decomposition (POD) model reduction method is applied to the mathematical model of the special valve, and the POD basis matrix is constructed through snapshots, thus obtaining a low-dimensional model and improving computational efficiency. The research results indicate that, while ensuring sufficient accuracy, this method can reduce the dimensionality of the special valve model, improve the computational speed of the simulator in calculating this case, and verify the effectiveness of the POD method.

A parallel power system linear model reduction method based on extended Krylov subspace

With the ever-increasing scale of power systems, stability analysis and control usually bear heavy storage and massive calculation burdens. In view of this, the model order reduction technique proves valuable by constructing a low-dimensional approximate model of the original system, which is crucial for efficiently handling large-scale systems. Balanced truncation (BT), a famous model reduction method, confronts practical limitations as it requires the systems to be stable and cannot deal with unstable models. Therefore, a parallel linear balanced truncation method for power systems based on extended Krylov subspace (EKS) is proposed in this work. Besides extending the BT method to unstable systems by α-shift, the key contribution also lies in strategies to enhance the convergence of the EKS method, whereupon the algorithm improvements include effective and efficient techniques for solving dual Lyapunov equations, and parallel acceleration of the singular value decomposition. The results of the simulation case verify that the proposed method can effectively improve the convergence of the EKS method by increasing α-shift, and the improvement work in this paper reduces the total time consumption of the BT method by about 26 %–33 % of the original, exhibiting better calculation efficiency. In addition, the case studies show that the simplified model still retains the time-domain and frequency-domain response characteristics of the original high-dimensional model.

Reduced-order modeling of modular, position-dependent systems with translating interfaces

Many complex mechatronic systems consist of multiple interconnected dynamical subsystems, which are designed, developed, analyzed, and manufactured by multiple independent teams. To support such a design approach, a modular model framework is needed to reduce computational complexity and, at the same time, enable multiple teams to develop and analyze the subsystems in parallel. In such a modular framework, the subsystem models are typically interconnected by means of a static interconnection structure. However, many complex dynamical systems exhibit position-dependent behavior (e.g., induced by translating interfaces) which cannot be captured by such static interconnection models. In this paper, a modular model framework is proposed, which allows to construct an interconnected system model, which captures the position-dependent behavior of systems with translating interfaces, such as linear guide rails, through a position-dependent interconnection structure. Additionally, this framework allows to apply model reduction on subsystem level, enabling a more effective reduction approach, tailored to the specific requirements of each subsystem. Furthermore, we show the effectiveness of this framework on an industrial wire bonder. Here, we show that including a position-dependent model of the interconnection structure (1) enables to accurately model the dynamics of a system over the operating range of the system and, (2) modular model reduction methods can be used to obtain a computationally efficient interconnected system model with guaranteed accuracy specifications.

Mode selection for component mode synthesis with guaranteed assembly accuracy

In this work, a modular approach is introduced to select the most important eigenmodes for each component of a composed structural dynamics system to obtain the required accuracy of the reduced-order assembly model. To enable the use of models of complex (structural) dynamical systems in engineering practice, e.g., in a design, optimization and/or control context, the complexity of the models needs to be reduced. When the model consist of an assembly of multiple interconnected structural components, component mode synthesis is often the preferred model reduction method. The standard approach to component mode synthesis for such system is to select the eigenmodes of a component that are most important to accurately model the dynamic behaviour of this component in a certain frequency range of interest. However, often, a more relevant goal is to obtain, in this frequency range, an accurate model of the assembly. In the proposed approach, accuracy requirements on the level of the assembly are translated to accuracy requirements on component level, by employing techniques from the field of systems and control. With these component-level requirements, the eigenmodes that are most important to accurately model the dynamic behaviour of the assembly can be selected in a modular fashion. We demonstrate with two structural dynamics benchmark systems that this method based on assembly accuracy allows for a computationally efficient selection of eigenmodes that (1) guarantees satisfaction of the assembly accuracy requirements and (2) results in most cases in reduced-order models of significantly lower order with respect to the industrial standard approach in which component eigenmodes are selected using a frequency criterion.

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.