Nonlinear Model Order Reduction Research Articles

Locally enhanced reduced order modeling for the nonlinear geometric response of structures with defects

This paper focuses on the reduced order modeling (ROM) of structures with local defects undergoing large deformations, i.e., within the nonlinear geometric range. At the contrary of prior investigations, it is desired here to construct such ROMs by enhancing the nonlinear reduced order model of the corresponding virgin structure, i.e., the one without the defect, not by carrying out a separate modeling effort. To this end, the first objective is on the formulation of local enrichments of the displacements basis functions that complement those of the virgin structure to accurately capture not only the displacements but also the stresses of the structure with defect. It is proposed that such enrichment functions can be constructed from the linear static analysis of the zone affected by the defect subjected to an imposed far field displacement derived from the virgin structure basis functions. The second objective of this investigation is on assessing which parameters of the reduced order model would need to be updated if the defect, and thus the enrichments, were to change. Validation results on the finite element model of a beam-like panel with a notch do confirm the appropriateness of the basis enrichments and, moreover, suggest that only the linear stiffness and stress coefficients relating to the enrichments would need to be updated if the defect changes, which represents a significant computational benefit. The implementation of the above process with a localized numerical model of the defect, e.g., using generalized finite element (GFEM), is briefly discussed.

Neural network closures for nonlinear model order reduction

Many reduced-order models are neither robust with respect to parameter changes nor cost-effective enough for handling the nonlinear dependence of complex dynamical systems. In this study, we put forth a robust machine learning framework for projection-based reduced-order modeling of such nonlinear and nonstationary systems. As a demonstration, we focus on a nonlinear advection-diffusion system given by the viscous Burgers equation, which is a prototypical setting of more realistic fluid dynamics applications due to its quadratic nonlinearity. In our proposed methodology the effects of truncated modes are modeled using a single layer feed-forward neural network architecture. The neural network architecture is trained by utilizing both the Bayesian regularization and extreme learning machine approaches, where the latter one is found to be more computationally efficient. A significant emphasis is laid on the selection of basis functions through the use of both Fourier bases and proper orthogonal decomposition. It is shown that the proposed model yields significant improvements in accuracy over the standard Galerkin projection methodology with a negligibly small computational overhead and provide reliable predictions with respect to parameter changes.

Nonlinear reduced-order modeling and effectiveness of electrically-actuated microbeams for bio-mass sensing applications

We apply perturbation techniques to develop a reduced-order model of an electrically-actuated microcantilever beam with a tip mass deployed as resonant sensor for bio-mass detection and sensing. This analytical model is validated against numerical model obtained by combining the differential quadrature method for space discretization and Runge–Kutta for time marching. The model is then employed to analyze the nonlinear dynamics and effectiveness of the bio-mass sensor under varying electric loading and explore novel concepts to quantify the mass of biological entities. The working principle of the present bio-mass sensor is based on inspecting the attenuation in the microbeam vibrations resulting from the biological element being deposited on its tip and then extracting the corresponding mass. The output parameter of the present bio-mass sensor is considered as the change in the maximum beam deflections at the tip with and without added mass. Calibration curves, showing the variations of the output parameter with the added mass, are generated to demonstrate the feasibility of the proposed sensing approach for mass detection of biological elements, particularly the Escherichia coli. Reducing the AC voltage when exciting the microbeam is observed to enhance the sensitivity of the output parameter for specific mass threshold. However, the operational range of the bio-mass sensor can be extended when applying higher DC and AC voltages.

Vibration Frequency Optimization of Jointed Structures with Contact Nonlinearities using Hyper-Reduction

We present a new method for the analysis and optimization of jointed structures with local contact nonlinearities. It is based on nonlinear model order reduction using the Energy-Conserving Sampling and Weighting (ECSW) method (Farhat et al., 2014) for the nonlinear parts of the model, and a Galerkin projection of the complete domain onto a basis spanned by eigenmodes. The analysis of a typical example problem case shows that accurate results and substantial speed-ups can be obtained, while retaining a high resolution and precision of the results. Using the reduced-order model, the dominant frequency of a step response of the two-beam example model was successfully optimized to match a certain target frequency ω⁎. One evaluation of the reduced-order model was on average 4.33 times faster than an evaluation of the full-order model. The optimum was successfully verified by the evaluation of the full-order model for the optimum design parameters which produced the same target frequency. This shows promise for the improvement of classical optimization approaches using reduced-order models also for different problem types.

Model Order Reduction for Managed Pressure Drilling Systems based on a Model with Local Nonlinearities

Automated Managed Pressure Drilling (MPD) is a method for fast and accurate pressure control in drilling operations. The achievable performance of automated MPD is limited, firstly, by the control system and, secondly, by the hydraulics model based on which this control system is designed. Hence, an accurate hydraulics model is needed that, at the same time, is simple enough to allow for the use of high performance controller design methods. This paper presents an approach for nonlinear Model Order Reduction (MOR) for MPD systems. For a single-phase flow MPD system, a nonlinear model is derived that can be decomposed into a feedback interconnection of a high-order linear subsystem and low-order nonlinear subsystem. This structure, under certain conditions, allows for a nonlinear MOR procedure that preserves key system properties such as stability and provides a computable error bound. The effectiveness of this MOR method for MPD systems is illustrated through simulations.

기하비선형 축소 시스템의 유전자 알고리즘을 통한 고유 모드 최적화

기하비선형 축소 시스템의 유전자 알고리즘을 통한 고유 모드 최적화

Veering and nonlinear interactions of a clamped beam in bending and torsion

Understanding the linear and nonlinear dynamic behaviour of beams is critical for the design of many engineering structures such as spacecraft antennae, aircraft wings, and turbine blades. When the eigenvalues of such structures are closely-spaced, nonlinearity may lead to interactions between the underlying linear normal modes (LNMs). This work considers a clamped-clamped beam which exhibits nonlinear behaviour due to axial tension from large amplitudes of deformation. An additional cross-beam, mounted transversely and with a movable mass at each tip, allows tuning of the primary torsion LNM such that it is close to the primary bending LNM. Perturbing the location of one mass relative to that of the other leads to veering between the eigenvalues of the bending and torsion LNMs. For a number of selected geometries in the region of veering, a nonlinear reduced order model (NLROM) is created and the nonlinear normal modes (NNMs) are used to describe the underlying nonlinear behaviour of the structure. The relationship between the ‘closeness’ of the eigenvalues and the nonlinear dynamic behaviour is demonstrated in the NNM backbone curves, and veering-like behaviour is observed. Finally, the forced and damped dynamics of the structure are predicted using several analytical and numerical tools and are compared to experimental measurements. As well as showing a good agreement between the predicted and measured responses, phenomena such as a 1:1 internal resonance and quasi-periodic behaviour are identified.

Modal Substructuring of Geometrically Nonlinear Plates

AbstractThis work considers model reduction of geometrically nonlinear finite element (FE) model of a plate structure developed in a commercial FE package. The structure is first divided into smaller substructures. Since there is normally no access to the nonlinear stiffness tensors in FE packages, a non‐intrusive method is used in this paper to reduce the order of each substructure separately. In order to generate the nonlinear reduced order model (NLROM), the reduced substructures are assembled using the coupling procedure of the Component Mode Synthesis (CMS) method. As a linear basis, truncated free and fixed interface modes are used here to check the efficiency of the developed NLROM based on them. A plate structure subjected to large deflections is considered in this study to implement the substructuring method. For the sake of validation, Nonlinear Normal Modes (NNMs) are employed to check the convergence of NLROMs in a broadband frequency range. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimal online selection of type 1 diabetes-glucose metabolism models

We address an optimal experimental design (OED) procedure for the online selection of type-1-diabetes (T1D) mellitus glucose metabolistic models. A fully observable reduced-order nonlinear dynamic model is presented and subsequently parameterised for Göttingen Minipigs and patients, that were both subject to an automatic insulin delivery. A bank of continuous–discrete unscented Kalman filters (CDUKF) is designed and parameterised for Göttingen Minipigs and patients. Based on this filter bank of CDUKF, a novel online OED design procedure is developed, that is used to identify the correct parameter set out of several available sets for measured blood glucose concentrations. The procedure utilises forward model simulations to calculate optimal system inputs. This leads to the identification of the correct parameter set under arbitrary conditions. Results are presented for both subgroups.

Design and Parametric Study of the Magnetic Sensor for Position Detection in Linear Motor Based on Nonlinear Parametric model order reduction.

This paper presents a design approach for a magnetic sensor module to detect mover position using the proper orthogonal decomposition-dynamic mode decomposition (POD-DMD)-based nonlinear parametric model order reduction (PMOR). The parameterization of the sensor module is achieved by using the multipolar moment matching method. Several geometric variables of the sensor module are considered while developing the parametric study. The operation of the sensor module is based on the principle of the airgap flux density distribution detection by the Hall Effect IC. Therefore, the design objective is to achieve a peak flux density (PFD) greater than 0.1 T and total harmonic distortion (THD) less than 3%. To fulfill the constraint conditions, the specifications for the sensor module is achieved by using POD-DMD based reduced model. The POD-DMD based reduced model provides a platform to analyze the high number of design models very fast, with less computational burden. Finally, with the final specifications, the experimental prototype is designed and tested. Two different modes, 90° and 120° modes respectively are used to obtain the position information of the linear motor mover. The position information thus obtained are compared with that of the linear scale data, used as a reference signal. The position information obtained using the 120° mode has a standard deviation of 0.10 mm from the reference linear scale signal, whereas the 90° mode position signal shows a deviation of 0.23 mm from the reference. The deviation in the output arises due to the mechanical tolerances introduced into the specification during the manufacturing process. This provides a scope for coupling the reliability based design optimization in the design process as a future extension.

Modal interactions and energy transfers in large-amplitude vibrations of functionally graded microcantilevers

Modal interactions and internal energy transfers are investigated in the large-amplitude oscillations of a functionally graded microcantilever with an intermediate spring-support. Based on the Mori–Tanaka homogenization technique and the modified couple stress theory, the energy terms of the functionally graded microsystem (kinetic and size-dependent potential energies) are developed and dynamically balanced. Large-amplitude deformations, due to having one end free, are modeled taking into account curvature-related nonlinearities and assuming an inextensibility condition. The continuous model of the functionally graded microsystem is reduced, by means of the Galerkin method, yielding an inertial- and stiffness-wise nonlinear model. Numerical simulations on this highly nonlinear reduced-order model of the functionally graded microcantilever are performed using a continuation method; a possible case of modal interactions is determined by obtaining the natural frequencies of the microsystem. The nonlinear oscillations of the microcantilever are examined, and it is shown how the energy fed to the functionally graded microsystem (from the base excitation) is transferred between different modes of oscillation.

Nonlinear Dynamic Model-Based Multiobjective Sensor Network Design Algorithm for a Plant with an Estimator-Based Control System

A novel sensor network design (SND) algorithm is developed for maximizing process efficiency while minimizing sensor network cost for a nonlinear dynamic process with an estimator-based control system. The multiobjective optimization problem is solved following a lexicographic approach where the process efficiency is maximized first followed by minimization of the sensor network cost. The partial net present value, which combines the capital cost due to the sensor network and the operating cost due to deviation from the optimal efficiency, is proposed as an alternative objective. The unscented Kalman filter is considered as the nonlinear estimator. The large-scale combinatorial optimization problem is solved using a genetic algorithm. The developed SND algorithm is applied to an acid gas removal (AGR) unit as part of an integrated gasification combined cycle (IGCC) power plant with CO2 capture. Due to the computational expense, a reduced order nonlinear model of the AGR process is identified and parallel ...

Semi-analytic contact technique in a non-linear parametric model order reduction method for gear simulations

In this work we present a novel method for the solution of gear contact problems in flexible multi-body. These problems are characterized by significant variation in the location and size of the contact area, typically requiring a high number of degrees of freedom to correctly capture deformation and stress fields. Therefore fully dynamic simulation is computationally prohibitive. To overcome these limitations, we exploit a combined analytic-numerical contact model within a parametric model order reduction (PMOR) scheme. The reduction space consists of a truncated set of eigenvectors augmented with a parameter dependent set of residual static shape vectors. Each static shape is computed by interpolating among a set of displacement modes of the interacting bodies, obtained from a series of precomputed static contact analyses. During the contact analyses, an analytic model based on the Hertz theory describes the teeth local deformation. We implement the proposed method in an in-house code and we apply it to spur and helical gears dynamic contact analyses. We compare the results with classical PMOR schemes highlighting how the combined use of the semi-analytic contact model allows to decrease further the model complexity as well as the computational burden, for both static and dynamic cases. Finally, we validate the methodology by means of a comparison with experimental data found in literature, showing that the numerical method is able to capture quantitatively the static transmission error measurements in case of both helical and spur geared transmission for different torque levels.

Parametric Excitation of a Microbeam-String With Asymmetric Electrodes: Multimode Dynamics and the Effect of Nonlinear Damping

The dynamic motion of a parametrically excited microbeam-string affected by nonlinear damping is considered asymptotically and numerically. It is assumed that the geometrically nonlinear beam-string, subject to only modulated alternating current voltage, is closer to one of the electrodes, thus resulting in an asymmetric dual gap configuration. A consequence of these novel assumptions is a combined parametric and hard excitation in the derived continuum-based model that incorporates both linear viscous and nonlinear viscoelastic damping terms. To understand how these assumptions influence the beam's performance, the conditions that lead to both principal parametric resonance and a three-to-one internal resonance are investigated. Such conditions are derived analytically from a reduced-order nonlinear model for the first three modes of the microbeam-string using the asymptotic multiple-scales method which requires reconstitution of the slow-scale evolution equations to deduce an approximate spatio-temporal solution. The response is investigated analytically and numerically and reveals a bifurcation structure that includes coexisting in-phase and out-of-phase solutions, Hopf bifurcations, and conditions for the loss of orbital stability culminating with nonstationary quasi-periodic solutions and chaotic strange attractors.

Nonlinear Reduced-Order Models for Aerodynamic Lift of Oscillating Airfoils2

For the present study, setting Strouhal number as the control parameter, we perform numerical simulations for the flow over oscillating NACA-0012 airfoil at a Reynolds number of 1000. This study reveals that aerodynamic forces produced by oscillating airfoils are independent of the initial kinematic conditions suggesting the existence of limit cycle. Frequencies present in the oscillating lift force are composed of the fundamental harmonics and its odd harmonics. Using these numerical simulations, we analyze the shedding frequencies close to the excitation frequencies. Hence, considering it as a primary resonance case, we model the unsteady lift force with a modified van der Pol oscillator. Using the method of multiple scales and spectral analysis of the steady-state computational fluid dynamics (CFD) solutions, we estimate the frequencies and the damping terms in the reduced-order model (ROM). We show the applicability of this model to all planar motions of the airfoil; heaving, pitching, and flapping. With increasing the Strouhal number, the nonlinear damping terms for all types of motion approach similar magnitudes. Another important aspect in one of the proposed model is its ability to capture the time-averaged value of the aerodynamic lift force. We also notice that increase in the magnitude of the lift force is due to the effect of destabilizing linear damping parameter.

Measurement-Directed Reduction of Dynamic Models in Power Systems

The paper describes a new model reduction procedure tailored to power systems. It uses measurement data to devise a family of reduced order nonlinear models while retaining physical interpretability of parameters and equations. The manifold boundary approximation method (MBAM) uses the Fisher information matrix calculated from measurements to identify the least relevant parameter combination in the original model. Next, it numerically constructs a geodesic on the corresponding statistical manifold originating from the initial parameters in the least relevant parameter direction until a manifold boundary is found. MBAM then identifies a limiting approximation in the mathematical form of the model and removes one parameter combination. The simplified model is recalibrated by fitting its behavior to that of the original model, and the process is repeated as appropriate. MBAM is demonstrated on the example of a synchronous generator (SG), which has been treated extensively in the literature. Implications of the proposed model reduction procedure on large power system models are illustrated on a 441-bus, 72-SG dynamical model.

Non-linear boiling water reactor stability with Shannon Entropy

The most common indicator currently used to assess boiling water reactor (BWR) instability is the Decay Ratio (DR), which is an indicator that assumes stationarity and linearity of BWR signals. However, it is well known that BWRs are complex dynamical systems that may exhibit chaotic behavior when an instability event occurs, jeopardizing the DR validity and reliability when the reactor is working at a specific operating point. Thus, it is required to study new stability indicators that satisfy as much as possible this complex dynamics of BWR systems. With this latter fundamental idea in mind, in this work, the non-linear Shannon Entropy (SE) is explored to study BWR instability. The SE measures the uncertainty of BWR signals to appraise for system stability, a low SE estimation indicates a predictable BWR operation (stable behavior) whereas a high SE estimation indicates an unpredictable BWR operation (unstable behavior). The SE estimation was validated with artificial signals from a non-linear Reduced Order Model (ROM), that represents qualitatively the dynamic behavior of a BWR system. The result comparison proves that the SE satisfies the BWR complex dynamics whereas the DR does not during the chaotic behavior. The SE was also compared with the Largest Lyapunov Exponent (LLE), which represents adequately the chaotic behavior of a BWR but, from the practical point of view, it cannot be applied to an online stability monitor, while the methodology presented in this work based on the SE, is a good candidate.

Modal Substructuring of Geometrically Nonlinear Finite Element Models with Interface Reduction

Substructuring methods have been widely used in structural dynamics to divide large, complicated finite element models into smaller substructures. For linear systems, many methods have been developed to reduce the subcomponents down to a low-order set of equations using a special set of component modes, and these are then assembled to approximate the dynamics of a large-scale model. In this paper, a substructuring approach is developed for coupling geometrically nonlinear structures, where each subcomponent is drastically reduced to a low-order set of nonlinear equations using a truncated set of fixed-interface and characteristic constraint modes. The method used to extract the coefficients of the nonlinear reduced-order model is nonintrusive, in that it does not require any modification to the commercial finite element code but computes the reduced-order model from the results of several nonlinear static analyses. The nonlinear reduced-order models are then assembled to approximate the nonlinear differential equations of the global assembly. The method is demonstrated on the coupling of two geometrically nonlinear plates with simple supports at all edges. The plates are joined at a continuous interface through the rotational degrees of freedom, and the nonlinear normal modes of the assembled equations are computed to validate the models. The proposed substructuring approach reduces a 12,861-degree-of-freedom model down to only 23 degrees of freedom while still accurately reproducing the nonlinear normal modes.

Nonlinear analysis of rotating nanocrystalline silicon microbeams for microgyroscope applications

In this paper, we examine the nonlinear dynamical responses of a microgyroscope consisting of a rotating microbeam made of nanocrystalline material with attached proof mass, subject to electric actuation, and operating at high frequency. The working principle of this inertial sensor is based on exploiting the transfer of the mechanical energy among two vibrations modes (drive and sense) via the Coriolis effect to measure the rotation rate. A nonlinear reduced-order model (ROM) governing the microbeam dynamics is developed by the application of the differential quadrature method and finite difference method for space and time discretization, respectively. The developed ROM is used to study the nonlinear behavior of the microbeam near the primary resonance for various grain sizes of the nanocrystalline material and under different electric actuation configurations. The operating DC voltage of the drive mode is selected to ensure that the microgyroscope operates away from the pull-in instability. A sensitivity analysis of the microsystem output parameter to the rotation rate when varying the material properties of the microbeam and electric actuation is then performed. The fringing field of the electrostatic force is found to reduce slightly the pull-in voltage and the natural frequency of the microsystem, amplify the motion in the sense direction, and enlarge the dynamic snap-through bandwidth. As for the effect of the material properties, considering a microbeam with bigger grain size of the constituent nanocrystalline silicon is observed to reduce the motion of the sense mode, increase the natural frequency, and shrink the snap-through bandwidth. Furthermore, operating at high base rotation rates while deploying microbeams with small nanocrystalline grain size is found to switch the dynamic behavior of the sense mode from the nearly-linear to the softening type. Finally, introducing bias in the DC voltage applied along the drive and sense directions is observed to degrade the performance of the electrically-actuated microgyroscope.

Nonlinear model order reduction for the fast solution of induction heating problems in time-domain

PurposeThis paper aims to propose a fast and accurate simulation of large-scale induction heating problems by using nonlinear reduced-order models.Design/methodology/approachA projection space for model order reduction (MOR) is quickly generated from the first kernels of Volterra’s series to the problem solution. The nonlinear reduced model can be solved with time-harmonic phasor approximation, as the nonlinear quadratic structure of the full problem is preserved by the projection.FindingsThe solution of induction heating problems is still computationally expensive, even with a time-harmonic eddy current approximation. Numerical results show that the construction of the nonlinear reduced model has a computational cost which is orders of magnitude smaller than that required for the solution of the full problem.Research limitations/implicationsOnly linear magnetic materials are considered in the present formulation.Practical implicationsThe proposed MOR approach is suitable for the solution of industrial problems with a computing time which is orders of magnitude smaller than that required for the full unreduced problem, solved by traditional discretization methods such as finite element method.Originality/valueThe most common technique for MOR is the proper orthogonal decomposition. It requires solving the full nonlinear problem several times. The present MOR approach can be built directly at a negligible computational cost instead. From the reduced model, magnetic and temperature fields can be accurately reconstructed in whole time and space domains.