Proper Orthogonal Decomposition Reduced-order Model Research Articles

POD reduced-order model for steady laminar flow based on the body-fitted coordinate

ABSTRACTCurrent reduced-order models (ROM) by the proper orthogonal decomposition (POD) for fluid flow are on a fixed domain. We expect to combine the POD model reduction and the body-fitted coordinate methods so that the POD reduced-order model for fast calculations of fluid flow can be expanded to domains with different shapes. In this paper, the principle is studied for obtaining basis functions of body-fitted POD model for fluid flow. Three methods to obtain basis functions are discussed theoretically. It is pointed out that contravariant components must be utilized to construct the sample matrix for the fast calculation of steady laminar flow in different-shape domains. Taking this principle into account, a standard POD ROM for steady laminar flow problems is established. It is found that the model appears instable when the geometric parameters vary a lot. To overcome this problem, supplementary matrix is introduced to establish a calibrated model with much better accuracy and stability. The test cases show that the calibrated POD reduced-order model, with higher accuracy and stability than the standard one, can fulfill the fast calculations of steady laminar flow on different-shape domains.

A proposed Fast algorithm to construct the system matrices for a reduced-order groundwater model

Past research has demonstrated that a reduced-order model (ROM) can be two-to-three orders of magnitude smaller than the original model and run considerably faster with acceptable error. A standard method to construct the system matrices for a ROM is Proper Orthogonal Decomposition (POD), which projects the system matrices from the full model space onto a subspace whose range spans the full model space but has a much smaller dimension than the full model space. This projection can be prohibitively expensive to compute if it must be done repeatedly, as with a Monte Carlo simulation. We propose a Fast Algorithm to reduce the computational burden of constructing the system matrices for a parameterized, reduced-order groundwater model (i.e. one whose parameters are represented by zones or interpolation functions). The proposed algorithm decomposes the expensive system matrix projection into a set of simple scalar-matrix multiplications. This allows the algorithm to efficiently construct the system matrices of a POD reduced-order model at a significantly reduced computational cost compared with the standard projection-based method. The developed algorithm is applied to three test cases for demonstration purposes. The first test case is a small, two-dimensional, zoned-parameter, finite-difference model; the second test case is a small, two-dimensional, interpolated-parameter, finite-difference model; and the third test case is a realistically-scaled, two-dimensional, zoned-parameter, finite-element model. In each case, the algorithm is able to accurately and efficiently construct the system matrices of the reduced-order model.

Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems

The proper orthogonal decomposition reduced-order model (POD-ROM) has been widely used as a computationally efficient surrogate model in large-scale numerical simulations of complex systems. However, when it is applied to a Hamiltonian system, a naive application of the POD method can destroy the Hamiltonian structure in the reduced-order model. In this paper, we develop a new reduced-order modeling approach for Hamiltonian systems, which modifies the Galerkin projection-based POD-ROM so that the appropriate Hamiltonian structure is preserved. Since the POD truncation can degrade the approximation of the Hamiltonian function, we propose to use a POD basis from shifted snapshots to improve the approximation to the Hamiltonian function. We further derive a rigorous a priori error estimate for the structure-preserving ROM and demonstrate its effectiveness in several numerical examples. This approach can be readily extended to dissipative Hamiltonian systems, port-Hamiltonian systems, etc.

POD-Galerkin method for finite volume approximation of Navier–Stokes and RANS equations

Numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications. Reduced Order Model (ROM) can be employed whenever fast simulations are required, or in general, whenever a trade-off between computational cost and solution accuracy is a preeminent issue as in process optimization and control. In this work, the efforts have been put to develop a ROM for Computational Fluid Dynamics (CFD) application based on Finite Volume approximation, starting from the results available in turbulent Reynold-Averaged Navier–Stokes simulations in order to enlarge the application field of Proper Orthogonal Decomposition-Reduced Order Model (POD-ROM) technique to more industrial fields. The approach is tested in the classic benchmark of the numerical simulation of the 2D lid-driven cavity. In particular, two simulations at Re=103 and Re=105 have been considered in order to assess both a laminar and a turbulent case. Some quantities have been compared with the Full Order Model in order to assess the performance of the proposed ROM procedure i.e., the kinetic energy of the system and the reconstructed quantities of interest (velocity, pressure and turbulent viscosity). In addition, for the laminar case, the comparison between the ROM steady-state solution and the data available in literature has been presented. The results have turned out to be very satisfactory both for the accuracy and the computational times. As a major outcome, the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent POD-Galerkin methods.

International journal of computational fluid dynamics real-time prediction of unsteady flow based on POD reduced-order model and particle filter

ABSTRACTAn integrated method consisting of a proper orthogonal decomposition (POD)-based reduced-order model (ROM) and a particle filter (PF) is proposed for real-time prediction of an unsteady flow field. The proposed method is validated using identical twin experiments of an unsteady flow field around a circular cylinder for Reynolds numbers of 100 and 1000. In this study, a PF is employed (ROM-PF) to modify the temporal coefficient of the ROM based on observation data because the prediction capability of the ROM alone is limited due to the stability issue. The proposed method reproduces the unsteady flow field several orders faster than a reference numerical simulation based on Navier–Stokes equations. Furthermore, the effects of parameters, related to observation and simulation, on the prediction accuracy are studied. Most of the energy modes of the unsteady flow field are captured, and it is possible to stably predict the long-term evolution with ROM-PF.

Parameter Estimation for a 2D Tidal Model with POD 4D VAR Data Assimilation

Combining the proper orthogonal decomposition (POD) reduced order method and 4D VAR (four-dimensional Variational) data assimilation method with a two-dimensional (2D) tidal model, a model is constructed to simulate theM2tide in the Bohai, Yellow, and East China Seas (BYECS). This model consists of two submodels: the POD reduced order forward model is used to simulate the tides, while its adjoint model is used to optimize the control variables. Numerical experiment is carried out to assimilate the harmonic constants, which are derived from TOPEX/Poseidon (T/P) altimeter data, into the 2D tidal model through optimizing the initial values and the temporally and spatially varying open boundary conditions (OBCs). The absolute mean difference between the model results and observations is 3.2 cm and 2.9∘for amplitude and phase-lag, respectively, better than the results of Lu and Zhang (2006), suggesting that the construction of the POD reduced order model and the inversion of control variables are successful.

Stability analysis of the POD reduced order method for solving the bidomain model in cardiac electrophysiology

In this paper we show the numerical stability of the Proper Orthogonal Decomposition (POD) reduced order method used in cardiac electrophysiology applications. The difficulty of proving the stability comes from the fact that we are interested in the bidomain model, which is a system of degenerate parabolic equations coupled to a system of ODEs representing the cell membrane electrical activity. The proof of the stability of this method is based on a priori estimates controlling the gap between the reduced order solution and the Galerkin finite element one. We present some numerical simulations confirming the theoretical results. We also combine the POD method with a time splitting scheme allowing a faster solving of the bidomain problem and show numerical results. Finally, we conduct numerical simulation in 2D illustrating the stability of the POD method in its sensitivity to the ionic model parameters. We also perform 3D simulation using a massively parallel code. We show the computational gain using the POD reduced order model. We also show that this method has a better scalability than the full finite element method.

A stabilized proper orthogonal decomposition reduced-order model for large scale quasigeostrophic ocean circulation

In this paper, a stabilized proper orthogonal decomposition (POD) reduced-order model (ROM) is presented for the barotropic vorticity equation. We apply the POD-ROM model to mid-latitude simplified oceanic basins, which are standard prototypes of more realistic large-scale ocean dynamics. A mode dependent eddy viscosity closure scheme is used to model the effects of the discarded POD modes. A sensitivity analysis with respect to the free eddy viscosity stabilization parameter is performed for various POD-ROMs with different numbers of POD modes. The POD-ROM results are validated against the Munk layer resolving direct numerical simulations using a fully conservative fourth-order Arakawa scheme. A comparison with the standard Galerkin POD-ROM without any stabilization is also included in our investigation. Significant improvements in the accuracy over the standard Galerkin model are shown for a four-gyre ocean circulation problem. This first step in the numerical assessment of the POD-ROM shows that it could represent a computationally efficient tool for large scale oceanic simulations over long time intervals.

A POD reduced order model for resolving angular direction in neutron/photon transport problems

This article presents the first Reduced Order Model (ROM) that efficiently resolves the angular dimension of the time independent, mono-energetic Boltzmann Transport Equation (BTE). It is based on Proper Orthogonal Decomposition (POD) and uses the method of snapshots to form optimal basis functions for resolving the direction of particle travel in neutron/photon transport problems. A unique element of this work is that the snapshots are formed from the vector of angular coefficients relating to a high resolution expansion of the BTE's angular dimension. In addition, the individual snapshots are not recorded through time, as in standard POD, but instead they are recorded through space. In essence this work swaps the roles of the dimensions space and time in standard POD methods, with angle and space respectively.It is shown here how the POD model can be formed from the POD basis functions in a highly efficient manner. The model is then applied to two radiation problems; one involving the transport of radiation through a shield and the other through an infinite array of pins. Both problems are selected for their complex angular flux solutions in order to provide an appropriate demonstration of the model's capabilities. It is shown that the POD model can resolve these fluxes efficiently and accurately. In comparison to high resolution models this POD model can reduce the size of a problem by up to two orders of magnitude without compromising accuracy. Solving times are also reduced by similar factors.

Feedback stabilization of an oscillating vertical cylinder by POD Reduced-Order Model

The objective is to demonstrate the use of reduced-order models (ROM) based on proper orthogonal decomposition (POD) to stabilize the flow over a vertically oscillating circular cylinder in the laminar regime (Reynolds number equal to 60). The 2D Navier-Stokes equations are first solved with a finite element method, in which the moving cylinder is introduced via an ALE method. Since in fluid-structure interaction, the POD algorithm cannot be applied directly, we implemented the fictitious domain method of Glowinski et al. [1] where the solid domain is treated as a fluid undergoing an additional constraint. The POD-ROM is classically obtained by projecting the Navier-Stokes equations onto the first POD modes. At this level, the cylinder displacement is enforced in the POD-ROM through the introduction of Lagrange multipliers. For determining the optimal vertical velocity of the cylinder, a linear quadratic regulator framework is employed. After linearization of the POD-ROM around the steady flow state, the optimal linear feedback gain is obtained as solution of a generalized algebraic Riccati equation. Finally, when the optimal feedback control is applied, it is shown that the flow converges rapidly to the steady state. In addition, a vanishing control is obtained proving the efficiency of the control approach.

Fast thermal simulation of a heated crude oil pipeline with a BFC-Based POD reduced-order model

A proper orthogonal decomposition (POD) Galerkin reduced-order model (ROM) for unsteady-state heat conduction problems is proposed based on body-fitted coordinate (BFC), and the discrete format of spatial derivative of basis function which could enhance the calculation precision is employed in the present paper. Applying the POD reduced-order model into the fast thermal calculation of oil pipeline, such as the batch transportation of cold and hot oil as well as pipeline shutdown, the computing speed of POD reduced-order model is about 259 times and 177 times faster than that of traditional finite volume method (FVM) respectively. And from the calculation results, the reduced-order model is proved to enjoy good precision. Thus the reduced-order model holds great engineering application value and would be a very helpful tool to solve the time-consuming problem involved in the optimal design and risk assessment of oil pipeline.

Adaptive POD basis computation for parametrized nonlinear systems using optimal snapshot location

The construction of reduced-order models for parametrized partial differential systems using proper orthogonal decomposition (POD) is based on the information of the so-called snapshots. These provide the spatial distribution of the nonlinear system at discrete parameter and/or time instances. In this work a strategy is used, where the POD reduced-order model is improved by choosing additional snapshot locations in an optimal way; see Kunisch and Volkwein (ESAIM: M2AN, 44:509–529, 2010). These optimal snapshot locations influences the POD basis functions and therefore the POD reduced-order model. This strategy is used to build up a POD basis on a parameter set in an adaptive way. The approach is illustrated by the construction of the POD reduced-order model for the complex-valued Helmholtz equation.

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.

Proper Orthogonal Decomposition Reduced-Order Model for Nonlinear Aeroelastic Oscillations

In this study, the proper orthogonal decomposition method in conjunction with a Galerkin projection scheme is employed to solve the title problem of a fluttering plate in both two and three dimensions undergoing supersonic flow using von Kármán’s large deflection plate theory and quasi-steady supersonic aerodynamic theory. Proper-orthogonal-decomposition-based reduced-order models are constructed by using the responses as snapshots from the conventional Galerkin method, which also plays a role as a reference method in this paper. Results for the buckled, limit cycle oscillation, and chaotic responses of the simply supported plate are presented and compared with the Galerkin solutions. Numerical examples demonstrate that the proper orthogonal decomposition reduced-order model permits a much lower-dimensional model as compared to that obtainable via the Galerkin approach. For example, for a plate length-to-width ratio equal to 4, only two proper orthogonal decomposition modes are required to describe the panel oscillation with good accuracy. This produces a reduction in the computational time to less than 3 s in comparison with almost 900 s when using the 16 modes required to obtain the same accuracy with the Galerkin approach.

Variational multiscale proper orthogonal decomposition: Navier‐stokes equations

We develop a variational multiscale proper orthogonal decomposition (POD) reduced‐order model (ROM) for turbulent incompressible Navier‐Stokes equations. Under two assumptions on the underlying finite element approximation and the generation of the POD basis, the error analysis of the full discretization of the ROM is presented. All error contributions are considered: the spatial discretization error (due to the finite element discretization), the temporal discretization error (due to the backward Euler method), and the POD truncation error. Numerical tests for a three‐dimensional turbulent flow past a cylinder at Reynolds number show the improved physical accuracy of the new model over the standard Galerkin and mixing‐length POD ROMs. The high computational efficiency of the new model is also showcased. Finally, the theoretical error estimates are confirmed by numerical simulations of a two‐dimensional Navier‐Stokes problem. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 641–663, 2014

Model reduction of a coupled numerical model using proper orthogonal decomposition

Summary Numerical models for variable-density flow and solute transport (VDFST) are widely used to simulate seawater intrusion and related problems. The mathematical model for VDFST is a coupled nonlinear dynamical system, so the numerical discretizations in time and space are usually required to be as fine as possible. As a result, fine-scale transient models require large computational time, which is a disadvantage for state estimation, forward prediction or model inversion. The purpose of this research is to develop mathematical and numerical methods to simulate VDFST via a model order reduction technique called Proper Orthogonal Decomposition (POD) designed for nonlinear dynamical systems. POD was applied to extract leading “model features” (basis functions) through singular value decomposition (SVD) from observational data or simulations (snapshots) of high-dimensional systems. These basis functions were then used in the Galerkin projection procedure that yielded low-dimensional (reduced-order) models. The original full numerical models were also discretized by the Galerkin Finite-Element Method (GFEM). The implementation of the POD reduced-order method was straightforward when applied to the full order model to the complex model. The developed GFEM-POD model was applied to solve two classic VDFST cases, the Henry problem and the Elder problem, in order to investigate the accuracy and efficiency of the POD model reduction method. Once the snapshots from full model results are obtained, the reduced-order model can reproduce the full model results with acceptable accuracy but with less computational cost in comparison with the full model, which is useful for model calibration and data assimilation problems. We found that the accuracy and efficiency of the POD reduced-order model is mainly determined by the optimal selection of snapshots and POD bases. Validation and verification experiments confirmed our POD model reduction procedure.

Nonlinear model reduction for unsteady discontinuous flows

We develop a new nonlinear reduced-order model (ROM) based on proper orthogonal decomposition (POD), which can be used for quantitative simulation of not only smooth flows, but also flows with strong discontinuities. The new model is derived using a Galerkin projection of the fully conservative, nonlinear discretized 2-D Euler equations onto the POD basis constructed for each conservative variable. This approach can be interpreted as a variant of the spectral method with a truncated set of basis functions. A system of ordinary differential equations (ODEs) derived using this model reduction technique resembles the major nonlinear and conservation properties of the original discretized Euler equations. The new reduced-order model also preserves the stability properties of the discrete full-order model equations, so that no additional stabilization is required unlike conventional POD-based models that are susceptible to numerical instabilities. The performance of the new POD ROM is evaluated for 2-D compressible unsteady inviscid flows over a wide range of Mach numbers including trans- and supersonic flows with strong shock waves.

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.

POD reduced-order unstructured mesh modeling applied to 2D and 3D fluid flow

A new scheme for implementing a reduced order model for complex mesh-based numerical models (e.g. finite element unstructured mesh models), is presented. The matrix and source term vector of the full model are projected onto the reduced bases. The proper orthogonal decomposition (POD) is used to form the reduced bases. The reduced order modeling code is simple to implement even with complex governing equations, discretization methods and nonlinear parameterizations. Importantly, the model order reduction code is independent of the implementation details of the full model code. For nonlinear problems, a perturbation approach is used to help accelerate the matrix equation assembly process based on the assumption that the discretized system of equations has a polynomial representation and can thus be created by a summation of pre-formed matrices.In this paper, by applying the new approach, the POD reduced order model is implemented on an unstructured mesh finite element fluid flow model, and is applied to 3D flows. The error between the full order finite element solution and the reduced order model POD solution is estimated. The feasibility and accuracy of the reduced order model applied to 3D fluid flows are demonstrated.

Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison

This paper puts forth two new closure models for the proper orthogonal decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of 3D turbulent flow past a circular cylinder at Re=1000. Five criteria are used to judge the performance of the proper orthogonal decomposition reduced-order models: the kinetic energy spectrum, the mean velocity, the Reynolds stresses, the root mean square values of the velocity fluctuations, and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models are the most accurate.