Nonlinear Model Order Reduction Research Articles

Generation of a reduced-order model of an Unmanned Combat Air Vehicle using indicial response functions

Within the NATO STO AVT-251 Task Group, a generic Unmanned Combat Air Vehicle (UCAV) planform was redesigned based on requirements derived from parts of the flight envelope of the defined mission. Because of the lambda-shape planform and associated flow phenomena including shocks and vortices, the aerodynamic design process relies heavily on high-fidelity predictions from computational fluid dynamics (CFD). These simulations enable accurate flight dynamics predictions, allowing for the identification of any potential performance issues early in the design process. Potentially, a complete flight dynamics model can be derived from CFD calculations provided that the underlying reduced order model (ROM) captures the underlying physics contained within the CFD results. This work investigates the creation of a nonlinear ROM using indicial response functions. The response functions are obtained using a grid motion approach that separates the effects of angle of attack and pitch rate. This approach is demonstrated using three different CFD codes from various organizations: ENSOLV at Royal Netherlands Aerospace Centre, USM3D at NASA Langley Research Center, and Kestrel at the United States Air Force Academy. ROM predictions were generated for a manoeuver resembling a low-speed pull-up. The predictions are found to be sensitive to the quality of the steady-state solutions from which step-response calculations were started, as well as the convergence of the step responses themselves. Nevertheless, the indicial response method is shown to provide aerodynamic predictions with acceptable accuracy including transient effects, and as such is a promising method for computationally efficient flight dynamics predictions.

Modeling and analysis of grid-synchronizing stability of a Type-IV wind turbine under grid faults

Transient behavior and response of wind turbines have been extensively studied in the context of the low-voltage-ride-through (LVRT) for many years, in which the identified transients are mostly associated with circuit responses, whereas underlying stability issues relevant to converter control yet remain concealed. Recently, small-signal analyses of grid-tied converters have shown that the phase-locked-loop (PLL) may interact with other converter controls when synchronized to nonideal grids, if not properly treated, it may lead to oscillations. Enlightened by this, this paper will explore the PLL effects on the stability of a Type-IV wind turbine, however, from a large-signal analysis viewpoint. To achieve this, a nonlinear reduced-order model focusing on the transient interaction of PLL and the faulty grid is developed rigorously from the model assumption to verification. Based on this, the grid-synchronizing stability (GSS) provoked by grid faults is identified and the mechanism of which is revealed through the equal-area-principle (EAP). Then, impacts of system parameters (e.g. PLL bandwidth) on GSS margin are quantitatively evaluated by calculating the critical-clearing-time (CCT), and the acquired knowledge could be useful guidelines for PLL parameter design. Finally, all the analyses are verified by a switching model of the Type-IV wind turbine system in PSCAD/EMTDC.

A nonlinear data-driven reduced order model for computational homogenization with physics/pattern-guided sampling

Developing an accurate nonlinear reduced order model from simulation data has been an outstanding research topic for many years. For many physical systems, data collection is very expensive and the optimal data distribution is not known in advance. Thus, maximizing the information gain remains a grand challenge. In a recent paper, Bhattacharjee and Matouš (2016) proposed a manifold-based nonlinear reduced order model for multiscale problems in mechanics of materials. Expanding this work here, we develop a novel sampling strategy based on the physics/pattern-guided data distribution. Our adaptive sampling strategy relies on enrichment of sub-manifolds based on the principal stretches and rotational sensitivity analysis. This novel sampling strategy substantially decreases the number of snapshots needed for accurate reduced order model construction (i.e., ∼5× reduction of snapshots over Bhattacharjee and Matouš (2016)). Moreover, we build the nonlinear manifold using the displacement rather than deformation gradient data. We provide rigorous verification and error assessment. Finally, we demonstrate both localization and homogenization of the multiscale solution on a large particulate composite unit cell.

Linear and nonlinear statistical response theories with prototype applications to sensitivity analysis and statistical control of complex turbulent dynamical systems.

Statistical response theory provides an effective tool for the analysis and statistical prediction of high-dimensional complex turbulent systems involving a large number of unresolved unstable modes, for example, in climate change science. Recently, the linear and nonlinear response theories have shown promising developments in overcoming the curse-of-dimensionality in uncertain quantification and statistical control of turbulent systems by identifying the most sensitive response directions. We offer an extensive illustration of using the statistical response theory for a wide variety of challenging problems under a hierarchy of prototype models ranging from simple solvable equations to anisotropic geophysical turbulence. Directly applying the linear response operator for statistical responses is shown to only have limited skill for small perturbation ranges. For stronger nonlinearity and perturbations, a nonlinear reduced-order statistical model reduction strategy guaranteeing model fidelity and sensitivity provides a systematic framework to recover the multiscale variability in leading order statistics. The linear response operator is applied in the training phase for the optimal nonlinear model responses requiring only the unperturbed equilibrium statistics. The statistical response theory is further applied to the statistical control of inherently high-dimensional systems. The statistical response in the mean offers an efficient way to recover the control forcing from the statistical energy equation without the need to run the expensive model. Among all the testing examples, the statistical response strategy displays uniform robust skill in various dynamical regimes with distinct statistical features. Further applications of the statistical response theory include the prediction of extreme events and intermittency in turbulent passive transport and a rigorous saturation bound governing the total statistical growth from initial and external uncertainties.

Simulating swing dynamics of a power system model using nonlinear model order reduction

PurposeThe purpose of this paper is to demonstrate the applicability of the Discrete Empirical Interpolation method (DEIM) for simulating the swing dynamics of benchmark power system problems. The authors demonstrate that considerable savings in computational time and resources are obtained using this methodology. Another purpose is to apply a recently developed modified DEIM strategy with a reduced on-line computational burden on this problem.Design/methodology/approachOn-line computational cost of the power system dynamics problem is reduced by using DEIM, which reduces the complexity of the evaluation of the nonlinear function in the reduced model to a cost proportional to the number of reduced modes. The on-line computational cost is reduced by using an approximate snap-shot ensemble to construct the reduced basis.FindingsConsiderable savings in computational resources and time are obtained when DEIM is used for simulating swing dynamics. The on-line cost implications of DEIM are also reduced considerably by using approximate snapshots to construct the reduced basis.Originality/valueApplicability of DEIM (with and without approximate ensemble) to a large-scale power system dynamics problem is demonstrated for the first time.

Bolt loosening detection in a jointed beam using empirical mode decomposition–based nonlinear system identification method

In this work, a state-of-art nonlinear system identification method based on empirical mode decomposition is utilized and extended to detect bolt loosening in a jointed beam. This nonlinear system identification method is based on identifying the multi-scale dynamics of the underlying system. Only structural dynamic response signals are needed to construct a reduced-order model to represent the system concerned. It makes the method easy to use in practice. A new bolt loosening identification procedure based on the constructed system nonlinear reduced-order model is proposed. A new damage feature to indicate bolt loosening is presented. Experimental works are carried out to validate the proposed method. The results show that the proposed damage detection method can detect bolt loosening effectively, and the proposed damage feature values increase with the increase of bolt torques. The damage feature calculated from the response solution of the reduced-order model can give robust and sensitive indication of bolt loosening.

Nonlinear reduced-order modeling of the forced and autonomous aeroelastic response of a membrane wing using Harmonic Balance methods

A nonlinear reduced-order model for a membrane wing at low Reynolds numbers is obtained by coupling a linear aerodynamic model with a nonlinear one-dimensional membrane equation. Steady-state solutions for the membrane’s aeroelastic response are obtained by means of Harmonic Balance methods in the frequency-domain and in the time-domain. Harmonic Balance methods are adopted to investigate the aeroelastic response of a membrane wing pitching about its leading-edge at Re=100, for different values of the pitching angle. Differences between the two Harmonic Balance approaches are discussed and validations of the methods are carried out using high-fidelity Direct Numerical Simulations. The effect of the Reynolds number on the wing’s lift response is taken into account by increasing the Reynolds number to Re=1000. Harmonic Balance methods are also used to analyze the limit-cycle autonomous oscillations of the membrane for linearly unstable cases. The frequency of the oscillations can be predicted by introducing a frequency-searching algorithm in the solution procedure.

Flutter and gust response analysis of a wing model including geometric nonlinearities based on a modified structural ROM

In this paper, a framework is established for nonlinear flutter and gust response analyses based on an efficient Reduced Order Model (ROM). The proposed method can be used to solve the aeroelastic response problems of wings containing geometric nonlinearities. A structural modeling approach presented herein describes the stiffness nonlinearities with a modal formulation. Two orthogonal spanwise modes describe the foreshortening effects of the wing. Dynamic linearization of the ROM under nonlinear equilibrium states is applied to a nonlinear flutter analysis, and the fully nonlinear ROM coupled with the non-planar Unsteady Vortex Lattice Method (UVLM) is applied to gust response analysis. Furthermore, extended Precise Integration Method (PIM) ensures accuracy of the dynamic equation solutions. To demonstrate applicability and accuracy of the method presented, a wind tunnel test is conducted and good agreements between theoretical and test results of nonlinear flutter speed and gust response deflection are reached. The method described in this paper is suitable for predicting the nonlinear flutter speed and calculating the gust responses of a large-aspect-ratio wing in time domain. Meanwhile, the results derived highlight the effects of geometric nonlinearities obviously.

Non-linear model order reduction for elastohydrodynamic lubrication simulations of polymer seals

Abstract In the field of elastohydrodynamic lubrication (EHL) simulations, the importance of considering non-linearities in the structure increases constantly. In most cases, the finite element method is used to discretize the non-linear continuum equations. Apart from this, model order reduction techniques are used in EHL simulations to decrease the number of unknowns and the computational time. However, the reduction methods typically used are limited to linear problems. The aim of this paper is to introduce a model order reduction technique to EHL simulations, that captures non-linear behavior. As an example, a hyperelastic 2D axially symmetric seal is reduced and coupled to the Reynolds equation. The simulation is accelerated by a factor of 10 and beyond while the non-linear properties are preserved.

Nonparametric probabilistic approach for uncertainty quantification of geometrically nonlinear mistuned bladed-disks.

The present research concerns the dynamical analysis of mistuned rotating bladed-disks for which nonlinear geometrical effects exist. The present methodology requires the construction of an adapted reduced-order basis from which a nonlinear reduced-order model is constructed. The mistuning phenomenon is taken into account by considering a nonparametric probabilistic approach based on the information theory. In the present context the uncertainty is introduced by replacing the reduced-order basis with a stochastic reduced-order basis (SROB). This latter one is obtained by using a new nonparametric probabilistic approach of model-form uncertainties so that each realization of the SROB respects some mathematical properties linked to the available information under constraints concerning the specified boundary conditions and the usual orthogonality properties. With such strategy, the computational effort is focused on the stochastic nonlinear reduced internal forces and the related tangential operator which are explicitly constructed using the SROB combined with the finite element method. The numerical application is a rotating mistuned bladed-disk subjected to a load for which geometrical nonlinearities effects occur. The uncertainty propagation in this nonlinear dynamical system is then analyzed.

Nonlinear analysis of a reduced-order model with relaxation effects for BWRs

The aim of this work is the derivation of a nonlinear reduced-order mathematical model for boiling water reactors with relaxation effects in the propagation phenomena (P-ROM). The nonlinear BWR system is modeled by a set of ordinary differential equations (ODEs) which describe the dynamics of neutron density, the precursor concentration of delayed neutrons, the fuel temperature and the void reactivity. The reduced-order model with relaxation effects is a novel approximation where the neutron density is a second order ODE derived with P1 approximation of transport theory. Furthermore, the fuel temperature is a second order ODE obtained by considering a non-Fourier type constitute law. The void reactivity corresponds to that used in the classic reduced-order model (C-ROM), as it considers voids propagation at a finite velocity. Essentially, the P-ROM considers delay times or relaxation times in the dynamic description of the neutron density, heat transfer in the fuel, and void fraction reactivity. Nonlinear stability analysis with the estimation of the largest Lyapunov exponents (LLE) of the novel P-ROM for different times of relaxation is discussed and compared with classic approximation. We report evidence that suggests that the relaxation effects strongly determine not only the dynamic evolution of BWRs but their global stability in terms of a bifurcation parameter.

Analysis of a reduced-order nonlinear model of a multi-physics beam

This article studies nonlinear behaviors of a two-dimensional beam with piezoelectric patches on it. Governing electromechanical equations of the beam taking into account several coupled piezoelectric patches, at different positions, are derived. Then, spatiotemporal variables of the system are separated. A methodology is proposed, for the detection of different mode functions and corresponding frequencies of the multi-physics beam, via using space equations of the system. As a representative example, a homogeneous beam with a single piezoelectric patch is considered. The paper is followed by consideration of two particular cases: (i) a single mode of the system is in resonance with the direct lateral-base excitation and (ii) two modes of the system present an internal resonance, while the first one is in resonance with the lateral-base excitation. For both cases, the electromechanical system equations are projected on its targeted mode(s). The temporal equations are treated via a multiple scale method leading to detections of its fixed points. The effects of one of the nonlinear coefficients of the piezoelectric patch, on the overall responses of the multi-physics beam, in terms of changing its behavior from hardening to softening (or vice versa), are discussed and commented upon. Moreover, it is shown that the piezoelectric patch is able to control the targeted mode(s) of the system.

A Trajectory Piecewise-Linear Approach to Nonlinear Model Order Reduction of Wind Farms

This paper presents a method to develop computationally efficient dynamic model of a wind farm suitable for large disturbance simulation. The method based on a trajectory piecewise linear approximation uses single and multiple training trajectory to develop a nonlinear reduced order model. Simulation results using a small demonstration wind farm system and a large practical wind farm system are discussed to demonstrate the effectiveness of the model in capturing dynamic behavior of wind farm following large disturbances.

Model order reduction for random nonlinear dynamical systems and low-dimensional representations for their quantities of interest

We examine nonlinear dynamical systems of ordinary differential equations or differential algebraic equations. In an uncertainty quantification, physical parameters are replaced by random variables. The state or inner variables as well as a quantity of interest are expanded into series with orthogonal basis functions like the polynomial chaos expansions, for example. On the one hand, the stochastic Galerkin method yields a large coupled dynamical system. On the other hand, a stochastic collocation method, which uses a quadrature rule or a sampling scheme, can be written in the form of a large weakly coupled dynamical system. We apply projection-based methods of nonlinear model order reduction to the large systems. A reduced-order model implies a low-dimensional representation of the quantity of interest. We focus on model order reduction by proper orthogonal decomposition. The error of a best approximation located in a low-dimensional subspace is analysed. We illustrate results of numerical computations for test examples.

Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition

This paper presents a structure-exploiting nonlinear model reduction method for systems with general nonlinearities. First, the nonlinear model is lifted to a model with more structure via variable transformations and the introduction of auxiliary variables. The lifted model is equivalent to the original model; it uses a change of variables but introduces no approximations. When discretized, the lifted model yields a polynomial system of either ordinary differential equations or differential-algebraic equations, depending on the problem and lifting transformation. Proper orthogonal decomposition (POD) is applied to the lifted models, yielding a reduced-order model for which all reduced-order operators can be precomputed. Thus, a key benefit of the approach is that there is no need for additional approximations of nonlinear terms, which is in contrast with existing nonlinear model reduction methods requiring sparse sampling or hyper-reduction. Application of the lifting and POD model reduction to the FitzHugh–Nagumo benchmark problem and to a tubular reactor model with Arrhenius reaction terms shows that the approach is competitive in terms of reduced model accuracy with state-of-the-art model reduction via POD and discrete empirical interpolation while having the added benefits of opening new pathways for rigorous analysis and input-independent model reduction via the introduction of the lifted problem structure.

An Error Indicator-Based Adaptive Reduced Order Model for Nonlinear Structural Mechanics—Application to High-Pressure Turbine Blades

The industrial application motivating this work is the fatigue computation of aircraft engines’ high-pressure turbine blades. The material model involves nonlinear elastoviscoplastic behavior laws, for which the parameters depend on the temperature. For this application, the temperature loading is not accurately known and can reach values relatively close to the creep temperature: important nonlinear effects occur and the solution strongly depends on the used thermal loading. We consider a nonlinear reduced order model able to compute, in the exploitation phase, the behavior of the blade for a new temperature field loading. The sensitivity of the solution to the temperature makes the classical unenriched proper orthogonal decomposition method fail. In this work, we propose a new error indicator, quantifying the error made by the reduced order model in computational complexity independent of the size of the high-fidelity reference model. In our framework, when the error indicator becomes larger than a given tolerance, the reduced order model is updated using one time step solution of the high-fidelity reference model. The approach is illustrated on a series of academic test cases and applied on a setting of industrial complexity involving five million degrees of freedom, where the whole procedure is computed in parallel with distributed memory.

Fixed‐precision randomized low‐rank approximation methods for nonlinear model order reduction of large systems

SummaryMany model order reduction (MOR) methods employ a reduced basis to approximate the state variables. For nonlinear models, V is often computed using the snapshot method. The associated low‐rank approximation of the snapshot matrix can become very costly as m,n grow larger. Widely used conventional singular value decomposition methods have an asymptotic time complexity of , which often makes them impractical for the reduction of large models with many snapshots. Different methods have been suggested to mitigate this problem, including iterative and incremental approaches. More recently, the use of fast and accurate randomized methods was proposed. However, most work so far has focused on fixed‐rank approximations, where rank k is assumed to be known a priori. In case of nonlinear MOR, stating a bound on the precision is usually more appropriate. We extend existing research on randomized fixed‐precision algorithms and propose a new heuristic for accelerating reduced basis computation by predicting the rank. Theoretical analysis and numerical results show a good performance of the new algorithms, which can be used for computing a reduced basis from large snapshot matrices, up to a given precision ε.

Linear iterative method for closed-loop control of quasiperiodic flows

This work proposes a feedback-loop strategy to suppress intrinsic oscillations of resonating flows in the fully nonlinear regime. The frequency response of the flow is obtained from the resolvent operator about the mean flow, extending the framework initially introduced by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to study receptivity mechanisms in turbulent flows. Using this linear time-invariant model of the nonlinear flow, modern control methods such as structured ${\mathcal{H}}_{\infty }$-synthesis can be used to design a controller. The approach is successful in damping self-sustained oscillations associated with specific eigenmodes of the mean-flow spectrum. Despite excellent performance, the linear controller is however unable to completely suppress flow oscillations, and the controlled flow is effectively attracted towards a new dynamical equilibrium. This new attractor is characterized by a different mean flow, which can in turn be used to design a second controller. The method can then be iterated on subsequent mean flows, until the coupled system eventually converges to the base flow. An intuitive parallel can be drawn with Newton’s iteration: at each step, a linearized model of the flow response to a perturbation of the input is sought, and a new linear controller is designed, aiming at further reducing the fluctuations. The method is illustrated on the well-known case of two-dimensional incompressible open-cavity flow at Reynolds number $Re=7500$, where the fully developed flow is initially quasiperiodic (2-torus state). The base flow is reached after five iterations. The present work demonstrates that nonlinear control problems may be solved without resorting to nonlinear reduced-order models. It also shows that physically relevant linear models can be systematically derived for nonlinear flows, without resorting to black-box identification from input–output data; the key ingredient being frequency-domain models based on the linearized Navier–Stokes equations about the mean flow. Applicability to amplifier flows and turbulent dynamics has, however, yet to be investigated.

Transient Stability Analysis of Grid-Tied Converters Considering PLL’s Nonlinearity

Transient instability events of grid-tied converters probably occur while riding through grid faults. During low-voltage ride through (LVRT) period, seeing from the converter terminal towards the grid, the Thevenin equivalent grid impedance becomes pretty significant, accordingly making the converter terminal voltage highly sensitive to the output current. Under such circumstances, it is challenging for the converter to resynchronize with the grid via a phase-locked loop (PLL). This paper develops a reduced-order nonlinear model to elaborate on the dynamic synchronization characteristic of the converters. By considering the impact of grid impedance and analysing spatial vector tracking relation, resynchronization principle of the converters during LVRT is revealed. Besides, the impacts of circuit parameters and controller parameters, including residual grid voltage, grid impedance, current references, and PLL parameters, on the transient stability of the converters are investigated. The results are verified by simulation and experimental results.

Nonlinear Unsteady Reduced-Order Modeling for Gust-Load Predictions

A tremendous number of gust-load cases need to be computed during the aircraft design and certification process. From an aerodynamic point of view, gust-load predictions in industry rely on linear potential flow methods, which are inappropriate at transonic flight conditions. Prediction accuracy can be enhanced by accounting for aerodynamic loads computed with computational fluid dynamics, eventually resulting in lighter, more efficient designs. However, full-order, unsteady time-marching simulations are still prohibitively expensive in an industrial environment. Therefore, different reduced-order modeling techniques have been proposed to decrease the computational cost. This paper focuses on an unsteady nonlinear reduced-order model based on least-squares residual minimization and a comparison to the linearized frequency-domain method. Although the latter is in line with current industrial practice of sampling aerodynamic forces in the frequency domain, it neglects dynamic nonlinearities, which are included in the former approach. Results are presented for an airfoil and the NASA Common Research Model at transonic flow conditions exhibiting shock-induced separation during the airframe–gust interaction. Essential quantities for the gust-load analysis, such as global coefficients, sectional forces, and distributed quantities, are evaluated and compared to highlight the strengths and weaknesses of both model reduction techniques.