Proper Orthogonal Decomposition Reduced-order Model Research Articles

A new proper orthogonal decomposition method with second difference quotients for the wave equation

Recently, researchers have investigated the relationship between proper orthogonal decomposition (POD), difference quotients (DQs), and pointwise in time error bounds for POD reduced order models of partial differential equations. In a recent work (Eskew and Singler, Adv. Comput. Math., 49, 2023, no. 2, Paper No. 13), a new approach to POD with DQs was developed that is more computationally efficient than the standard DQ POD approach and it also retains the guaranteed pointwise in time error bounds of the standard method. In this work, we extend this new DQ POD approach to the case of second difference quotients (DDQs). Specifically, a new POD method utilizing DDQs and only one snapshot and one DQ is developed and used to prove ROM error bounds for the damped wave equation. This new approach eliminates data redundancy in the standard DDQ POD approach that uses all of the snapshots, DQs, and DDQs. We show that this new DDQ approach also has pointwise in time data error bounds similar to DQ POD and use it to prove pointwise and energy ROM error bounds. We provide numerical results for the POD ROM errors to demonstrate the theoretical results. We also explore an application of POD to simulate ROMs past the training interval for collecting the snapshot data for the standard POD approach and the DDQ POD method.

Optimization of air distribution and coal blending in pulverized coal boilers for high-temperature corrosion prevention based on POD reduced-order modeling

High-temperature corrosion in coal-fired boilers poses a significant threat to safe operation. However, there is currently a lack of effective online monitoring and optimization methods for high-temperature corrosion. Therefore, this study proposes a novel approach to rapidly predict the distribution of chemical species and evaluate high-temperature corrosion degrees, along with optimizing operating conditions to prevent high-temperature corrosion. Firstly, a total of 564 Computational Fluid Dynamics (CFD) simulations are performed on a 330 MW tangentially fired boiler, covering various operating conditions, including coal blending, air distribution, boiler load, etc., to obtain a database of O2, CO, and H2S distributions within the boiler. Then a method based on Proper Orthogonal Decomposition (POD) and Support Vector Regression (SVR) is used to process the database to realize real-time prediction of boiler chemical species distribution, which is next utilized as inputs of a high-temperature corrosion evaluation to obtain in-situ corrosion degree distribution. Finally, Particle Swarm Optimization (PSO) is used to optimize coal blending and air distribution schemes, effectively reducing severe corrosion ratio from 36.34 % to 10.04 % in a typical case by improving the atmosphere near the water walls. This study thus provides a new perspective for online boiler diagnostics and digital twin construction, particularly by achieving online monitoring of high-temperature corrosion and optimizing operating conditions to prevent corrosion.

POD method for solving elastodynamics problems of functionally gradient materials based on radial integration boundary element method

Functionally graded materials (FGMs) are widely utilized due to their excellent mechanical properties, prompting a need to explore the dynamic response of FGMs. To accurately and efficiently analyze elastodynamics problems, the radial integration boundary element method (RIBEM) is coupled with the proper orthogonal decomposition (POD) method in this paper. The POD method can transform a high-dimensional model into a low-dimensional system with high precision. Using the basic solution (Kelvin solution), the governing partial differential equation for the linear elastic dynamics problem of FGMs is transformed into a boundary-domain integral equation. The radial integration method is employed to convert the domain integral arising from material inhomogeneity and inertia terms into an equivalent boundary integral. This establishes a solution for the elastic dynamics problem of FGMs without the need for internal grid RIBEM. The Newmark time integration scheme is applied to solve the second-order ordinary differential equations. The displacement field obtained by RIBEM is utilized to construct the snapshots matrix, from which the POD reduced-order model is established. Numerical examples validate that the POD method significantly reduces the model's order, enhancing computational efficiency while also demonstrating the high accuracy of the POD method.

Air distribution and coal blending optimization to reduce slagging on coal-fired boiler water wall based on POD reduced order modeling for CFD

Slagging is a common phenomenon interfering with the safe and efficient operation of coal-fired boilers. Due to the lack of monitoring methods for species and temperature distribution near the water wall, which is critical for slagging formation, it is a tricky problem to detect and reduce the slagging in real time. A method for slagging distribution prediction and anti-slagging optimization of a 330 MW tangentially coal-fired boiler is developed in this paper. Proper orthogonal decomposition (POD) reduced order modeling and support vector machine (SVM) are used to predict the distribution of temperature and O2 near the water wall by taking 15 variables as input parameters (load, coal types, primary air rate, secondary air rate, excess air coefficient, and secondary air velocities). Next, fuzzy comprehensive evaluation (FCE) is employed to quantify the slagging occurrence and distribution on the water wall. Furthermore, the real time slagging prediction is realized with the combination of POD, SVM, and FCE. Finally, the conventional genetic algorithm (CGA) and simulated annealing genetic algorithm (SAGA) are adopted to optimize the air distribution schemes and coal blending schemes to reduce the slagging ratio, and their performances are compared. The results indicate that the predictions of temperature and O2 with POD show a good agreement with CFD simulations. In the optimization process, whether air distribution is optimized alone or both coal blending and air distribution are optimized simultaneously, the performance of SAGA is better than that of CGA. After optimizing the air distribution with SAGA, the slagging ratio of a typical operating condition with a load of 330 MW is reduced from 14.21% to 7.16%. The slagging ratio is further reduced to 5.36% after optimizing air distribution and coal blending schemes simultaneously. The proposed slagging prediction and optimization for anti-slagging opens a new perspective to alleviate slagging of water walls effectively.

A Fast Reduced-Order Model for Radial Integration Boundary Element Method Based on Proper Orthogonal Decomposition in the Non-Uniform Coupled Thermoelastic Problems

To efficiently address the challenge of thermoelastic coupling in functionally graded materials, we propose an approach that combines the radial integral boundary element method (RIBEM) with proper orthogonal decomposition (POD). This integration establishes a swift reduced-order model to transform the high-dimensional system of equations into a more manageable, low-dimensional counterpart. The implementation of this reduced-order model offers the potential for rapid numerical simulations of functionally graded materials (FGMs) under thermal shock loading. Initially, the RIBEM is utilized to resolve the thermal coupling issue within the FGMs. From these solutions, a snapshot matrix is constructed, capturing the solved temperature and displacement fields. Subsequently, the POD modes are established and a POD reduced-order model is constructed for the boundary element format of the thermally coupled problem. Finally, a system of low-order discrete differential equations is solved. Numerical experiments demonstrate that the results obtained from the reduced-order model closely align with those of the full-order model, even when considering variations in structural parameters or impact loads. Thus, the introduction of the reduced-order model not only guarantees solution accuracy but also significantly enhances computational efficiency.

A data-driven online prediction method for surface-deformed liquid level in vessels under ocean conditions

Measurement of liquid level in vessels under ocean conditions, such as in the marine nuclear reactor pressure vessel, pressurizer and so on, suffers from the problem of strong free surface distortion. In order to solve such problem, an online prediction method based on reduced order model (ROM) and compressive sensing theory is proposed in this paper. The whole field online prediction of complex fluid flow characteristics can be realized by combining the observation data from the known sparse sensor with the ROM. In this paper, the three-dimensional flow field of sloshing flow in a cavity is numerically simulated to establish abundant numerical database. Taking the liquid level of the flow field as the prediction object, the ROM is established by using the proper orthogonal decomposition (POD) method to predict the liquid level. The results show that POD ROM can accurately predict the unknown case, the smaller the rotating speed is, the better the prediction result is, and increasing the number of measuring points and the number of basis functions can reduce the prediction error to a certain extent. The reduced model constructed by using QR decomposition to obtain measuring points can effectively reduce the error of prediction results.

Fast Prediction of the Temperature Field Surrounding a Hot Oil Pipe Using the POD-BP Model

The heat transfer assessment of a buried hot oil pipe is essential for the economical and safe transportation of the pipeline, where the basis is to determine the temperature field surrounding the pipe quickly. This work proposes a novel method to efficiently predict the temperature field surrounding a hot oil pipe, which combines the proper orthogonal decomposition (POD) method and the backpropagation (BP) neural network, named the POD-BP model. Specifically, the BP neural network is used to establish the mapping relationship between spectrum coefficients and the preset parameters of the sample. Compared with the classical POD reduced-order model, the POD-BP model avoids solving the system of reduced-order governing equations with spectrum coefficients as variables, thus improving the prediction speed. Another advantage is that it is easy to implement and does not require tremendous mathematical derivation of reduced-order governing equations. The POD-BP model is then used to predict the temperature field surrounding the hot oil pipe, and the sample matrix is obtained from the numerical results using the finite volume method (FVM). In validation cases, both steady and unsteady states are investigated, and multiple boundary conditions, thermal properties, and even geometry parameters (different buried depths and pipe diameters) are tested. The mean errors of steady and unsteady cases are 0.845~3.052% and 0.133~1.439%, respectively. Appealingly, almost no time, around 0.008 s, is consumed in predicting unsteady situations using the proposed POD-BP model, while the FVM requires a computational time of 70 s.

A reduced-order modeling for thermo-mechanical coupling analyses by using radial integration boundary element method

In order to efficiently solve thermo-mechanical coupling problems, this work combines the radial integration boundary element method (RIBEM) and the proper orthogonal decomposition (POD) to construct a fast reduced-order model. This model transforms high-dimensional systems into low-dimensional ones, enabling faster computation of thermo-mechanical coupling problems under thermal loads. Firstly, RIBEM is employed to solve coupled thermoelastic dynamic problems under thermal shock loads. Domain integrals are converted to boundary integrals using the radial integration method, establishing a purely boundary element algorithm without internal cells. Subsequently, finite difference techniques are applied to calculate the discretized algebraic equations, and the obtained field variables are used as a snapshot matrix to establish POD modes and construct the POD reduced-order model. Numerical examples demonstrate that the results of the reduced-order model are essentially consistent with the full-order model across various physical parameters and boundary conditions. Therefore, this reduced-order model significantly improves computational efficiency while maintaining high accuracy, especially when solving large-scale problems.

A data-driven reduced-order model for rotor optimization

Abstract. For rotor design applications, such as wind turbine rotors or urban air mobility (UAM) rotorcraft and flying-car design, there is a significant challenge in quickly and accurately modeling rotors operating in complex, turbulent flow fields. One potential path for deriving reasonably accurate but low-cost rotor performance predictions is available through the application of data-driven surrogate modeling. In this study, an initial investigation is undertaken to apply a proper orthogonal decomposition (POD)-based reduced-order model (ROM) for predicting rotor distributed loads. The POD ROM was derived based on computational fluid dynamics (CFD) results and utilized to produce distributed-pressure predictions on rotor blades subjected to topology change due to variations in the twist and taper ratio. Rotor twist, θ, was varied between 0, 10, 20, and 30∘, while the taper ratio, λ, was varied as 1.0, 0.9, 0.8, and 0.7. For a demonstration of the approach, all rotors consisted of a single blade. The POD ROM was validated for three operation cases: a high-pitch or a high-thrust rotor in hover, a low-pitch or a low-thrust rotor in hover, and a rotor in forward flight at a low speed resembling wind turbine operation with wind shear. Results showed that reasonably accurate distributed-load predictions could be achieved and the resulting surrogate model can predict loads at a minimal computational cost. The computational cost for the hovering blade surface pressure prediction was reduced from 12 h on 440 cores required for CFD to a fraction of a second on a single core required for POD. For rotors in forward flight, cost was reduced from 20 h on 440 cores to less than a second on a single core. The POD ROM was used to carry out a design optimization of the rotor such that the figure of merit was maximized for hovering-rotor cases and the lift-to-drag effective ratio was maximized in forward flight.

Research on digital twin modeling method of transformer temperature field based on POD

For the calculation of transient fluid–solid coupling temperature field of transformer, the traditional Galerkin finite element full order model has the defect that the calculation time is up to hours-level, which is difficult to meet the timeliness requirements of the digital twin model. Firstly, the implementation architecture of transformer digital twin based on IoT is established, and the necessity of multiphysical field simulation and model reduction technology for building the digital twin model is introduced. Then, based on the method of transformer digital twin implementation,it is proposed to combine the proper orthogonal decomposition (POD) with the finite element (FE) to establish the Galerkin FE full order model for the governing equation of the transient temperature field, the reduced order mode composed of the first several bases is intercepted to build the reduced order model of transient fluid–solid coupling temperature field. Finally, the POD reduced order model is applied to calculate the temperature rise of a transformer. According to the actual transformer temperature rise test, the temperature values of different measuring points are selected to compare the calculation error and time of the reduced order model and the full order model. The results show that the error between the third order model and full order model meets the criterion of POD truncation error, and the calculation time is shortened by 192 times, which can meet the seconds-level calculation requirements.

Fast Vibration Reduction Optimization Approach for Complex Thin-Walled Shells Accelerated by Global Proper Orthogonal Decomposition Reduced-Order Model

A fast vibration reduction optimization approach accelerated by the global proper orthogonal decomposition (POD) reduced-order model (ROM) is proposed, aiming at increasing the efficiency of frequency response analysis and vibration reduction optimization of complex thin-walled shells. At the offline stage, the global POD ROM is adaptively updated using the sample configurations generated by the CV (cross validation)–Voronoi sequence sampling method. In comparison to the traditional direct sampling method, the proposed approach achieves higher global prediction accuracy. At the online stage, the fast vibration reduction optimization is performed by combining the surrogate-based efficient global optimization (EGO) method and the proposed ROM. Two representative examples are carried out to verify the effectiveness and efficiency of the proposed approach, including examples of an aerospace S-shaped curved stiffened shell and a Payload Attach Fitting. The results indicate that the proposed approach achieves high prediction accuracy and efficiency through the verification by FOM and obtains better optimization ability over the direct optimization method based on FOM.

Spectral Proper Orthogonal Decomposition Reduced-Order Model for Analysis of Aerothermoelasticity

Analysis and simulation of fluid–thermal–structure coupling play an essential role in hypersonic flight vehicle design. In this study, the spectral proper orthogonal decomposition (SPOD) method is employed to establish the reduced-order model (ROM) for a hypersonic panel with a thermal protection system. First, the subsystem of the aerothermoelastic (AETI) system is modeled. Specifically, the aerodynamic heating is calculated with Eckert’s reference enthalpy method. The heat transfer is carried out with the finite difference method, and the aeroelastic equation is built by integrating the von Kármán plate theory and the third-order piston theory. Additionally, the two-way coupling between aerothermal and aeroelastic subsystems is used to construct the AETI system. Second, the optimal snapshots with the resultant local SPOD modes (SPOMs) are evaluated in terms of accuracy. Third, the global SPOMs for variable flight parameters are promoted to develop the robustness of the proposed SPOD/ROM. The findings indicate that taking the transient chaotic response as snapshots can produce global SPOMs. SPOD/ROM for the present AETI system improves the efficiency by 10–50 times compared to the classic Galerkin method. Moreover, the global SPOMs to variable Mach number and flight altitude are essentially the natural modes of the AETI panel, and thus are dependent on the temperature of the fluttering panel, which is our new finding in the present study. Overall, the current research will provide methods for accurately and efficiently predicting thermal loads, flutter boundary, and dynamic response for hypersonic flight vehicles.

Augmented reduced order models for turbulence

The authors introduce an augmented-basis method (ABM) to stabilize reduced-order models (ROMs) of turbulent incompressible flows. The method begins with standard basis functions derived from proper orthogonal decomposition (POD) of snapshot sets taken from a full-order model. These are then augmented with divergence-free projections of a subset of the nonlinear interaction terms that constitute a significant fraction of the time-derivative of the solution. The augmenting bases, which are rich in localized high wavenumber content, are better able to dissipate turbulent kinetic energy than the standard POD bases. Several examples illustrate that the ABM significantly out-performs L2-, H1- and Leray-stabilized POD ROM approaches. The ABM yields accuracy that is comparable to constraint-based stabilization approaches yet is suitable for parametric model-order reduction in which one uses the ROM to evaluate quantities of interests at parameter values that differ from those used to generate the full-order model snapshots. Several numerical experiments point to the importance of localized high wavenumber content in the generation of stable, accurate, and efficient ROMs for turbulent flows.

On the stability of POD basis interpolation on Grassmann manifolds for parametric model order reduction

Proper Orthogonal Decomposition (POD) basis interpolation on Grassmann manifolds has been successfully applied to problems of parametric model order reduction (pMOR). In this work we address the necessary stability conditions for the interpolation, all defined from strong mathematical background. A first condition concerns the domain of definition of the logarithm map. Second, we show how the stability of interpolation can be lost if certain geometrical requirements are not satisfied by making a concrete elucidation of the local character of linearization. To this effect, we draw special attention to the Grassmannian exponential map and the optimal injectivity condition of this map, related to the cut–locus of Grassmann manifolds. From this, an explicit stability condition is established and can be directly used to determine the loss of injectivity in practical pMOR applications. A third stability condition is formulated when increasing the number p of POD modes, deduced from the principal angles of subspaces of different dimensions p. Definition of this condition leads to an understanding of the non-monotonic oscillatory behavior of the Reduced Order Model (ROM) error-norm with respect to the number of POD modes, and on the contrary, the well-behaved monotonic decrease of the error-norm in the two numerical examples presented herein. We have chosen to perform pMOR in hyperelastic structures using a non-intrusive approach for inserting the interpolated spatial POD ROM basis in a commercial FEM code. The accuracy is assessed by a posteriori error norms defined using the ROM FEM solution and its high-fidelity counterpart simulation. Numerical studies successfully ascertained and highlighted the implication of stability conditions which are general and can be applied to a variety of other linear or nonlinear problems involving parametrized ROMs generation based on POD basis interpolation on Grassmann manifolds.

A novel model reduction technique for mistuned blisks based on proper orthogonal decomposition in frequency domain

Aiming at the problem of dynamic model reduction of mistuned blisks, this study proposes a model reduction method based on the proper orthogonal decomposition (POD) in the frequency domain and establishes the reduced-order model (ROM). The mistuning amount of blisk is characterized as generalized mistuned load, and then a mistuned load interpolation algorithm is introduced to solve the forced vibration response. POD ROM is used to analyze the modal characteristics and vibration response characteristics of a mistuned blisk with 20 sectors, and the results are compared with the full-order model (FOM) and the subset nominal mode (SNM) reduced model. The results show that the model reduction method proposed in this paper can improve the calculation efficiency while ensuring calculation accuracy. In the aspect of modal characteristics of the mistuned blisk, its calculation accuracy is better than the SNM model reduction method, the calculation error of natural frequency is less than 0.05%, and the spatial distribution characteristics of mistuned modes are closer to the results of FOM. In terms of response characteristics, the forced response accuracy is affected by the integration time step and relaxation factor, and the steady-state amplitude error in the time domain can be controlled below 5%.

Proper orthogonal decomposition for reduced order dynamic modeling of vapor compression systems

A computationally efficient but accurate dynamic modeling approach for vapor compression systems is important for many applications. Nonlinear model order reduction techniques which generate reduced order models based on high fidelity vapor compression cycle (VCC) models are attractive for the purposes. In this paper, a number of technical challenges of applying model order reduction methods to VCCs are described and corresponding solution approaches are presented. It starts with a reformulation of a standard finite volume heat exchanger model for matching the baseline model reduction structure. Reduced order models for evaporator and condenser are constructed from numerical snapshots of the high fidelity models using Proper Orthogonal Decomposition (POD). Methodologies for system stability and numerical efficiency of POD reduced order models are described. The reduced order heat exchanger models are then coupled with quasi-static models of other components to form a reduced order cycle model. Transient simulations were conducted over a wide range of operating conditions and results were compared with the full order model as well as measurements. The validation results indicate that the reduced order model can execute much faster than a high-fidelity finite volume model with negligible prediction errors.

Fast calculation of latent heat storage process in the direct steam generation solar thermal power system using a POD reduced-order model

A Proper Orthogonal Decomposition (POD) reduced-order model for the latent heat storage process in a direct steam generation solar thermal power (DSG-STP) system is established based on the numerical simulation combined with the Lee model and enthalpy-porosity approach. Then, the computational accuracy and speed of the POD reduced-order model are examined by two unsteady-state cases. The research results show that the POD reduced-order model has good precision and fast computational speed. In the working conditions tested in this study, the relative mean error (RME) of POD-predicted temperature does not exceed 0.1% compared with the finite volume method’s results. Meantime, the POD calculation can improve the computational efficiency by hundreds of times (decrease from about 4 h to 45.865 s, about 314 times, for Case A). Thus, the POD reduced-order model has significant engineering value and is a promising means to accurately and efficiently solve the latent heat storage process in the DSG-STP system.

Numerical investigations on model order reduction to SEM based on POD-DEIM to linear/nonlinear heat transfer problems

The motivation in this article is to foster and conduct an in-depth investigation on reduced-order modeling (ROM) techniques via proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) in conjunction with high-order Legendre spectral element method (SEM) applied for time-dependent linear and nonlinear heat conduction problems. This approach significantly relieves the CPU burden, yet preserves the numerical accuracy of high-order schemes; this is especially pronounced for linear transient problems. For nonlinear cases, we use DEIM to save CPU time for nonlinear counterparts. For this purpose, we first obtain training data by solving the system of full-order models (FOM) using snapshot techniques to construct the POD basis. The well-known generalized single step single solve (GSSSS-1) framework is next employed for the first-order system for the time discretization. Numerical results for both linear and nonlinear heat conduction problems such as in a nuclear reactor, etc., show excellent agreement between FOM solutions and ROM solutions, and the CPU advantages via POD ROM techniques are also prominent for high-order Legendre SEM.

Comparison of the Atmospheric 200 hPa Jet’s Analyses between Proper Orthogonal Decomposition and Advanced Dynamic Mode Decomposition Method

In this paper, a frequently employed technique named the sparsity-promoting dynamic mode decomposition (SPDMD) is proposed to analyze the velocity fields of atmospheric motion. The dynamic mode decomposition method (DMD) is an effective technique to extract dynamic information from flow fields that is generated from direct experiment measurements or numerical simulation and has been broadly employed to study the dynamics of the flow, to achieve a reduced-order model (ROM) of the complex high dimensional flow field, and even to predict the evolution of the flow in a short time in the future. However, for standard DMD, it is hard to determine which modes are the most physically relevant, unlike the proper orthogonal decomposition (POD) method which ranks the decomposed modes according to their energy content. The advanced modal decomposition method SPDMD is a variant of the standard DMD, which is capable of determining the modes that can be used to achieve a high-quality approximation of the given field. It is novel to introduce the SPDMD to analyze the atmospheric flow field. In this study, SPDMD is applied to extract essential dynamic information from the 200 hPa jet flow, and the decomposed results are compared with the POD method. To further demonstrate the extraction effect of POD/SPDMD methods on the 200 hPa jet flow characteristics, the POD/SPDMD reduced-order models are constructed, respectively. The results show that both modal decomposition methods successfully extract the underlying coherent structures from the 200 hPa jet flow. And the DMD method provides additional information on the modal properties, such as temporal frequency and growth rate of each mode which can be used to identify the stability of the modes. It is also found that a fewer order of modes determined by the SPDMD method can capture nearly the same dynamic information of the jet flow as the POD method. Furthermore, from the quantitative comparisons between the POD and SPDMD reduced-order models, the latter provides a higher precision than the former, especially when the number of modes is small.

A POD reduced-order model for resolving the neutron transport problems of nuclear reactor

Due to the high-dimensional integro-differential properties of the neutron transport equation (NTE) and the large-scale property of the nuclear reactor system, the detailed neutron transport simulation can be very time-consuming, which urges researchers to develop a fast and accurate technique. To improve this condition, this work develops a set of reduced-order models for the NTE (NTEROM) by combining the proper orthogonal decomposition (POD) and the Galerkin projection techniques to efficiently solve the neutron transport problems. Results of some typical benchmarks show that the proposed POD based NTEROM can estimate the neutron transport behaviors with high fidelity and outstanding computing efficiency. Furthermore, the prediction function of the proposed NTEROMs can be extended to approximately calculate the neutron transport problems with the parameters outside of the snapshots’ parameter range. This work can provide new perspectives and ideas on the issue of improving the efficiency of numerical calculations in nuclear reactor engineering.