POD Reduced-order Model Research Articles

Rapid Prediction of the In Situ Pyrolysis Performance of Tar-Rich Coal Using the POD Method

In this paper, a POD reduced-order interpolation model for solving the in situ pyrolysis process of tar-rich coal is employed to predict the flow and heat transfer performance in the porous media region so as to save computational resources and realize fast calculations. Numerical simulation using the finite volume method (FVM) is firstly used to obtain sample data, based on the samples through the primary function and spectral coefficients of the solutions. The physical field information and parameter distribution under different conditions of inlet temperature, inlet velocity and permeability are predicted. The results are compared with those of FVM to verify the accuracy of the calculated results. The relative mean deviation (RME) of the results of the POD prediction of each parameter for each working condition was synthesized to be no more than 5%. The performance of in situ pyrolysis of tar-rich coal is then investigated, and the oil and gas production are predicted. As the inlet velocity increases from 0.3 m/s to 0.9 m/s, the fraction of high-quality oil and gas production reaches 0.47 and then decreases to 0.38. Increasing the inlet temperature and permeability has a negative effect on the fraction of high-quality hydrocarbon production, after which the quality fraction of high-quality oil and gas dropped sharply to about 0.22. Porosity has a positive impact on the oil and gas production. When the porosity reaches 0.3, the quality fraction of high-quality oil and gas can reach 0.27.

Error analysis of a residual-based stabilization-motivated POD-ROM for incompressible flows

This article presents error bounds for a velocity–pressure segregated POD reduced order model discretization of the Navier–Stokes equations. The stability is proven in L∞(L2) and energy norms for velocity, with bounds that do not depend on the viscosity, while for pressure it is proven in a semi-norm of the same asymptotic order as the L2 norm with respect to the mesh size. The proposed estimates are calculated for the two flow problems, the flow past a cylinder and the lid-driven cavity flow. Their quality is then assessed in terms of the predicted logarithmic slope with respect to the velocity POD contribution ratio. We show that the proposed error estimates allow a good approximation of the real errors slopes and thus a good prediction of their rate of convergence.

Assessing the wall energy efficiency design under climate change using POD reduced order model

Within the environmental context, numerical modeling is a promising approach to assess the energy efficiency of building. Resilient buildings need to be designed, capable of adapting to future extreme heat. Simulations are required assuming a one-dimensional heat transfer problem through walls and a simulation horizon of several years (nearly 30). The computational cost associated with such modeling is quite significant and model reduction methods are worth investigating. The objective is to propose a reliable reduced-order model for such long-term simulations. For this, an alternative model reduction approach is investigated, assuming a known Proper Orthogonal Decomposition reduced basis for time, and not for space as usually. The model enables computing parametric solutions using basis interpolation on the tangent space of the Grassmann manifold. Three study cases are considered to verify the efficiency of the reduced-order model. Results highlight that the model has a satisfying accuracy of 10-3 compared to reference solutions. The last case study focuses on the wall energy efficiency design under climate change according to a four-dimensional parameter space. The latter is composed of the load material emissivity, heat capacity, thermal conductivity and thickness insulation layer. Simulations are carried over 30 years considering climate change. The solution minimizing the wall work rate is determined with a computational ratio of 0.1% compared to standard approaches.

A POD reduced-order model based on spectral Galerkin method for solving the space-fractional Gray–Scott model with error estimate

This paper deals with developing a fast and robust numerical formulation to simulate a system of fractional PDEs. At the first stage, the time variable is approximated by a finite difference method with first-order accuracy. At the second stage, the spectral Galerkin method based upon the fractional Jacobi polynomials is employed to discretize the spatial variables. We apply a reduced-order method based upon the proper orthogonal decomposition technique to decrease the utilized computational time. The unconditional stability property and the order of convergence of the new technique are analyzed in detail. The proposed numerical technique is well known as the reduced-order spectral Galerkin scheme. Furthermore, by employing the Newton–Raphson method and semi-implicit schemes, the proposed method can be used for solving linear and nonlinear ODEs and PDEs. Finally, some examples are provided to confirm the theoretical results.

Study on a POD reduced-order model for steady-state flows in fractured porous media

Abstract Traditional modeling methods of the reduced-order model can not be directly applied to fracture-dominated flow in porous media, violating the mass-conservation nature. In this paper, a global reduced-order model for the steady-state flow in fractured porous media based on the embedded discrete fracture model is established for the first time by using the proper orthogonal decomposition combined with Galerkin projection method. Differing from the typical modeling method of the reduced-order model commonly used in the literature, the discrete matrix equations instead of the primary governing equations are projected onto the low-dimensional space formed by the basis functions. The proposed reduced-order model can successfully handle the coupled terms between the matrix and fracture media. The accuracy and robustness of the proposed model are verified for three complex fracture cases with different boundary conditions. The results show that the maximum relative deviation in these three cases is 8.07 × 10−6%, which can be negligible in practical applications. Additionally, the constructed global reduced-order model is more effective than the finite volume method (at least speed-up 10 times) among these test cases.

A novel POD reduced-order model based on EDFM for steady-state and transient heat transfer in fractured geothermal reservoir

A new proper orthogonal decomposition reduced-order model (POD-ROM) for fluid flow and heat transfer processes in a fractured geothermal reservoir is proposed. First, the primary governing equations are established based on the embedded discrete fracture model. Second, the POD-ROM is developed by projecting the discrete matrix equations instead of the primary governing equations onto the low-dimensional space spanned by the pressure and temperature basis functions. Compared with the typical POD modeling method, the new model reduction technique can easily deal with mass and energy transfer between the matrix and fractures, thus ensuring the mass and energy conservation of the whole system in the projection process. The new model does not require complex mathematical deduction and is easy to implement in practical applications. The accuracy and robustness of the model were carefully studied for three complex heat transfer cases. Calculation results indicated that the computational efficiency can be improved by at least 10–15 times while high reconstruction and prediction accuracy is maintained.

BFC-POD-ROM Aided Fast Thermal Scheme Determination for China’s Secondary Dong-Lin Crude Pipeline with Oils Batching Transportation

Since the transportation task of China’s Secondary Dong-Lin crude pipeline has been changed from Shengli oil to both Shengli and Oman oils, its transportation scheme had to be changed to “batch transportation”. To determine the details of batch transportation, large amounts of simulations should be performed, but massive simulation times could be costly (they can take hundreds of days with 10 computers) using the finite volume method (FVM). To reduce the intolerable time consumption, the present paper adopts a “body-fitted coordinate-based proper orthogonal decomposition reduced-order model” (BFC-POD-ROM) to obtain faster simulations. Compared with the FVM, the adopted method reduces the time cost of thermal simulations to 2.2 days from 264 days. Subsequently, the details of batch transportation are determined based on these simulations. The Dong-Lin crude oil pipeline has been safely operating for more than two years using the determined scheme. It is found that the field data are well predicted by the POD reduced-order model with an acceptable error in crude oil engineering.

2D Burgers equation with large Reynolds number using POD/DEIM and calibration

SummaryModel order reduction of the two‐dimensional Burgers equation is investigated. The mathematical formulation of POD/discrete empirical interpolation method (DEIM)‐reduced order model (ROM) is derived based on the Galerkin projection and DEIM from the existing high fidelity‐implicit finite‐difference full model. For validation, we numerically compared the POD ROM, POD/DEIM, and the full model in two cases of Re = 100 and Re = 1000, respectively. We found that the POD/DEIM ROM leads to a speed‐up of CPU time by a factor of O(10). The computational stability of POD/DEIM ROM is maintained by means of a careful selection of POD modes and the DEIM interpolation points. The solution of POD/DEIM in the case of Re = 1000 has an accuracy with error O(10−3) versus O(10−4) in the case of Re = 100 when compared with the high fidelity model. For this turbulent flow, a closure model consisting of a Tikhonov regularization is carried out in order to recover the missing information and is developed to account for the small‐scale dissipation effect of the truncated POD modes. It is shown that the computational results of this calibrated ROM exhibit considerable agreement with the high fidelity model, which implies the efficiency of the closure model used. Copyright © 2016 John Wiley & Sons, Ltd.

Assessment of probability density function based on POD reduced-order model for ensemble-based data assimilation

An integrated method of a proper orthogonal decomposition based reduced-order model (ROM) and data assimilation is proposed for the real-time prediction of an unsteady flow field. In this paper, a particle filter (PF) and an ensemble Kalman filter (EnKF) are compared for data assimilation and the difference in the predicted flow fields is evaluated focusing on the probability density function (PDF) of the model variables. The proposed method is demonstrated using identical twin experiments of an unsteady flow field around a circular cylinder at the Reynolds number of 1000. The PF and EnKF are employed to estimate temporal coefficients of the ROM based on the observed velocity components in the wake of the circular cylinder. The prediction accuracy of ROM-PF is significantly better than that of ROM-EnKF due to the flexibility of PF for representing a PDF compared to EnKF. Furthermore, the proposed method reproduces the unsteady flow field several orders faster than the reference numerical simulation based on the Navier–Stokes equations.

Maximum likelihood estimation of missing data applied to flow reconstruction around NACA profiles

In this paper, we investigate the maximum likelihood estimation for missing data in fluid flows series. The maximum likelihood estimation is provided with the expectation-maximization (EM) algorithm applied to the linear and quadratic proper orthogonal decomposition POD-Galerkin reduced-order models (ROMs) for various sub-samplings of large data sets. The flows around a NACA0012 profile at Reynolds numbers of 103 and angle of incidence of and a NACA0015 profile at Reynolds numbers of 105 and angle of incidence of are first investigated using time-resolved particle image velocimetry measurements and sub-sampled according to different ratios of missing data. The EM algorithm is then applied to the POD ROMs constructed from the sub-sampled data sets. The results show that, depending on the sub-sampling used, the EM algorithm is robust with respect to the Reynolds number and can reproduce the velocity fields and the main structures of the missing flow fields for 50% and 75% of missing data.

POD reduced-order model for steady natural convection based on a body-fitted coordinate

In this article, a POD reduced-order model for steady-state natural convection based on a body-fitted coordinate system is established. The velocity basis functions and temperature basis functions are generated, respectively. Based on the basis functions, the governing equations for velocity spectrum coefficient and temperature spectrum coefficient are deduced, which are coupled pluralistic quadratic nonlinear equations and solved iteratively by the Newton–Raphson method. Compared with the existing natural convection POD reduced-order models, the major advantage of the proposed POD model is its capability to calculate natural convections in different-shape domains having identical geometry characters. Two typical examples are given to show the model implementation procedure as well as to illustrate its good performance in terms of accuracy and robustness. It is found that even though the geometries and the physical conditions of the test cases differ greatly from those of the sampling cases, the reduced-order model can acquire accurate results efficiently.

A minimum residual projection to build coupled velocity–pressure POD–ROM for incompressible Navier–Stokes equations

The pressure term which appears in the ROM (reduced order model) associated to the incompressible Navier–Stokes equations, in particular for the shear flows, plays an important role on the velocity. The aim of this paper is to propose a Proper Orthogonal Decomposition based reduced order model (POD–ROM) to obtain both the velocity and pressure fields for incompressible flows. Two PODs are performed, one for the velocity and the other for the pressure. Contrary to existing projection methods available in the literature, the temporal velocity and pressure coefficients are sought by minimizing the residual of the momentum equation only, without the need of a Poisson equation. For the numerical test cases considered in this paper, the proposed minimum residual projection enables to obtain accurately the pressure field, and in turn to slightly improve the velocity one. The method is tested on two fluid flows: a transient mixed-convection flow and a periodic flow around a circular cylinder. In this last case, the drag, lift and pressure coefficients, as well as the Strouhal number are properly recovered compared to those of the full model.

Regularization method for calibrated POD reduced-order models

In this work we present a regularization method to improve the accuracy of reduced-order models based on Proper Orthogonal Decomposition. The bench mark configuration retained corresponds to a case of relatively simple dynamics: a two-dimensional flow around a cylinder for a Reynolds number of 200. Finally, we show for this flow configuration that this procedure is efficient in term of reduction of errors.

Fast Calculation of the Soil Temperature Field around a Buried Oil Pipeline using a Body-Fitted Coordinates-Based POD-Galerkin Reduced-Order Model

In order to calculate the soil temperature field around a buried oil pipeline of different geometric parameters and boundary conditions fast and accurately, the POD-Galerkin reduced-order model is established based on body-fitted coordinates (BFC). The most exciting feature of the established BFC-based reduced-order model is that it is independent of not only boundary conditions but also geometrical shape; that is, the reduced-order model can accurately calculate the specific physical problems which have different boundary conditions or different geometrical shapes from those of the sample cases. To some extent, this study expands the application scope of the POD reduced-order model.

A nonlinear POD reduced order model for limit cycle oscillation prediction

As the amplitude of the unsteady flow oscillation is large or large changes occur in the mean background flow such as limit cycle oscillation, the traditional proper orthogonal decomposition reduced order model based on linearized time or frequency domain small disturbance solvers can not capture the main nonlinear features. A new nonlinear reduced order model based on the dynamically nonlinear flow equation was investigated. The nonlinear second order snapshot equation in the time domain for proper orthogonal decomposition basis construction was obtained from the Taylor series expansion of the flow solver. The NLR 7301 airfoil configuration and Goland+ wing/store aeroelastic model were used to validate the capability and efficiency of the new nonlinear reduced order model. The simulation results indicate that the proposed new reduced order model can capture the limit cycle oscillation of aeroelastic system very well, while the traditional proper orthogonal decomposition reduced order model will lose effectiveness.