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Proper Orthogonal Decomposition (POD) basis interpolation on Grassmann manifolds has been successfully applied to problems of parametric model order reduction (pMOR). In this work we address the necessary stability conditions for the interpolation, all defined from strong mathematical background. A first condition concerns the domain of definition of the logarithm map. Second, we show how the stability of interpolation can be lost if certain geometrical requirements are not satisfied by making a concrete elucidation of the local character of linearization. To this effect, we draw special attention to the Grassmannian exponential map and the optimal injectivity condition of this map, related to the cut–locus of Grassmann manifolds. From this, an explicit stability condition is established and can be directly used to determine the loss of injectivity in practical pMOR applications. A third stability condition is formulated when increasing the number p of POD modes, deduced from the principal angles of subspaces of different dimensions p . Definition of this condition leads to an understanding of the non-monotonic oscillatory behavior of the Reduced Order Model (ROM) error-norm with respect to the number of POD modes, and on the contrary, the well-behaved monotonic decrease of the error-norm in the two numerical examples presented herein. We have chosen to perform pMOR in hyperelastic structures using a non-intrusive approach for inserting the interpolated spatial POD ROM basis in a commercial FEM code. The accuracy is assessed by a posteriori error norms defined using the ROM FEM solution and its high-fidelity counterpart simulation. Numerical studies successfully ascertained and highlighted the implication of stability conditions which are general and can be applied to a variety of other linear or nonlinear problems involving parametrized ROMs generation based on POD basis interpolation on Grassmann manifolds.

Parametric model order reduction techniques often struggle to accurately represent transport-dominated phenomena due to a slowly decaying Kolmogorov n -width. To address this challenge, we propose a non-intrusive, data-driven methodology that combines the shifted proper orthogonal decomposition (POD) with deep learning. Specifically, the shifted POD technique is utilized to derive a high-fidelity, low-dimensional model of the flow, which is subsequently utilized as input to a deep learning framework to forecast the flow dynamics under various temporal and parameter conditions. The efficacy of the proposed approach is demonstrated through the analysis of one- and two-dimensional wildland fire models with varying reaction rates, and its error is compared with the error of other similar methods. The results indicate that the proposed approach yields reliable results within the percent range, while also enabling rapid prediction of system states within seconds.

The analysis of the unsteady flow field in axial compressor cascade is conducted using methods such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). Data on the unsteady flow field of the Stage-35 compressor cascade are acquired via computational fluid dynamics (CFD) simulations and subsequently processed using POD and DMD for dimensionality reduction. Using singular value decomposition, the POD technique identifies the dominant modes, showing that the first nine modes account for 99% of the energy in the flow field, thus highlighting the primary flow structures. On the other hand, the DMD approach isolates the periodic and high-frequency dynamics within the flow field by decomposing the dynamic modes, effectively identifying fine variations in the unsteady flow. The study examines the flow field at three distinct moments within an unsteady cycle, specifically at 1/4T, 1/2T, and 3/4T, reconstructing the flow field at each instance and performing root mean square error analysis. Reconstruction results and error analysis demonstrate that the POD method excels at reconstructing low-frequency features, whereas the DMD method accurately identifies the unsteady dynamic aspects of the flow field, excelling in resolving high-frequency details. Both methods demonstrate high feasibility regarding the accuracy and efficiency of flow field reconstruction.

In this paper, an approach based on the synergistic use of proper orthogonal decomposition and Kalman filtering is proposed for the online health monitoring of damaged structures. The reduced-order model of a structure is obtained during an (offline) initial training stage of monitoring; afterward, effective estimations of a possible structural damage are provided online by tracking the evolution in time of stiffness parameters and projection bases handled in the model order reduction procedure. Such tracking is accomplished via two Kalman filters: a first (extended) one to deal with the time evolution of a joint state vector, gathering the reduced-order state and the stiffness terms degraded by damage; a second one to deal with the update of the reduced-order model in case of damage evolution. Both filters exploit the information conveyed by measurements of the structural response to the external excitations. Results are reported for a (pseudo-experimental) benchmark test on an eight-story shear building. Capability and performance of the proposed approach are assessed in terms of tracked variation of the stiffness terms of the reduced-order model, identified damage location and speed-up of the whole health monitoring procedure.

The objective is to study the vortical structural behaviors of a transient pitching hydrofoil and their effects on the hydrodynamic performance. The pitching motion of the hydrofoil is set to pitch up with an almost constant rate from 5° to 15° and then back to 5°, with the Reynolds number 4.4×10 5 and the frequency 2 Hz. The results show that the main coherent structures around the pitching hydrofoil include small-scale laminar separation bubble (LSB), large-scale second vortex (SV) and trailing edge vortex (TEV) which are all vortical. The relationship between the vortical structure and the lift is investigated with the finite-domain impulse theory. It indicates that the major part of the lift is contributed by the LSB, whereas the shedding and the formation of the SV and TEV cause the fluctuation of the lift. The proper orthogonal decomposition (POD) method is applied to capture the most energetic modes, revealing that the LSB mode occupies a large amount of energy in the flow field. The dynamic mode decomposition (DMD) method accurately extracts the dominant frequency and modal characteristics, with the first mode corresponding to the mean flow, the second mode corresponding to the LSB structure and the third and fourth modes corresponding to the vortex shedding.

We propose a technique for performing spectral (in time) analysis of spatially-resolved flowfield data, without needing any temporal resolution or information. This is achieved by combining projection-based reduced-order modeling with spectral proper orthogonal decomposition. In this method, space-only proper orthogonal decomposition is first performed on velocity data to identify a subspace onto which the known equations of motion are projected, following standard Galerkin projection techniques. The resulting reduced-order model is then utilized to generate time-resolved trajectories of data. Spectral proper orthogonal decomposition (SPOD) is then applied to this model-generated data to obtain a prediction of the spectral content of the system, while predicted SPOD modes can be obtained by lifting back to the original velocity field domain. This method is first demonstrated on a forced, randomly generated linear system, before being applied to study and reconstruct the spectral content of two-dimensional flow over two collinear flat plates perpendicular to an oncoming flow. At the range of Reynolds numbers considered, this configuration features an unsteady wake characterized by the formation and interaction of vortical structures in the wake. Depending on the Reynolds number, the wake can be periodic or feature broadband behavior, making it an insightful test case to assess the performance of the proposed method. In particular, we show that this method can accurately recover the spectral content of periodic, quasi-periodic, and broadband flows without utilizing any temporal information in the original data. To emphasize that temporal resolution is not required, we show that the predictive accuracy of the proposed method is robust to using temporally-subsampled data.

This paper deals with developing a fast and robust numerical formulation to simulate a system of fractional PDEs. At the first stage, the time variable is approximated by a finite difference method with first-order accuracy. At the second stage, the spectral Galerkin method based upon the fractional Jacobi polynomials is employed to discretize the spatial variables. We apply a reduced-order method based upon the proper orthogonal decomposition technique to decrease the utilized computational time. The unconditional stability property and the order of convergence of the new technique are analyzed in detail. The proposed numerical technique is well known as the reduced-order spectral Galerkin scheme. Furthermore, by employing the Newton–Raphson method and semi-implicit schemes, the proposed method can be used for solving linear and nonlinear ODEs and PDEs. Finally, some examples are provided to confirm the theoretical results.

The PODMT3DMS-Tool, which consists of proper orthogonal decomposition (POD) linked to the Modular Transport 3-Dimensional Multi Species (MT3DMS) code for nitrate simulation in groundwater, is introduced. POD, as a statistical technique, reduces a large amount of information produced by the MT3DMS model to provide the main components of the PODMT3DMS-Tool, i.e., space- and time-dependent terms of nitrate. The low-dimensional components represent time- and space-dependent factors in the aquifer response such as hydraulic, hydrogeological and water quality variables represented in the simulation using the MT3DMS model. The PODMT3DMS-Tool is thus a combined statistical and conceptual model with a simple structure and comparable accuracy to MT3DMS. Practical application of the PODMT3DMS-Tool to the Karaj Aquifer in Iran for a period of 6 years revealed agreement between nitrate concentrations simulated by the PODMT3DMS-Tool and MT3DMS, with a mean absolute error of less than 0.5 mg/L in most parts of the aquifer. Moreover, the PODMT3DMS-Tool needed only about 10% of the calculation time required by MT3DMS. The PODMT3DMS-Tool can be used to predict nitrate concentration in the Karaj Aquifer, while its simplicity also makes it potentially useful for other water resources problems.

The comparison of different time-varying three-dimensional hemodynamic data (4D) is a formidable task. The purpose of this study is to investigate the potential of the proper orthogonal decomposition (POD) for a quantitative assessment. The complex spatial-temporal flow information was analyzed using proper orthogonal decomposition to reduce the complexity of the system. PC-MRI blood flow measurements and computational fluid dynamic simulations of two subject-specific IAs were used to compare the different flow modalities. The concept of Modal Assurance Criterion (MAC) provided a further detailed objective characterization of the most energetic individual modes. The most energetic flow modes were qualitatively compared by visual inspection. The distribution of the kinetic energy on the modes was used to quantitatively compare pulsatile flow data, where the most energetic mode was associated to approximately 90% of the total kinetic energy. This distribution was incorporated in a single measure, termed spectral entropy, showing good agreement especially for Case 1. The proposed quantitative POD-based technique could be a valuable tool to reduce the complexity of the time-dependent hemodynamic data and to facilitate an easy comparison of 4D flows, e.g., for validation purposes.

PurposeThe space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.Design/methodology/approachThe fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.FindingsA fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.Originality/valueThe spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.