POD-Galerkin Reduced-order Model Research Articles

Projection-based reduced-order modelling of time-periodic problems, with application to flow past flapping hydrofoils

Simulating forced time-periodic flows in industrial applications presents significant computational challenges, partly due to the need to overcome costly transients before achieving time-periodicity. Reduced-order modelling emerges as a promising method to speed-up computations. We extend upon the work of Lotz et al. (2024) where a time-periodic space–time model is introduced. We present a time-periodic reduced-order model that directly finds the time-periodic solution without requiring extensive time integration. The reduced-order model gives a reduction in variables in both space and time. Our approach involves a POD-Galerkin reduced-order model based on a time-periodic full-order model that employs isogeometric analysis, residual-based variational multiscale turbulence modelling and weak boundary conditions. The projection-based reduced-order model inherits these features. We evaluate the reduced-order model with numerical experiments on moving hydrofoils. The motion is known a priori and we restrict ourselves to two spatial dimensions. In these experiments we vary the Strouhal and Reynolds numbers, and the motion profile respectively. Reduced-order model solutions agree well with those of the full-order model. The errors over the entire time period of thrust and lift forces are less than 0.2%. This includes complex scenarios such as the transition from drag to thrust production with increasing Strouhal number. Our time-periodic reduced-order model offers speed-ups ranging from O(102) to O(103) compared to the full-order model, depending upon the basis size. This makes it an appealing solution for prescribed time-periodic problems, with potential for additional speedup through nonlinear reduction techniques such as hyper-reduction.

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.

Application of data-driven model reduction techniques in reactor neutron field calculations

High-order harmonic techniques can be used to recreate neutron flux distributions in reactor cores using the neutron diffusion equation. However, traditional source iteration and source correction iteration techniques have sluggish convergence rates and protracted calculation periods. The correctness of the implicitly restarted Arnoldi method (IRAM) in resolving the eigenvalue problems of the one-dimensional and two-dimensional neutron diffusion equations was confirmed by computing the benchmark problems SLAB_1D_1G and two-dimensional steady-state TWIGL using IRAM. By integrating Galerkin projection with Proper Orthogonal Decomposition (POD) techniques, a POD-Galerkin reduced-order model was developed and the IRAM model was used as the full-order model. For 14 macroscopic cross-section values, the TWIGL benchmark problem was perturbed within a 20% range. We extracted 100 sample points using the Latin hypercube sampling method, and 70% of the samples were used as the testing set to assess the performance of the reduced-order model The remaining 30% were utilized as the training set to develop the reduced-order model, which was employed to rebuild the TWIGL benchmark problem. The reduced-order model demonstrates good flexibility and can efficiently and accurately forecast the effective multiplication factor and neutron flux distribution in the core. The reduced-order model predicts keff and neutron flux distribution with a high degree of agreement compared to the full-order model. Additionally, the reduced-order model's computation time is only 10.18% of that required by the full-order model.The neutron flux distribution of the steady-state TWIGL benchmark was recreated using the reduced-order model. The obtained results indicate that the reduced-order model can accurately predict the keff and neutron flux distribution of the steady-state TWIGL benchmark.Overall, the proposed technique not only has the potential to accurately project neutron flux distributions in transient settings, but is also relevant for reconstructing neutron flux distributions in steady-state conditions; thus, its applicability is bound to increase in the future.

POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equations

We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier–Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A POD-Galerkin PINN ROM is then derived by introducing deep neural networks which approximate the reduced outputs with the input being time and/or parameters of the model. The neural networks incorporate the physical equations (the POD-Galerkin reduced equations) into their structure as part of the loss function. Using this approach, the reduced model is able to approximate unknown parameters such as physical constants or the boundary conditions. A demonstration of the applicability of the proposed ROM is illustrated by three cases which are the steady flow around a backward step, the flow around a circular cylinder and the unsteady turbulent flow around a surface mounted cubic obstacle.

A POD-Galerkin reduced-order model for two-dimensional Rayleigh-Bénard convection with viscoelastic fluid

Reduced-order model (ROM) based on proper orthogonal decomposition (POD) has been widely used in different fields, such as optimization design and flow control, etc., which needs to get calculation results quickly. Most of present studies about POD-ROM focus on the Newtonian fluid flows, whereas, few studies are related with viscoelastic fluid flows. In this paper, POD-ROM is the first time developed for viscoelastic fluid flow by considering a typical two-dimensional thermal fluid problem: Rayleigh-Bénard convection (RBC). Three different cases are chosen to simulate viscoelastic RBC flow at Rayleigh number of 108, Prandtl number of 7.0 and Weissenberg number of 0.1, 0.2 and 0.3, respectively. It is found that phase portraits under different POD modes share the same trend as that for DNS. By introducing ‘intrinsic stabilization scheme’ closure model, it shows that the relative errors of fluctuation velocity, temperature and conformation tensor fields decrease greatly, i.e., POD-ROM with closure model can perform better than POD-ROM without closure model for long-time prediction.

Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism

Closure modeling based on the Mori-Zwanzig formalism has proven effective to improve the stability and accuracy of projection-based model order reduction. However, closure models are often expensive and infeasible for complex nonlinear systems. Towards efficient model reduction of general problems, this paper presents a recurrent neural network (RNN) closure of parametric POD-Galerkin reduced-order model. Based on the short time history of the reduced-order solutions, the RNN predicts the memory integral which represents the impact of the unresolved scales on the resolved scales. A conditioned long short term memory (LSTM) network is utilized as the regression model of the memory integral, in which the POD coefficients at a number of time steps are fed into the LSTM units, and the physical/geometrical parameters are fed into the initial hidden state of the LSTM. The reduced-order model is integrated in time using an implicit-explicit (IMEX) Runge-Kutta scheme, in which the memory term is integrated explicitly and the remaining right-hand-side term is integrated implicitly to improve the computational efficiency. Numerical results demonstrate that the RNN closure can significantly improve the accuracy and efficiency of the POD-Galerkin reduced-order model of nonlinear problems. The POD-Galerkin reduced-order model with the RNN closure is also shown to be capable of making accurate predictions, well beyond the time interval of the training data.

POD model order reduction with space-adapted snapshots for incompressible flows

We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity-reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem.

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

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

Extrapolating POD reduced-order model based on temperature gradient for unsteady heat conduction

A new proper orthogonal decomposition (POD) reduced-order model based on temperature gradient is established for heat conduction differential equation. In the most of present studies of heat conduction POD reduced-order model, the POD basis functions are extracted from temperature fields by adopting proper orthogonal decomposition or singular value decomposition (SVD). Different from conventional POD reduced-order method, this paper optimize the gradient of POD solutions by extracting the optimal orthogonal basis from temperature gradient. After getting POD basis, the POD reduced-order model is solved by using Galerkin projection. POD-Galerkin reduced-order model of two dimensional, constant property and non-steady heat conduction differential equations is established, and the accuracy of the reduced-order model under different boundary conditions is further studied. The results suggests that the relative error of POD solution is less than 0.1% while the average computing time reduces from 1.64 s to 0.02 s.

Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD–Galerkin method and a vascular shape parametrization

In this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD-Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are discussed, highlighting the computational performance of the proposed framework, as well as its capability to carry out sensitivity analysis studies, so far out of reach. In particular, a reduced-order simulation takes only a few minutes to run, resulting in computational savings of 99 % of CPU time with respect to the full-order discretization. Moreover, the error between full-order and reduced-order solutions is also studied, and it is numerically found to be less than 1 % for reduced-order solutions obtained with just O(100) online degrees of freedom.

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.

Study on a BFC-Based POD-Galerkin Reduced-Order Model for the Unsteady-State Variable-Property Heat Transfer Problem

A proper orthogonal decomposition–reduced-order model (POD-ROM) method for unsteady-state heat conduction problems with variable physical properties is developed based on the body-fitted coordinate. Two mathematically equivalent forms of the ROM are derived, but their performance is quite different for finite volume. It is found that to obtain accurate prediction by the POD-ROM, the operators of the Jacobi factor and other variables in the ROM should be consistent with those in the finite-volume method (FVM). Validated by a complicated unsteady-state heat conduction problem, the established ROM can obtain accurate prediction of the unsteady-state heat conduction problem with variable properties in which the maximum ratio of thermal conductivity reaches 333.

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.

Unsteady-state thermal calculation of buried oil pipeline using a proper orthogonal decomposition reduced-order model

The POD-Galerkin reduced-order model is established for the thermal process of the buried oil pipeline, which carries complex and inhomogeneous boundary conditions, including Dirichlet (or first-type), Neumann (or second-type), and Robin (or third-type) boundary conditions. Especially, the treatment of different boundary conditions is given in detail. The implementation of POD-Galerkin reduced-order model on unstructured mesh is also introduced. Subsequently, the thermal processes of batching transportation and commissioning in an oil pipeline are calculated by the established reduced-order model, indicating that the reduced-order model is accurate and efficient. The application of this reduced-order model may provide a convenient and rapid approach for the design and ensuring safe and economic operation of the oil transportation pipeline.

A nonlinear POD-Galerkin reduced-order model for compressible flows taking into account rigid body motions

The construction of a nonlinear reduced-order model for fluid–structure interaction problems is investigated in this paper for unsteady compressible flows excited by the rigid body motion of a structure. The reduction is achieved by means of a Galerkin projection of the Navier–Stokes equations on the first POD modes resulting from the proper orthogonal decomposition. In the first part of the paper, the projection technique is carried out on a purely aerodynamic case in order (i) to validate an efficient iterative technique based on an updated QR decomposition to compute the POD modes, and (ii) to discuss the merits of different correction methods introduced to improve the long-term stability of the reduced-order model. The second and most original part of the paper deals with the construction of the reduced set of equations which arise from the projection of the compressible Navier–Stokes equations formulated in a suitable moving frame representing the rigid body motion. The expressions of the resulting non-autonomous terms appearing in the reduced-order model have also been optimized to reduce the computational costs.