توسعة مدل رتبه‌کاسته پارامتری و وابسته به زمان برای مسائل نفوذ و نفوذ-جابجایی بر مبنای روش تجزیه متعامد بهینه

نوع مقاله : مقاله پژوهشی

نویسندگان

1 بخش مهندسی مکانیک، دانشگاه قم

2 گروه مهندسی مکانیک، دانشگاه قم

چکیده

مدلسازی و شبیه‌سازی عددی به عنوان یک ابزار مناسب جهت تحلیل رفتار دینامیکی سیستم‌های مهندسی به‌شمار‌می‎رود. استفاده از این روش‌ها، به ویژه برای مسائل ناپایا، معمولا نیازمند صرف زمان زیادی است. به همین دلیل، توسعه روش‌هایی با سرعت بالاتر و افزایش راندمان محاسباتی همواره به عنوان یک موضوع مهم مورد توجه پژوهشگران بوده‌است. روش کاهش مرتبه روشی است که در سالیان اخیر برای کاهش زمان محاسبات به طور گسترده‌ای مورد استفاده قرار گرفته‌است. در این روش، با کاستن از قیود سیستم، بدون تغییر در ویژگی‌های ذاتی مسئله، سرعت محاسبات به طرز چشم‎گیری افزایش می‌یابد. در این پژوهش، با بهره‌گیری از مفاهیم پایه‌ای سیستم‌های دینامیکی، دو مسئله نفوذ حرارتی و نفوذ-جابجایی به صورت مستقل مورد بررسی قرار گرفته و با استفاده از روش تجزیه متعامد بهینه، الگوی رتبه کاسته برای معادلات حاکم بر این پدیده‌ها ایجاد شده‌است. بر همین اساس، برای هر یک از مسائل، مبتنی بر تصویرسازی معادله حاکم در فضای برداری مودهای میدان، با استفاده از مودهای پرانرژی‌تر، مدل رتبه کاسته با توجه به ویژگی پایه‌های متعامد به دست می‌آید. مدل به‌دست‌آمده به منظور شبیه‌سازی تغییرات زمانی فرآیند، به درستی می‌تواند جایگزین معادلة اصلی شده و با دقت بسیار مناسبی رفتار سیستم موردنظر را پیش‌بینی کند.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Development of parametric and time dependent reduced order model for diffusion and convection-diffusion problems based on proper orthogonal decomposition method

نویسندگان [English]

  • Mohammad Kazem Moayyedi 1
  • Farshad Sabaghzadeghan 2
1 Department of Mechanical Engineering, University of Qom
2 Department of Mechanical Eng. Univ. Qom
چکیده [English]

Simulation and numerical analysis of physical phenomena, especially for unstable problems, due to dependency of the numerical algorithms on the computer hardware to the increasing of the number of computational nodes, is the most important feature of their solutions. For this reason, increases the number of computations then increased computational costs. The order reduction method has been widely used in recent years to reduce computational time. In this way, by reducing the constraints of the system, without changing the inherent features of the problem, the computational efficiency will dramatically increase. In this study, using the basic concepts of dynamical systems, two problems of thermal diffusion and convection-diffusion are investigated independently and by using the proper orthogonal analysis method, a reduced order model is established for the equations governing these phenomena created. Accordingly, for each of the problems, based on the projection of the governing equation in the vector space of modes, by using more energetic modes, a reduced order model is obtained with respect to the orthogonal basis properties. The model obtained in order to simulate the process time variations can properly replace the original equation and predict the behavior of the system with very good accuracy.

کلیدواژه‌ها [English]

  • Proper orthogonal decomposition
  • Diffusion equation
  • Convection-diffusion equation
  • Reduced order model
  • Surrogate model
[1] Chen, Y., Model order reduction for nonlinear systems, M. Sc. Thesis, Dept. of Elec. Eng. Comput. Sci., Massachusetts Institute of Technology, (1999).
[2] Schilders, W. H., et al, Model order reduction: theory research aspects and applications, Springer, (2008)
[3] Liang, Y., et al, Proper orthogonal decomposition and its applications—Part I: Theory, Journal of Sound and vibration, 252(3) (2002) 527-544.
[4] Karhunen, K, Zur spektraltheorie stochastischer prozesse, Ann. Acad. Sci. Fennicae, AI 34 (1946)
[5] Yaglom, A. and V. Tatarski, The structure of inhomogeneous turbulence Atmospheric Turbulence and Radio Wave Propagation, Nauka, (1967) 166-178.
[6] Sirovich, L. and M. Kirby, Low-dimensional procedure for the characterization of human faces, Josa a, 4(3) (1987) 519-524.
[7] Taeibi-Rahni, M., et al, Fast Estimation of Aerodynamics Data Using Proper Orthogonal Decomposition, The 10th Fluid Dynamics Conference, Yazd, Iran, (2006). (In Persian)
[8] Bui-Thanh, T., et al, Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition, AIAA journal, 42(8) (2004) 1505-1516.
[9] Willcox, K, Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition, Computers & fluids, 35(2) (2006) 208-226.
[10] Sabetghadam, F., et al, Gappy Low-dimensional POD, A Powerful Tool of Data Reconstruction of the Unsteady Flow Fields, CFD Journal, 17(3) (2008) 156-164.
[11] LeGresley, P. and J. Alonso, Investigation of non-linear projection for pod based reduced order models for aerodynamics, 39th Aerospace Sciences Meeting and Exhibit, (2001)
[12] Moayyedi, M., K., et al, Development of Inverse Aerodynamics Design Model Based on Proper Orthogonal Decomposition Method, the first Annual Conference on Aerodynamics and Hydrodynamics, Tehran, Iran, (2011) (In Persian)
[13] Lieu, T. and C. Farhat, Adaptation of aeroelastic reduced-order models and application to an F-16 configuration, AIAA Journal, 45(6) (2007) 1244-1257.
[14] Tabib, M. V. and J. B. Joshi, Analysis of dominant flow structures and their flow dynamics in chemical process equipment using snapshot proper orthogonal decomposition technique, Chemical Engineering Science, 63(14) (2008) 3695-3715.
[15] Druault, P. and C. Chaillou, Use of proper orthogonal decomposition for reconstructing the 3D in-cylinder mean-flow field from PIV data, Comptes Rendus Mécanique, 335(1) (2007) 42-47.
[16] Hilberg, D., et al, The application of classical POD and snapshot POD in a turbulent shear layer with periodic structures, Applied scientific research, 53(3-4) (1994) 283-290.
[17] Pastur, L., et al, Determining the spectral signature of spatial coherent structures in an open cavity flow, Physical review, E 72(6) (2005) 065301.
[18] Rempfer, D. and H. F. Fasel, Evolution of three-dimensional coherent structures in a flat-plate boundary layer, Journal of Fluid Mechanics, 260 (1994) 351-375.
[19] Ravindran, S, Proper orthogonal decomposition in optimal control of fluids, Technical report NASA/TM-1999-209113, NASA Langley Research Center, Hampton, Virginia, (1999).
[20] Ostrowski, Z., et al, Advances in application of proper orthogonal decomposition in inverse problems, Proc. 5th Int. conf. on inverse problems in engineering: theory and practice, Cambridge, UK, (2005)
[21] Shinde, V., et al, Galerkin-free model reduction for fluid-structure interaction using proper orthogonal decomposition,  Journal of Computational Physics, 396 (2019) 579-595.
[22] Blanc, T. J., et al, Reduced-Order Modeling of Conjugate Heat Transfer Processes, Journal of Heat Transfer, 138(5) (2016)
[23] Fagiano, L. and R. Gati, On the order reduction of the radiative heat transfer model for the simulation of plasma arcs in switchgear devices, Journal of Quantitative Spectroscopy and Radiative Transfer, 169 (2016) 58-78.
[24] Carlberg, K., et al, Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction, Journal of Computational Physics 330 (2017) 693-734.
[25] Choi, Y. and K. Carlberg, Space--Time Least-Squares Petrov--Galerkin Projection for Nonlinear Model Reduction, SIAM Journal on Scientific Computing, 41(1) (2019) A26-A58.
[26] Esfahanian, V., et al, Simulation of lead-acid battery using model order reduction, Journal of Power Sources, (2015) 279: 294-305.
[27] Tian, Z. F. and P. Yu, A high-order exponential scheme for solving 1D unsteady convection–diffusion equations, Journal of computational and applied mathematics, 235(8) (2011) 2477-2491.