def frontogenesis_timestep(N, days, tstepsize, Heun=None):
tf = 60 * 60 * 24 * days #final time
#initialise parameters from SG Eady model
H = 1.e4
L = 1.e6
g = 10.
f = 1.e-4
theta0 = 300.
#create new directory to store results
if Heun:
newdir = "/scratchcomp04/cvr12/Results_" + str(days) + "D_" + str(
N) + "N_" + str(int(tstepsize)) + "_heun"
else:
newdir = "/scratchcomp04/cvr12/Results_" + str(days) + "D_" + str(
N) + "N_" + str(int(tstepsize)) + "_euler"
os.mkdir(newdir)
#initialise source density with periodic BCs in x
bbox = np.array([-L, 0., L, H])
Xdens = pdx.sample_rectangle(bbox)
f0 = np.ones(4) / (2 * H * L)
rho = np.zeros(Xdens.shape[0])
T = ma.delaunay_2(Xdens, rho)
dens = pdx.Periodic_density_in_x(Xdens, f0, T, bbox)
#initialise points in geostrophic space
[Y, thetap] = initialise_points(N, bbox, RegularMesh=True)
Y = dens.to_fundamental_domain(Y)
if Heun:
#timestep using Heun's Method
[Y, w, t] = heun_sg(Y,
dens,
tf,
bbox,
newdir,
h=tstepsize,
add_data=True)
t.tofile(newdir + '/time.txt', sep=" ", format="%s")
else:
#timestep using forward euler scheme
[Y, w, t] = forward_euler_sg(Y,
dens,
tf,
bbox,
newdir,
h=tstepsize,
add_data=True)
t.tofile(newdir + '/time.txt', sep=" ", format="%s")
return ('complete results ' + str(days) + 'D_' + str(N) + 'N_' +
str(int(tstepsize)))
import matplotlib
matplotlib.use('Agg')
import numpy as np
from periodic_densities import Periodic_density_in_x, sample_rectangle, periodicinx_draw_laguerre_cells_2
import MongeAmpere as ma
import matplotlib.pyplot as plt
import matplotlib.tri as tri
# source: uniform measure on the square with sidelength 1
bbox = [0., 0., 1., 1.]
Xdens = sample_rectangle(bbox)
f0 = np.ones(4)
rho = np.zeros(Xdens.shape[0])
T = ma.delaunay_2(Xdens, rho)
dens = Periodic_density_in_x(Xdens, f0, T, bbox)
print "mass=%g" % dens.mass()
# target is a random set of points, with random weights
N = 6
Y = np.random.rand(N, 2)
nu = np.ones(N)
nu = (dens.mass() / np.sum(nu)) * nu
print "mass(nu) = %f" % sum(nu)
print "mass(mu) = %f" % dens.mass()
w = ma.optimal_transport_2(dens, Y, nu)
#x,y = periodicinx_draw_laguerre_cells_2(dens,Y,w)
import matplotlib
matplotlib.use('Agg')
import numpy as np
import periodic_densities as pdx
import MongeAmpere as ma
import matplotlib.pyplot as plt
import matplotlib.tri as tri
from PIL import Image
N = 10000
L = 1.e6
H = 1.e4
bbox = np.array([-L, 0., L, H])
Xdens = pdx.sample_rectangle(bbox)
f0 = np.ones(4) / (2 * L * H)
rho = np.zeros(Xdens.shape[0])
T = ma.delaunay_2(Xdens, rho)
dens = pdx.Periodic_density_in_x(Xdens, f0, T, bbox)
Y = np.random.rand(N, 2)
xY[:, 0] = Y[:, 0] * 2 * L - L
Y[:, 1] *= H
#print(Y)
print(Y.shape)
nu = np.ones(Y[:, 0].size)
nu = (dens.mass() / np.sum(nu)) * nu
w = ma.optimal_transport_2(dens, Y, nu, verbose=True)
def validity_analysis_results(N, days, tstepsize, t0=0., Heun=None):
#initialise parameters
H = 1.e4
L = 1.e6
g = 10.
f = 1.e-4
theta0 = 300
C = 3e-6
if Heun:
print('heun')
datadir = "/scratchcomp04/cvr12/Results_" + str(days) + "D_" + str(
N) + "N_" + str(int(tstepsize)) + "_heun/data"
resultdir = "/scratchcomp04/cvr12/Results_" + str(days) + "D_" + str(
N) + "N_" + str(int(tstepsize)) + "_heun"
else:
print('euler')
datadir = "/scratchcomp04/cvr12/Results_" + str(days) + "D_" + str(
N) + "N_" + str(int(tstepsize)) + "_euler/data"
resultdir = "/scratchcomp04/cvr12/Results_" + str(days) + "D_" + str(
N) + "N_" + str(int(tstepsize)) + "_euler"
os.mkdir(datadir)
#set up uniform density for domain Gamma
bbox = np.array([-L, 0., L, H])
Xdens = pdx.sample_rectangle(bbox)
f0 = np.ones(4) / 2 / L / H
rho = np.zeros(Xdens.shape[0])
T = ma.delaunay_2(Xdens, rho)
dens = pdx.Periodic_density_in_x(Xdens, f0, T, bbox)
#set final time(t), step size (h) and total number of time steps(N)
tf = 60 * 60 * 24 * days
h = tstepsize
N = int(np.ceil((tf - t0) / h))
#initialise arrays to store data values
KEmean = np.zeros(N + 1)
energy = np.zeros(N + 1)
vgmax = np.zeros(N + 1)
PE = np.zeros(N + 1)
KE = np.zeros(N + 1)
rmsv = np.zeros(N + 1)
w = np.fromfile(resultdir + '/weights_results_' + str(int(h)) +
'/weights_0.txt',
sep=" ")
ke = np.zeros((int(w.size), 2))
vg = np.zeros((int(w.size), 2))
t = np.array([t0 + n * h for n in range(N + 1)])
for n in range(0, N + 1):
Y = np.fromfile(resultdir + '/points_results_' + str(int(h)) +
'/points_' + str(n) + '.txt',
sep=" ")
w = np.fromfile(resultdir + '/weights_results_' + str(int(h)) +
'/weights_' + str(n) + '.txt',
sep=" ")
l = int(w.size)
Y = Y.reshape((l, 2))
#calculate centroids and mass of cells
[Yc, m] = dens.lloyd(Y, w)
mtile = np.tile(m, (2, 1)).T
#calculate second moments to find KE and maximum of vg
[m1, I] = dens.moments(Y, w)
#find kinetic energy, maximum value of vg and total energy
ke[:, 0] = 0.5 * f * f * (m * Y[:, 0]**2 - 2 * Y[:, 0] * m1[:, 0] +
I[:, 0])
vg[:, 0] = f * (Y[:, 0] - Yc[:, 0])
#map back to fundamental domain
ke = dens.to_fundamental_domain(ke)
vg = dens.to_fundamental_domain(vg)
E = ke[:,
0] - f * f * Y[:, 1] * m1[:, 1] + 0.5 * f * f * H * Y[:, 1] * m
pe = -f * f * Y[:, 1] * m1[:, 1] + 0.5 * f * f * H * Y[:, 1] * m
energy[n] = np.sum(E)
KE[n] = np.sum(ke[:, 0])
KEmean[n] = np.sum(ke[:, 0]) * float(l)
rmsv[n] = np.sqrt(KEmean[n])
PE[n] = np.sum(pe)
vgmax[n] = np.amax(vg[:, 0])
energy.tofile(datadir + '/energy.txt', sep=" ", format="%s")
KEmean.tofile(datadir + '/KEmean.txt', sep=" ", format="%s")
vgmax.tofile(datadir + '/vgmax.txt', sep=" ", format="%s")
t.tofile(datadir + '/time.txt', sep=" ", format="%s")
PE.tofile(datadir + '/PE.txt', sep=" ", format="%s")
KE.tofile(datadir + '/KE.txt', sep=" ", format="%s")
rmsv.tofile(datadir + '/rmsv.txt', sep=" ", format="%s")
return ('complete results ' + str(days) + 'D_' + str(N) + 'N_' +
str(int(tstepsize)))
def initialise_points(N, bbox, RegularMesh = None):
'''Function to initialise a mesh over the domain [-L,L]x[0,H]
and transform to geostrophic coordinates
args:
N: number of grid points in z direct, total number of points
will be 2*N*N
RegularMesh: controls whether the mesh is regular or an optimised
sample of random points in the domain
returns:
A numpy array of coordinates of points in geostrophic space
'''
H = bbox[3]
L = bbox[2]
#set up regular mesh or optimised sample of random points
if RegularMesh:
npts = N
dx = 1./npts
s = np.arange(dx*0.5,1.,dx)
s1 = np.arange(-1. + dx*0.5,1.,dx)
x,z = np.meshgrid(s1*L,s*H)
nx = x.size
x = np.reshape(x,(nx,))
z = np.reshape(z,(nx,))
y = np.array([x,z]).T
else:
npts = 2*N*N
bbox = np.array([0., -0.5, 2., 0.5])
Xdens = sample_rectangle(bbox);
f0 = np.ones(4)/2;
w = np.zeros(Xdens.shape[0]);
T = ma.delaunay_2(Xdens,w);
Sdens = Periodic_density_in_x(Xdens,f0,T,bbox)
X = ma.optimized_sampling_2(Sdens,npts,niter=5)
X = Sdens.to_fundamental_domain(X)
x = X[:,0]*L - L
z = (X[:,1]+0.5)*H
y = np.array([x,z]).T
#define thetap from CULLEN 2006
g = 10.
f = 1.e-4
theta0 = 300.
C = 3e-6
CP = 0.1
Nsq = 40/H*g/theta0
buoyancy_stratification = (z-H/2)*Nsq # b = g*theta/theta_0
thetap = buoyancy_stratification/g*theta0 + CP*np.sin(np.pi*(x/L + z/H))
X = x
Z = g*thetap/theta0/f/f
Y = np.array([X,Z]).T
return Y, thetap
def initialise_points(N, bbox, RegularMesh=None):
'''Function to initialise a mesh over the domain [-L,L]x[0,H]
and transform to geostrophic coordinates
args:
N: number of grid points in z direct, total number of points
will be 2*N*N
RegularMesh: controls whether the mesh is regular or an optimised
sample of random points in the domain
returns:
A numpy array of coordinates of points in geostrophic space
'''
H = bbox[3]
L = bbox[2]
#set up regular mesh or optimised sample of random points
if RegularMesh:
npts = N
dx = 1. / npts
s = np.arange(dx * 0.5, 1., dx)
s1 = np.arange(-1. + dx * 0.5, 1., dx)
x, z = np.meshgrid(s1 * L, s * H)
nx = x.size
x = np.reshape(x, (nx, ))
z = np.reshape(z, (nx, ))
y = np.array([x, z]).T
else:
npts = 2 * N * N
bbox = np.array([0., -0.5, 2., 0.5])
Xdens = sample_rectangle(bbox)
f0 = np.ones(4) / 2
w = np.zeros(Xdens.shape[0])
T = ma.delaunay_2(Xdens, w)
Sdens = Periodic_density_in_x(Xdens, f0, T, bbox)
X = ma.optimized_sampling_2(Sdens, npts, niter=5)
x = X[:, 0] * L - L
z = (X[:, 1] + 0.5) * H
y = np.array([x, z]).T
#define thetap from CULLEN 2006
Nsq = 2.5e-5
g = 10.
f = 1.e-4
theta0 = 300.
C = 3e-6
B = 0.25
thetap = Nsq * theta0 * z / g + B * np.sin(np.pi * (x / L + z / H))
vg = B * g * H / L / f / theta0 * np.sin(
np.pi *
(x / L + z / H)) - 2 * B * g * H / np.pi / L / f / theta0 * np.cos(
np.pi * x / L)
X = vg / f + x
Z = g * thetap / f / f / theta0
Y = np.array([X, Z]).T
thetap = f * f * theta0 * Z / g
return Y, thetap
def validity_analysis_results(N, days, tstepsize, t0 = 0., Heun = None):
#initialise parameters
H = 1.e4
L = 1.e6
g = 10.
f = 1.e-4
theta0 = 300
C = 3e-6
if Heun:
print('heun')
datadir = "Results_"+str(days)+"D_"+str(N)+"N_"+str(int(tstepsize))+"_heun/data"
resultdir = "Results_"+str(days)+"D_"+str(N)+"N_"+str(int(tstepsize))+"_heun"
plotsdir = "Results_"+str(days)+"D_"+str(N)+"N_"+str(int(tstepsize))+"_heun/plots"
else:
print('euler')
datadir = "Results_"+str(days)+"D_"+str(N)+"N_"+str(int(tstepsize))+"_euler/data"
resultdir = "Results_"+str(days)+"D_"+str(N)+"N_"+str(int(tstepsize))+"_euler"
plotsdir = "Results_"+str(days)+"D_"+str(N)+"N_"+str(int(tstepsize))+"_euler/plots"
os.mkdir(datadir)
os.mkdir(plotsdir)
#set up uniform density for domain Gamma
bbox = np.array([-L, 0., L, H])
Xdens = pdx.sample_rectangle(bbox)
f0 = np.ones(4)/2/L/H
rho = np.zeros(Xdens.shape[0])
T = ma.delaunay_2(Xdens,rho)
dens = pdx.Periodic_density_in_x(Xdens,f0,T,bbox)
#set up plotting uniform density for domain Gamma
bbox = np.array([-L, 0., L, H])
Xdens = pdx.sample_rectangle(bbox)
npts = 100
Xrnd = 2*L*(np.random.rand(npts)-0.5)
Zrnd = H*np.random.rand(npts)
Xdens = np.concatenate((Xdens, np.array((Xrnd, Zrnd)).T))
f0 = np.ones(npts+4)/2/L/H
rho = np.zeros(Xdens.shape[0])
T = ma.delaunay_2(Xdens,rho)
pdens = pdx.Periodic_density_in_x(Xdens,f0,T,bbox)
#set final time(t), step size (h) and total number of time steps(N)
tf = 60*60*24*days
h = tstepsize
N = int(np.ceil((tf-t0)/h))
#initialise arrays to store data values
KEmean = np.zeros(N+1)
energy = np.zeros(N+1)
vgmax = np.zeros(N+1)
PE = np.zeros(N+1)
KE = np.zeros(N+1)
rmsv = np.zeros(N+1)
w = np.fromfile(resultdir+'/weights_results_'+str(int(h))+'/weights_0.txt', sep = " ")
ke = np.zeros((int(w.size),2))
vg = np.zeros((int(w.size),2))
t = np.array([t0 + n*h for n in range(N+1)])
Ndump = 100
for n in range(0,N+1, Ndump):
print(n,N)
try:
Y = np.fromfile(resultdir+'/points_results_'+str(int(h))+'/points_'+str(n)+'.txt', sep = " ")
w = np.fromfile(resultdir+'/weights_results_'+str(int(h))+'/weights_'+str(n)+'.txt', sep = " ")
l = int(w.size)
Y = Y.reshape((l,2))
except IOError:
break
#Plots
C = Y[:,1].reshape((l,1))
print(C.shape, w.shape)
img = periodic_laguerre_diagram_to_image(pdens,Y,w,C,bbox,100,100)
plt.pcolor(img)
plt.savefig(plotsdir+'/c_'+str(n)+'.jpg')
plt.clf()
plt.plot(Y[:,0], Y[:,1], '.')
plt.savefig(plotsdir+'/Y_'+str(n)+'.jpg')
plt.clf()
#calculate centroids and mass of cells
[Yc, m] = dens.lloyd(Y, w)
mtile = np.tile(m,(2,1)).T
#calculate second moments to find KE and maximum of vg
[m1, I] = dens.moments(Y, w)
#find kinetic energy, maximum value of vg and total energy
ke[:,0] = 0.5*f*f*(m*Y[:,0]**2 - 2*Y[:,0]*m1[:,0] + I[:,0])
vg[:,0] = f*(Y[:,0] - Yc[:,0])
#map back to fundamental domain
ke = dens.to_fundamental_domain(ke)
vg = dens.to_fundamental_domain(vg)
E = ke[:,0] - f*f*Y[:,1]*m1[:,1] + 0.5*f*f*H*Y[:,1]*m
pe = - f*f*Y[:,1]*m1[:,1] + 0.5*f*f*H*Y[:,1]*m
energy[n] = np.sum(E)
KE[n] = np.sum(ke[:,0])
KEmean[n] = np.sum(ke[:,0])*float(l)
rmsv[n] = np.sqrt(KEmean[n])
PE[n] = np.sum(pe)
vgmax[n] = np.amax(vg[:,0])
energy.tofile(datadir+'/energy.txt',sep = " ",format = "%s")
KEmean.tofile(datadir+'/KEmean.txt',sep = " ",format = "%s")
vgmax.tofile(datadir+'/vgmax.txt',sep = " ",format = "%s")
t.tofile(datadir+'/time.txt',sep = " ",format = "%s")
PE.tofile(datadir+'/PE.txt',sep = " ",format = "%s")
KE.tofile(datadir+'/KE.txt',sep = " ",format = "%s")
rmsv.tofile(datadir+'/rmsv.txt',sep = " ",format = "%s")
return('complete results '+str(days)+'D_'+str(N)+'N_'+str(int(tstepsize)))