JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (2): 109-117.doi: 10.6040/j.issn.1671-9352.0.2019.669

Previous Articles    

Application of local non-intrusive reduced basis method in Rayleigh-Taylor instability

WEN Xiao1, LIU Qi2*, GAO Zhen2, DON Wai-sun2, LYU Xian-qing1   

  1. 1. Key Laboratory of Physical Oceanography, Ocean University of China, Qingdao 266100, Shandong, China;
    2. School of Mathematical Sciences, Ocean University of China, Qingdao 266100, Shandong, China
  • Published:2020-02-14

PDF (PC)

533

Abstract: A local non-intrusive reduced basis method(RBM)is proposed to simulate the evolution of Rayleigh-Taylor instability(RTI), in which the amplitude and time of initial small perturbations are considered as free parameters. RBM regards the solution as a linear combination of a set of reduced basis functions, which can be obtained by the proper orthogonal decomposition. Furthermore, the artificial neural network is used to establish the mapping relationship between the parameters and the coefficients of the reduced basis functions. Due to the structures of RTI becoming more and more complex with the increasing time, especially the rollup structures of small-scale vortices in the late stage, RTI is considered being divided into the early stage(linear)and the middle and late stage(quasi-non-linear and weak non-linear systems), i.e. time parameter is considered in stages. The time parameter is divided into three, five and six segments and the local RBM allows a potential speedup up to a factor of about four times faster than the global RBM with similar accuracy.

Key words: Rayleigh-Taylor instability, local non-intrusive reduced basis method, artificial neural network

CLC Number: 

  • O242
[1] RAYLEIGH L. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density[J]. Proceedings of the London Mathematical Society, 1882, 14(1):170-177.
[2] TAYLOR G. The instability of liquid surfaces when accelerated in a direction perpendicular to their planes(I)[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1950, 201(1065):192-196.
[3] NITTMANN J, FALLE S A E G, GASKELL P H. The dynamical destruction of shocked gas clouds[J]. Monthly Notices of the Royal Astronomical Society, 1982, 201(4):833-847.
[4] ISOBE H, MIYAGOSHI T, SHIBATA K, et al. Filamentary structure on the Sun from the magnetic Rayleigh-Taylor instability[J]. Nature, 2005, 434(7032):478-481.
[5] YE W H, ZHANG W Y, CHEN G N, et al. Numerical simulations of the FCT method on Rayleigh-Taylor and Richtmyer-Meshkov instabilities[J]. Chinese Journal of Physics, 1998, 15(3):278-282.
[6] LI X L, JIN B X, GLIMM J. Numerical study for the three dimensional Rayleigh-Taylor instability through the TVD/AC scheme and parallel computation[J]. Journal of Computational Physics, 1996, 126(2):343-355.
[7] TRYGGVASON G, BUNNER B, ESMAEELI A, et al. A Front-Tracking method for the computations of multiphase flow[J]. Journal of Computational Physics, 2001, 169(2):708-759.
[8] TARTAKOVSKY A M, MEAKIN P. A smoothed particle hydrodynamics model for miscible flow in three-dimensional fractures and the two-dimensional Rayleigh-Taylor instability[J]. Journal of Computational Physics, 2005, 207(2):610-624.
[9] DUCHEMIN L, JOSSERAND C, CLAVIN P, et al. Asymptotic behavior of the Rayleigh-Taylor instability[J]. Physical Review Letters, 2005, 94(22): 224501.
[10] COOK A W, DIMOTAKIS P E. Transition stages of Rayleigh-Taylor instability between miscible fluids[J]. Journal of Fluid Mechanics, 2001, 443:69-99.
[11] ZHANG J, WANG L F, YE W H, et al. Weakly nonlinear incompressible Rayleigh-Taylor instability in spherical and planar geometries[J]. Physics of Plasmas, 2018, 25(2):022701.
[12] ZHANG J, WANG L F, YE W H, et al. Weakly nonlinear incompressible Rayleigh-Taylor instability in spherical geometry[J]. Physics of Plasmas, 2017, 24(6):062703.
[13] ZHANG J, WANG L F, YE W H, et al. Weakly nonlinear multi-mode Rayleigh-Taylor instability in two-dimensional spherical geometry[J]. Physics of Plasmas, 2018, 25(8):082713.
[14] ZHAO K G, XUE C, WANG L F, et al. Two-dimensional thin shell model for the nonlinear Rayleigh-Taylor instability in spherical geometry[J]. Physics of Plasmas, 2019, 26(2):022710.
[15] ITO K, RAVINDRAN S S. A reduced-order method for simulation and control of fluid flows[J]. Journal of Computational Physics, 1998, 143(2):403-425.
[16] BUI T T, WILLCOX K, GHATTAS O. Model reduction for large-scale systems with high-dimensional parametric input space[J]. SIAM Journal on Scientific Computing, 2008, 30(6):3270-3288.
[17] HESTHAVEN J S, UBBIALI S. Non-intrusive reduced order modeling of nonlinear problems using neural networks[J]. Journal of Computational Physics, 2018, 363:55-78.
[18] AUDOUZE C, VUYST F D, NAIR P B. Reduced-order modeling of parameterized PDEs using time-space-parameter principal component analysis[J]. International Journal for Numerical Methods in Engineering, 2009, 80(8):1025-1057.
[19] HESTHAVEN J S, ROZZA G, STAMM B. Certified reduced basis methods for parametrized partial differential equations[M]. Cham, Switzerland: Springer International Publishing, 2016.
[20] BORGES R, CARMONA M, COSTA B, et al. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws[J]. Journal of Computational Physics, 2008, 227(6):3191-3211.
[21] BULJAK V. Inverse analyses with model reduction: proper orthogonal decomposition and radial basis functions for fast simulations[M]. Berlin: Springer, 2012: 85-139.
[22] KOHONEN T. An introduction to neural computing[J]. Neural Networks, 1988, 1(1):3-16.
[23] 刘冰,郭海霞. MATLAB神经网络超级学习手册[M]. 北京:人民邮电出版社,2014: 156-168. LIU Bing, GUO Haixia. Neural network super learning manual based on MATLAB[M]. Beijing: Posts and Telecom Press, 2014: 159-168.
[24] CENATE C F, RANI B S, RAMADEVI R, et al. Optimization of the cascade feed forward back propagation network for defect classification in ultrasonic images[J]. Russian Journal of Nondestructive Testing, 2016, 52(10):557-568.
[25] CASTRO M, COSTA B, DON W S. High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws[J]. Journal of Computational Physics, 2011, 230(5):1766-1792.
[1] DU Xi-hua, SHI Xiao-qin, FENG Chang-jun, LI Liang. rediction of chromatograph retention index by artificial neural  network by #br# study on volatile constituents of wild chinese chives [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(1): 50-53.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd