Lattice Boltzmann D2Q9

Python Numpy

from __future__ import print_function

"""
The Lattice Boltzmann Methods D2Q9
------------

Channel flow past a cylindrical obstacle, using a LB method
Copyright (C) 2006 Jonas Latt
Address: Rue General Dufour 24,  1211 Geneva 4, Switzerland
E-mail: Jonas.Latt@cui.unige.ch
"""
from benchpress import util
import numpy as np

# D2Q9 Lattice constants
t = [4 / 9., 1 / 9., 1 / 9., 1 / 9., 1 / 9., 1 / 36., 1 / 36., 1 / 36., 1 / 36.]
cx = [0, 1, 0, -1, 0, 1, -1, -1, 1]
cy = [0, 0, 1, 0, -1, 1, 1, -1, -1]
opp = [0, 3, 4, 1, 2, 7, 8, 5, 6]
uMax = 0.02  # maximum velocity of Poiseuille inflow
Re = 100  # Reynolds number


def cylinder(height, width, obstacle=True):
    assert (height > 2)
    assert (width > 2)

    lx = width
    ly = height
    state = {"lx": lx, "ly": ly}
    obst_x = lx / 5. + 1  # position of the cylinder; (exact
    obst_y = ly / 2. + 0  # y-symmetry is avoided)
    obst_r = ly / 10. + 1  # radius of the cylinder

    nu = uMax * 2. * obst_r / Re  # kinematic viscosity
    omega = 1. / (3 * nu + 1. / 2.)  # relaxation parameter

    col = np.arange(1, ly - 1)

    # Place obstacle
    if obstacle:
        y, x = np.meshgrid(np.arange(ly), np.arange(lx))
        obst = (x - obst_x) ** 2 + (y - obst_y) ** 2 <= obst_r ** 2
        obst[:, 0] = obst[:, ly - 1] = 1
        bbRegion = np.asarray(obst, dtype=np.bool)
    else:
        bbRegion = None

    # We need some of the constants as 3D numpy arrays
    t_3d = np.asarray(t)[:, np.newaxis, np.newaxis]
    cx_3d = np.asarray(cx)[:, np.newaxis, np.newaxis]
    cy_3d = np.asarray(cy)[:, np.newaxis, np.newaxis]

    # Initial condition: (rho=0, u=0) ==> fIn[i] = t[i]
    fIn = np.array(t_3d.repeat(lx, axis=1).repeat(ly, axis=2))

    state['fIn'] = fIn
    state['cx_3d'] = cx_3d
    state['cy_3d'] = cy_3d
    state['col'] = col
    state['omega'] = float(omega)
    state['bbRegion'] = bbRegion

    # We save the visualize limits in order to make it constant
    # We find the limit by calculating the first iteration if `ux`
    ux = np.sum(cx_3d * fIn, axis=0) / np.sum(fIn, axis=0)
    L = ly - 2.0
    y = col - 0.5
    ux[0, 1:ly - 1] = 4 * uMax / (L ** 2) * (y * L - y ** 2)
    state['viz_max'] = ux.max()
    state['viz_min'] = ux.min()
    return state


def solve(B, state, timesteps, visualize=False):
    # load the ready only state
    ly = int(state['ly'])
    lx = int(state['lx'])
    col = state['col']
    cx_3d = state['cx_3d']
    cy_3d = state['cy_3d']
    bbRegion = state['bbRegion']
    omega = state['omega']

    def loop_body(fIn):

        # Macroscopic variables
        rho = np.sum(fIn, axis=0)
        ux = np.sum(cx_3d * fIn, axis=0) / rho
        uy = np.sum(cy_3d * fIn, axis=0) / rho

        # Macroscopic (Dirichlet) boundary conditions

        # Inlet: Poisseuille profile
        L = ly - 2.0
        y = col - 0.5
        ux[0, 1:ly - 1] = 4 * uMax / (L ** 2) * (y * L - y ** 2)
        uy[0, 1:ly - 1] = 0

        # Using a loop instead of "index of indexes"
        # t1 = fIn[[0, 2, 4], 0, 1:ly-1].sum(axis = 0)
        # t1 = np.zeros(ly-2, bohrium=fIn.bohrium)
        t1 = np.zeros(ly - 2)
        for i in [0, 2, 4]:
            t1 += fIn[i, 0, 1:ly - 1]

        # Using a loop instead of "index of indexes"
        # t2 = 2 * fIn[[3, 6, 7], 0, 1:ly-1].sum(axis = 0)
        # t2 = np.zeros(ly-2, bohrium=fIn.bohrium)
        t2 = np.zeros(ly - 2)
        for i in [3, 6, 7]:
            t2 += fIn[i, 0, 1:ly - 1]
        t2 *= 2

        rho[0, 1:ly - 1] = 1 / (1 - ux[0, 1:ly - 1]) * (t1 + t2)

        # Outlet: Zero gradient on rho/ux
        rho[lx - 1, 1:ly - 1] = 4.0 / 3 * rho[lx - 2, 1:ly - 1] - \
                                1.0 / 3 * rho[lx - 3, 1:ly - 1]
        uy[lx - 1, 1:ly - 1] = 0
        ux[lx - 1, 1:ly - 1] = 4.0 / 3 * ux[lx - 2, 1:ly - 1] - \
                               1.0 / 3 * ux[lx - 3, 1:ly - 1]

        # fEq = np.zeros((9, lx, ly), bohrium=fIn.bohrium)
        fEq = np.zeros((9, lx, ly))
        # fOut = np.zeros((9, lx, ly), bohrium=fIn.bohrium)
        fOut = np.zeros((9, lx, ly))
        for i in range(0, 9):
            cu = 3 * (cx[i] * ux + cy[i] * uy)
            fEq[i] = rho * t[i] * (1 + cu + 0.5 * cu ** 2 - \
                                   1.5 * (ux ** 2 + uy ** 2))
            fOut[i] = fIn[i] - omega * (fIn[i] - fEq[i])

        # Microscopic boundary conditions
        for i in range(0, 9):
            # Left boundary:
            fOut[i, 0, 1:ly - 1] = fEq[i, 0, 1:ly - 1] + 18 * t[i] * cx[i] * cy[i] * \
                                   (fIn[7, 0, 1:ly - 1] - fIn[6, 0, 1:ly - 1] - fEq[7, 0, 1:ly - 1] + fEq[6, 0,
                                                                                                      1:ly - 1])
            # Right boundary:
            fOut[i, lx - 1, 1:ly - 1] = fEq[i, lx - 1, 1:ly - 1] + 18 * t[i] * cx[i] * cy[i] * \
                                        (fIn[5, lx - 1, 1:ly - 1] - fIn[8, lx - 1, 1:ly - 1] - \
                                         fEq[5, lx - 1, 1:ly - 1] + fEq[8, lx - 1, 1:ly - 1])
            # Bounce back region:
            # fOut[i,bbRegion] = fIn[opp[i],bbRegion]
            # Using a explict mask
            if bbRegion is not None:
                masked = fIn[opp[i]].copy() * bbRegion
                fOut[i] = fOut[i] * ~bbRegion + masked

        # Streaming step
        for i in range(0, 9):
            # fIn[i] = np.roll(np.roll(fOut[i], cx[i], axis = 0), cy[i], axis = 1)
            # Replacing the np.roll() call with:
            if cx[i] == 1:
                # t1 = np.empty_like(fOut[i], bohrium=fIn.bohrium)
                t1 = np.empty_like(fOut[i])
                t1[1:] = fOut[i][:-1]
                t1[0] = fOut[i][-1]
                fOut[i] = t1
            elif cx[i] == -1:
                # t1 = np.empty_like(fOut[i], bohrium=fIn.bohrium)
                t1 = np.empty_like(fOut[i])
                t1[:-1] = fOut[i][1:]
                t1[-1] = fOut[i][0]
                fOut[i] = t1
            if cy[i] == 1:
                # t1 = np.empty_like(fOut[i], bohrium=fIn.bohrium)
                t1 = np.empty_like(fOut[i])
                t1[:, 1:] = fOut[i][:, :-1]
                t1[:, 0] = fOut[i][:, -1]
                fIn[i] = t1
            elif cy[i] == -1:
                # t1 = np.empty_like(fOut[i], bohrium=fIn.bohrium)
                t1 = np.empty_like(fOut[i])
                t1[:, :-1] = fOut[i][:, 1:]
                t1[:, -1] = fOut[i][:, 0]
                fIn[i] = t1
            else:
                fIn[i] = fOut[i]
        if visualize:
            util.plot_surface(ux.T, "2d", 0, state['viz_max'], state['viz_min'])

    B.do_while(loop_body, timesteps, state['fIn'])


def main():
    B = util.Benchmark()
    H = B.size[0]
    W = B.size[1]
    I = B.size[2]
    state = B.load_data()
    if state is None:
        state = cylinder(H, W)
    B.start()
    solve(B, state, I, B.visualize)
    B.stop()
    B.save_data(state)
    B.pprint()
    if B.verbose:
        print("Iterations=%s, State: %s." % (I, state))
    if B.visualize:
        util.confirm_exit()


if __name__ == "__main__":
    main()