POD-Galerkin Projection Research Articles

Application of model-order reduction of non-linear time-dependent neutronics via POD-Galerkin projection and matrix discrete empirical interpolation

Presented here is an application of model-order reduction to non-linear neutronic transients. The effort described extends previous work (Elzohery and Roberts, 2021b) aimed to develop and assess a Reduced-order model (ROM) framework that can be used for different purposes, such as design optimization and propagating input parameter uncertainties like nuclear data. Here, the target applications is non-linear. The ROM was built with the use of the POD-Galerkin projection method and the matrix version of the Discrete Empirical Interpolation Method (MDEIM) for improved efficiency. Where applications of the ROM to the diffusion equation have been introduced before, this work focuses on treating the non-linearity induced by coupling with different physics. Previous work has used DEIM to approximate non-linear functions resulting from multiphysics coupling such as the temperature and the velocity field (German et al., 2019, 2022). Here, the MDEIM is used to approximate the projection of the problem operator that otherwise must be computed at each non-linear iteration within each time step, which increases the ROM cost. By applying the method to the LRA benchmark, speed ups from 6 to 40 were achieved depending on the full-order model (FOM) spatial fidelity. The maximum relative error in the assemblies power was in the order of 1×10−5%. Moreover, the ROM was parameterized by POD-greedy sampling, and a case study was performed where the kinetic data, i.e., the precursor decay constants and delayed-neutron fraction were assumed to be the model parameters of interest. For the parameterized ROM, the maximum relative error in the assembly power was 1×10−3%, The computational gain from the MDEIM-ROM was compared with that of a direct projection ROM and showed that employing the MDEIM is computationally advantageous, and the advantage grows with dimension of the problem. The method was implemented in the open-source deterministic transport code Detran (Roberts, 2014) where only access to the system operator was needed to construct the ROM, and; hence, the method is trivially invasive and could easily be implemented in many modern deterministic diffusion codes.

Model-Order Reduction of Non-Linear Time-Dependent Neutronics Via Pod-Galerkin Projection and Deim

Model-Order Reduction of Non-Linear Time-Dependent Neutronics Via Pod-Galerkin Projection and Deim

Global POD-Galerkin ROMs for Fluid Flows with Moving Solid Structures

Traditional proper orthogonal decomposition (POD)-Galerkin projection for reduced-order models (ROMs) of fluid flows is based on a fixed domain. The current method removes this limitation by considering a single combined domain of fluid and solid, whereas the original solid boundary conditions are reinforced by additional ROM terms. The combined domain requires a new inner product defined in the same combined domain to compute POD modes and the projection of equations. Solid motion is considered first as a continuous motion, then a decomposed motion, represented by a few solid modes to further reduce the computational cost. This new global approach was applied first on a two-dimensional direct numerical simulation (DNS) database of the flow past an oscillatory cylinder, and then on a three-dimensional DNS database of the flow past an oscillatory sphere. Last, the approach was applied on a high-resolution particle image velocimetry database from an experiment at higher Reynolds number of the flow past a pitching-up NACA0012 airfoil. The ROMs derived by the global POD-Galerkin projection approach, for all three moving-boundary cases, have shown adequate accuracy in the reconstruction of flow fields and the prediction of key aerodynamic features while keeping very low computational costs.

Modeling neutronic transients with Galerkin projection onto a greedy-sampled, POD subspace

Presented here is an application of POD-Galerkin projection to develop a reduced-order-model (ROM) that approximates the solution of the multi-group, time-dependent, diffusion equation with a lower computational cost. Such reduced models are often sought in the context of parametric studies and uncertainty quantification, which require successive evaluation of the full-order model (FOM). To construct the ROM efficiently, greedy sampling was used to generate a reduced subspace that represents the entire parameter (here, cross-sections) domain. By using this subspace, the ROM is expected to yield accurate predictions when used in place of the FOM for uncertainty quantification conducted, e.g., with direct Monte-Carlo sampling. The TWIGL benchmark was used here for illustration. A ROM constructed with a POD basis of rank 10 was able to reproduce the core power with a maximum RMS error on the order of 10–6, and for the precursors concentration it was 10–9. Moreover, the statistical moments obtained from the ROM were in good agreement with those of the full-order-model. The mean relative error of the power sample mean prediction was about 4 ×10–6 for both the ramp and the step perturbations. In terms of the computational efficiency, the ROM time, including the projection, is approximately four-times faster than the full-order-model.

Data-driven POD-Galerkin reduced order model for turbulent flows

In this work we present a Reduced Order Model which is specifically designed to deal with turbulent flows in a finite volume setting. The method used to build the reduced order model is based on the idea of merging/combining projection-based techniques with data-driven reduction strategies. In particular, the work presents a mixed strategy that exploits a data-driven reduction method to approximate the eddy viscosity solution manifold and a classical POD-Galerkin projection approach for the velocity and the pressure fields, respectively. The newly proposed reduced order model has been validated on benchmark test cases in both steady and unsteady settings with Reynolds up to Re=O(105).

Parametric study of flow-induced vibrations in cylinder arrays under single-phase fluid cross flows using POD-ROM

Modeling numerically Flow-Induced Vibrations in heat exchangers at a microscopic scale requires high computational resources and time which are still unreachable. Therefore model reduction is investigated in the present work in order to address the issue of simulation computational time reduction. In the framework of POD-Galerkin projection methods, the purpose is to propose optimal a posteriori reduction strategies enabling error control on approximation as well as Reduced-Order Model (ROM) interpolation to deal with sensitivity analysis of solutions to parameter perturbations. A multi-phase fluid–solid POD-Galerkin-based method is proposed for modeling flows and vibrations in cylinder arrangements under single-phase fluid cross-flows. Moreover a single-POD basis method is evaluated in the context of ROM interpolation. This work is a first step in the development of robust ROM describing fluid and solid dynamics in the presence of turbulence, heat transfer effects and large magnitude structure displacements and deformations.

Reduced order modelling of flow and mixing in an automobile HVAC system using proper orthogonal decomposition

Proper Orthogonal Decomposition (POD) is applied for the development of a Reduced Order Model (ROM) of the mixing chamber of an automobile Heating, Ventilating, and Air-conditioning (HVAC) unit to accurately predict the enthalpy flows at the outlets for control purposes.The proposed method collects snapshots of different parametric variations of the mixing chamber state to construct the POD subspace. The data is gathered from numerical simulations using the open source tool OpenFOAM. A generic demonstrator is modelled, which represents a simplified HVAC geometry and consists of a two-dimensional cavity, two inlets and three outlets. Two approaches to obtain the weighting coefficients for unknown operating conditions are compared with each other, namely the POD-Galerkin projection (POD-G) and a coupled POD and flow resistance (cPOD-FR) model. Within the cPOD-FR procedure, the weighting coefficients are estimated using the calculated airflow from the flow resistance model (FR). Subsequently, the approximated coefficients are applied to reconstruct the enthalpy flow rates.It could be shown that the accuracy of the cPOD-FR procedure delivers most promising results for a ROM of the enthalpy flow rates. The procedure is capable to predict the integrated enthalpy flow rates at the outlets for 90% of the evaluated test cases within an error margin of less than 5%.As a major outcome, the applicability of POD methods to generate ROMs is shown and a comprehensive comparison of weighting procedures for HVAC applications is given. This approach may provide a systematic framework to accurately predict the enthalpy flow rates at the outlets of HVAC units for real-time model-based control at low computational effort.

Reduced-Order Model for Unsteady Compressible Flow Based on POD-Galerkin Projection

Italic textIn this paper, the reduced-order model (ROM) of unsteady compressible flow based on POD-Galerkin projection has been investigated. The Euler equation formulated with the conservative variables for compressible flow has been reformulated using modified primitive variables. The POD modes are computed using snapshot method and then an explicit quadratic ROM is constructed by applying the Galerkin projection to the modified Euler equation. Because of lacking any dissipation in POD-Galerkin projection, the flow calibration method is introduced to account for the numerical dissipation to stabilize the ROM. At last, the NACA0012 airfoil undergoing pitch harmonic oscillating in Mach number Mach = 0.755 is calculated as a test case. For this flow configuration, the calibrated coefficients of the ROM are almost the same as the initial guess which comes from POD-Galerkin projection; the differences are mostly focused on the linear Lij coefficients and quadratic Qijk coefficients.

Fast Prediction Method for Steady‐State Heat Convection

AbstractA reduced model by proper orthogonal decomposition (POD) and Galerkin projection methods for steady‐state heat convection is established on a nonuniform grid. It was verified by thousands of examples that the results are in good agreement with the results obtained from the finite volume method. This model can also predict the cases where model parameters far exceed the sample scope. Moreover, the calculation time needed by the model is much shorter than that needed for the finite volume method. Thus, the nonuniform POD‐Galerkin projection method exhibits high accuracy, good suitability, and fast computation. It has universal significance for accurate and fast prediction. Also, the methodology can be applied to more complex modeling in chemical engineering and technology, such as reaction and turbulence.

Low-dimensional analysis, using POD, for two mixing layer–wake interactions

The mixing layer–wake interaction is studied experimentally in the framework of two flow configurations. For the first one, the initial conditions of the mixing layer are modified by using a thick trailing edge, a wake effect is therefore superimposed to the mixing layer from its beginning (blunt trailing edge). In the second flow configuration, a canonical mixing layer is perturbed in its asymptotic region by the wake of a cylinder arranged perpendicular to the plane of the mixing layer. These interactions are analyzed mainly by using two-point velocity correlations and the proper orthogonal decomposition (POD). These two flow configurations differ by the degree of complexity they involve: the former is mainly 2D while the latter is highly 3D. The blunt trailing edge configuration is analyzed by using rakes of hot wire probes. This flow configuration is found to be considerably different when compared to a conventional mixing layer. It appears in particular that the scale of the large structures depends only on the trailing edge thickness and does not grow in its downstream evolution. A criterion, based on POD, is proposed in order to separate wake–mixing layer dominant areas of the downstream evolution of the flow. The complex 3D dynamical behaviour resulting from the interaction between the canonical plane mixing layer and the wake of a cylinder is investigated using data arising from particle image velocimetry measurements. An analysis of the velocity correlations shows different length scales in the regions dominated by wake like structures and shear layer type structures. In order to characterize the particular organization in the plane of symmetry, a POD-Galerkin projection of the Navier–Stokes equations is performed in this plane. This leads to a low-dimensional dynamical system that allows the analysis of the relationship between the dominant frequencies to be performed. A reconstruction of the dominant periodic motion suspected from previous studies is then retrieved.