Explicit synchronous partitioned scheme for coupled reduced order models based on composite reduced bases
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Bochev, Pavel B.↗
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Reduced order models for Lagrangian hydrodynamics
It is reported, as a mathematical model of high-speed flow and shock wave propagation in a complex multimaterial setting, Lagrangian hydrodynamics is characterized by moving meshes, advection-dominated solutions, and moving shock fronts with sharp gradients. These challenges hinder the existing projection-based model reduction schemes from being practical. We develop several variations of projection-based reduced order model techniques for Lagrangian hydrodynamics by introducing three different reduced bases for position, velocity, and energy fields. A time-windowing approach is also developed to address the challenge imposed by the advection-dominated solutions. Lagrangian hydrodynamics is formulated as a nonlinear problem, which requires a proper hyper-reduction technique. Therefore, we apply the over-sampling DEIM and SNS approaches to reduce the complexity due to the nonlinear terms. Finally, we also present both a posteriori and a priori error bounds associated with our reduced order model. We compare the performance of the spatial and time-windowing reduced order modeling approaches in terms of accuracy and speed-up with respect to the corresponding full order model for several numerical examples, namely Sedov blast, Gresho vortices, Taylor-Green vortices, and triple-point problems.
97 MATHEMATICS AND COMPUTING↗
Investigation of Reduced-Order Modeling for Aircraft Stability and Control Prediction
High-fidelity computational fluid dynamics tools offer the potential to approximate increments for ground-to-flight scaling effects, as well as to augment the dynamic damping derivative data for motion-based flight simulators. Unfortunately, the computational expense is currently prohibitive for populating a complete simulator database. This work investigates an existing surrogate-based, indicial response reduced-order model methodology as a means to efficiently augment a flight simulator database with high-fidelity nonlinear aerodynamic damping derivatives. Creation of the reduced-order model is based on the superposition integrals of the step response with the derivative of its corresponding input signal. Step responses are calculated using a computational grid motion approach that separates the effects of angle of attack and sideslip angle from angular rates, and rates from angle of attack and sideslip. It is demonstrated that the transients produced during the start of a forced-oscillation motion are captured by the reduced-order model to the level of fidelity of a comparable computational solution. Aerodynamic coefficients computed within minutes by the reduced-order model for an aircraft undergoing an 18-second half Lazy-8 maneuver and a 25-second Immelmann turn maneuver are compared with those from full computational flight solutions that required days to complete. Finally, a cost-benefit assessment is included that demonstrates a compelling advantage for this approach. d for maneuvering, flexible vehicles.
Frink, Neal T.↗
Enhancing high-fidelity nonlinear solver with reduced order model
Abstract 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.
97 MATHEMATICS AND COMPUTING↗
Interpretable and flexible non-intrusive reduced-order models using reproducing kernel Hilbert spaces
This paper develops an interpretable, non-intrusive reduced-order modeling technique using regularized kernel interpolation. Existing non-intrusive approaches approximate the dynamics of a reduced-order model (ROM) by solving a data-driven least-squares regression problem for low-dimensional matrix operators. Our approach instead leverages regularized kernel interpolation, which yields an optimal approximation of the ROM dynamics from a user-defined reproducing kernel Hilbert space. We show that our kernel-based approach can produce interpretable ROMs whose structure mirrors full-order model structure by embedding judiciously chosen feature maps into the kernel. The approach is flexible and allows a combination of informed structure through feature maps and closure terms via more general nonlinear terms in the kernel. We also derive a computable a posteriori error bound that combines standard error estimates for intrusive projection-based ROMs and kernel interpolants. In conclusion, the approach is demonstrated in several numerical experiments that include comparisons to operator inference using both proper orthogonal decomposition and quadratic manifold dimension reduction.
Data-driven model reduction↗
Mean-square-error bounds for reduced-order linear state estimators
The mean-square error of reduced-order linear state estimators for continuous-time linear systems is investigated. Lower and upper bounds on the minimal mean-square error are presented. The bounds are readily computable at each time-point and at steady state from the solutions to the Ricatti and the Liapunov equations. The usefulness of the error bounds for the analysis and design of reduced-order estimators is illustrated by a practical numerical example.
Baram, Y.↗
A Parametric, Data-Driven, Non-Intrusive Reduced-Order Model Framework for Crystal Plasticity Simulations of Voids
The influence of the internal structure at micrometer length scales on the deformation of polycrystalline materials can be effectively captured using crystal plasticity finite element methods (CPFEM). However, the complexity and nonlinearity of the deformation equations CPFEM solves demand significant computational power and resources to achieve accurate predictions, limiting its broader application. To address this challenge, we have identified a reduced-order representation of the complex data in order to establish a computationally efficient reduced-order models (ROM) and drastically reduce the computational expense of CPFEM. Specifically, in this work, we developed a parametric, data-driven, and non-intrusive ROM framework for CPFEM using proper orthogonal decomposition (POD) and sparse variational Gaussian process (SVGP) regression for single-crystal microstructures under tensile loading conditions. The developed protocol enables one to compress field into a latent/low-dimensional space described by principal component analysis (PCA) via the singular value decomposition (SVD) algorithm. As a result, the high-dimensional data are reduced to a significantly smaller amount of dimensions with POD bases and POD coefficients. Furthermore, we deployed an ensemble of SVGPs—extended from the classical Gaussian process (GP) regression for scalability and handling big data—in a massively parallel manner to train and predict latent POD coefficients using known POD bases from a set of previously obtained simulations results. Lastly, using the predicted POD coefficients, we reconstructed the full-field results and showed reasonable agreement compared with the true values obtained from running CPFEM. The developed framework is validated with a set of CPFEM simulations of a single embedded void in single-crystal aluminum alloy. While the framework is broadly applicable, this work specifically focuses on single-crystal microstructures, a single load case (e.g., tensile), and a specific void geometry (spherical).
Anisotropy↗
A Nonlinear Reduced Order Method for Prediction of Acoustic Fatigue
The goal of this investigation is to assess the quality of high-cycle-fatigue life estimation via a reduced order method, for structures undergoing geometrically nonlinear random vibrations. Modal reduction is performed with several different suites of basis functions. After numerically solving the reduced order system equations of motion, the physical displacement time history is obtained by an inverse transformation and stresses are recovered. Stress ranges obtained through the rainflow counting procedure are used in a linear damage accumulation method to yield fatigue estimates. Fatigue life estimates obtained using various basis functions in the reduced order method are compared with those obtained from numerical simulation in physical degrees-of-freedom.
Przekop, Adam↗
Reduced-Order Structure-Preserving Model for Parallel-Connected Three-Phase Grid-Tied Inverters
Next-generation power networks will contain large numbers of grid-connected inverters satisfying a significant fraction of system load. Since each inverter model has a relatively large number of dynamic states, it is impractical to analyze complex system models where the full dynamics of each inverter are retained. To address this challenge, we derive a reduced-order structure-preserving model for parallel-connected grid-tied three-phase inverters. Here, each inverter in the system is assumed to have a full-bridge topology, LCL filter at the point of common coupling, and the control architecture for each inverter includes a current controller, a power controller, and a phase-locked loop for grid synchronization. We outline a structure-preserving reduced-order inverter model with lumped parameters for the setting where the parallel inverters are each designed such that the filter components and controller gains scale linearly with the power rating. By structure preserving, we mean that the reduced-order three-phase inverter model is also composed of an LCL filter, a power controller, current controller, and PLL. We show that the system of parallel inverters can be modeled exactly as one aggregated inverter unit and this equivalent model has the same number of dynamical states as any individual inverter in the system. Numerical simulations validate the reduced-order model.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Reduced Order Model of Transactive Bidding Loads
Transactive energy (TE) has been identified to provide better grid efficiency and reliability by market-based transactive exchanges between energy producers and energy consumers. Simulations of TE systems are crucial to evaluate the benefits and impacts of different transactive mechanisms. However, such simulations can be time consuming due to the information exchange between various participants and complex co-simulation environments. In this paper, we develop a reduced order model to speed up the simulation of transactive systems in TE simulation platform (TESP) while achieving very low error between the reduced order and full model results. Specifically, the developed reduced order model consists of an aggregate responsive load agent which utilizes two Recurrent Neural Networks (RNNs) with Long Short-Term Memory units (LSTMs) to enable transactive elements to collectively participate in the TE system. The proposed aggregate responsive load (ARL) agent is able to produce similar transactive behaviors to the full simulation model while achieving significant simulation time reduction. Finally, we also show that the developed model enables generalization of simulation results across different dates and across different number of loads included in the simulations.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Preconditioning for Hyper-reduction in Reduced Order Models
Many projection-based reduced order models (pROM) that utilize the governing equation and data to accelerate physical simulations can be applied to nonlinear dynamical systems. To avoid full order model (FOM) scale update for each time step, hyperreduction techniques are developed to sample high dimensional nonlinear terms. Our study aims to investigate if preconditioning the least-squares problem used for the nonlinear approximation can improve the robustness of the condition number of the problem while achieving high accuracy. In our study, we use the row-normalization matrix motivated by the Christoffel function as the preconditioner and solve the corresponding weighted least-squares problem. Numerical results for Lagrangian hydrodynamics examples are analyzed to explore how the preconditioner works compared to existing hyper-reduction techniques.
97 MATHEMATICS AND COMPUTING↗
Simplified Predictive Models for CO 2 Sequestration Performance Assessment: Research Topical Report on Task #4 - Reduced-Order Method (ROM) Based Models
Reduced-order models provide a means for greatly accelerating the detailed simulations that will be required to manage CO 2 storage operations. In this work, we investigate the use of one such method, POD-TPWL, which has previously been shown to be effective in oil reservoir simulation problems. This method combines trajectory piecewise linearization (TPWL), in which the solution to a new (test) problem is represented through a linearization around the solution to a previously-simulated (training) problem, with proper orthogonal decomposition (POD), which enables solution states to be expressed in terms of a relatively small number of parameters. We describe the application of POD-TPWL for CO 2 -water systems simulated using a compositional procedure. Stanford’s Automatic Differentiation-based General Purpose Research Simulator (AD-GPRS) performs the full-order training simulations and provides the output (derivative matrices and system states) required by the POD-TPWL method. A new POD-TPWL capability introduced in this work is the use of horizontal injection wells that operate under rate (rather than bottom-hole pressure) control. Simulation results are presented for CO 2 injection into a synthetic aquifer and into a simplified model of the Mount Simon formation. Test cases involve the use of time-varying well controls that differ from those used in training runs. Results of reasonable accuracy are consistently achieved for relevant well quantities. Runtime speedups of around a factor of 370 relative to full- order AD-GPRS simulations are achieved, though the preprocessing needed for POD-TPWL model construction corresponds to the computational requirements for about 2.3 full-order simulation runs. A preliminary treatment for POD-TPWL modeling in which test cases differ from training runs in terms of geological parameters (rather than well controls) is also presented. Results in this case involve only small differences between training and test runs, though they do demonstrate that the approach is able to capture basic solution trends. The impact of some of the detailed numerical treatments within the POD-TPWL formulation is considered in an Appendix.
Mishra, Srikanta↗
A Reduced-Order Aggregated Model for Parallel Inverter Systems with Virtual Oscillator Control
This paper introduces a reduced-order aggregated model for parallel-connected inverters controlled with virtual oscillator control (VOC). The premise of VOC is to modulate inverter dynamics to emulate those of nonlinear oscillators with the goal of realizing a stable ac microgrid in the absence of communication, synchronous generation, or a stiff grid. To obtain a reduced-order model for a system of parallel-connected inverters with VOC, we first formulate a set of scaling laws that describe how the controller and filter parameters of a given inverter depend on its voltage and power rating. Subsequently, we show that N parallel inverters which adhere to this scaling law can be modeled with the same structure and hence the same computational burden of the model of a single inverter. The proposed aggregate model is experimentally validated on a system of three parallel inverters with heterogeneous power ratings.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Reduced-Order Structure-Preserving Model for Parallel-Connected Three-Phase Grid-Tied Inverters: Preprint
Given that next-generation infrastructures will contain large numbers of grid-connected inverters and these interfaces will be satisfying a growing fraction of system load, it is imperative to analyze the impacts of power electronics on such systems. However, since each inverter model has a relatively large number of dynamic states, it would be impractical to execute complex system models where the full dynamics of each inverter are retained. To address this challenge, we derive a reduced-order structure-preserving model for parallel-connected grid-tied three-phase inverters. Here, each inverter in the system is assumed to have a full-bridge topology, LCL filter at the point of common coupling, and the control architecture for each inverter includes a current controller, a power controller, and a phase-locked loop for grid synchronization. We outline a structure-preserving reduced-order inverter model for the setting where the parallel inverters are each designed such that the filter components and controller gains scale linearly with the power rating. By structure preserving, we mean that the reduced-order three-phase inverter model is also composed of an LCL filter, a power controller, current controller, and PLL. That is, we show that the system of parallel inverters can be modeled exactly as one aggregated inverter unit and this equivalent model has the same number of dynamical states as an individual inverter in the paralleled system. Numerical simulations validate the reduced-order models.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Reduced Order Methods for Prediction of Thermal-Acoustic Fatigue
The goal of this investigation is to assess the quality of high-cycle-fatigue life estimation via a reduced order method, for structures undergoing random nonlinear vibrations in a presence of thermal loading. Modal reduction is performed with several different suites of basis functions. After numerically solving the reduced order system equations of motion, the physical displacement time history is obtained by an inverse transformation and stresses are recovered. Stress ranges obtained through the rainflow counting procedure are used in a linear damage accumulation method to yield fatigue estimates. Fatigue life estimates obtained using various basis functions in the reduced order method are compared with those obtained from numerical simulation in physical degrees-of-freedom.
Przekop, A.↗
Local reduced-order modeling for electrostatic plasmas by physics-informed solution manifold decomposition
Despite advancements in high-performance computing and modern numerical algorithms, computational cost remains prohibitive for multi-query kinetic plasma simulations. Here, in this work, we develop data-driven reduced-order models (ROMs) for collisionless electrostatic plasma dynamics, based on the kinetic Vlasov-Poisson equation. Our ROM approach projects the equation onto a linear subspace defined by the proper orthogonal decomposition (POD) modes. We introduce an efficient tensorial method to update the nonlinear term using a precomputed third-order tensor. We capture multiscale behavior with a minimal number of POD modes by decomposing the solution manifold into multiple time windows and creating temporally local ROMs. We consider two strategies for decomposition: one based on the physical time and the other based on the electric field energy. Applied to the 1D1V Vlasov–Poisson simulations, that is, prescribed E-field, Landau damping, and two-stream instability, we demonstrate that our ROMs accurately capture the total energy of the system both for parametric and time extrapolation cases. The temporally local ROMs are more efficient and accurate than the single ROM. In addition, in the two-stream instability case, we show that the energy-windowing reduced-order model (EW-ROM) is more efficient and accurate than the time-windowing reduced-order model (TW-ROM). With the tensorial approach, EW-ROM solves the equation approximately 90 times faster than Eulerian simulations while maintaining a maximum relative error of 7.5% for the training data and 11% for the testing data.
Electrostatic plasmas↗
Application of an Affine Nonlinear Galerkin Reduced-order Model to Compressible Fluid Flows
Galerkin reduced-order models (ROMs) often struggle to accurately capture multiscale fluid physics in challenging flow regimes such as flows experiencing compressibility effects. This in part stems from the global nature of both the basis construction problem and the spectral formulation itself. In this work, a multi-basis ROM is developed in an affine space based on proper orthogonal decomposition (POD) by projecting the full Navier-Stokes equations expressed in terms of the specific volume, velocity, and pressure primitive variables. The model is applied to high-fidelity numerical simulation datasets obtained for a canonical compressible flow configuration: the flow over a backward facing step at different subsonic Mach numbers. It is observed that application of an eigenvalue reassignment (ER) stabilization method is required to avoid early divergence of the ROM predictions for this configuration in the three Mach numbers tested. The sensitivity of the POD-ROM results to the choice of parameters in the stabilization algorithm is discussed.
reduced-order model↗
Application of an Affine Nonlinear Galerkin Reduced-order Model to Compressible Fluid Flows
Galerkin reduced-order models (ROMs) often struggle to accurately capture multiscale fluid physics in challenging flow regimes such as flows experiencing compressibility effects. This in part stems from the global nature of both the basis construction problem and the spectral formulation itself. In this work, a multi-basis ROM is developed in an affine space based on proper orthogonal decomposition (POD) by projecting the full Navier-Stokes equations expressed in terms of the specific volume, velocity, and pressure primitive variables. The model is applied to high-fidelity numerical simulation datasets obtained for a canonical compressible flow configuration: the flow over a backward facing step at different subsonic Mach numbers. It is observed that application of an eigenvalue reassignment (ER) stabilization method is required to avoid early divergence of the ROM predictions for this configuration in the three Mach numbers tested. The sensitivity of the POD-ROM results to the choice of parameters in the stabilization algorithm is discussed.
reduced-order model↗