Iranian Journal of Mechanical Engineering Transactions of the ISME

Iranian Journal of Mechanical Engineering Transactions of the ISME

Stabilization of Intrusive DMD based Reduced Order Model of the Convection-diffusion Equations

Document Type : Research Paper

Authors
Mohammad Kazem Moayyedi
Fatemeh Bigdeloo
Zohreh Khakzari
CFD Turbulence and Combustion Research Lab., Department of Mechanical Engineering, University of Qom, Qom, Iran / Institute of Aerospace Studies, University of Qom, Qom, Iran
10.30506/jmee.2024.2007857.1317
Abstract
Simulation of complex and non-linear dynamical systems needs numerical algorithms which are time-consuming due to hardware limitations. Therefore, many studies were focused on developing models with high speed of computation and good relative accuracy. Reduced-order model is the method that could be an alternative approach for simulating complex dynamical systems using some appropriate data from the system responses. Usually, these models are developed based on the dominant features of the desired system. Dynamic Mode Decomposition is one of the methods for calculating these basis functions. In this study, using this method and based on the concepts of dynamical systems, a reduced-order model has been developed for Burgers equation. Also, due to the incompleteness of the related modal space which is changed to low-dimensional space in order reduction procedure, the dissipation level of the surrogate model is decreased. Therefore, by using an artificial viscosity which is called the eddy viscosity approach, the stability of the model is improved. Eventually, comparing the results obtained by the stabilized reduced order model and direct numerical simulation data, the accuracy of the model with an average error of 0.4% is proven.
Keywords
Dynamic Mode Decomposition (DMD)
Reduced order model
Eddy viscosity approach
Burgers equation
Stabilization approach

Subjects

  • Chen, "Model Order Reduction for Nonlinear Systems," Ph.D. Thesis, Dept. of Mathematics, Massachusetts Institute of Technology, Univ. Massachusetts, 1999. Available: http://hdl.handle.net/1721.1/9381.

 

  • H. Schilders, H.A. Van der Vorst, and J. Rommes, "Model Order Reduction: Theory, Research Aspects and Applications," Springer Berlin, Heidelberg, Vol. 13, 2008, doi: 10.1007/978-3-540-78841-6_3.

 

  • Bang, H.S. Abdel-Khalik, and J.M. Hite, "Hybrid Reduced Order Modeling Applied to Nonlinear Models," Int. J. Numer. Meth. Eng., Vol. 91, No. 9, pp. 929-949, 2012, doi: 10.1002/nme.4298.

 

  • C L. Callaham, S.L. Brunton, and J.-C. Loiseau, "On the Role of Nonlinear Correlations in Reduced-order Modelling," J. Fluid. Mech., No. 1, 2022, doi: 10.1017/jfm.2021.

 

 

  • Liang, H. Lee, S. Lim, W. Lin, K. Lee, and C. Wu, "Proper Orthogonal Decomposition and Its Applications—Part I: Theory," J. Sound. Vib., Vol. 252, No. 3, pp. 527-544, 2002, doi: 10.1006/jsvi.2001.4041.

 

  • Higham, M. Shahnam, and A. Vaidheeswaran, "Using a Proper Orthogonal Decomposition to Elucidate Features in Granular Flows," Granul. Matter., Vol. 22, pp. 1-13, 2020, doi: 10.1007/s10035-020-01037-7.

 

  • K. Moayyedi and F. Sabaghzadeghan, "Development of Parametric and Time Dependent Reduced Order Model for Diffusion and Convection-diffusion Problems Based on Proper Orthogonal Decomposition Method," Amirkabir J. Mech. Eng., Vol. 53, No. 7, pp. 8-8, 2021, doi: 10.22060/mej.2020.16936.6483.

 

  • Edwards, L.S. Tuckerman, R.A. Friesner, and D. Sorensen, "Krylov Methods for the Incompressible Navier-Stokes Equations," J. Comput. Phys., Vol. 110, No. 1, pp. 82-102, 1994, doi: 10.1006/jcph.1994.1007.

 

  • B. Lehoucq, "Implicitly Restarted Arnoldi Methods and Subspace Iteration," Siam J. Matrix. Anal. A., Vol. 23, No. 2, pp. 551-562, 2001, doi: 10.1137/s0895479899358595.

 

  • J. Schmid, "Dynamic Mode Decomposition of Numerical and Experimental Data," J. Fluid. Mech., Vol. 656, pp. 5-28, 2010, doi: 10.1017/s0022112010001217.

 

  • Wu, D. Laurence, S. Utyuzhnikov, and I. Afgan, "Proper Orthogonal Decomposition and Dynamic Mode Decomposition of Jet in Channel Crossflow," Nucl. Eng. Des., Vol. 344, pp. 54-68, 2019, doi: 10.1016/j.nucengdes.2019.01.015.

 

  • W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D.S. Henningson, "Spectral Analysis of Nonlinear Flows," J. Fluid Mech., Vol. 641, pp. 115-127, 2009, doi: 10.1017/s0022112009992059.

 

 

  • Sabaghzadeghan and M. Moayyedi, "Reduced Order Model of Conduction Heat Transfer in a Solid Plate Based on Dynamic Mode Decomposition," Sharif J. Mech. Eng., Vol. 37, No. 2, pp. 3-12, 2021, dor: 20.1001.1.26764725.1400.373.2.2.3.

 

  • Hu, C. Yang, W. Yi, K. Hadzic, L. Xie, R. Zou, and M. Zhou, "Numerical Investigation of Centrifugal Compressor Stall with Compressed Dynamic Mode Decomposition," Aerosp. Sci. Technol., Vol. 106, 2020, doi: 10.1016/j.ast.2020.106153.

 

  • Sun, T. Tian, X. Zhu, O. Hua, and Z. Du, "Investigation of the Near Wake of a Horizontal-axis Wind Turbine Model by Dynamic Mode Decomposition," Energy, Vol. 227, 2021, doi: 10.1016/j.energy.2021.120418.

 

  • L. Brunton, J.L. Proctor, J.H. Tu, and J.N. Kutz, "Compressed Sensing and Dynamic Mode Decomposition," J. Comput. Dynam., Vol. 2, No. 2, pp. 165-191, 2016, doi: 10.3934/jcd.2015002.

 

  • Seena and H.J. Sung, "Spatiotemporal Representation of the Dynamic Modes in Turbulent Cavity Flows," Int. J. Heat. Fluid. Fl., Vol. 44, pp. 1-13, 2013, doi: 10.1016/j.ijheatfluidflow.2013.02.011.

 

  • Bai, E. Kaiser, J.L. Proctor, J.N. Kutz, and S.L. Brunton, "Dynamic Mode Decomposition for Compressive System Identification," AIAA J., Vol. 58, No. 2, pp. 561-574, 2020, doi: 10.2514/1.j057870.

 

  • K. Moayyedi, F. Bigdeloo, and F. Sabaghzadeghan, "Stabilization of Reduced Order Model for Convection-diffusion Problems Based on Dynamic Mode Decomposition at High Reynolds Numbers using Eddy Viscosity Approach," Amirkabir J. Mech. Eng., Vol. 54, No. 11, pp. 2479-2498, 2023, doi: 10.22060/MEJ.2022.20801.7327.

 

  • K. Moayyedi, Z. Khakzari, and F. Sabaghzadeghan, "Studying the Effect of Eddy Viscosity Closure on the Calibration of the DMD Based Reduced-order Model to Predict the Long-term Behavior of Convection-diffusion Equations," Fluid Mech Aero. J., Vol. 11, No. 1, pp. 83-96, 2022, dor: https://dorl.net/dor/20.1001.1.23223278.1401.11.1.6.8.

 

  • J. Schmid, "Dynamic Mode Decomposition and its Variants," Annu. Rev. Fluid. Mech., Vol. 54, pp. 225-254, 2022, doi: 10.1146/annurev-fluid-030121-015835.

 

  • E. Ahmed, P. H. Dabaghian, O. San, D. A. Bistrian, and I. M. Navon, "Dynamic Mode Decomposition with Core Sketch," Phys. Fluids, Vol. 34, No. 6, 2022, doi: 10.1063/5.0095163.

 

  • Krake, D. Klötzl, and B. Weiskopf, "Constrained Dynamic Mode Decomposition," IEEE. T. Vis. Comput. Gr., Vol. 29, No. 1, pp. 182-192, 2022, doi: 10.1109/tvcg.2022.3209437.

 

  • J. Baddoo, B. Herrmann, B. J. McKeon, J. Nathan Kutz, and S. L. Brunton, "Physics-informed Dynamic Mode Decomposition," Proceedings of Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 479, No. 2271, 2023, doi: 10.1098/rspa.2022.0576.

 

  • Sabaghzadeghan, "Development the Reduce Order Model for Convection-diffusion and Diffusion Problems Based on Proper Orthogonal Decomposition and Dynamic Mode Decomposition," M.Sc. Thesis, Mechanical Engineering, University of Qom, 2019, Available: B2n.ir/a69815, (In Persian).

 

  • F. Tian and P. Yu, "A High-order Exponential Scheme for Solving 1D Unsteady Convection–diffusion Equations," J. Compute. Appl. Math., Vol. 235, No. 8, pp. 2477-2491, 2011, Available: https://doi.org/10.1016/j.cam.2010.11.001.

 

  • W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson, "Spectral Analysis of Nonlinear Flows," J. Fluid Mech., Vol. 641, pp. 115-127, 2009, doi: 10.1017/s0022112009992059.

 

  • J. Schmid, "Dynamic Mode Decomposition of Numerical and Experimental Data," J. Fluid Mech., Vol. 656, pp. 5-28, 2010, doi: 10.1017/s0022112010001217.

 

  • L. Proctor and P. A. Eckhoff, "Discovering Dynamic Patterns from Infectious Disease Data using Dynamic Mode Decomposition," Int. Health, Vol. 7, No. 2, pp. 139-145, 2015, doi: 10.1093/inthealth/ihv009.

 

  • W. Brunton, L. A. Johnson, J. G. Ojemann, and J. N. Kutz, "Extracting Spatial–temporal Coherent Patterns in Large-scale Neural Recordings using Dynamic Mode Decomposition," J. Neurosci. Meth., Vol. 258, pp. 1-15, 2016, doi: 10.1016/j.jneumeth.2015.10.010.

 

  • Berger, M. Sastuba, D. Vogt, B. Jung, and H. Ben Amor, "Estimation of Perturbations in Robotic Behavior using Dynamic Mode Decomposition," Adv. Robotics, Vol. 29, No. 5, pp. 331-343, 2015, doi: 10.1080/01691864.2014.981292.

 

 

  • Mann and J. N. Kutz, "Dynamic Mode Decomposition for Financial Trading Strategies," Quant. Financ., Vol. 16, No. 11, pp. 1643-1655, 2016, doi: 10.1080/14697688.2016.1170194.

 

  • Aubry, P. Holmes, J. L. Lumley, and E. Stone, "The Dynamics of Coherent Structures in the Wall Region of a Turbulent Boundary Layer," J. Fluid Mech., Vol. 192, pp. 115-173, 1988, doi: 10.1017/s0022112088001818.

 

  • Rempfer, "Kohärente Strukturen und Chaos Beim Laminar-turbulenten Grenzschichtumschlag," Ph.D Thesis, Univ. Stuttgart, 1997, Available: B2n.ir/b51161, (In German).

 

  • San and T. Iliescu, "Proper Orthogonal Decomposition Closure Models for Fluid Flows: Burgers Equation," Int. J. Numer. Anal. Mod., Vol. 5, pp. 217-237, 2013, Available: https://doi.org/10.48550/arXiv.1308.3276.

 

  • Tadmor, "Convergence of Spectral Methods for Nonlinear Conservation Laws," Siam. J. Numer. Anal., Vol. 26, No. 1, pp. 30-44, 1989, doi: 10.1137/0726003.

 

  • Sirisup and G. E. Karniadakis, "A Spectral Viscosity Method for Correcting the Long-term Behavior of POD Models," J. Comput. Phys., Vol. 194, No. 1, pp. 92-116, 2004, doi: 10.1016/j.jcp.2003.08.021.

 

  • Chollet, "Two-point Closure Used for a Sub-grid Scale Model in Large Eddy Simulations," in Turbulent Shear Flows 4: Selected Papers from the Fourth International Symposium on Turbulent Shear Flows, University of Karlsruhe, Karlsruhe, FRG, September 12–14, 1983, Springer, 1985, pp. 62-72, doi: 10.1007/978-3-642-69996-2_11.

 

  • Lesieur and O. Metais, "New Trends in Large-eddy Simulations of Turbulence," Annu. Rev. Fluid Mech., Vol. 28, No. 1, pp. 45-82, 1996, doi: 10.1146/annurev.fl.28.010196.000401.