Proper Orthogonal Decomposition Basis Research Articles

A reduced-dimension method of Crank-Nicolson finite element solution coefficient vectors for the unsteady Burgers equation

This paper primarily focuses on the dimensionality reduction of finite element (FE) solution coefficient vectors for the unsteady Burgers equation, solved using the Crank-Nicolson FE (CNFE) method. The proper orthogonal decomposition (POD) basis is constructed from the snapshot matrix, which is formed using the first L solutions, where L is significantly smaller than the total number of time steps N of the CNFE method. By reconstructing the matrix form of the CNFE method, a reduced-dimension Crank-Nicolson finite element (RDCNFE) method is proposed and stability analysis and error estimates are discussed. Numerical tests are implemented to verify the theoretical results and demonstrate the high efficiency of the RDCNFE method.

A projection-based time-segmented reduced order model for fluid-structure interactions

In this paper, a type of novel projection-based, time-segmented reduced order model (ROM) is proposed for dynamic fluid-structure interaction (FSI) problems based upon the arbitrary Lagrangian–Eulerian (ALE)-finite element method (FEM) in a monolithic frame, where spatially, each variable is separated from others in terms of their attribution (fluid/structure), category (velocity/pressure) and component (two/three dimension) while temporally, the proper orthogonal decomposition (POD) bases are constructed in some deliberately partitioned time segments tailored through extensive numerical trials. By the combination of spatial and temporal decompositions, the developed ROM approach enables prolonged simulations under prescribed accuracy thresholds. Numerical experiments are carried out to compare numerical performances of the proposed ROM with corresponding full-order model (FOM) by solving a two-dimensional FSI benchmark problem that involves a vibrating elastic beam in the fluid, where the performance of offline ROM on perturbed physical parameters in the online phase is investigated as well. Extensive numerical results demonstrate that the proposed ROM has a comparable accuracy to while much higher efficiency than the FOM. The developed ROM approach is dimension-independent and can be seamlessly extended to solve high dimensional FSI problems.

Proper orthogonal decomposition reduced-order model of the global oceans

AbstractA reduced-order model (ROM) of the global oceans is developed by projecting the hydrostatic Boussinesq equations of motion onto a proper orthogonal decomposition (POD) basis. Three-dimensional POD modes are calculated from the ocean fields of an ensemble climate reanalysis dataset. The coefficients in the POD ROM are calculated using a regression approach. The performance of various POD ROM configurations are assessed. Each configuration is derived from an alternate sea-water equation of state, linking the density and temperature fields. POD ROM variants incorporating an equation of state in which density is a quadratic function of temperature, are able to reproduce the statistics of the large-scale structures at a fraction of the computational cost required to numerically simulate this flow. Due to the speed and efficiency of calculation, such reduced-order models of the global geophysical system will enable researchers and policy makers to assess the physical risk for a broader range of potential future climate scenarios.

Application of proper orthogonal decomposition to flow fields around various geometries and reduced-order modeling

This study is focused on a reduced-order model (ROM) based on proper orthogonal decomposition (POD) for unsteady flow around a stationary object, which allows prediction with different object geometry as a parameter. The conventional POD method is applicable only to data with the same computational grid for all snapshots. This study proposed a novel POD methodology that performs on flow snapshots, including some time-series data of flow fields around objects of different shapes and numerically computed by different computational grids. The concept of the proposed POD involved mapping the flow fields computed on different grids in computational space. Consequently, the optimal POD basis for minimizing reconstruction errors in physical space was obtained in the computational space. The proposed POD was applied to the flow around ellipses and airfoils generated via conformal mapping to a cylinder. The ROM formulated using the proposed POD bases reconstructed the flow fields around the ellipses with different aspect ratios and airfoils with varying shapes. Using the modes obtained by the proposed POD, the ROM was demonstrated to stably predict the time evolution of the flow around objects, which is not included in the snapshots. In the ROM, the difference between the frequency of the flow field in the POD snapshot and that of the reconstructed flow field resulted in a phase error owing to the time evolution. The mean squared error between the flow fields obtained via the ROM and the directly solved Navier–Stokes equations was under 10−7 when the reconstructed flow and the flow included in the snapshot had the same frequency as that of Kármán vorticities behind the objects. Based on these observations, the proposed POD is suitable for constructing an ROM to reconstruct the flow around various geometries.

Order reduction of special valve models based on pod method

Abstract The opening size of the disc-type special regulating valve (referred to as the special valve) can be used to regulate the fluid pressure and flow inside the cabin, and is an important equipment in the intake system of the high-altitude simulation test bench (referred to as the high-altitude bench). Due to the high dimensionality and complex solution of the special valve model, the calculation time is relatively long. In this paper, the Proper Orthogonal Decomposition (POD) model reduction method is applied to the mathematical model of the special valve, and the POD basis matrix is constructed through snapshots, thus obtaining a low-dimensional model and improving computational efficiency. The research results indicate that, while ensuring sufficient accuracy, this method can reduce the dimensionality of the special valve model, improve the computational speed of the simulator in calculating this case, and verify the effectiveness of the POD method.

Reduced-order methods for neutron transport kinetics problem based on proper orthogonal decomposition and dynamic mode decomposition

This work proposes two reduced-order methods to address the high computing resource consumption of neutron transport kinetics problems. The first method is based on proper orthogonal decomposition (POD), which integrates the POD basis with the governing equation. The second method is based on dynamic mode decomposition (DMD), which constructs an approximate linear relationship to predict the time-dependent physical quantities. Numerical results demonstrate that both two reduced-order methods can achieve high accuracy and efficient prediction. For the selected cases, POD can obtain a speed-up ratio of 105 – 429, while DMD can obtain a speed-up ratio of 2300 – 12500. For step transient situations, POD and DMD predict power errors do not exceed 0.42% and 0.05%. For ramp transient situations, POD and DMD predict power errors do not exceed 0.69% and 1.9%. This work can provide some useful suggestions for applications and further development in neutron transport kinetics reduced-order model.

Automated optimal experimental design strategy for reduced order modeling of aerodynamic flow fields

Aerodynamic flow fields reveal essential physical insights (such as shocks) that substantially affect aerodynamic performance. However, conventional flow field computations require time-consuming simulations. Alternatively, reduced-order models (ROMs) allow fast flow field predictions, enabling real-time decision-making. Efficient sampling is required to train ROMs without incuring prohibitive costs. In this paper, we propose a fully automated optimal experimental design (Auto-OED) strategy on proper orthogonal decomposition (POD) for ROM-based rapid flow field predictions. Auto-OED uses two individual optimal experimental design (OED) strategies, automatically selects the number of POD bases for the first sampling strategy, and intelligently switches to the second strategy on the fly. We showcased the Auto-OED strategy on airfoil flow field predictions in the transonic regime. OED-based ROMs were constructed over 20 trials for robustness tests with a total of 40 training samples in each trial, including 16 random initial samples by Latin hypercube sampling (LHS) and 24 OED samples. The results demonstrated that the ROM predictions completely based on LHS had an error of 1.6×10−3 while the existing OED strategies-based ROMs over 20 trials achieved a mean error (μerr) of 7.5×10−4 with a standard deviation (σerr) of 9.0×10−5. In contrast, the best Auto-OED ROM achieved the lowest μerr of 7.5×10−4 and lowest σerr of 6.2×10−5. These error reductions confirm the viability of the proposed Auto-OED-based ROM in fluid field predictions and potentially other engineering applications of a similar type.

A domain decomposing reduced-order method for solving the multi-domain transient heat conduction problems

In this article, a domain decomposition algorithm combined with a reduced-order model (ROM) is proposed to deal with the transient heat conduction problems in multi-domain structures. Using the domain decomposition algorithm and the finite element method (FEM), the original problem is decomposed into multiple non-overlapping subdomains, which can be solved independently under appropriate boundary conditions. However, additional inter-regional connected calculation will be brought through the simple geometry decomposition. Thus, the proper orthogonal decomposition (POD), a projection-based ROM technique, is utilized to further reduce the number of unknowns and the calculation time for each subdomain. To be more specific, the local POD bases of each subdomain are obtained by decomposing a snapshot matrix that gathered according to time by the application of the virtual boundary conditions on interfaces of the adjacent subdomains and other boundaries. Furthermore, the problems under different boundary conditions can be linked by the variable condensation technique on the interfaces, which enhances the efficiency of the numerical solver dramatically. Finally, the accuracy and efficiency of the proposed method are demonstrated by comparing the results calculated using both the proposed algorithm and the direct finite element simulation.

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective gray low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. The ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

Adaptive space-time model order reduction with dual-weighted residual (MORe DWR) error control for poroelasticity

In this work, the space-time MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) framework is extended and further developed for single-phase flow problems in porous media. Specifically, our problem statement is the Biot system which consists of vector-valued displacements (geomechanics) coupled to a Darcy flow pressure equation. The MORe DWR method introduces a goal-oriented adaptive incremental proper orthogonal decomposition (POD) based-reduced-order model (ROM). The error in the reduced goal functional is estimated during the simulation, and the POD basis is enriched on-the-fly if the estimate exceeds a given threshold. This results in a reduction of the total number of full-order-model solves for the simulation of the porous medium, a robust estimation of the quantity of interest and well-suited reduced bases for the problem at hand. We apply a space-time Galerkin discretization with Taylor-Hood elements in space and a discontinuous Galerkin method with piecewise constant functions in time. The latter is well-known to be similar to the backward Euler scheme. We demonstrate the efficiency of our method on the well-known two-dimensional Mandel benchmark and a three-dimensional footing problem.

Fast reconstruction of boiler numerical physical field based on proper orthogonal decomposition and conditional deep convolutional generative adversarial networks

ABSTRACT Obtaining real-time physical field data inside the boiler furnace is crucial for combustion diagnostics and optimization. In this work, we propose a fast reconstruction technique for the physical fields in a tangentially-fired coal boiler. The method combines the Proper Orthogonal Decomposition (POD) and Conditional Deep Convolutional Generative Adversarial Networks (cDCGAN) based on the boiler physical field obtained by computational fluid dynamics (CFD). Firstly, CFD simulations are performed using data from the boiler Distributed Control System (DCS). After verifying the CFD results, we expanded the simulation to 105 operational condition cases sample sets. Then, the physical fields of the sample set undergo POD to determine the appropriate number of POD bases, thereby ensuring the accuracy of the reconstruction. Finally, the cDCGAN method is employed to establish the correlation between the POD bases and the operational conditions of the boiler. Results show that the POD-cDCGAN method achieves rapid and accurate reconstruction of temperature and velocity fields, with mean relative error (MRE) below 1.5% and root relative squared error (RRSE) below 5.7%. The reconstruction time is reduced to 10.57s, significantly faster than direct CFD methods. The POD-cDCGAN method offers an efficient approach for online operational diagnosis and digital twin construction of boiler equipment.

Dissipation-based proper orthogonal decomposition of turbulent Rayleigh–Bénard convection flow

We present a formulation of proper orthogonal decomposition (POD) producing a velocity–temperature basis optimized with respect to an H1 dissipation norm. This decomposition is applied, along with a conventional POD optimized with respect to an L2 energy norm, to a dataset generated from a direct numerical simulation of Rayleigh–Bénard convection in a cubic cell (Ra=107, Pr=0.707). The dataset is enriched using symmetries of the cell, and we formally link symmetrization to degeneracies and to the separation of the POD bases into subspaces with distinct symmetries. We compare the two decompositions, demonstrating that each of the 20 lowest dissipation modes is analogous to one of the 20 lowest energy modes. Reordering of modes between the decompositions is limited, although a corner mode known to be crucial for reorientations of the large-scale circulation is promoted in the dissipation decomposition, indicating suitability of the dissipation decomposition for capturing dynamically important structures. Dissipation modes are shown to exhibit enhanced activity in boundary layers. Reconstructing kinetic and thermal energy, viscous and thermal dissipation, and convective heat flux, we show that the dissipation decomposition improves overall convergence of each quantity in the boundary layer. Asymptotic convergence rates are nearly constant among the quantities reconstructed globally using the dissipation decomposition, indicating that a range of dynamically relevant scales is efficiently captured. We discuss the implications of the findings for using the dissipation decomposition in modeling and argue that the H1 norm allows for a better modal representation of the flow dynamics.

Multi-fidelity reduced-order surrogate modelling

High-fidelity numerical simulations of partial differential equations (PDEs) given a restricted computational budget can significantly limit the number of parameter configurations considered and/or time window evaluated. Multi-fidelity surrogate modelling aims to leverage less accurate, lower-fidelity models that are computationally inexpensive in order to enhance predictive accuracy when high-fidelity data are scarce. However, low-fidelity models, while often displaying the qualitative solution behaviour, fail to accurately capture fine spatio-temporal and dynamic features of high-fidelity models. To address this shortcoming, we present a data-driven strategy that combines dimensionality reduction with multi-fidelity neural network surrogates. The key idea is to generate a spatial basis by applying proper orthogonal decomposition (POD) to high-fidelity solution snapshots, and approximate the dynamics of the reduced states—time-parameter-dependent expansion coefficients of the POD basis—using a multi-fidelity long short-term memory network. By mapping low-fidelity reduced states to their high-fidelity counterpart, the proposed reduced-order surrogate model enables the efficient recovery of full solution fields over time and parameter variations in a non-intrusive manner. The generality of this method is demonstrated by a collection of PDE problems where the low-fidelity model can be defined by coarser meshes and/or time stepping, as well as by misspecified physical features.

Flow fields prediction for data-driven model of parallel twin cylinders based on POD-RBFNN and POD-BPNN surrogate models

Flow around cylinders is an important phenomenon in many different engineering fields. In this paper, the fast prediction of the pressure fields of parallel twin cylinders is implemented based on a data-driven algorithm. Firstly, the pressure fields of parallel twin cylinders with a low Reynolds number are obtained through the Computational Fluid Dynamics (CFD) method. The pressure fields at different time steps are collected to form a snapshot matrix. The Proper Orthogonal Decomposition (POD) algorithm is then applied to obtain the POD basis vectors of the snapshot matrix, enabling the reconstruing the pressure fields. Subsequently, two reduced-order models (ROM) called the POD-RBFNN and POD-BPNN surrogate models are proposed in this paper. The POD-RBFNN surrogate model uses the Radial Basis Function Neural Network (RBFNN) to train the POD mode coefficients obtained from the POD algorithm, while the POD-BPNN surrogate model uses the Backpropagation Neural Network (BPNN) for the same purpose. Linearly combining the POD mode coefficients predicted by the POD-RBFNN or POD-BPNN surrogate models with the POD basis vectors obtained from the POD algorithm enables fast and efficient prediction of pressure fields for non-sample points. Finally, comparisons are made between the predicted pressure fields obtained from these two surrogate models and the actual values obtained through CFD simulations. It is found that both the POD-RBFNN and POD-BPNN surrogate models proposed in this paper not only significantly improve efficiency but also maintain a high level of accuracy. However, the training time of the POD-RBFNN surrogate model is significantly shorter than that of the POD-BPNN surrogate model. Additionally, the POD-RBFNN surrogate model exhibits smaller Root Mean Square Errors (RMSE) and Mean Absolute Error (MAE). For the data-driven model of parallel twin cylinders described in this paper, the POD-RBFNN surrogate model is more suitable to predict the pressure fields. The research results in this paper are believed to hold significant value for CFD calculations of parallel twin cylinder models, offering essential guidance for a deeper understanding of cylinder flow problems and optimizing engineering design.

Surrogate Modelling of Field Performance Analysis and Optimization via Hierarchical Proper Orthogonal Decomposition and Adaptive Sampling Radial Basis Function

Surrogate Modelling of Field Performance Analysis and Optimization via Hierarchical Proper Orthogonal Decomposition and Adaptive Sampling Radial Basis Function

Non-Intrusive Reduced-Order Modeling Based on Parametrized Proper Orthogonal Decomposition

A new non-intrusive reduced-order modeling method based on space-time parameter decoupling for parametrized time-dependent problems is proposed. This method requires the preparation of a database comprising high-fidelity solutions. The spatial bases are extracted from the database through first-level proper orthogonal decomposition (POD). The algebraic relationship between the time trajectory/parameter positions and the projection coefficient is described by the linear superposition of the second-level POD bases (temporal bases) and the second-level projection coefficients (parameter-dependent coefficients). This decomposition strategy decouples the space-time parameter effects, providing a stable foundation for fast predictions of parametrized time-dependent problems. The mappings between the parameter locations and the parameter-dependent coefficients are approximated as Gaussian process regression (GPR) models. The accuracy and efficiency of the PPOD-ROM are demonstrated through two numerical examples: flows past a cylinder and turbine flows with a clocking effect.

A data-driven analysis of short and long laminar separation bubbles

This work investigates the statistical response of short and long laminar separation bubbles to external flow parameters, such as Reynolds number, free-stream turbulence intensity and streamwise pressure gradient, known to govern bubble formation and characteristics. A parametric experimental campaign has been performed using particle image velocimetry on a flat plate to provide a comprehensive database for the characterization of separation-induced transition in both short and long separation bubbles. The proper orthogonal decomposition (POD) was applied to the data set of all dividing streamlines commonly used to identify a laminar separation bubble. This provides an optimal state-space basis for the data-driven classification of the state of a laminar separation bubble, with the leading modes capturing the change in length and height of the laminar separation bubble in response to changes in the flow parameters. When projected onto the POD subspace constituted by the first three leading modes, the normalized data from the present study and the results from prior investigations not used in the modal analysis collapse on the same trajectory in the low-dimensional space. The present POD basis can be therefore adopted for the description of the general response of the time-mean shape of a laminar separation bubble to changes in the main influencing parameters. A well-defined pattern was observed in the case of short laminar separation bubbles in the reduced-order space defined by the first three POD coefficients, whereas a higher dispersion in the long-bubble regime indicates an increased sensitivity of long bubbles to the external flow characteristics.

Reduced-order model-based variational inference with normalizing flows for Bayesian elliptic inverse problems

We propose a reduced-order model-based variational inference method with normalizing flows for Bayesian elliptic inverse problems. The coefficient of the elliptic PDE is represented by a finite number of parameters. We aim to estimate the posterior distributions of these parameters in the framework of variational inference, and the approximation of the posterior distribution is constructed through a normalizing flow. Moreover, as data of inputs and outputs of the forward problem are accumulated during the training of the normalizing flow, we can naturally exploit the low-dimensional intrinsic structure of the forward elliptic PDE using reduced-order models based on these data, in which information of the true posterior is incorporated. We construct a low-dimensional set of data-driven basis functions in the solution space using proper orthogonal decomposition (POD) and train a neural network that maps the parameters to the coefficients of these data-driven basis functions. The surrogate forward map, which is the combination of the reduced-order model and the parameter-to-coefficient neural network, is then applied to the Bayesian elliptic inverse problem, where in each iteration the forward problem is evaluated in the space spanned by the POD basis functions using the surrogate forward map. Thus the inversion will be significantly accelerated as the surrogate forward map is essentially a low-dimensional model. We present numerical examples to show the accuracy and efficiency of the proposed method.

A predictive surrogate model for hemodynamics and structural prediction in abdominal aorta for different physiological conditions

Background and objectiveThis study investigates the application of a Predictive Surrogate Model (PSM) for the prediction of the fluid and solid variables in the abdominal aorta by integrating Proper Orthogonal Decomposition (POD) and Long Short-Term Memory (LSTM) techniques. MethodsThe Fluid-Structure Interaction (FSI) solver, which serves as the Full-Order Model (FOM), can capture the blood hemodynamics and structural mechanics precisely for a variety of physiological states, namely the rest and exercise conditions. ResultsDetailed analyses have been conducted on velocity components, pressure, Wall Shear Stress (WSS), and Oscillatory Shear Index (OSI) variables. Firstly, the reconstruction error has been derived based on a specific number of POD bases to assess the Reduced Order Model (ROM). Notably, the reconstruction error for velocity components in the rest condition is one order of magnitude higher than that in the exercise condition, yet both remained below 10%. This error for pressure is even more minimal, being less than 1%. ConclusionsThe PSM is evaluated against rest and exercise conditions, exhibiting promising results despite the inherent complexities of the physiological conditions. Despite the inherent complexities of phenomena in the aorta, the predictive model demonstrates consistent error magnitudes for velocity components and wall-related indices, while solid variables show slightly higher errors.

Accelerating inverse inference of ensemble Kalman filter via reduced-order model trained using adaptive sparse observations

The implementation of data assimilation often struggles with low computational efficiency due to the complexity of high-fidelity numerical modeling and the large size of the observation vectors. To address this challenge, it is necessary to employ reduced-order strategies for both the full-order dynamical model and observation operator. Proper orthogonal decomposition (POD) based reduced-order model (ROM) and discrete empirical interpolation method (DEIM) based sparse observation identification are often considered as primary solutions. However, their effective combination requires the determination of high-quality POD reduced-order basis vectors. The current research focuses on incorporating the ROM and DEIM into a Kalman filter framework, resulting in a reduced-order Kalman filter with sparse observations. The key aspect is identifying the optimal POD basis, which can be achieved iteratively by introducing a relative root-mean-square-error (RMSE) for a near-optimal exploration of both the desired low-order POD basis and observation locations. The singular value decomposition (SVD) of the snapshots is performed in a low-dimensional subspace, leading to significant computational savings. This improved POD basis not only enhances the reconstruction of the ROM, but also helps build the DEIM observation matrix, resulting in simultaneous dimensionality reduction of the state and observation space. The effectiveness of this algorithm is demonstrated through the retrieval of initial conditions for one-dimensional tsunami wave equations with real data scenarios. Experiments are conducted for both smooth and nonsmoothed initial conditions. The results show that when the observation data is available at 3 and 6 spatial observation locations, our method only requires fewer vectors with the first 45 and 90 dominant modes derived using the RMSE, compared to as many as 210 and 252 derived from the relative energy criterion. Our method enables a low-modal ROM to effectively reconstruct the wave height field, leading to considerable prediction capability for potential long-wave dynamics with comparable accuracy. The CPU time savings achieved through our method is at least one to two orders of magnitude compared to the CPU time required by the full-order model.