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Beyond linear analysis: Exploring stability of multiple-relaxation-time lattice Boltzmann method for nonlinear flows using decision trees and evolutionary algorithms

  • *Corresponding author: Mohamed Mahdi Tekitek

    *Corresponding author: Mohamed Mahdi Tekitek
Abstract / Introduction Full Text(HTML) Figure(9) / Table(4) Related Papers Cited by
  • This study investigates the numerical stability of the two-dimensional lattice Boltzmann (LB) scheme for nonlinear flows. While the BGK (single relaxation time) scheme is known for its simplicity, it can be unstable for complex flows. Conversely, the LB scheme with multiple relaxation times (MRT) offers greater stability for such nonlinear problems. However, traditional stability analysis methods like von Neumann local analysis are limited to the linear case. Here, we examine the stability of a specific nonlinear test case with fixed viscosity. We employ a decision tree, a machine learning technique valued for its interpretability, to explore and characterize the stability zone for the free relaxation parameters. To further investigate, a simple global optimization method (evolutionary algorithm) is used to identify a set of stable relaxation parameters for various test cases, including the doubly periodic shear layers, Taylor-Green vortex, and lid-driven cavity. Notably, this method enables the discovery of stable, non-trivial LB parameter sets for the nonlinear case. Finally, to assess numerical stability or instability with these non-trivial parameters, a global stability analysis is conducted over the entire lattice, encompassing the boundary conditions of the linearized scheme.

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  • Figure 1.  D2Q9 lattice Boltzmann scheme; discrete velocities $ \, v_j \, $ for $ \, 0 \leq j \leq 8 $

    Figure 2.  Contours of vorticity for the doubly periodic shear layers flow, where $ \nu = 10^{-4} $ at time step $ t = 15000 $, using the BGK LB scheme with $ \delta = 0.05 $, $ w = 80 $, and $ U_0 = 0.008 $

    Figure 3.  Contours of vorticity for the doubly periodic shear layers flow, where $ \nu = 10^{-4} $ at time step $ t = 3400 $, using the BGK LB scheme with $ \delta = 0.05 $, $ w = 80 $, and $ U_0 = 0.023 $

    Figure 4.  Contours of vorticity at timestep $ t = 3400 $ using the MRT LB scheme, with $ \delta = 0.05 $, $ w = 80 $, and $ U_0 = 0.023 $

    Figure 5.  Decision Tree for predicting stable parameters for the doubly periodic shear layers test case with $ N_x = N_y = 128 $, $ \nu = 10^{-4} $, and $ U_0 = 0.1 $

    Figure 6.  The doubly periodic shear layers flow simulated using MRT LB scheme at time step $ t = 20000 $

    Figure 7.  The lid driven cavity flow at $ Re = 1280 $ simulated using MRT LB scheme at time step $ t = 20000 $

    Figure 8.  The lid driven cavity flow at $ Re = 1280 $ simulated using MRT LB scheme at time step $ t = 400000 $

    Figure 9.  Modulus of eigenvalue of matrix $ G $ vs. timestep. (Left) Stable case. (Right) Unstable case. (See parameter values in the text)

    Table 1.  Equilibrium of non-conserved moments and associated relaxation coefficients for the D2Q9 Lattice Boltzmann method in fluid problems

    moment equilibrium relaxation coefficient/parameter
    $ m_3= e $ $ m_3^{eq}={e}^{eq} = -2 \rho + \frac{3}{ \lambda^2 \rho} (j_x^2+j_y^2) $ $ s_e=s_3 $
    $ m_4= \epsilon $ $ m_4^{eq}={\epsilon}^{eq} = \rho - \frac{3}{\lambda^2\rho} (j_x^2+j_y^2) $ $ s_\epsilon =s_4 $
    $ m_5= q_x $ $ m_5^{eq}={q_x}^{eq} = -\frac{j_x}{\lambda} $ $ s_{q} =s_5 $
    $ m_6= q_y $ $ m_6^{eq}={q_y}^{eq} = -\frac{j_y}{\lambda} $ $ s_{q} =s_6 $
    $ m_7= p_{xx} $ $ m_7^{eq}={p_{xx}} = \frac{1}{\lambda^2 \rho} (j_x^2-j_y^2) $ $ s_\nu=s_7 $
    $ m_8= p_{xy} $ $ m_8^{eq}= p_{xy} = \frac{1}{\lambda^2 \rho} j_x j_y $ $ s_\nu=s_8 $
     | Show Table
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    Table 2.  Stable parameters found by the global optimization

    Problem "doubly periodic shear layers", $ N_t=20000 $, $ N_x=N_y=128 $
    $ \nu = 10^{-4} $ $ \nu = 10^{-3} $ $ \nu = 10^{-3} $ $ \nu = 5. 10^{-4} $ $ \nu = 10^{-3} $
    $ U_0 = 0.1 $ $ U_0 = 0.08 $ $ U_0 = 0.08 $ $ U_0 = 0.08 $ $ U_0 = 0.16 $
    Constraints $ s_e \leq 1.9999 $, $ (s_\varepsilon, \, s_q) \in [\frac{1}{\sqrt{2}}\, , 1.9999]^2 $
    $ s_e \geq 1.99 $ $ s_e \geq \frac{1}{\sqrt{2}} $ $ s_e \geq 1.99 $ $ s_e \geq 1.99 $ $ s_e \geq 1.99 $
    Stable parameters
    $ s_e^* $ 1.9962 0.9483 1.9992 1.990 1.991
    $ s_\varepsilon^* $ 1.9877 0.7525 1.9943 1.983 1.950
    $ s_q^* $ 1.2599 1.2242 1.7882 1.628 1.958
     | Show Table
    DownLoad: CSV

    Table 3.  Stable parameters found by the global optimization

    Problem "Taylor-Green", $ N_t=20000 $, $ N_x=N_y=128 $
    $ \nu = 10^{-4} $ $ \nu = 10^{-3} $ $ \nu = 10^{-3} $ $ \nu = 5. 10^{-4} $ $ \nu = 10^{-3} $
    $ U_0 = 0.1 $ $ U_0 = 0.08 $ $ U_0 = 0.08 $ $ U_0 = 0.08 $ $ U_0 = 0.16 $
    Constraints $ s_e \leq 1.9999 $, $ (s_\varepsilon, \, s_q) \in [\frac{1}{\sqrt{2}}\, , 1.9999]^2 $
    $ s_e \geq 1.99 $ $ s_e \geq \frac{1}{\sqrt{2}} $ $ s_e \geq 1.99 $ $ s_e \geq 1.99 $ $ s_e \geq 1.99 $
    Stable parameters
    $ s_e^* $ 1.9961 1.8724 1.9911 1.9936 1.9960
    $ s_\varepsilon^* $ 1.9818 1.4284 1.9608 1.9640 1.9732
    $ s_q^* $ 1.9182 1.7044 1.8628 1.7100 1.8515
     | Show Table
    DownLoad: CSV

    Table 4.  Stable parameters found by the global optimization

    Problem "lid driven cavity", $ N_t=20000 $, $ N_x=N_y=128 $
    $ \nu = 10^{-3} $ $ \nu = 10^{-3} $
    $ U_0 = 0.1 $ $ U_0 = 0.08 $
    Constraints $ s_e \leq 1.9999 $, $ (s_\varepsilon, \, s_q) \in [\frac{1}{\sqrt{2}}\, , 1.9999]^2 $
    $ s_e \geq 1.99 $ $ s_e \geq \frac{1}{\sqrt{2}} $
    Stable parameters
    $ s_e^* $ 1.9907 1.3551
    $ s_\varepsilon^* $ 1.9704 0.8489
    $ s_q^* $ 0.7626 1.3210
     | Show Table
    DownLoad: CSV
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