Full-order Model Research Articles

Active-learning-based nonintrusive model order reduction

Abstract Model order reduction (MOR) can provide low-dimensional numerical models for fast simulation. Unlike intrusive methods, nonintrusive methods are attractive because they can be applied even without access to full order models (FOMs). Since nonintrusive MOR methods strongly rely on snapshots of the FOMs, constructing good snapshot sets becomes crucial. In this work, we propose a novel active-learning-based approach for use in conjunction with nonintrusive MOR methods. It is based on two crucial novelties. First, our approach uses joint space sampling to prepare a data pool of the training data. The training data are selected from the data pool using a greedy strategy supported by an error estimator based on Gaussian process regression. Second, we introduce a case-independent validation strategy based on probably approximately correct learning. While the methods proposed here can be applied to different MOR methods, we test them here with artificial neural networks and operator inference.

A POD-based reduced order model applied to 1D shallow water equations

Many environmental problems involving free surface flow can be solved using the shallow water equations (SWE) often involving high computational costs due to the large spatial and temporal scales of the events. In recent times, reduced order models (ROM) techniques are increasingly used to improve the computational efficiency of simulation models. The Proper Orthogonal Decomposition (POD) method provides an orthogonal basis for representing a given set of data and constructing the ROM by means of the method of snapshots. In this work, a POD-based intrusive ROM strategy is applied to the 1D SWE. The main goal of this work is to build a simulation model able to reproduce realistic scenarios. We analyse the computational improvement and the accuracy of the ROM results with respect to those of the full-order model (FOM).

Retrieval of initial condition for Burgers' equation using reduced-order EnKF via POD-based sparse observations

<p style='text-indent:20px;'>The data assimilation (DA) is a popular method to solve uncertainty quantification problem which has attracted more and more attention in disaster assessment and climate change research. However, the DA usually faces the computational problem associated with "the curse of dimensionality'' when the high-dimensional inverse problem is involved. For this, the usual strategy is to combine the dimensionality reduction method with the DA to mitigate the computational demand in practical application. In this paper, we develop a reduced-order model (ROM)-based EnKF used for retrieving the initial condition, where the prediction model and the observation model are all POD-dependent. In addition to the reduced-order model (ROM) being prediction model, the idea of compressed sensing motivates the acquisition of the observation model that allows an efficient combination of the EnKF with QD algorithm to obtain the optimal sparse observation locations. In this way, computational acceleration is gained in retrieving the initial condition. The effectiveness of this algorithm is demonstrated through retrieving the initial condition of an one-dimensional (1-D) Burgers' equation. Experimental results show that a satisfactory retrieval result is obtained using the current ROM-based EnKF with the computational (CPU) time, <inline-formula><tex-math id="M1">\begin{document}$ 42.64\rm{s} $\end{document}</tex-math></inline-formula>, nearly 25 times faster, and the error result <inline-formula><tex-math id="M2">\begin{document}$ 7.47\times 10^{-3} $\end{document}</tex-math></inline-formula>, two order of magnitude <inline-formula><tex-math id="M3">\begin{document}$ O(10^2) $\end{document}</tex-math></inline-formula> more accurate, than the implementation of the traditional full-order model (FOM)-based EnKF where the computational instability is observed at each iteration though in the same condition as set in the ROM-based EnKF, leading to a poor result (the CPU time <inline-formula><tex-math id="M4">\begin{document}$ 1057.97\rm{s} $\end{document}</tex-math></inline-formula>, and the error accuracy <inline-formula><tex-math id="M5">\begin{document}$ 4.06\times 10^{-1}) $\end{document}</tex-math></inline-formula>. The present study extends the application of ROM to efficiently dealing with ill-posed problem as a regularization strategy through replacing the FOM with the ROM such that the low-order truncation of the POD basis leads to the ROM with as few degrees of freedom as possible, which can efficiently inhibit the higher frequency errors, and hence achieve stability of the proposed approach.</p>

Stochastic production scheduling for water-oil displacement by reduced-order models generated using liquid production rates

In this work, we show an effective approach for application of reduced order model (ROMs) constructed with proper orthogonal decomposition (POD) using liquid production rates for snapshot generation. These ROMs are used to maximise oil production while controlling associated water production. Using ROMs, we perform a parallel genetic algorithm (PGA) and consider liquid production rates as decision variables. The balanced rates are used for injection wells, based on open-flow potential and total amount of produced reservoir volume. The net present value (NPV) is selected as objective function to ensure project profitability and to penalise water production. The NPV by ROM optimisation is approached to within 98% of the NPV obtained by optimisation using the full-order model demonstrating acceptable accuracy. A synthetic model and a real field sector are used for evaluation. The optimisation runtime reduces by 55% in the synthetic model and 71% for optimisation with ROM in the sector case. [Received: April 4, 2022; Accepted: July 14, 2022]

Torque-Minimizing Control Allocation for Overactuated Quadrupedal Locomotion

In this paper, we improve upon a method for optimal control of quadrupedal robots which utilizes a full-order model of the system. The original method utilizes offline nonlinear optimal control to synthesize a control scheme which exponentially orbitally stabilizes the closed-loop system. However, it is not able to handle the overactuated phases which frequently occur during quadrupedal locomotion as a result of the multi-contact nature of the system. We propose a modified method, which handles overactuated gait phases in a way that utilizes the full range of available actuators to minimize torque expenditure without requiring output trajectories to be modified. It is shown that the system under the proposed controller exhibits the same properties, i.e. exponential orbital stability, with the same or lower point-wise torque magnitude. A simulation study demonstrates that the reduction in torque may in certain cases be substantial.

Thermo-rheological reduced order models for non-Newtonian fluid flows with power-law viscosity via the modal identification method

In the framework of melted polymer flows characterization, this study deals with the formulation, construction and validation of thermo-rheological Reduced Order Models (ROMs) for incompressible flows of pseudoplastic fluids. The dynamic viscosity is described by a shear rate power law defined by consistency index K and pseudoplastic index n. The flow dynamics are assumed to be quasi-static whereas the thermal state is unsteady. Viscous dissipation acts as a heat source term in the energy equation. ROMs are built through the Modal Identification Method (MIM). First, their general form is derived from governing local conservation equations and the viscosity power-law, with an original approach to handle the issues related to the pseudoplastic index n. Then ROMs are identified using Particle Swarm Optimization and Ordinary Least Squares, from simulations coming from a reference Full Order Model (FOM). The approach is applied to a polymer flow in an annular duct, corresponding to an experimental lab apparatus. ROMs allow computing temperature at chosen locations of interest whatever the applied inputs, here parameters K, n and inlet flowrate as well as a time-varying volumetric heat source power in the central axis. Compared to the reference Finite Elements FOM, computing time is considerably reduced with limited loss of accuracy.

Parametric Aeroelastic Reduced-Order Model with State-Consistence Enforcement

State-space reduced-order models (ROMs) constructed by traditional system identification methods suffer from the state-inconsistence issue and poor ROM interpolatability for varying flight conditions. This paper presents a novel system identification method and a state-consistence enforcement (SCE) algorithm to generate a state-space aeroelastic (AE)-ROM within a broad flight parameter space. A new regularization term is proposed to modify the traditional autoregressive exogenous formulation and specifically penalize state inconsistence between ROMs at varying flight conditions. The new formulation cast in the form of the generalized Tikhonov regularization problem can be computed very efficiently and produce fundamentally state-consistent ROMs. Rigorous numerical analysis that compares the state-consistent AE-ROM with the full-order model and the ROM without SCE is also conducted. The results convincingly prove that the proposed approach significantly improves the state consistence of ROMs at varying flight conditions. To the best of our knowledge, this research represents the first effort of developing a state-consistent data-driven AE-ROM and enabling aeroelasticity analysis within a broad flight parameter space.

Adaptive sparse interpolation for accelerating nonlinear stochastic reduced-order modeling with time-dependent bases

Stochastic reduced-order modeling based on time-dependent bases (TDBs) has proven successful for extracting and exploiting low-dimensional manifold from stochastic partial differential equations (SPDEs). The nominal computational cost of solving a rank-r reduced-order model (ROM) based on time-dependent basis, a.k.a. TDB-ROM, is roughly equal to that of solving the full-order model for r random samples. As of now, this nominal performance can only be achieved for linear or quadratic SPDEs—at the expense of a highly intrusive process. On the other hand, for problems with non-polynomial nonlinearity, the computational cost of solving the TDB evolution equations is the same as solving the full-order model. In this work, we present an adaptive sparse interpolation algorithm that enables stochastic TDB-ROMs to achieve nominal computational cost for generic nonlinear SPDEs. Our algorithm constructs a low-rank approximation for the right hand side of the SPDE using the discrete empirical interpolation method (DEIM). The presented algorithm does not require any offline computation and as a result the low-rank approximation can adapt to any transient changes of the dynamics on the fly. We also propose a rank-adaptive strategy to control the error of the sparse interpolation. Our algorithm achieves computational speedup by adaptive sampling of the state and random spaces. We illustrate the efficiency of our approach for two test cases: (1) one-dimensional stochastic Burgers’ equation, and (2) two-dimensional compressible Navier–Stokes equations subject to one-hundred-dimensional random perturbations. In all cases, the presented algorithm results in orders of magnitude reduction in the computational cost.

Control-oriented computational fuel cell dynamics modeling – Model order reduction vs. computational speed

Many 1D computational fuel cell dynamics models are established to predict the performance of proton exchange membrane (PEM) fuel cells for control purposes. The 1D models are of high fidelity over a wide range of operating conditions in comparison with empirical and analytical models and less dependent on a large quantity of experimental data in comparison with data-driven models. However, the relation between the various simplified 1D models and their computing efficiency has not been fully understood. Therefore, the present study investigates the trade-off relation between the model order reduction and computational speed of control-oriented 1D models. A full-order model, considering all essential physical phenomena, is validated against experimental data. Seven reduced-order models, depending on whether convection, liquid water, and thermal effects are simulated, are compared with the full-order model in terms of accuracy and computational speed under the same boundary and initial conditions. The results indicate that the two-phase flow has a more significant impact on the cell performance than non-isothermal and convection effects at high relative humidity and current density conditions. The computing speed of the reduced-order model, in which the convection, liquid water, and temperature variation are omitted, can be 140 times faster than the full-order model.

Multi-fidelity surrogate modeling using long short-term memory networks

When evaluating quantities of interest that depend on the solutions to differential equations, we inevitably face the trade-off between accuracy and efficiency. Especially for parametrized, time-dependent problems in engineering computations, it is often the case that acceptable computational budgets limit the availability of high-fidelity, accurate simulation data. Multi-fidelity surrogate modeling has emerged as an effective strategy to overcome this difficulty. Its key idea is to leverage many low-fidelity simulation data, less accurate but much faster to compute, to improve the approximations with limited high-fidelity data. In this work, we introduce a novel data-driven framework of multi-fidelity surrogate modeling for parametrized, time-dependent problems using long short-term memory (LSTM) networks, to enhance output predictions both for unseen parameter values and forward in time simultaneously — a task known to be particularly challenging for data-driven models. We demonstrate the wide applicability of the proposed approaches in a variety of engineering problems with high- and low-fidelity data generated through fine versus coarse meshes, small versus large time steps, or finite element full order versus deep learning reduced-order models. Numerical results show that the proposed multi-fidelity LSTM networks not only improve single-fidelity regression significantly, but also outperform the multi-fidelity models based on feed-forward neural networks.

Data-driven stability analysis via the superposition of reduced-order models for the flutter of circular cylinder submerged in three-dimensional spanwise shear inflow at subcritical Reynolds number

In this paper, we present a novel data-driven theory for the stability analysis of a flow-induced vibration (FIV) system consisting of an elastically mounted circular cylinder submerged in three-dimensional (3D) spanwise shear inflow at a subcritical Reynolds number. The presented data-driven theory separates the cylinder into several elements along the spanwise direction and treats the aerodynamics of each element as a two-dimensional (2D) situation subject to a uniform inflow. An eigensystem realization algorithm is constructed to obtain the separate 2D flow reduced-order model (ROM) for each element, and then, the superposition of those 2D ROMs (SROM) is processed to obtain the simplified 3D flow ROM. The simplified 3D flow ROM is coupled with the structural model to perform a linear stability analysis of the FIV system under study. The proposed data-driven technique demonstrates high consistency with the high-fidelity full-order model (FOM) with regard to the prediction of flutter lock-in boundaries while being more time-efficient, whereas the traditional direct 3D data-driven analysis involves significant errors. The growth rate obtained using SROM is negatively correlated with the lagging time (reflected in the FOM calculation) for the FIV system to evolve from the initial stationary state to the final equilibrium state. The evolution of the structural instability range with the variation in the mass ratio is analyzed/predicted by the proposed data-driven theory. The determination of the lock-in regime using the FOM is accompanied by a careful discussion of the associated dynamical responses, including phase differences, structural oscillation frequencies, lift coefficients, and wake patterns.

Finite element based Model Order Reduction for parametrized one-way coupled steady state linear thermo-mechanical problems

This contribution focuses on the development of Model Order Reduction (MOR) for one-way coupled steady state linear thermo-mechanical problems in a finite element setting. We apply Proper Orthogonal Decomposition (POD) for the computation of reduced basis space. On the other hand, for the evaluation of the modal coefficients, we use two different methodologies: the one based on the Galerkin projection (G) and the other one based on Artificial Neural Network (ANN). We aim to compare POD-G and POD-ANN in terms of relevant features including errors and computational efficiency. In this context, both physical and geometrical parametrization are considered. We also carry out a validation of the Full Order Model (FOM) based on customized benchmarks in order to provide a complete computational pipeline. The framework proposed is applied to a relevant industrial problem related to the investigation of thermo-mechanical phenomena arising in blast furnace hearth walls. • Computational pipeline for fast and accurate thermo-mechanical FEM simulations. • Comparison between POD-Galerkin and POD-ANN model order reduction. • Parametric setting involving both physical properties and geometrical features. • Assessment of MOR approach with an industrial problem related to ironmaking process.

Small-Signal Stability Boundary of Heterogeneous Multi-Converter Power Systems Dominated By the Phase-Locked Loops’ Dynamics

While renewable resources are increasingly integrated into the electric power system, the small-signal instability risk may be induced by grid-following converters using phase-locked loops (PLLs) for grid synchronization, especially in low short-circuit strength systems. The stability analysis complexity is further increased in such a heterogeneous multi-converter system (HMCS), where multiple converters may have different PLLs’ control parameters provided by various vendors. To understand how the different PLLs’ dynamics collectively affect the small-signal stability of the HMCS, we transform the HMCS into an equivalent homogeneous multi-converter system for the small-signal stability analysis. Then, we analytically derives the small-signal stability boundary condition of the HMCS dominated by the PLLs’ dynamics (HMCS-DPLL). The derived stability boundary allows us to obtain analytical results about how the small-signal stability of the HMCS-DPLL is collectively affected by network structure, converter operating conditions, and different PLLs’ control parameters. Based on the stability boundary condition, a computationally efficient method is also proposed to identify the design rationality of PLL control parameters as well as the small-signal stability and stability margin of the HMCS-DPLL. The analytical results and proposed method are validated by modal analysis and electromagnetic transient simulation with the full-order models on a 9-converter heterogeneous system.

On the informativity of direct identification experiments in dynamical networks

Data informativity is a crucial property to ensure the consistency of the prediction error estimate. This property has thus been extensively studied in the open-loop and in the closed-loop cases, but has only been briefly touched upon in the dynamic network case. In this paper, we consider the prediction error identification of the modules in a row of a dynamic network using the full input approach. Our main contribution is to propose a number of easily verifiable data informativity conditions for this identification problem. Among these conditions, we distinguish a sufficient data informativity condition that can be verified based on the topology of the network and a necessary and sufficient data informativity condition that can be verified via a rank condition on a matrix of coefficients that are related to a full-order model structure of the network. These data informativity conditions allow to determine different situations (i.e., different excitation patterns) leading to data informativity. In order to be able to distinguish between these different situations, we also propose an optimal experiment design problem that allows to determine the excitation pattern yielding a certain pre-specified accuracy with the least excitation power.

Nonlinear mapping of non-conservative forces for reduced-order modelling

Non-intrusive or indirect reduced-order modelling strategies, such as the implicit condensation and expansion method, are applicable to geometrically nonlinear structures modelled using commercial finite-element (FE) software. Traditionally, the non-conservative forces acting on the structure are reduced via a linear projection onto the space spanned by the reduced modeshapes. As such, only the forces acting directly on these reduced modes can be captured, while any energy gained or dissipated by the statically condensed modes is neglected. This can lead to significant inaccuracies in the reduced-order model (ROM) predictions, which is demonstrated here using a 2-degrees-of-freedom (DOF) oscillator, and an FE model of a cantilever beam. It is shown that the non-conservative forces acting on the statically condensed modes can be captured using a nonlinear mapping of the physical DOFs into the reduced coordinates. This introduces additional terms in the reduced equations of motion, which we describe as force compensation . Excellent agreement is observed between the forced response curves of the full-order models and those of our proposed ROMs, both for the oscillator as well as the cantilever beam under different external excitation conditions (i.e. a constant-direction force and a follower force).

Regression Trees on Grassmann Manifold for Adapting Reduced-Order Models

Low-dimensional and computationally less-expensive reduced-order models (ROMs) have been widely used to capture the dominant behaviors of high-4dimensional systems. An ROM can be obtained, using the well-known proper orthogonal decomposition (POD), by projecting the full-order model to a subspace spanned by modal basis modes that are learned from experimental, simulated, or observational data, i.e., training data. However, the optimal basis can change with the parameter settings. When an ROM, constructed using the POD basis obtained from training data, is applied to new parameter settings, the model often lacks robustness against the change of parameters in design, control, and other real-time operation problems. This paper proposes to use regression trees on Grassmann manifold to learn the mapping between parameters and POD bases that span the low-dimensional subspaces onto which full-order models are projected. Motivated by the observation that a subspace spanned by a POD basis can be viewed as a point in the Grassmann manifold, we propose to grow a tree by repeatedly splitting the tree node to maximize the Riemannian distance between the two subspaces spanned by the predicted POD bases on the left and right daughter nodes. Five numerical examples are presented to comprehensively demonstrate the performance of the proposed method, and compare the proposed tree-based method to the existing interpolation method for POD basis and the use of global POD basis. The results show that the proposed tree-based method is capable of establishing the mapping between parameters and POD bases, and thus adapt ROMs for new parameters.

Enhancing high-fidelity nonlinear solver with reduced order model

We propose the use of reduced order modeling (ROM) to reduce the computational cost and improve the convergence rate of nonlinear solvers of full order models (FOM) for solving partial differential equations. In this study, a novel ROM-assisted approach is developed to improve the computational efficiency of FOM nonlinear solvers by using ROM’s prediction as an initial guess. We hypothesize that the nonlinear solver will take fewer steps to the converged solutions with an initial guess that is closer to the real solutions. To evaluate our approach, four physical problems with varying degrees of nonlinearity in flow and mechanics have been tested: Richards’ equation of water flow in heterogeneous porous media, a contact problem in a hyperelastic material, two-phase flow in layered porous media, and fracture propagation in a homogeneous material. Overall, our approach maintains the FOM’s accuracy while speeding up nonlinear solver by 18–73% (through suitable ROM-assisted FOMs). More importantly, the proximity of ROM’s prediction to the solution space leads to the improved convergence of FOMs that would have otherwise diverged with default initial guesses. We demonstrate that the ROM’s accuracy can impact the computational efficiency with more accurate ROM solutions, resulting in a better cost reduction. We also illustrate that this approach could be used in many FOM discretizations (e.g., finite volume, finite element, or a combination of those). Since our ROMs are data-driven and non-intrusive, the proposed procedure can easily lend itself to any nonlinear physics-based problem.

Adaptive Learning and Optimization of High-Speed Sailing Maneuvers for America's Cup

Abstract This paper presents an adaptive control strategy to optimize the sailing maneuvers of an AC75 foiling sailboat competing in America's Cup. Foiling yachts have nonlinear, high-dimensional, and unstable dynamics due to several articulations for fast motions and maneuverability. Achieving aggressive and optimal maneuvers requires taking these complex dynamics into account instead of analytical optimizations using reduced-order models. We compared extremum-seeking and Jacobian learning (JL) control approaches on a full-order model to achieve optimal maneuvers and used JL to optimize articulations. The controllers were integrated with a high-fidelity sailboat simulator for safe and efficient maneuver optimization. The optimal solutions were subject to physical/actuator constraints and those enforced to ensure the feasibility of the maneuvers by humans (sailors). The close-hauled and tacking maneuvers were optimized to achieve maximum velocity made good (VMG) and minimum loss of VMG, respectively. The optimal maneuvers boast a marginal VMG loss of less than 1.5%, which enables exploiting areas of good wind conditions in the racing environment.

Fast $L^2$ Optimal Mass Transport via Reduced Basis Methods for the Monge--Ampère Equation

Repeatedly solving the parameterized optimal mass transport (pOMT) problem is a frequent task in applications such as image registration and adaptive grid generation. It is thus critical to develop a highly efficient reduced solver that is equally accurate as the full order model. In this paper, we propose such a machine learning--like method for pOMT by adapting a new reduced basis (RB) technique specifically designed for nonlinear equations, the reduced residual reduced over-collocation (R2-ROC) approach, to the parameterized Monge--Ampère equation. It builds on top of a narrow-stencil finite difference method (FDM), a so-called truth solver, which we propose in this paper for the Monge--Ampère equation with the transport boundary. Together with the R2-ROC approach, it allows us to handle the strong and unique nonlinearity pertaining to the Monge--Ampère equation achieving online efficiency without resorting to any direct approximation of the nonlinearity. Several challenging numerical tests demonstrate the accuracy and high efficiency of our reduced solver for solving the parameterized Monge--Ampère equation, effectively transporting the nontrivial boundaries.

Robust error assessment for reduced order vibro-acoustic problems

Robust and efficient automatic model order reduction is important in the design process of vibro-acoustic structures. For correct assessment of the reduction error without requiring the solution of the full order model, reliable error estimation techniques are required. Such error estimators can be used in iterative algorithms to automatically determine a reasonable order and expansion point distribution for reduced-order models. An accurate estimation of the approximation error is important in order to design an efficient adaptive model order reduction process requiring as little a priori knowledge of the system response as possible. In this contribution, different techniques for estimating the approximation error of reduced-order models are compared and employed in adaptive algorithms aiming at generating reduced-order models of vibro-acoustic systems. For many applications, vibro-acoustic systems need to be evaluated in a specific frequency range, as certain system characteristics are only observed in this region. Therefore, the employed error estimation methods are assessed regarding their effectiveness estimating the error in a particular frequency region.