Nonlinear Model Order Reduction Research Articles

An instantaneous-baseline multi-indicial nonlinear ultrasonic resonance spectral correlation technique for fatigue crack detection and quantification

This article presents an instantaneous-baseline multi-indicial nonlinear ultrasonic resonance spectral correlation technique for fatigue crack detection and quantification. A reduced-order nonlinear oscillator model is tailored to illuminate the contact acoustic nonlinearity (CAN) and the nonlinear resonance phenomena. The analytical formulation considers the rough surface condition of the fatigue cracks, with a crack open–close transitional range for the effective modeling of the variable-stiffness nonlinear mechanism. Multiple damage indices (DIs) associated with the degree of nonlinearity of the interrogated structures are then proposed by correlating the ultrasonic resonance spectra. Three perspectives of the nonlinear resonance phenomena are investigated to detect and monitor the fatigue crack growth: (1) time-history dependence, which evolves different resonance states depending on the loading history; (2) amplitude dependence, which renders significantly different nonlinear responses under various levels of excitation amplitudes; (3) breakage of superposition, which effectively distinguishes nonlinear resonant responses from the linear counterparts. These DIs are established using instantaneous baselines, facilitating the fatigue damage monitoring without the prior knowledge of a pristine structure. Fatigue tests on a thin aluminum plate with a rivet hole are conducted to induce fatigue cracks in the specimen. The experimental results demonstrate that the proposed technique shows remarkable sensitivity to the nucleation and growth of the fatigue cracks. This paper differs from the existing literature on nonlinear resonance-based techniques in that it focuses on the resonance phenomenon aroused by the contact acoustic nonlinearity from localized fatigue cracks, rather than the diffused material nonlinearity. The novelty of the paper resides in the establishment of an instantaneous baseline technique utilizing the nonlinear resonance features without the need of referring to a pristine baseline situation. The paper finishes with discussion, concluding remarks, and suggestions for future work.

Nonlinear model order reduction of continuous-time image reconstruction systems

Nonlinear model order reduction of continuous-time image reconstruction systems

Nonlinear Modeling of Multi-Converter Systems Within DC-Link Timescale

The emergence of various power electronic devices, including voltage source converters (VSCs), has greatly changed the dynamics of power systems. It is a great challenging to put forward new theories for modeling, analysis, and understanding of power-electronic-based power systems. In this paper, we first establish a detailed nonlinear model of multiple VSC systems, and then, by using the singular perturbation technique, obtain a reduced-order nonlinear model within the DC-link timescale. The latter model consists of differential algebraic equations (DAEs) and keeps slow dynamics in transient processes, showing a balance between computational accuracy and system complexity. Furthermore, the model is compared with the traditional electromechanical dynamical model of power systems, which are dominated by synchronous generators (SGs). Our model of multi-converter systems can be put into a similar diagram to the traditional power system dynamics, with which most of power electrical engineers are familiar. Finally, wide simulation results verify the efficiency of the proposed model.

Balanced Reduced-Order Models for Iterative Nonlinear Control of Large-Scale Systems

We propose a new framework to design controllers for high-dimensional nonlinear systems. The control is designed through the iterative linear quadratic regulator (ILQR), an algorithm that computes control by iteratively applying the linear quadratic regulator on the local linearization of the system at each time step. Since ILQR is computationally expensive, we propose to first construct reduced-order models (ROMs) of the high-dimensional nonlinear system. We derive nonlinear ROMs via projection, where the basis is computed via balanced truncation (BT) and LQG balanced truncation (LQG-BT). Numerical experiments are performed on a semi-discretized nonlinear Burgers equation. We find that the ILQR algorithm produces good control on ROMs constructed either by BT or LQG-BT, with BT-ROM based controllers outperforming LQG-BT slightly for very low-dimensional systems.

Linear-Frequency-Domain Method for the Determination of First- and Second-Order Volterra Kernels

In this work, a novel approach in determining the first- and second-order frequency-domain Volterra kernels for weakly nonlinear partial differential equations (PDEs) with a forcing term in semidiscrete form based on the application of the harmonic probing method is presented. This represents a formal extension of the linearized frequency-domain (LFD) methods to a nonlinear framework, leading to a so-called LFD method for a second-order functional series expansion method. The method allows for the representation of weak nonlinearities by solving two input-independent linear algebraic systems of equations in the frequency domain, and thus circumvents the solution of the nonlinear PDE by numerical integration for each different input, representing a nonlinear reduced-order model (ROM) for the physical phenomena. The general form of the equations is derived, and several applications are discussed. The proposed method overcomes two constraints present in other methods for the solution of nonlinear PDEs, namely, the consideration of exclusively periodic solutions as in the harmonic balance method and the dependency of the kernels with the input signal as in the Volterra kernel identification methods.

Nonlinear modeling and vibration mitigation of combined vortex-induced and base vibrations through energy harvesting absorbers

This study derives a nonlinear reduced-order model in order to evaluate the efficacy of an energy harvesting absorber at controlling base excitations, vortex-induced vibrations, and simultaneous base excitations and vortex-induced vibrations. Vibration absorbers are secondary systems that couple with a primary structure to dissipate the mechanical energy of vibrations. An energy harvesting absorber turns a portion of that dissipated energy into useful electrical energy which can be used to power small systems. The effects of various design parameters of the energy harvesting absorber on the behavior of the system is examined under different excitations. Next, an investigation of increasing the stiffness of the absorber is carried out with a goal of increasing the range in which meaningful energy can be harvested. Designs for the energy harvesting absorber are compared for their control of the system and their energy output when the system is subjected to concurrent base excitations and vortex-induced vibrations. Using an energy harvesting absorber tuned for vortex-induced vibrations provides the greatest reduction in amplitude and the most even power output over the range of wind speeds and base excitations.

DEIM reduced order model constructed by hybrid snapshot simulation

This paper presents a methodology that combines the hybrid snapshot simulation (Bai and Wang in Int J Comput Methods 18:2050029, 2020) and the discrete empirical interpolation method (DEIM) to reduce the computational cost of constructing a DEIM-based nonlinear reduced order model (ROM) while preserving its accuracy. In distinct contrast to its traditional counterpart, the time span of the hybrid snapshot simulation is divided into multiple intervals, and a majority of the intervals are simulated by local ROMs, while a fraction by the full order model (FOM). To tackle the challenge associated with limited FOM data for ROM-DEIM construction, a new approach is proposed to reconstruct and enrich the snapshot data of nonlinear model terms using solution data in ROM intervals. A formulation and procedure based on the cell-centered finite volume method (FVM) scheme are also developed to take into account two types of nonlinearities in ROM-DEIM: the componentwise and the transport-related, non-componentwise. A global ROM-DEIM is produced immediately at the end of the snapshot simulation and can be used to accelerate online simulation for different scenarios. The proposed methodology demonstrates excellent computational performance. Specifically, the computational time of the snapshot simulation is reduced by \(\sim\) 50% without compromising ROM-DEIM accuracy (relative error less than 0.8% compared with FOM). The results prove feasibility of combining hybrid snapshot simulation and DEIM for robust and efficient ROM-DEIM development.

Thermoelastic modeling and comparative analysis of biomass sensors under rippling deformation and magnetic field

This study is intended for investigating the linear and nonlinear responses of carbon nanotube-based mass sensors in thermal and magnetic environment. For accurate modeling, the carbon nanotube rippling deformation effect is considered and various representations for the thermal expansion are examined and compared to each other. By utilizing Euler-Bernoulli beam theory assumptions and Eringen's nonlocal elasticity theory, the nonlinear reduced-order model is developed on the basis of the extended Hamilton's principle. The results show that the natural and resonant frequencies and frequency shifts of the system are strongly dependent on the magnetic field and thermal expansion representation. The method of multiple scale is used to determine the modulation equation including the von Karman and rippling nonlinearities. The results show a very good agreement between the perturbation solution and the numerical integration results for specific conditions of the forcing, temperature difference, and quality factor. A comparative study between the linear and nonlinear mass sensing approaches is performed to show their limits of applicability. It is demonstrated that the linear approach may result in erroneous detection of the deposited mass. The obtained results indicate that the longitudinal magnetic field enhances the dynamic stability of the carbon nanotube mechanical resonator in the pre-buckling oscillation regime, while the dynamic stability of the nanoscale resonator is decreased in the presence of the magnetic field for the post-buckling configuration. Also, the ripple-based nonlinearity is accompanied by an increase in the mass responsivity of the resonator. On the contrary, mass sensitivity of the carbon nanotube resonator is diminished by considering the von Kármán geometric nonlinearity. This study shows the importance of considering the nonlinear effects on the system's sensitivity from frequency and amplitude sensing techniques.

Nonlinear Model Order Reduction via Nonlinear Moment Matching with Dynamic Mode Decomposition

In this manuscript, we propose a novel reduction framework for obtaining Reduced Order Models (ROMs) of large-scale, nonlinear dynamical systems. We advocate the use of Nonlinear Moment Matching (NLMM) with the Dynamic Mode Decomposition (DMD) to get a much efficient dimensionality reduction scheme. While NLMM does not require the expensive computation of the time-displaced snapshot ensemble of the Full Order Model (FOM) in the offline stage, DMD avoids the evaluation of the nonlinear term in the online stage. Thus, significant savings in terms of the overall CPU-time is achieved. To substantiate our observations, we test the proposed scheme on a variety of benchmark models and compare the results with POD–DEIM for reference.

A framework for parametric reduction in large-scale nonlinear dynamical systems

This manuscript presents a general parametric model order reduction (pMOR) framework for nonlinear dynamical systems. At first, a family of local nonlinear reduced order models (ROMs) is generated for different parametric space vectors using the nonlinear moment matching (NLMM) scheme along-with the discrete empirical interpolation method (DEIM). Then, the nonlinear reduced order model for any new parameter is obtained by interpolating the neighbouring reduced order models after projecting them onto a universal subspace. The advantage of such a scheme is that the parametric dependency is maintained in the reduced nonlinear models. Finally, we substantiate our observations by a suite of numerical tests.

Forward sensitivity approach for estimating eddy viscosity closures in nonlinear model reduction.

In this paper, we propose a variational approach to estimate eddy viscosity using forward sensitivity method (FSM) for closure modeling in nonlinear reduced order models. FSM is a data assimilation technique that blends model's predictions with noisy observations to correct initial state and/or model parameters. We apply this approach on a projection based reduced order model (ROM) of the one-dimensional viscous Burgers equation with a square wave defining a moving shock, and the two-dimensional vorticity transport equation formulating a decay of Kraichnan turbulence. We investigate the capability of the approach to approximate an optimal value for eddy viscosity with different measurement configurations. Specifically, we show that our approach can sufficiently assimilate information either through full-field or sparse noisy measurements to estimate eddy viscosity closure to cure standard Galerkin reduced order model (GROM) predictions. Therefore, our approach provides a modular framework to correct forecasting error from a sparse observational network on a latent space. We highlight that the proposed GROM-FSM framework is promising for emerging digital twin applications, where real-time sensor measurements can be used to update and optimize surrogate model's parameters.

Deep neural networks for nonlinear model order reduction of unsteady flows

Unsteady fluid systems are nonlinear high-dimensional dynamical systems that may exhibit multiple complex phenomena in both time and space. Reduced Order Modeling (ROM) of fluid flows has been an active research topic in the recent decade with the primary goal to decompose complex flows into a set of features most important for future state prediction and control, typically using a dimensionality reduction technique. In this work, a novel data-driven technique based on the power of deep neural networks for ROM of the unsteady fluid flows is introduced. An autoencoder network is used for nonlinear dimension reduction and feature extraction as an alternative for singular value decomposition (SVD). Then, the extracted features are used as an input for a long short-term memory (LSTM) network to predict the velocity field at future time instances. The proposed autoencoder-LSTM method is compared with non-intrusive reduced order models based on dynamic mode decomposition (DMD) and proper orthogonal decomposition. Moreover, an autoencoder-DMD algorithm is introduced for ROM, which uses the autoencoder network for dimensionality reduction rather than SVD rank truncation. The results show that the autoencoder-LSTM method is considerably capable of predicting fluid flow evolution, where higher values for the coefficient of determination R2 are obtained using autoencoder-LSTM compared to other models.

Coupling of model order reduction and multirate techniques for coupled dynamical systems

In refined network simulation one is faced with simulating coupled dynamical systems, as for instance circuits are coupled to heat or electromagnetics. Simulation costs of such coupled systems can be reduced by both model order reduction and multirate techniques. In this paper we discuss the coupling of both approaches by linking the slowest-first multirate technique to the nonlinear proper orthogonal decomposition model order reduction. For a nonlinear coupled thermal–electrical test example numerical results show the proven convergence of the combined reduced order multirate scheme.

Reduced order models for wind turbine blades with large deflections

Non-intrusive nonlinear reduced order modeling (ROM) techniques have been applied by researchers to obtain computationally cheap and yet accurate structural responses of aircraft panels. However, its application to wind turbine blades is new and challenging due to much larger deflections of wind turbine blades. This study improves a non-intrusive nonlinear ROM method for wind turbine blades going through large deflections. In the nonlinear ROM, the nonlinear stiffness is described by the quadratic and cubic functions, and the secondary motions induced by the primary large deflections are described by the modal derivative vectors in the reduction basis. The non-intrusive nature of the method requires a geometrically nonlinear solver, and HAWC2 is chosen in this study for the computation of nonlinear stiffness terms. Two examples, including a cantilever beam example and the NREL 5MW wind turbine blade model, are used to evaluate the accuracy and computational effectiveness of the nonlinear ROMs. The cantilever beam example shows that the nonlinear ROM can accurately capture the axial displacements due to large deflections reaching 20 % of span length as well as the torsion coupled with flapwise and edgewise motions. The NREL blade example shows that the nonlinear ROM is accurate for the tip displacements more than 5.9 m. Because the size of the nonlinear ROM is much smaller than that of HAWC2 model, a speedup factor of 8.5 for computational time is observed for the NREL blade example.

Reduced-order modeling of inverter-based generation using hybrid singular perturbation

This paper contributes to the field of inverter modelling for large-scale simulations by introducing a novel Model-Order Reduction (MOR) method based on singular perturbation. Motivated by the timescale separation between the fast and slow dynamics in an inverter-based power system, the proposed nonlinear MOR concept extends on the existing zero- and first-order reduction methods by combining the low computational burden of the former approach with the higher accuracy of the latter one. As a result, such hybrid MOR technique preserves the slow system dynamics of the full-order model, while simultaneously capturing the impact of the removed fast states on slow variables. Moreover, we introduce several improvements to the existing first-order MOR in order to make it tractable and more efficient when applied to a realistic full-order inverter model. The novel hybrid approach is applied to both grid-forming and grid-following inverter control schemes, and compared against existing reduction methods from the literature. The results showcase a better time-domain performance of the hybrid method during transients, while having a negligible increase in computational requirements compared to the traditional zero-order approach.

Control-Oriented Modeling for Managed Pressure Drilling Automation Using Model Order Reduction

Automation of managed pressure drilling (MPD) enables fast and accurate pressure control in drilling operations. The performance that can be achieved by automated MPD is determined by, first, the controller design and, second, the hydraulics model that is used as a basis for controller design. On the one hand, such hydraulics model should be able to accurately capture essential flow dynamics, for example, wave propagation effects, for which typically complex models are needed. On the other hand, a suitable model should be simple enough to allow for extensive simulation studies supporting scenario analysis and high-performance controller design well. In this paper, we develop a model order reduction approach for the derivation of such a control-oriented model for single-phase flow MPD operations. In particular, a nonlinear model order reduction procedure is presented that preserves key system properties such as stability and provides guaranteed (accuracy) bounds on the reduction error. To demonstrate the quality of the derived control-oriented model, comparisons with field data and both open-loop and closed-loop simulation-based case studies are presented.

On the stable mode selection for efficient component mode synthesis of geometrically nonlinear beams

A large-scale structural system containing a substantial amount of degrees of freedom requires many computations to achieve the optimum of the system. In particular, nonlinear structural analysis compels a recursive calculation in limited computational resources. Thus, a powerful tool, which improves the efficiency of analysis in the limitations, may be required. In structural dynamics, various methods related to model order reduction have been developed in the past decades [1–15]. In addition, several mode selection methods to enhance the rate of reduction have been reported in the Refs. [16–19]. Most of them are focused on dynamic analysis. In this paper, we focus on the nonlinear static response of reduced order model with a proper mode selection criterion. To construct the nonlinear reduced order model, modal derivatives [10–14], which are the quadratic enhancement of modal basis, have been employed. The degrees of freedom used to the model reduction have been selected via the singular value decomposition (SVD). Finally, the performance of the selection scheme for the static problem is verified via numerical examples.

On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flows

In the literature on nonlinear projection-based model order reduction for computational fluid dynamics problems, it is often claimed that due to modal truncation, a projection-based reduced-order model (PROM) does not resolve the dissipative regime of the turbulent energy cascade and therefore is numerically unstable. Efforts at addressing this claim have ranged from attempting to model the effects of the truncated modes to enriching the classical subspace of approximation in order to account for the truncated phenomena. The objective of this paper is to challenge this claim. Exploring the relationship between projection-based model order reduction and semi-discretization and using numerical evidence from three relevant flow problems, this paper argues in an orderly manner that the real culprit behind most if not all reported numerical instabilities of PROMs for turbulence and convection-dominated turbulent flow problems is the Galerkin framework that has been used for constructing the PROMs. The paper also shows that alternatively, a Petrov-Galerkin framework can be used to construct numerically stable and accurate PROMs for convection-dominated laminar as well as turbulent flow problems, without resorting to additional closure models or tailoring of the subspace of approximation. It also shows that such alternative PROMs deliver significant speed-up factors.

Updating Geometrically Nonlinear Reduced-Order Models Using Nonlinear Modes and Harmonic Balance

Nonlinear reduced-order models are being investigated for use as digital twins for advanced aircraft and spacecraft. In concept, the digital twin would be tuned to accurately describe the behavior of the vehicle, and then it would be updated as the vehicle is modified or ages. Unfortunately, the linear model correlation and validation techniques that are commonly used in the aerospace industry are not valid for nonlinear models, so a new set of tools is needed. This work presents an algorithm to update geometrically nonlinear reduced-order models using nonlinear normal modes as a correlation metric. The nonlinear normal modes serve as a strong metric to correlate the numerical models because they can be extracted from experiments, and they describe the dynamics of the nonlinear system over a range of amplitudes and are independent of the loading applied to the system. This paper presents a novel method of computing analytical gradients of nonlinear normal mode solutions with respect to system parameters using the multiharmonic balance method. The procedure is first tested using numerical simulations and then applied to tune the reduced-order model of a curved beam based on nominal blueprints and idealized boundary conditions to match a nonlinear normal mode that has been measured experimentally from a three-dimensional printed test specimen.

Modelling method and mistuning amplification analysis of the forced response for coated blisks in the presence of coatings-material nonlinearity

Abstract Hard coatings have been employed to achieve the goal of vibration reduction for blisks in recent years. Nevertheless, the modelling method for randomly mistuned blisks with coatings-material nonlinearity and the effects of this nonlinearity on the amplification factors have not been investigated in previous studies. In this work, a linear reduced order model (ROM) for randomly mistuned blisks with linear coatings is developed by using the Craig-Bampton (C–B) method and an interface reduction technology firstly. Furthermore, the transformation matrices employed in the linear ROM is also used to reduce the dimension of perturbation matrices caused by coatings-material nonlinearity, and then the nonlinear ROM is developed and solved by an improved Newton-Raphson iterative process. Statistical analysis is performed to validate the developed models by comparing with the full models and to investigate the effects of coatings-material nonlinearity on the amplification factors for randomly mistuned blisks. The results suggest that the developed ROMS are in good consistency with the full models, and the amplification factors for the nonlinear analysis are decreased significantly with the force amplitudes increasing.