Article
Mathematics
Jan S. Hesthaven, Cecilia Pagliantini, Gianluigi Rozza
Summary: The article discusses the latest advances in reduced basis methods for time-dependent problems, including structure-preserving reduced order models, localized and adaptive methods based on nonlinear approximations, and data-driven techniques based on non-intrusive reduced order models. The article provides a comparative discussion that offers insights into the advantages, disadvantages, and potential issues of these methods.
Article
Computer Science, Interdisciplinary Applications
Elizabeth H. Krath, Forrest L. Carpenter, Paul G. A. Cizmas, David A. Johnston
Summary: This study introduces a novel, more efficient reduced-order model for compressible flows based on proper orthogonal decomposition (POD). By using specific volume instead of density, the coefficients of the system of ODEs in the reduced-order model were pre-computed. Various methods were used to enhance ODE solver stability. Validation was done for two cases, showing a speedup exceeding four orders of magnitude compared to the full-order model.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Sourav Dutta, Matthew W. Farthing, Emma Perracchione, Gaurav Savant, Mario Putti
Summary: This work develops Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with Radial Basis Function (RBF) interpolation method for efficient reduced order models in time-dependent problems. The performance is compared with traditional nonlinear POD (NPOD) model, different greedy algorithms are studied, and a new psr-greedy algorithm is proposed. Experimental results are presented for realistic 2D shallow water flow applications.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Engineering, Multidisciplinary
Margarita Chasapi, Pablo Antolin, Annalisa Buffa
Summary: This work presents a reduced order modeling framework for parameterized second-order linear elliptic partial differential equations on unfitted geometries. Efficient projection-based models utilizing reduced basis method and discrete empirical interpolation are proposed, which can handle geometrical parameters in unfitted domain discretizations. The proposed method is computationally efficient and accurate, agnostic to the underlying discretization choice. Numerical experiments on benchmark problems demonstrate significant reduction of online computational cost compared to standard ROMs with the same level of accuracy. The methodology is also applicable to three-dimensional geometries of linear elastic problems.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Xi Li, Yan Luo, Minfu Feng
Summary: In this paper, an efficient proper orthogonal decomposition based reduced-order model (POD-ROM) for nonstationary Stokes equations is proposed. The new scheme combines the classical projection method with POD technique, resulting in low computational costs and improved efficiency.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Engineering, Mechanical
Giorgio Gobat, Andrea Opreni, Stefania Fresca, Andrea Manzoni, Attilio Frangi
Summary: In this study, the Proper Orthogonal Decomposition (POD) method is applied to efficiently simulate the nonlinear behavior of Micro-Electro-Mechanical-Systems (MEMS) in various scenarios involving geometric and electrostatic nonlinearities. The POD method reduces the polynomial terms up to cubic order associated with large displacements through exact projection onto a low-dimensional subspace spanned by the Proper Orthogonal Modes (POMs). Electrostatic nonlinearities are modeled using precomputed manifolds based on the amplitudes of the electrically active POMs. The reliability of the assumed linear trial space is extensively tested in challenging applications such as resonators, micromirrors, and arches with internal resonances. Comparisons are made between the periodic orbits computed with POD and the invariant manifold approximated with Direct Normal Form approaches, highlighting the reliability and remarkable predictive capabilities of the technique, particularly in terms of estimating the frequency response function of selected output quantities of interest.
MECHANICAL SYSTEMS AND SIGNAL PROCESSING
(2022)
Article
Energy & Fuels
Jemimah-Sandra Samuel, Ann Helen Muggeridge
Summary: This study presents and compares two new non-intrusive reduced order frameworks for faster modeling of gas reservoirs with time-varying production. One method extends POD with RBF, while the other uses an autoencoder to estimate flow dynamics in hyperspace. Both frameworks are trained using sample outputs from commercial reservoir simulations and can provide reasonable forecasts up to 300 times faster than conventional simulations.
JOURNAL OF PETROLEUM SCIENCE AND ENGINEERING
(2022)
Article
Chemistry, Multidisciplinary
Yongxin Shi, Zhao Ke, Wei Sun, Peng Zhang, Qiang Yang, Kuo Tian
Summary: This study proposes a fast vibration reduction optimization approach accelerated by the global proper orthogonal decomposition (POD) reduced-order model (ROM) to increase the efficiency of frequency response analysis and vibration reduction optimization of complex thin-walled shells. The global POD ROM is adaptively updated using the CV-Voronoi sequence sampling method to achieve higher global prediction accuracy. The fast vibration reduction optimization is performed by combining the surrogate-based efficient global optimization (EGO) method and the proposed ROM, achieving high prediction accuracy and efficiency.
APPLIED SCIENCES-BASEL
(2023)
Article
Mathematics
Vladimir Temlyakov
Summary: This paper suggests a new criterion for evaluating the theoretical efficiency of a greedy algorithm and proves results on the rate of convergence of greedy algorithms that provide expansions. The study considers both Hilbert spaces and the more general case of Banach spaces. The new aspect of this research is bounding the approximation error using the product of two norms - the norm of f and the A1-norm of f, whereas typically only the A1-norm of f is used. The results show that certain greedy algorithms (Pure Greedy Algorithm (PGA) and its modifications) perform as well as the Orthogonal Greedy Algorithm (OGA) in terms of convergence rate, despite PGA being known to be much worse than OGA in the standard sense. The new results provide better accuracy bounds than known results for small parallel to f parallel to.
Article
Engineering, Multidisciplinary
Fahad Alsayyari, Zoltan Perko, Marco Tiberga, Jan Leen Kloosterman, Danny Lathouwers
Summary: This approach combines POD with Smolyak hierarchical interpolation model to build reduced-order models for parametrized partial differential equations in a nonintrusive manner. The adaptive sampling of time helps capture important dynamics of the system effectively.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mechanics
Bo Zhang
Summary: In this study, a physics-assimilated convolutional autoencoder (CAE) neural network was developed for nonlinear mode decomposition. The results showed that this method is capable of extracting the dominant features of flow fields and considering the underlying nonlinear dynamics.
Article
Mathematics, Applied
Xiang Sun, Xiaomin Pan, Jung-Il Choi
Summary: The proposed method utilizes POD and PCE to construct an efficient stochastic representation model through non-intrusive methods, significantly reducing computational costs and storage requirements for high-dimensional physical and random spaces, while demonstrating similar accuracy in predicting statistical quantities as classical sparse PCE.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Thermodynamics
Jiacheng Ma, Donghun Kim, James E. Braun
Summary: This paper presents a computationally efficient and accurate dynamic modeling approach for vapor compression systems using model order reduction techniques. By reformulating the heat exchanger model and applying POD, reduced order models for evaporator and condenser are constructed with system stability and numerical efficiency in mind. Transient simulations conducted under various operating conditions show that the reduced order model can execute faster with negligible prediction errors compared to the high-fidelity finite volume model.
INTERNATIONAL JOURNAL OF REFRIGERATION
(2021)
Article
Mathematics, Applied
Birgul Koc, Tomas Chacon Rebollo, Samuele Rubino
Summary: In this paper, we provide evidence of uniform error bounds for proper orthogonal decomposition (POD) reduced order modeling (ROM) of the Burgers equation with the inclusion of difference quotients (DQs). Our study focuses on the behavior of DQ ROM error bounds using different POD spaces and error measures. Numerical tests show that DQ ROM errors are significantly smaller than noDQ errors, and the addition of DQs in the POD process leads to an optimality/super-optimality behavior.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mechanics
Xuyi Jia, Chunna Li, Wen Ji, Chunlin Gong
Summary: This paper proposes a new modeling method based on a hybrid reduced-order model to improve the accuracy and robustness of predicting complex and strongly nonlinear flow structures. The method combines DMD and POD to capture different flow properties and achieves better accuracy and forecasting capability for shock waves and vortex.
