single-particle-motion/boris.py

#!/usr/bin/python3
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import rc
#rc('font',**{'family':'serif','serif':['Computer Modern']})
#rc('text', usetex=True)
from scipy.special import ellipe,ellipk

e = 1.602176565e-19 #osnovni naboj [C]
m_pr = 1.672621777e-27 #masa protona [kg]
m_el = 9.10938291e-31 #masa elektrona [kg]
c = 299792458 #hitrost svetlobe [m/s]

def boris(x0, v0, E, B, dt, tdur, q=1, m=1):
    ''' Borisov algoritem zabjega skoka za
    integracijo diferencialne enačbe

    x0 = [x,y,z] : vektor zacetne pozicije
    v0 = [vx,vy,vz] : vektor zacetne hitrosti
    E : funkcija, ki daje jakost polja E(x)
    B : funkcija, ki daje jakost polja B(x)
    dt: casovni korak '''

    duration = int(tdur/dt)

    X = np.zeros((duration,3))
    V = np.zeros((duration,3))
    x = x0
    v = v0
    u = v0 / np.sqrt(1-(np.sum(v0**2)/c**2))

    q_prime = dt * q * 0.5 / m
    
    # https://en.wikipedia.org/wiki/Particle-in-cell
    # https://warpx.readthedocs.io/en/latest/theory/picsar_theory.html
    for time in range(duration):
        u_min = u + q_prime*E(x)
        t = q_prime*B(x)/np.sqrt(1+np.sum(u_min**2)/c**2)
        u_prime = u_min + np.cross(u_min,t)
        u_plu = u_min + np.cross(u_prime,2*t)/(1+np.sum(t**2))
        u = u_plu + q_prime*E(x)
        v = u/np.sqrt(1+np.sum(u_min**2)/c**2)
        x = x + dt*v
        V[time,:] = v
        X[time,:] = x


        #implementacija po wikipediji - nerelativisticno
        #h = q_prime*B(x)
        #s = 2*h / (1+np.dot(h,h))
        #u = v + q_prime*E(x)
        #u_ = u + np.cross(u+np.cross(u,h),s)
        #
        #v = u_ + q_prime * E(x)
        #x = x + dt*v
        #V[time,:] = v
        #X[time,:] = x

    return X, V

def vay(x0, v0, E, B, dt, tdur, q=1, m=1):
    ''' doi: 19,1963/1.2837054 '''
    print("ni implementiran")
    return 0



def B_zemlja(x):
    ''' model zemljinega magnentnega polja aproksimiran z 
    magnetnim dipolom '''
    B0 = 3.07e-5 # [T]
    Rz = 6378137 # radii zemlje [m]

    r = np.sqrt(np.sum(x**2))
    k = -B0*Rz**3/r**5
    return k*np.array([3*x[0]*x[2],3*x[1]*x[2],2*x[2]**2-x[0]**2-x[1]**2])

def B_loop(p, a):
    ''' polje magnetne zanke polmera a v xy ravnini, normiran tok 1A
    glej: Sinigoj, ELMG polje, stran 200'''
    mu = 2e-7 # mu/2/pi
    x,y,z = p
    r = np.sqrt(x**2+y**2)
    if r < 1e-15:
        Bz = 2*np.pi*1e-7*a**2/(a**2+z**2)**1.5
        return np.array([0.,0.,Bz])
    m = np.sqrt(4*a*r/( (r+a)**2+z**2))
    K = ellipk(m**2)
    E = ellipe(m**2)

    C = 1./np.sqrt((a+r)**2+z**2)

    Br = -mu * z/r*C * (-K + (a**2+r**2+z**2)*E/((a-r)**2+z**2))
    Bz = mu*C * (K - (a**2-r**2-z**2)*E/((a-r)**2+z**2))

    return np.array([Br*x/r, Br*y/r, Bz])



def B_bottle(p, a, h, I):
    ''' polje magnetne steklenice visine 2h in polmera a
    '''
    B1 = B_loop(p-np.array([0.,0.,h]),a)*I
    B2 = B_loop(p+np.array([0.,0.,h]),a)*I
    return B1+B2
    

def tokamak(p, a, b, I, n):
    ''' polje v modelu tokamaka, zanke polmera a
    nataknjene na toroid polmera b '''
    Rx = np.array([[1.,0.,0.],[0.,0.,-1.],[0.,1.,0.]])
    B = np.zeros(3)
    premik = np.array([b,0.,0.])

    for i in range(n):
        theta = i*2.*np.pi/n
        Rz = np.array([[np.cos(theta), np.sin(theta),0.],
                [-np.sin(theta),np.cos(theta), 0.],
                [0.,0.,1.]])
        #print(Rz)
        p_ = np.matmul(Rx.T, np.matmul(Rz.T,p)) + premik 
        b_ = I*B_loop(p_,a)
        b = np.matmul(Rz, np.matmul(Rx,b_))
        B = B+b
    
    return B

def Wk(m,V):
    return 0.5*m*np.sum(V**2,1)

def plot3(X,enote='m',zemlja=False,lim=[],bottle=[],tokamak=[]):
    fig = plt.figure()
    ax = fig.add_subplot(projection='3d')
    if zemlja:
        u = np.linspace(0, 2 * np.pi, 100)
        v = np.linspace(0, np.pi, 100)
        x = np.outer(np.cos(u), np.sin(v))
        y = np.outer(np.sin(u), np.sin(v))
        z = np.outer(np.ones(np.size(u)), np.cos(v))
        ax.plot_surface(x,y,z,zorder=-4)
        #ax.set_box_aspect(np.ptp(X,axis=0))

    ax.plot(X[:,0],X[:,1],X[:,2],zorder=4)
    ax.set_xlabel(r'$x$ [{}]'.format(enote))
    ax.set_ylabel(r'$y$ [{}]'.format(enote))
    ax.set_zlabel(r'$z$ [{}]'.format(enote))
    if lim:
        ax.set_xlim(lim)
        ax.set_ylim(lim)
        ax.set_zlim(lim)

    if bottle:
        r = bottle[0]
        h = bottle[1]
        fi = np.linspace(0,2*np.pi,300)
        a = r*np.cos(fi)
        b = r*np.sin(fi)
        ax.plot(a,b,+h,color='red',zorder=100)
        ax.plot(a,b,-h,color='red',zorder=100)
    if tokamak:
        ax=plot_tokamak(tokamak[0],tokamak[1],tokamak[2],ax)


    plt.show()

def plot_tokamak(a,b,n,ax=None):
    fi = np.linspace(0,2.*np.pi,300)
    x = a*np.cos(fi)
    y = a*np.sin(fi)
    z = np.zeros(300)
    X=np.zeros([300,3,n])

    x = x-b
    Rx = np.array([[1.,0.,0.],[0.,0.,-1.],[0.,1.,0.]])

    if ax == None:
        fig = plt.figure()
        ax = fig.add_subplot(projection='3d')

    for i in range(n):
        theta = i*2.*np.pi/n
        Rz = np.array([[np.cos(theta),-np.sin(theta),0.],
            [np.sin(theta),np.cos(theta),0.],
            [0.,0.,1]])
        for j in range(300):
            p = np.array([x[j],y[j],z[j]])
            #X=np.zeros([300,3])
            X[j,:,i] = np.matmul(Rz, np.matmul(Rx,p.T))
        ax.plot(X[:,0,i],X[:,1,i],X[:,2,i],color='red',zorder=1)
        plt.draw()
    plt.show()

def rplot(X, enote='m', xy=[0,1],os='xy', fname=None):
    fig = plt.figure()
    ax = fig.subplots()
    ax.plot(X[:,xy[0]],X[:,xy[1]], 'k')
    ax.grid()
    dx = X[-1,xy[0]]-X[-2,xy[0]]
    dy = X[-1,xy[1]]-X[-2,xy[1]]
    #ax.arrow(X[-1,xy[0]],X[-1,xy[1]],dx,dy,head_width=0.1,head_length=0.2,
    #        overhang=0.3, zorder=10,color='k')
    ax.set_xlabel(r'${}$ [{}]'.format(os[0],enote))
    ax.set_ylabel(r'${}$ [{}]'.format(os[1],enote))
    if fname:
        plt.savefig(fname,bbox_inches='tight')
    plt.show()



if __name__ == "__main__": # testni del kode
    E = lambda x: np.array([0.0,0.2,0.3])
    B = lambda x: np.array([0.,0.0,1.*x[0]])

    x0 = np.array([-1.,0.,0.])
    v0 = np.array([0.,1.,0.])

    dt = 1e-2

    #X, V = boris(x0,v0,E,B,dt,20)

    #plot3(X)
    plot_tokamak(1,2,16)