forked from barbagroup/DirectForcingMethod
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NavierStokesSolver.py
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NavierStokesSolver.py
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import numpy as np
import scipy.sparse as sp
import scipy.sparse.linalg as sla
import numpy.linalg as la
import matplotlib.pyplot as plt
import os
import pyamg
import errno
class NavierStokesSolver:
def __init__(self, N=4, alphaImplicit=1., alphaExplicit=0., gamma=1., zeta=0., nu=0.1, dt=-1.0, folder="."):
self.nu = nu
self.N = N
self.h = 2*np.pi/N
self.dt = self.h*self.h/self.nu/10.
if dt>0:
self.dt = dt
print "nu =", self.nu
print "h =", self.h
print "dt =", self.dt
self.alphaExplicit = alphaExplicit
self.alphaImplicit = alphaImplicit
self.gamma = gamma
self.zeta = zeta
self.folder = folder
def initVecs(self):
N = self.N
self.phi = np.zeros(N*N)
self.q = np.zeros(2*N*N)
self.qStar = np.zeros(2*N*N)
self.rn = np.zeros(2*N*N)
self.H = np.zeros(2*N*N)
self.initFluxes()
def initFluxes(self):
h = self.h
N = self.N
index = 0
for j in xrange(N):
for i in xrange(N):
# u
x = (i+1)*h
y = (j+0.5)*h
self.q[index] = np.sin(x)*np.cos(y)*h
index+=1
# v
x = (i+0.5)*h
y = (j+1)*h
self.q[index] = -np.cos(x)*np.sin(y)*h
index+=1
def exactSolutionTaylorGreen(self, t):
h = self.h
N = self.N
self.exactSolution = np.zeros(2*N*N)
index = 0
for j in xrange(N):
for i in xrange(N):
# u
x = (i+1)*h
y = (j+0.5)*h
self.exactSolution[index] = np.exp(-2*self.nu*t)*np.sin(x)*np.cos(y)*h
index+=1
# v
x = (i+0.5)*h
y = (j+1)*h
self.exactSolution[index] = -np.exp(-2*self.nu*t)*np.cos(x)*np.sin(y)*h
index+=1
def initMatrices(self):
self.generateA()
self.generateBNQ()
self.generateQTBNQ()
def generateBNQ(self):
N = self.N
rows = np.zeros(4*N*N, dtype=np.int)
cols = np.zeros(4*N*N, dtype=np.int)
vals = np.zeros(4*N*N, dtype=np.float)
index = 0
row_index = 0
for j in xrange(N):
for i in xrange(N):
# u
rows[index] = row_index
cols[index] = j*N+i
vals[index] = -1.
index+=1
rows[index] = row_index
cols[index] = j*N+i+1 if i<N-1 else j*N
vals[index] = 1.
index+=1
row_index+=1
# v
rows[index] = row_index
cols[index] = j*N+i
vals[index] = -1.
index+=1
rows[index] = row_index
cols[index] = (j+1)*N+i if j<N-1 else i
vals[index] = 1.
index+=1
row_index+=1
self.BNQ = sp.csr_matrix((vals, (rows, cols)), shape=(2*N*N, N*N))
self.QT = self.BNQ.transpose(copy=True)
self.BNQ = self.dt*self.BNQ
def generateQTBNQ(self):
self.QTBNQ = self.QT*self.BNQ
idx = list(self.QTBNQ.indices).index(0)
self.QTBNQ.data[idx] *=2.0
self.ml = pyamg.ruge_stuben_solver(self.QTBNQ)
def generateA(self):
h = self.h
dt = self.dt
N = self.N
alpha = self.alphaImplicit
nu = self.nu
rows = np.zeros(2*5*N*N, dtype=np.int)
cols = np.zeros(2*5*N*N, dtype=np.int)
vals = np.zeros(2*5*N*N, dtype=np.float)
row_index = 0
index = 0
for j in xrange(N):
for i in xrange(N):
# u
rows[index] = row_index
cols[index] = 2*((j-1)*N+i) if j>0 else 2*((N-1)*N+i)
vals[index] = -alpha*nu*1./h**2
index+=1
rows[index] = row_index
cols[index] = 2*(j*N+(i-1)) if i>0 else 2*(j*N+(N-1))
vals[index] = -alpha*nu*1./h**2
index+=1
rows[index] = row_index
cols[index] = 2*(j*N+i)
vals[index] = 1./dt + alpha*nu*4./h**2
index+=1
rows[index] = row_index
cols[index] = 2*(j*N+(i+1)) if i<N-1 else 2*(j*N+0)
vals[index] = -alpha*nu*1./h**2
index+=1
rows[index] = row_index
cols[index] = 2*((j+1)*N+i) if j<N-1 else 2*(0*N+i)
vals[index] = -alpha*nu*1./h**2
index+=1
row_index+=1
# v
rows[index] = row_index
cols[index] = 2*((j-1)*N+i)+1 if j>0 else 2*((N-1)*N+i)+1
vals[index] = -alpha*nu*1./h**2
index+=1
rows[index] = row_index
cols[index] = 2*(j*N+(i-1))+1 if i>0 else 2*(j*N+(N-1))+1
vals[index] = -alpha*nu*1./h**2
index+=1
rows[index] = row_index
cols[index] = 2*(j*N+i)+1
vals[index] = 1./dt + alpha*nu*4./h**2
index+=1
rows[index] = row_index
cols[index] = 2*(j*N+(i+1))+1 if i<N-1 else 2*(j*N+0)+1
vals[index] = -alpha*nu*1./h**2
index+=1
rows[index] = row_index
cols[index] = 2*((j+1)*N+i)+1 if j<N-1 else 2*(0*N+i)+1
vals[index] = -alpha*nu*1./h**2
index+=1
row_index+=1
self.A = sp.csr_matrix((vals, (rows, cols)), shape=(2*N*N, 2*N*N))
def calculateRN(self):
alpha = self.alphaExplicit
N = self.N
h = self.h
index = 0
for j in xrange(N):
for i in xrange(N):
# u
center = index
south = (index-2*N)%(2*N*N)
west = index-2 if i>0 else index+2*(N-1)
east = index+2 if i<N-1 else index-2*(N-1)
north = (index+2*N)%(2*N*N)
diffusion_term = alpha*self.nu*(1./h**2)*(self.q[north] + self.q[south] + self.q[east] + self.q[west] - 4.*self.q[center])/h
southwest = south+1
southeast = southwest+2 if i<N-1 else southwest-2*(N-1)
northwest = center+1
northeast = northwest+2 if i<N-1 else northwest-2*(N-1)
convection_term = (0.25/h**2)*((self.q[center]+self.q[east])*(self.q[center]+self.q[east]) - (self.q[center]+self.q[west])*(self.q[center]+self.q[west]))/h \
+ (0.25/h**2)*((self.q[center]+self.q[north])*(self.q[northwest]+self.q[northeast]) - (self.q[center]+self.q[south])*(self.q[southwest]+self.q[southeast]))/h
self.H[index] = self.gamma*convection_term + self.zeta*self.H[index]
self.rn[index] = (-self.H[index] + diffusion_term + self.q[index]/h/self.dt)*h
index+=1
# v
center+= 1
south += 1
west += 1
east += 1
north += 1
diffusion_term = alpha*self.nu*(1./h**2)*(self.q[north] + self.q[south] + self.q[east] + self.q[west] - 4.*self.q[center])/h
northeast = north-1
northwest = northeast-2 if i>0 else northeast+2*(N-1)
southeast = center-1
southwest = southeast-2 if i>0 else southeast+2*(N-1)
convection_term = (0.25/h**2)*((self.q[center]+self.q[east])*(self.q[northeast]+self.q[southeast]) - (self.q[center]+self.q[west])*(self.q[northwest]+self.q[southwest]))/h \
+ (0.25/h**2)*((self.q[center]+self.q[north])*(self.q[center]+self.q[north]) - (self.q[center]+self.q[south])*(self.q[center]+self.q[south]))/h
self.H[index] = self.gamma*convection_term + self.zeta*self.H[index]
self.rn[index] = (-self.H[index] + diffusion_term + self.q[index]/h/self.dt)*h
index+=1
def stepTime(self):
# solve for intermediate velocity
self.calculateRN()
self.qStar, _ = sla.bicgstab(self.A, self.rn, tol=1e-8)
# solve for pressure
self.rhs2 = self.QT*self.qStar
#self.phi, _ = sla.cg(self.QTBNQ, self.rhs2)
self.phi = self.ml.solve(self.rhs2, tol=1e-8)
# projection step
self.q = self.qStar - self.BNQ*self.phi
def writeData(self, n):
h = self.h
N = self.N
U = np.zeros(N*N)
U[:] = self.q[::2]/h
U = np.reshape(U, (N,N))
x = np.linspace(h, 2*np.pi, N)
y = np.linspace(0.5*h, 2*np.pi-0.5*h, N)
X, Y = np.meshgrid(x, y)
plt.ioff()
CS = plt.contour(X, Y, U, levels=np.linspace(-1., 1., 11))
plt.colorbar(CS)
plt.axis([0, 2*np.pi, 0, 2*np.pi])
plt.gca().set_aspect('equal', adjustable='box')
plt.savefig("%s/u%07d.png" % (self.folder,n))
plt.clf()
V = np.zeros(N*N)
V[:] = self.q[1::2]/h
V = np.reshape(V, (N,N))
x = np.linspace(0.5*h, 2*np.pi-0.5*h, N)
y = np.linspace(h, 2*np.pi, N)
X, Y = np.meshgrid(x, y)
CS = plt.contour(X, Y, V, levels=np.linspace(-1., 1., 11))
plt.colorbar(CS)
plt.axis([0, 2*np.pi, 0, 2*np.pi])
plt.gca().set_aspect('equal', adjustable='box')
plt.savefig("%s/v%07d.png" % (self.folder, n))
plt.clf()
def runSimulation(self, nt=20, nsave=1, plot=True):
try:
os.makedirs(self.folder)
except OSError as exc:
if exc.errno == errno.EEXIST and os.path.isdir(self.folder):
pass
else:
raise
self.initVecs()
self.initMatrices()
if plot:
self.writeData(0)
for n in xrange(1,nt+1):
self.stepTime()
if n%nsave==0 and plot:
self.writeData(n)
if __name__ == "__main__":
NT = 20
solver = NavierStokesSolver(N=6, alphaExplicit=0., alphaImplicit=1., dt=0.1, folder="NS06")
solver.runSimulation(nt=NT)
solver = NavierStokesSolver(N=12, alphaExplicit=0., alphaImplicit=1., dt=0.1, folder="NS12")
solver.runSimulation(nt=NT)
solver = NavierStokesSolver(N=24, alphaExplicit=0., alphaImplicit=1., dt=0.1, folder="NS24")
solver.runSimulation(nt=NT)