POD-based Reduced-order Model Research Articles

Reduction of experimental effort in conventional engine calibration process by using reduced order model

The calibration process of the internal combustion engine becomes more costly and time-consuming than ever due to complexity of the engines and stringent emission regulations. The accurate and fast calibration of the engine requires a lot of performance tests which demands time, cost, and expert manpower as well as test facilities that must be taken into considerations. This process needs a large number of test points which should be examined to get an optimum map of the engine. Although some works have been done on reducing test time for the engine calibration, still the process timing remains one of the challenges for the automotive industry. The present study aims to decrease the calibration cost and time by employing the proper orthogonal decomposition (POD) method. The POD-based reduced order model is applied to reduce the number of experimental engine tests to regenerate the entire calibration table by using only a few operating points accurately. The low, mid-range and high RPM test points of a small single-cylinder, two-stroke engine are chosen as the input to the mathematical model to regenerate the missing calibration data. The comparison between the generated and the corresponding experimental data indicates that they are in good agreement (<10% difference) which implies the POD model is capable to decrease the number of test points and consequently the run time of dynamometer. Furthermore, although this approach is implemented for a conventional calibration process, however, the methodology can be extended to complex calibration procedures.

Reduced-order modelling of radiative transfer effects on Rayleigh–Bénard convection in a cubic cell

This paper presents a reduced-order modelling strategy for Rayleigh-Benard convection of a radiating gas, based on the proper orthogonal decomposition (POD). Direct numerical simulation (DNS) of coupled natural convection and radiative transfer in a cubic Rayleigh-Benard cell is performed for an air/H 2 O/CO 2 mixture at room temperature and at a Rayleigh number of 10 7. It is shown that radiative transfer between the isothermal walls and the gas triggers a convection growth outside the boundary-layers. Mean and turbulent kinetic energy increase with radiation, as well as temperature fluctuations to a lesser extent. As in the uncoupled case, the large-scale circulation (LSC) settles in one of the two diagonal planes of the cube with a clockwise or anticlockwise motion, and experiences occasional brief reorientations which are rotations of π/2 of the LSC in the horizontal plane. A POD analysis is conducted and reveals that the dominant POD eigenfunctions are preserved with radiation while POD eigenvalues are increased. Two POD-based reduced-order models including radiative transfer effects are then derived: the first one is based on coupled DNS data while the second one is an a priori model based on uncoupled DNS data. Owing to the weak temperature differences, radiation effects on mode amplitudes are linear in the models. Both models capture the increase in energy with radiation and are able to reproduce the low-frequency dynamics of reorientations and the high-frequency dynamics associated with the LSC velocity observed in the coupled DNS.

Deep-learning-based surrogate model for reservoir simulation with time-varying well controls

A new deep-learning-based reduced-order modeling (ROM) framework is proposed for application in subsurface flow simulation. The reduced-order model is based on an existing embed-to-control (E2C) framework and includes an auto-encoder, which projects the system to a low-dimensional subspace, and a linear transition model, which approximates the evolution of the system states in low dimension. In addition to the loss function for data mismatch considered in the original E2C framework, we introduce a physics-based loss function that penalizes predictions that are inconsistent with the governing flow equations. The loss function is also modified to emphasize accuracy in key well quantities of interest (e.g., fluid production rates). The E2C ROM is shown to be analogous to an existing ROM, POD-TPWL, which has been extensively developed for subsurface flow simulation. The new ROM is applied to oil-water flow in a heterogeneous reservoir, with flow driven by nine wells operating under time-varying control specifications. A total of 300 high-fidelity training simulations are performed in the offline stage, and the network training requires 10-12 minutes on a Tesla V100 GPU node. Online (runtime) computations achieve speedups of over a factor of 1000 relative to full-order simulations. Extensive test case results, with well controls varied over large ranges, are presented. Accurate ROM predictions are achieved for global saturation and pressure fields at particular times, and for injection and production well responses as a function of time. Error is shown to increase when 100 or 200 (rather than 300) training runs are used to construct the E2C ROM. Overall the E2C ROM is shown to provide reliable predictions with levels of perturbations in the well controls that are much larger than those used with existing POD-TPWL treatments. The current model is however limited to 2D systems, and the required number of training simulations is much larger than that for POD-based ROMs.

A proper orthogonal decomposition analysis method for transient nonlinear heat conduction problems. Part 1: Basic algorithm

The present work conducts a systematic and in-depth algorithm investigation for transient nonlinear heat conduction problems solved by using the proper orthogonal decomposition (POD) method. Part 1 of this two-part articles presents the process and characteristics of basic algorithms, including POD explicit and implicit time-marching methods. The accuracy and efficiency are verified by several numerical examples with various boundary conditions and element types. The results show that the POD-based reduced order model (ROM) can provide high quality temperature prediction of the transient nonlinear heat conduction problems when using implicit method. However, the computational time of implicit method is much longer than that of explicit one. The acceleration effect of POD-based ROM on the calculation of the transient nonlinear heat conduction problems is one order of magnitude lower than that of the corresponding transient linear heat conduction problems. The improvement of computational efficiency is not pronounced. Further studies of the more efficient advanced algorithms to deal with POD-based ROM for transient nonlinear heat conduction problems will be presented in Part 2. Additionally, an approximate POD-based ROM for transient nonlinear heat conduction problem is proposed, which can be constructed quickly by using POD modes obtained from the corresponding transient linear heat conduction system. It is confirmed to be feasible for allowing nonlinear behavior to be modeled at an acceptable level of accuracy. It has significant application potential of solving practical engineering problems.

Proper orthogonal decomposition analysis and modelling of large-scale flow reorientations in a cubic Rayleigh–Bénard cell

This paper investigates the large-scale flow reorientations of Rayleigh–Bénard convection in a cubic cell using proper orthogonal decomposition (POD) analysis and modelling. A direct numerical simulation is performed for air at a Rayleigh number of $10^{7}$ and shows that the flow is characterized by four quasi-stable states, corresponding to a large-scale circulation lying in one of the two diagonal planes of the cube with a clockwise or anticlockwise motion, with occasional brief reorientations. Proper orthogonal decomposition is applied to the joint velocity and temperature fields of an enriched database which captures the statistical symmetries of the flow. We found that each quasi-stable state consists of a superposition of four spatial modes representing three types of structures: (i) a mean-flow mode consisting of two stacked counter-rotating torus-like structures; (ii) two large-scale two-dimensional rolls (pair of degenerated modes) which form large-scale diagonal rolls when combined together; and (iii) an eight-roll mode that transports fluid from one corner to the other and strengthens the circulation along the diagonal. In addition, we identified three other modes that play a role in the reorientation process: two boundary-layer modes (pair of degenerated modes) that connect the core region with the horizontal boundary layers and one mode associated with corner rolls. The symmetries of the different POD modes are discussed, as well as their temporal dynamics. A description of the reorientation process in terms of POD modes is provided and compared with other modal approaches available in the literature. Finally, Galerkin projection is used to derive a POD-based reduced-order model. Unresolved modes are accounted for in the model by an extra dissipation term and the addition of noise. A seven-mode model is able to reproduce the low-frequency dynamics of the large-scale reorientations as well as the high-frequency dynamics associated with the large-scale circulation rotation. Linear stability analysis and sensitivity analysis confirm the role of the boundary-layer modes and the corner-rolls mode in the reorientation process.

A nonintrusive reduced order modelling approach using Proper Orthogonal Decomposition and locally adaptive sparse grids

Large-scale complex systems require high fidelity models to capture the dynamics of the system accurately. The complexity of these models, however, renders their use to be expensive for applications relying on repeated evaluations, such as control, optimization, and uncertainty quantification. Proper Orthogonal Decomposition (POD) is a powerful Reduced Order Modelling (ROM) technique developed to reduce the computational burden of high fidelity models. In cases where the model is inaccessible, POD can be used in a nonintrusive manner. The accuracy and efficiency of the nonintrusive reduced model are highly dependent on the sampling scheme, especially for high dimensional problems.To that end, we study integrating the locally adaptive sparse grids with the POD method to develop a novel nonintrusive POD-based reduced order model. In our proposed approach, the locally adaptive sparse grid is used to adaptively control the sampling scheme for the POD snapshots, and the hierarchical interpolant is used as a surrogate model for the POD coefficients. An approach to efficiently update the surpluses of the sparse grids with each POD snapshots update is also introduced. The robustness and efficiency of the locally adaptive algorithm are increased by introducing a greediness parameter, and a strategy to validate the reduced model after convergence. The proposed validation algorithm can also enrich the reduced model around regions of detected discrepancies. Three numerical test cases are presented to demonstrate the potential of the proposed POD-Adaptive algorithm. The first is a nuclear reactor point kinetics, the second is a general diffusion problem, and the last is a variation of the analytical Morris function. The results show that the developed algorithm reduced the number of model evaluations compared to the classical sparse grid approach. The built reduced models captured the dynamics of the reference systems with the desired tolerances. The non-intrusiveness and simplicity of the method provide great potential for a wide range of practical large scale applications.

POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell's equations

In this work we report on a reduced-order model (ROM) for the system of time-domain Maxwell's equations discretized by a discontinuous Galerkin (DG) method. We leverage previous results on proper orthogonal decomposition (POD) [1,2], in particular for the wave equation [3], to propose a POD-based ROM with an adaptive snapshot selection strategy where the snapshots are produced by a high order discontinuous Galerkin time-domain (DGTD) solver. The latter is formulated on an unstructured simplicial mesh, and combines a centered scheme for the definition of the numerical fluxes of the electric and magnetic fields at element interfaces with a second order leap-frog (LF2) time scheme for the time integration of the associated semi-discrete equations. The POD-based ROM is established by projecting (Galerkin projection) the global semi-discrete DG scheme onto a low-dimensional space generated by the POD basis vectors. Inspired from the approach followed in [2,3], we derive error bounds for the POD-based ROM that is adapted to our particular modeling and discretization settings. The adaptive snapshot selection algorithm exploits the results of this analysis to measure the control error. A snapshot choosing rule aiming at keeping the error estimate close to a target selection error tolerance is proposed, which is similar to the standard rules found in adaptive time-stepping ordinary differential equations (ODEs) solvers. An incremental singular value decomposition (ISVD) algorithm is used to update the SVD on-the-fly when a new snapshot is available. The purpose of this adaptive selection strategy is to save memory without storing snapshots, while producing a smaller error. Numerical experiments for the 2-D time-domain Maxwell's equations nicely illustrate the performance of the resulting POD-based ROM with adaptive snapshot selection.

POD-based reduced order model for flows induced by rigid bodies in forced rotation

This paper deals with the construction of reduced order models (ROMs) for the simulation of the interaction between a fluid and a rigid body with imposed rotation velocity. The approach is a follows. First, we derive a monolithic description of the fluid–structure interaction by extending the Navier–Stokes equations from the fluid domain to the solid domain (rotor) similarly to the fictitious-domain approach. Second, we build a ROM by a proper orthogonal decomposition (POD) of the resulting multi-phase flow. This method consists in (i) constructing an optimal albeit empirical spatial basis for a very small subspace of the solution space, and (ii) projecting the governing equations on this reduced basis. Third, we cope with the reconstruction of the high-dimensional velocity field needed to evaluate the imposed velocity constraint by a POD of the rigid body membership function. This provides a substantial computation time saving compared to existing approaches. Fourth, we use a novel approach to interpolate between available POD bases to build the proposed POD-ROM for a range of parameters values. The complete procedure is applied to a simple configuration and proves efficient in the reconstruction of the velocity in both the fluid domain and the solid domain, while substantially reducing the computational cost.

Data-Driven Time Parallelism via Forecasting

This work proposes a data-driven method for enabling the efficient, stable time-parallel numerical solution of systems of ordinary differential equations (ODEs). The method assumes that low-dimensional bases that accurately capture the time evolution of the dynamical-system state are available. The method adopts the parareal framework for time parallelism, which is defined by an initialization method, a coarse propagator that advances solutions on a coarse time grid, and a fine propagator that operates on an underlying fine time grid. Rather than employing usual approaches for initialization and coarse propagation, we propose novel data-driven techniques that leverage the available time-evolution bases. The coarse propagator is defined by a forecast (proposed in [K. Carlberg, J. Ray, and B. van Bloemen Waanders, Comput. Methods Appl. Mech. Engrg., 289 (2015), pp. 79--103]) applied locally within each coarse time interval, which comprises the following steps: (1) apply the fine propagator for a small number of time steps, (2) approximate the state over the entire coarse time interval using gappy proper orthogonal decomposition (POD) with the local time-evolution bases, and (3) select the approximation at the end of the time interval as the propagated state. We also propose both local-forecast and global-forecast initialization techniques. The method is particularly well suited for POD-based reduced-order models (ROMs), as the time-evolution bases can be extracted from readily available data, i.e., the right singular vectors arising during POD computation. In addition to performing analyses related to the method's accuracy, speedup, stability, and convergence, we also numerically demonstrate the method's performance. Here, numerical experiments on ROMs for a nonlinear convection--reaction problem demonstrate the method's ability to realize near-ideal speedups.

A POD-based reduced-order model for uncertainty analyses in shallow water flows

ABSTRACTThis work presents an intrusive reduced-order model (IROM) for uncertainty propagation analyses for flood flows. The 2D shallow water equations are reduced using Galerkin’s projection onto bases obtained from the snapshot-based proper orthogonal decomposition technique. To speed up the computations, the non-polynomial and nonlinear momentum and friction terms are judiciously approximated and the time accuracy issues are addressed using the principal interval decomposition technique. The performance of the IROM is investigated in some test cases. Also, this model is applied to the study of uncertainty propagation for a hypothetical flood in a real river, to derive a probabilistic flood map. The upstream discharge and the Manning roughness coefficient are considered as the uncertain parameters. For relatively small variations around the mean of the inputs, the comparisons of the statistical moments (mean and standard deviation) of the water depth show errors, between the reduced and full models, less than 0.72%. These simulations were completed at up to 50 times faster using the proposed reduced model.

A POD reduced-order model for wake steering control

Wake steering by active yawing of upstream wind turbines is a promising wind plant control technique. To enable the development of model-based wind plant control methods, there is a need for models that can marry the contrasting requirements of good fidelity and low computational cost. This paper presents a reduced-order model (ROM) obtained by directly compressing high-fidelity computational fluid dynamics (CFD) simulation data using the proper orthogonal decomposition (POD) method. At first, simulations of wake-interacting wind turbines are obtained for time-varying yaw settings using the lifting-line large-eddy simulation (LES) code SOWFA. Next, a ROM is synthesized from the CFD transient simulations, obtaining a discrete-time state-space model that captures the dominant dynamics of the underlying high-fidelity model with only a reduced number of states. The ROM is optionally augmented with a Kalman filter, which feeds back turbine power measurements from the plant to the model, enhancing its accuracy. Results obtained in realistic turbulent conditions show a good agreement between high-fidelity CFD solutions and the proposed POD-based ROM in terms of wake behavior and power output of waked turbines. Additionally, the ROM presents acceptable results when compared to wind tunnel experiments, including the capability of the model to partially correct for an intentionally built-in model mismatch.

Model order reduction for quasi-magnetostatic problems

PurposeTo reduce the computational scale for quasi-magnetostatic problems, model order reduction is a good option. Reduced-order modelling techniques based on proper orthogonal decomposition (POD) and centroidal Voronoi tessellation (CVT) have been used to solve many engineering problems. The purpose of this paper is to investigate the computational principle, accuracy and efficiency of the POD-based and the CVT-based reduced-order method when dealing with quasi-magnetostatic problems.Design/methodology/approachThe paper investigates computational features of the reduced-order method based on POD and CVT methods for quasi-magnetostatic problems. Firstly the construction method for the POD and the CVT reduced-order basis is introduced. Then, a reduced model is constructed using high-fidelity finite element solutions and a Galerkin projection. Finally, the transient quasi-magnetostatic problem of the TEAM 21a model is studied with the proposed reduced-order method.FindingsFor the TEAM 21a model, the numerical results show that both POD-based and CVT-based reduced-order approaches can greatly reduce the computational time compared with the full-order finite element method. And the results obtained from both reduced-order models are in good agreement with the results obtained from the full-order model, while the computational accuracy of the POD-based reduced-order model is a little higher than the CVT-based reduced-order model.Originality/valueThe CVT method is introduced to construct the reduced-order model for a quasi-magnetostatic problem. The computational accuracy and efficiency of the presented approaches are compared.

POD-Galerkin reduced-order model for viscoelastic turbulent channel flow

ABSTRACTIn this work, with elasticity governed by the Giesekus constitutive equation, a proper orthogonal decomposition (POD) reduced-order model of viscoelastic turbulent channel flow is established for the first time. The established reduced-order model is based on small sets of basis functions from the POD of the sampling data obtained by direct numerical simulation (DNS) for the studied flow. The POD reduced-order model is tested on cases with that are different from the samplings for viscoelastic turbulent channel flow. The results show that the errors for root-mean-square (rms) velocity fluctuations are significant at the top and bottom walls. It is found that each basis function plays an important role in describing the studied turbulence which makes it unmanageable to obtain accurate velocity field (including mean velocity and velocity fluctuations) through solving the reduced-order model. It is of necessity to take all the basis functions into consideration to depict the flows more accurately. However, the mean velocity obtained from the reduced-order model is of high precision, which states that the POD-based reduced-order model is a potential approach to obtain an accurate mean velocity field for viscoelastic turbulent flow, which has great significance in academic study as well as engineering. The calculation speed of the established reduced-order model is much faster than that of DNS, which indicates that the POD is a highly efficient way of obtaining the statistic characteristics, such as mean velocity in turbulent channel flow.

Targeted energy transfer in laminar vortex-induced vibration of a sprung cylinder with a nonlinear dissipative rotator

We computationally investigate the dynamics of a linearly-sprung circular cylinder immersed in an incompressible flow and undergoing transverse vortex-induced vibration (VIV), to which is attached a rotational nonlinear energy sink (NES) consisting of a mass that freely rotates at constant radius about the cylinder axis, and whose motion is restrained by a rotational linear viscous damper. The inertial coupling between the rotational motion of the attached mass and the rectilinear motion of the cylinder is “essentially nonlinear”, which, in conjunction with dissipation, allows for one-way, nearly irreversible targeted energy transfer (TET) from the oscillating cylinder to the nonlinear dissipative attachment. At the intermediate Reynolds number Re=100, the NES-equipped sprung cylinder undergoes repetitive cycles of slowly decaying oscillations punctuated by intervals of chaotic instabilities. During the slowly decaying portion of each cycle, the dynamics of the cylinder is regular and, for large enough values of the ratio ε of the NES mass to the total mass (i.e., NES mass plus cylinder mass), can lead to significant vortex street elongation with partial stabilization of the wake. As ε approaches zero, no such vortex elongation is observed and the wake patterns appear similar to that for a sprung cylinder with no NES. We apply proper orthogonal decomposition (POD) to the velocity flow field during a slowly decaying portion of the solution and show that, in situations where vortex elongation occurs, the NES, though not in direct contact with the surrounding fluid, has a drastic effect on the underlying flow structures, imparting significant and continuous passive redistribution of energy among POD modes. We construct a POD-based reduced-order model for the lift coefficient to characterize energy transactions between the fluid and the cylinder throughout the slowly decaying cycle. We introduce a quantitative signed measure of the work done by the fluid on the cylinder over one quasi-period of the slowly decaying response and find that vortex elongation is associated with a sign change of that measure, indicating that a reversal of the direction of energy transfer, with the cylinder “leaking energy back” to the flow, is responsible for partial stabilization and elongation of the wake. We interpret these findings in terms of the spatial structure and energy distribution of the POD modes, and relate them to the mechanism of transient resonance capture into a slow invariant manifold of the fluid–structure interaction dynamics.

Optimal flow control using a POD-based reduced-order model

ABSTRACTThe flow control constitutes a challenging research topic with numerous applications in aeronautics, aerodynamics, fluid mechanics, civil engineering, and so on. Due to the use of iterative algorithms, it is costly in both CPU time and memory requirements. In this paper, an optimal control strategy with reduced-order model built with proper orthogonal decomposition approach is proposed. The methodology is developed to control an anisothermal flow in a lid-driven cavity with one control parameter and then extended to at two control parameters. The control is realized in quasi-real time, which opens the way to promising practical applications.

Simulation of lead-acid battery using model order reduction

In this study, a reduced order model (ROM) based on proper orthogonal decomposition (POD) method has been applied to the coupled one-dimensional electrochemical transport equations in order to efficiently simulate lead-acid batteries, numerically. The governing equations, including conservation of charge in solid and liquid phases and conservation of species are solved simultaneously. The POD-based method for a lead-acid cell is used to simulate a discharge process to show the capability of the present method. The obtained results show that not only the POD-based ROM of lead-acid battery significantly decreases the computational time but also there is an excellent agreement with the results of previous computational fluid dynamics (CFD) models.

POD-based reduced order model of a thermoacoustic heat engine

In this study, numerical simulation of a thermo-acoustic heat engine is performed and a reduced order model of the system based on the Proper Orthogonal Decomposition method is obtained. The governing equations are solved with a finite volume-based solver. The reduced order model is constructed using the snapshots when the system reaches the limit cycle. A quadratic polynomial type ODE system with 11-modes is constructed from the Galerkin projection for the high fidelity CFD computations of the coupled thermo-acoustic system.

Mode decomposition methods for flows in high-contrast porous media. A global approach

We apply dynamic mode decomposition (DMD) and proper orthogonal decomposition (POD) methods to flows in highly-heterogeneous porous media to extract the dominant coherent structures and derive reduced-order models via Galerkin projection. Permeability fields with high contrast are considered to investigate the capability of these techniques to capture the main flow features and forecast the flow evolution within a certain accuracy. A DMD-based approach shows a better predictive capability due to its ability to accurately extract the information relevant to long-time dynamics, in particular, the slowly-decaying eigenmodes corresponding to largest eigenvalues. Our study enables a better understanding of the strengths and weaknesses of the applicability of these techniques for flows in high-contrast porous media. Furthermore, we discuss the robustness of DMD- and POD-based reduced-order models with respect to variations in initial conditions, permeability fields, and forcing terms.

Investigation of computational and accuracy issues in POD-based reduced order modeling of dynamic structural systems

In this paper, we investigate the performance of reduced order modeling of dynamic structural systems based on the proper orthogonal decomposition (POD) technique. Singular value decomposition of the so-called snapshot matrix is adopted to generate the reduced space, onto which the system equations of motion are projected to speedup the computations.To get insights into the achievable speedup and the capability of POD to provide an input-independent reduced model, we consider the 39-story Pirelli tower in Milan-Italy. First, we assume that a shear model of the building is excited by the May 18-1940, Mw 7.1, El Centro earthquake, and generate the data ensemble necessary to build the reduced model. Second, we assess the local and global accuracies of the same reduced model in tracking the dynamics of the building, if excited by the May 6-1976, Mw 6.4, Friuli earthquake and by the January 17-1995, Mw 6.8, Kobe earthquake, which differ from the El Centro one in terms of excited vibration frequencies. We show that POD allows to attain a speedup approaching 250, when the reduced order model is asked to feature a high accuracy; moreover, POD tends to outperform a standard modal analysis at increasing number of modes retained in the model.

Improved extrapolation of steady turbulent aerodynamics using a non-linear POD-based reduced order model

AbstractA reduced-order modelling (ROM) approach for predicting steady, turbulent aerodynamic flows based on computational fluid dynamics (CFD) and proper orthogonal decomposition (POD) is presented. Model-order reduction is achieved by parameter space sampling, solution space representation via POD and restriction of a CFD solver to the POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimisation problem. The method will be referred to as LSQ-ROM method. Two approaches of extracting POD basis information from CFD snapshot data are discussed: POD of the full state vector (global POD) and POD of each of the partial states separately (variable-by-variable POD). The method at hand is demonstrated for a 2D aerofoil (NACA 64A010) as well as for a complete industrial aircraft configuration (NASA Common Research Model) in the transonic flow regime by computing ROMs of the compressible Reynolds-averaged Navier-Stokes equations, pursuing both the global and the variable-by-variable POD approach. The LSQ-ROM approach is tried for extrapolatory flow conditions. Results are juxtaposed with those obtained by POD-based extrapolation using Kriging and the radial basis functions spline method. As a reference, the full-order CFD solutions are considered. For the industrial aircraft configuration, the cost of computing the reduced-order solution is shown to be two orders of magnitude lower than that of computing the reference CFD solution.