POD-based Reduced-order Model Research Articles

Data-driven surrogate modeling and optimization of supercritical jet into supersonic crossflow

For the design and optimization of advanced aero-engines, the prohibitively computational resources required for numerical simulations pose a significant challenge, due to the extensive exploration of design parameters across a vast design space. Surrogate modeling techniques offer a viable alternative for efficiently emulating numerical results within a notably compressed timeframe. This study introduces parametric Reduced-Order Models (ROMs) based on Convolutional AutoEncoders (CAE), Fully Connected AutoEncoders (FCAE), and Proper Orthogonal Decomposition (POD) to fast emulate spatial distributions of physical variables for a supercritical jet into a supersonic crossflow under different operating conditions. To further accelerate the decision-making process, an optimization model is developed to enhance fuel-oxidizer mixing efficiency while minimizing total pressure loss. Results indicate that CAE-based ROMs exhibit superior prediction accuracy while FCAE-based ROMs show inferior predictive accuracy but minimal uncertainty. The latter may be ascribed to the markedly greater number of hyperparameters. POD-based ROMs underperform in regions of strong nonlinear flow dynamics, coupled with higher overall prediction uncertainties. Both AE- and POD-based ROMs achieve online predictions approximately 9 orders of magnitude faster than conventional simulations. The established optimization model enables the attainment of Pareto-optimal frontiers for spatial mixing deficiencies and total pressure recovery coefficient.

Parameter identification and uncertainty propagation of hydrogel coupled diffusion-deformation using POD-based reduced-order modeling

AbstractThis study explores reduced-order modeling for analyzing time-dependent diffusion-deformation of hydrogels. The full-order model describing hydrogel transient behavior consists of a coupled system of partial differential equations in which the chemical potential and displacements are coupled. This system is formulated in a monolithic fashion and solved using the finite element method. We employ proper orthogonal decomposition as a model order reduction approach. The reduced-order model performance is tested through a benchmark problem on hydrogel swelling and a case study simulating co-axial printing. Then, we embed the reduced-order model into an optimization loop to efficiently identify the coupled problem’s material parameters using full-field data. Finally, a study is conducted on the uncertainty propagation of the material parameter.

A linear filter regularization for POD-based reduced-order models of the quasi-geostrophic equations

We propose a regularization for reduced-order models (ROMs) of the quasi-geostrophic equations (QGE) to increase accuracy when the proper orthogonal decomposition (POD) modes retained to construct the reduced basis are insufficient to describe the system dynamics. Our regularization is based on the so-called BV-α model, which modifies the nonlinear term in the QGE and adds a linear differential filter for the vorticity. To show the effectiveness of the BV-α model for ROM closure, we compare the results computed by a POD-Galerkin ROM with and without regularization for the classical double-gyre wind forcing benchmark. Our numerical results show that the solution computed by the regularized ROM is more accurate, even when the retained POD modes account for a small percentage of the eigenvalue energy. Additionally, we show that, although computationally more expensive than the ROM with no regularization, the regularized ROM is still a competitive alternative to full-order simulations of the QGE.

Feasibility Analysis of a POD-Based Reduced Order Model with Application in Eulerian–Lagrangian Simulations

Feasibility Analysis of a POD-Based Reduced Order Model with Application in Eulerian–Lagrangian Simulations

Data-driven physical fields reconstruction of supercritical-pressure flow in regenerative cooling channel using POD-AE reduced-order model

In order to effectively estimate the physical fields of active cooling channel with supercritical pressure hydrocarbon fuel, a novel data-driven reduced-order model framework is firstly proposed in the present work, which is established via combining reduced-order modeling method based on proper orthogonal decomposition and autoencoder neural network (POD-AE) with Kriging surrogate model. This framework mainly contains offline stage and online stage. Preliminary dimension reduction and feature capture of high-dimensional physical fields data is conducted using POD in the offline stage, and low-dimensional reduced-order models (ROMs) are constructed by the linear combination of orthogonal bases and feature coefficients. Specially, feature coefficients of ROMs are used to trained an AE, and then lower-dimensional secondary reduced-order models (2nd-ROMs) are established by secondary compression for ROMs using the trained AE. Furthermore, the Kriging model is trained for implicitly mapping boundary conditions or temperature data from the sensors on the bottom surface to the established 2nd-ROMs. In the online stage, physical fields estimation under new boundary conditions or wall temperature monitoring based on temperature data of sensors can be quickly fulfilled using this framework. Fifty groups of test conditions are considered to demonstrate the accuracy and efficiency of the POD-AE ROM framework. The results prove that it shows good efficiency, accuracy in estimating and monitoring the physical fields of the cooling channel with supercritical pressure n-dodecane within sample space. Compared with the traditional POD-based ROM framework, the proposed framework still achieved acceptable accuracy under all testing conditions when the original data was compressed to extremely low dimensions. And the calculation time only slightly increased.

Artificial neural network based correction for reduced order models in computational fluid mechanics

In this paper we propose a reduced order model (ROM) for incompressible flows based on proper orthogonal decomposition (POD) and having a finite element approximation as full order model (FOM). The main idea is to start from a purely POD-based ROM, projecting the equations onto the ROM space, and then add a nonlinear correction that depends on the ROM unknowns to enhance the final ROM model. This correction is based on the fact that we do have some available high fidelity data, namely, the snapshots. Thus, the correcting term is built as an artificial neural network (ANN) constructed with the snapshots as the training set. This correction is then introduced in the fully discrete ROM system to achieve more accurate solutions. A further feature of our approach is that we construct both the ROM and the FOM using the variational multi-scale (VMS) concept, and this allows us to understand the correction term as an approximation to the scales that are lost when passing from the FOM to the ROM. The resulting corrected ROM has a significant higher accuracy than the original one.

POD-based ROM applied to unsteady free surface water flow

The numerical resolution of shallow water equations (SWE) is required in many environmental problems involving free surface flows. The upwind augmented Roe’s method [2] is widely used due to the robust and stable solutions it offers in realistic scenarios if properly corrected to deal with entropy fix and wet–dry fronts. On the other hand, reduced-order models (ROMs) are known to achieve more efficiency in terms of computational cost without losing accuracy. In this work, we analyse the properties and performance of a proper orthogonal decomposition (POD) based ROMs for this type of flows.

POD-based reduced-order modeling study for thermal analysis of gas-cooled microreactor core

Small modular reactors require multi-physics coupling calculations to balance economy and stability, due to their compact structures. Traditional tools used for light water reactors are not effective in addressing the several modeling challenges posed by these calculations. The lumped parameter method is commonly used in the thermal analysis for its high computational speed, but it lacks accuracy due to the thermal model is one-dimensional. While computational fluid dynamics software (CFD) can provide high-precision and high-resolution thermal analysis, its low calculation efficiency making it challenging to be coupled with other programs. Proper Orthogonal Decomposition (POD) is one of the Reduced Order Model (ROM) methods employed in this study to reduce the dimensionality of sample data and to improve the thermal modelling of gas-cooled microreactors. In this work, a non-inclusive POD with neural network method is proposed and verified using a transient heat conduction model for a two-dimensional plate. The method is then applied to build a reduced order model of the gas-cooled micro-reactor core for rapid thermal analysis. The results show that the root mean square error of the reactor core temperature is less than 1.02% and the absolute error is less than 8.2°C while the computational cost is reduced by several orders of magnitude, shortening the calculation time from 1.5-hour to real-time display. These findings proved the feasibility of using POD and neural network in the development of ROMs for gas-cooled microreactor, providing a novel approach for achieving precise thermal calculation with minimized computational costs.

Development of POD-based Reduced Order Models applied to shallow water equations using augmented Riemann solvers

Reduced-order models (ROMs) based on the proper orthogonal decomposition have been proposed to reduce the computational resources required by the full-order models (FOMs) to approximate partial differential equations. In this paper a Roe-based ROM is developed to solve the shallow water equations in presence of source terms more efficiently than the Roe-based FOM. The well-balanced property and other numerical corrections such as the entropy fix and the wet–dry treatment are taken into account using augmented Riemann solvers to build the Roe-based FOM. In addition to this, a time averaging approach is necessary to develop the Roe-based ROM. This approach is validated by solving some cases and the computed solutions are compared with those ones of Lax–Friedrichs-based ROMs. It is also studied whether the ROM preserves or not the well-balancing, the entropy fix and the wet–dry treatment.

POD-based reduced order model for the prediction of global and local elastic responses of fibre-reinforced polymer considering varying fibre distribution

POD-based reduced order model for the prediction of global and local elastic responses of fibre-reinforced polymer considering varying fibre distribution

A POD-based reduced order model applied to 1D shallow water equations

Many environmental problems involving free surface flow can be solved using the shallow water equations (SWE) often involving high computational costs due to the large spatial and temporal scales of the events. In recent times, reduced order models (ROM) techniques are increasingly used to improve the computational efficiency of simulation models. The Proper Orthogonal Decomposition (POD) method provides an orthogonal basis for representing a given set of data and constructing the ROM by means of the method of snapshots. In this work, a POD-based intrusive ROM strategy is applied to the 1D SWE. The main goal of this work is to build a simulation model able to reproduce realistic scenarios. We analyse the computational improvement and the accuracy of the ROM results with respect to those of the full-order model (FOM).

A POD-based ROM strategy for the prediction in time of advection-dominated problems

The use of reduced-order models (ROMs) for the numerical approximation of the solution of partial differential equations is a topic of current interest, being motivated by the high computational efficiency of ROMs when compared to full-order models (FOMs). To construct a ROM to approximate the solution of transport equations, the use of the proper orthogonal decomposition (POD) method is a common choice. POD-based ROMs rely on the snapshot method, which consists in the off-line computation of a set of values corresponding to the solution up to the training time by means of the FOM. Then, the ROM is constructed and solved, up to the training time. When considering parabolic equations, the method is able to compute the solution beyond the training time. However, when considering hyperbolic problems, POD-based ROMs fail when computing the solution beyond the training time, this being one of the strongest limitations of POD-based ROMs. In this work, a strategy in the framework of POD-based ROMs to predict solutions in time is introduced. This method, called CT-ROM, is based on a coordinate transformation and allows to compute the solution of advection-dominated problems beyond the training time. The performance of this strategy is assessed using a variety of test cases, showing promising results in all of them. The extension of the CT-ROM to higher spatial dimensions by means of the Radon transform is also presented. The results obtained are encouraging and motivate the application of this idea to more complex problems.

Proper orthogonal decomposition based reduced-order modeling of flux-Limited gray thermal radiation

In this work, a proper orthogonal decomposition (POD) based reduced-order model (ROM) is developed to solve gray, flux-limited thermal radiation diffusion. We focus on the variable opacity radiation penetration benchmark posed by Olson, Auer, and Hall. The T−3 relationship for opacity in conjunction with high-temperature radiation penetrating an initially cold material produces a strong thermal radiation shock. This class of problems is particularly challenging for standard POD-based reduced-order modeling due to the nonlinearities presented by 1) the T4 source term, and 2) flux-limited diffusion operator. To address these challenges and develop a cost competitive ROM, we employ a “hyper-reduction” technique through discrete empirical interpolation (DEIM) and allow for adaptive reduced-order projections through principal interval decomposition (PID). Performance of the proposed methodology is quantified by comparing the cost savings and accuracy relative to a full-order computation. Reference solutions and snapshot data are obtained through a full-order calculation performed by the University of Chicago maintained astrophysics code, FLASH. For consistency and potential extensibility, the developed ROM is also implemented in FLASH. We find that in the initialization regime, where the thermal radiation wave is initially created by the warming of the material, this class of problems is highly reducible and suitable for POD-based ROMs. However, the strong convective nature of the wave propagation regime is less reducible and more challenging to create an efficient ROM.

Reduced Order Modeling of Deformable Tire-Soil Interaction With Proper Orthogonal Decomposition

AbstractIn this study, model order reduction of high-fidelity off-road mobility models is explored to address the computational intensity of nonlinear finite element deformable tire–soil interaction models. To this end, a model order reduction procedure for the tire–soil interaction model is developed with the proper orthogonal decomposition (POD), and it is integrated into the off-road mobility simulation framework, leveraging high-performance computing. The POD is, however, limited in that the modes are dependent on snapshot data collected during the running of a full order model, limiting the modes to being accurate only for the specific scenario from which they were collected. Due to this limitation, a method of mode adaptation through interpolation on a tangent space of the Grassmann manifold is investigated to allow modes to be predicted for cases in which a full order model has not been run. It is demonstrated by several numerical examples that the POD modes are effective at retaining predictive accuracy while reducing computational time. The results show that adapted POD modes are more capable of characterizing the behavior of the model than modes produced at a different value of the simulation parameter. The POD-based reduced order modeling approach is further extended to the full vehicle simulation on deformable terrain through the co-simulation coupling algorithm by leveraging the high-performance computing technique.

An iterative algorithm for POD basis adaptation in solving parametric convection–diffusion equations

In this paper, we develop an iterative algorithm for proper orthogonal decomposition (POD) basis adaptation in solving linear parametric PDEs. Specifically, we consider the convection–diffusion equations with diffusivity as a parameter. To construct POD basis functions for the convection–diffusion equation with a small diffusivity, we need a fine-grid solver to obtain accurate solution snapshots, which leads to a large amount of computation and memory costs. Meanwhile, a coarse-grid solver is sufficient for obtaining accurate solution snapshots for the convection–diffusion equation with a large diffusivity. We aim to adapt the POD basis functions extracted from the solution snapshots of a large diffusivity for the construction of a reduced-order model at a small diffusivity without resorting to a fine-grid solver. Our POD basis adaptation method exploits the implicit dependence of solutions on diffusivity. The POD basis functions are adapted through an iterative algorithm, where the full-order model simulation at a large diffusivity and the POD-based reduced-order model simulation at a small diffusivity are implemented, alternatively. We also provide convergence analysis for our POD basis adaptation method. The algorithm and convergence analysis can be generalized to other types of linear parametric PDEs without any difficulty. Finally, we present numerical results to demonstrate the performance and accuracy of the proposed method.

A reduced order method for nonlinear parameterized partial differential equations using dynamic mode decomposition coupled with k-nearest-neighbors regression

Accurately constructing a reduced order model (ROM) of nonlinear parameterized partial differential equations (PDEs) has always been a challenging problem in engineering and applied sciences. Dynamic mode decomposition (DMD) is a popular and efficient data-driven method for ROM, however, it is proposed for the model order reduction of time-dependent problems that it is inapplicable for the parameterized problems. In this paper, a new ROM is proposed based on the k-nearest-neighbors (KNN) regression and DMD, namely, KNN-DMD. The KNN can approximate the solution at any given parameter value by choosing and averaging the nearest k DMD solutions based on the distance between the given parameter value and other parameter values, leading to the applicability of DMD to parameterized problems. We apply the proposed method to various nonlinear parameterized PDEs, i.e., heat equations, reaction-diffusion equations, Burgers equations, and Navier-Stokes equations for the two-dimensional fluid flow over a cylinder. The results demonstrate the applicability and efficiency of the proposed KNN-DMD as a real-time ROM for parameterized PDEs. Furthermore, KNN-DMD shows better predictive ability than the POD-based ROMs outside of the training time region.

Improved Nonlinear Analysis of a Propeller Blade Based on Hyper-Reduction

In this study, an improved nonlinear-analysis framework capable of predicting geometric nonlinearity and high-speed rotation in rotating structures was developed. A nonlinear time-transient simulation requires large computations owing to an iterative solution algorithm. To reduce the anticipated computational cost, a proper orthogonal decomposition (POD)-based reduced-order modeling (ROM) combined with hyper-reduction is applied. To efficiently perform computations during the online stage, three hyper-reduction techniques were employed to approximate the nonlinear finite-element matrices: discrete empirical interpolation method (DEIM), Gauss–Newton with approximated tensors (GNAT), and energy-conserving sampling and weighting (ECSW). The present frameworks are applied to the time-transient simulation of a propeller, including parametric variations. Compared with the DEIM method, the GNAT and ECSW methods exhibited better enhancement of the accuracy and robustness of the reduced-order representation. Additionally, the computational efficiency of the ECSW method was improved significantly compared with that of other POD-based ROM approaches.

Thermal-electrochemical simulation of lead-acid battery using reduced-order model based on proper orthogonal decomposition for real-time monitoring purposes

The main contributions of this study are, first, developing an efficient reduced-order model (ROM) to fast and accurately simulate the temperature rise and electrochemical behavior of a lead-acid cell during discharge and, second, presenting a comprehensive analysis for the dominant thermal modes of the cell. For this purpose, the governing transport equations including conservation of potential in solid and liquid phases, conservation of species, and conservation of energy are solved simultaneously using proper orthogonal decomposition (POD) method. The basis of reduced region are created from singular value decomposition (SVD) of the snapshots which are captured from the simulation results of finite-volume method (FVM). The presented model is used to simulate the discharge process of a typical lead-acid cell, taken from the literature, at different initial temperatures and discharge current densities. The simulation results show that the discharge time is highly influenced by the temperature rise. Moreover, since the dynamic behavior of lead-acid battery is severely affected by the cell temperature, an improved POD-based ROM has been developed to compensate the drawbacks of original ROM and simulate a wide range of operating conditions. Finally, the obtained results show that the presented ROM can both accurately simulate the lead-acid battery and reduce the computational cost (speed-up factor of 17) which makes it suitable for real-time monitoring, control, and optimization purposes.

Reduced order model for nonlinear multi-directional ocean wave propagation

In this paper, we introduce a reduced order model (ROM) for the propagation of nonlinear multi-directional ocean wave-fields. The ROM relies on Galerkin projection of Zakharov equations embedded in the high-order spectral (HOS) method, which describes the evolution of nonlinear waves. The dominant flow features of wave evolution are computed from proper orthogonal decomposition (POD) and these modes are used for the projection. The HOS scheme to compute the vertical velocity is treated in a novel way for an efficient implementation of POD-based ROM. We refer to this alternative formalism of HOS as HOS-simple. The final reduced order model (ROM) is derived from the Galerkin projection of HOS-simple. For the case of irregular waves, where the number of modes required are in the range of 200, the ROM has no significant advantage since both HOS and HOS-simple are much faster than real-time. The real advantage is demonstrated in multi-directional (or short-crested) irregular waves, where the ROM is the only model capable of achieving real-time computation, a major improvement to the standard HOS method. The potential use of the ROM in propagating short-crested waves from far-field to near-field for real-world applications involving wave probes in a wave tank/controlled environment as well as X-band radar in open ocean is also demonstrated.

Hull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing

In the field of parametric partial differential equations, shape optimization represents a challenging problem due to the required computational resources. In this contribution, a data-driven framework involving multiple reduction techniques is proposed to reduce such computational burden. Proper orthogonal decomposition (POD) and active subspace genetic algorithm (ASGA) are applied for a dimensional reduction of the original (high fidelity) model and for an efficient genetic optimization based on active subspace property. The parameterization of the shape is applied directly to the computational mesh, propagating the generic deformation map applied to the surface (of the object to optimize) to the mesh nodes using a radial basis function (RBF) interpolation. Thus, topology and quality of the original mesh are preserved, enabling application of POD-based reduced order modeling techniques, and avoiding the necessity of additional meshing steps. Model order reduction is performed coupling POD and Gaussian process regression (GPR) in a data-driven fashion. The framework is validated on a benchmark ship.