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Stokes.py
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Stokes.py
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import numpy as np
from scipy.sparse import linalg
from scipy import sparse
def add_to_sparse(x,y,value,row,col,data):
row.append(x)
col.append(y)
data.append(value)
def return_sparse_matrix_Stokes(j_res, i_res, dx, dy, eta_s, eta_n, rho, gx_0, gy_0, kbond, kcont, p0cell,
lower_boundary="slip", upper_boundary="slip",
left_boundary="slip", right_boundary="slip"):
# Constructing sparse matrix for solving: x-Stokes, y-Stokes and continuity equations
# x-Stokes: ETA(d2vx/dx2+d2vx/dy2)-dP/dx=0
# y-Stokes: ETA(d2vy/dx2+d2vy/dy2)-dP/dy=gy*RHO
# continuity: dvx/dx+dvy/dy=0
# Boundary conditions: free slip
row = []
col = []
data = []
# define grid for messing with indexes
k = np.linspace(0,(j_res*i_res-1),(j_res*i_res)).astype('int')
k.shape = ((i_res,j_res))
vector = np.ones((3*i_res,j_res))
vector = vector.reshape((3*j_res*i_res,1))
P = lambda k: 3*k
Vx = lambda k: 3*k+1
Vy = lambda k: 3*k+2
dx2,dy2 = dx**2, dy**2
for j in range(0,j_res):
for i in range(0,i_res):
# Continuity equation
# Ghost pressure unknowns (i=0, j=0): P(i,j)=0
if i==0 or j==0:
add_to_sparse(P(k[i][j]),P(k[i][j]),1*kbond,row,col,data)
vector[P(k[i][j])] = 0
# Upper and lower left corners dP/dx=0 => P(i,j)-P(i,j+1)=0
elif (i==1 and j==1) or (i==i_res-1 and j==1):
add_to_sparse(P(k[i][j]),P(k[i][j]) , 1*kbond ,row,col,data)
add_to_sparse(P(k[i][j]),P(k[i][j+1]),-1*kbond ,row,col,data)
vector[P(k[i][j])] = 0
# Upper and lower right corners dP/dx=0 => P(i,j)-P(i,j-1)=0
elif (i==1 and j==j_res-1) or (i==i_res-1 and j==j_res-1):
add_to_sparse(P(k[i][j]),P(k[i][j]) , 1*kbond ,row,col,data)
add_to_sparse(P(k[i][j]),P(k[i][j-1]),-1*kbond ,row,col,data)
vector[P(k[i][j])] = 0
# One cell
elif i==1 and j==2:
add_to_sparse(P(k[i][j]),P(k[i][j]),1*kbond ,row,col,data) # Coefficient for P(i,j)
vector[P(k[i][j])] = p0cell # Right-hand-side part
else:
# Internal nodes: dvx/dx+dvy/dy=0
# dvx/dx=(vx(i-1,j)-vx(i-1,j-1))/dx
add_to_sparse(P(k[i][j]),Vx(k[i-1][j]) ,kcont/dx ,row,col,data) # Coefficient for P(i-1,j)
add_to_sparse(P(k[i][j]),Vx(k[i-1][j-1]) ,-kcont/dx ,row,col,data) # Coefficient for P(i-1,j-1)
# dvy/dy=(vy(i,j-1)-vy(i-1,j-1))/dy
add_to_sparse(P(k[i][j]),Vy(k[i ][j-1]) ,kcont/dy ,row,col,data) # Coefficient for P(i,j-1)
add_to_sparse(P(k[i][j]),Vy(k[i-1][j-1]) ,-kcont/dy ,row,col,data) # Coefficient for P(i-1,j-1)
vector[P(k[i][j])] = 0 # Right-hand-side part
# x-Stokes equation
# Ghost Vx unknowns (i=i_res) and boundary nodes (i=0, i=i_res-1, j=0, j=j_res-1)
# Ghost Vx unknowns (i=i_res: Vx(i,j)=0
if i == i_res-1:
add_to_sparse(Vx(k[i][j]),Vx(k[i][j]) ,kbond ,row,col,data) # Coefficient for Vx(i,j)
vector[Vx(k[i][j])] = 0 # Right-hand-side part
# Left and Right boundaries (j=0, j=j_res)
elif (j==0 or j==j_res-1) and i<i_res-1:
# Free slip, No slip: vx(i,j)=0
add_to_sparse(Vx(k[i][j]),Vx(k[i][j]) , kbond ,row,col,data) # Coefficient for Vx(i,j)
vector[Vx(k[i][j])] = 0 # Right-hand-side part
# Upper boundary, iner points (i=0, 0<j<j_res)
elif i==0 and 0<j<j_res-1:
# Free slip dVx/dy=0: Vx(i,j)-Vx(i+1,j)=0
if upper_boundary=="sleep":
add_to_sparse(Vx(k[i][j]),Vx(k[i][j]) , kbond ,row,col,data) # Coefficient for Vx(i,j)
add_to_sparse(Vx(k[i][j]),Vx(k[i+1][j]) ,-kbond ,row,col,data) # Coefficient for Vx(i+1,j)
vector[Vx(k[i][j])] = 0 # Right-hand-side part
## No slip vx=0: vx(i,j)-1/3*vx(i+1,j)=0
else:
add_to_sparse(Vx(k[i][j]),Vx(k[i][j]) , kbond ,row,col,data) # Coefficient for Vx(i,j)
add_to_sparse(Vx(k[i][j]),Vx(k[i+1][j]) ,-(1.0/3)*kbond ,row,col,data) #Coefficient for Vx(i+1,j)
vector[Vx(k[i][j])] = 0 # Right-hand-side part
# Lower boundary, iner points (i=i_res-1, 0<j<j_res)
elif i==i_res-2 and 0<j<j_res-1:
# Free slip dvx/dy=0: vx(i,j)-vx(i-1,j)=0
if lower_boundary=="sleep":
add_to_sparse(Vx(k[i][j]),Vx(k[i][j]) , kbond ,row,col,data) # Coefficient for Vx(i,j)
add_to_sparse(Vx(k[i][j]),Vx(k[i-1][j]) ,-kbond ,row,col,data) # Coefficient for Vx(i-1,j)
vector[Vx(k[i][j])] = 0 # Right-hand-side part
## No slip vx=0: vx(i,j)-1/3*vx(i-1,j)=0
else:
add_to_sparse(Vx(k[i][j]),Vx(k[i][j]) , kbond ,row,col,data) # Coefficient for Vx(i,j)
add_to_sparse(Vx(k[i][j]),Vx(k[i-1][j]) ,-(1.0/3)*kbond ,row,col,data) #Coefficient for Vx(i-1,j)
vector[Vx(k[i][j])] = 0 # Right-hand-side part
else:
# Internal nodes: dSxx/dx+dSxy/dy-dP/dx=0
# dSxx/dx=2*etan(i+1,j+1)*(vx(i,j+1)-vx(i,j))/dx^2-2*etan(i+1,j)*(vx(i,j)-vx(i,j-1))/dx^2
add_to_sparse(Vx(k[i,j]),Vx(k[i,j+1]), 2*eta_n[i+1,j+1]/dx2 ,row,col,data) #Coefficient for Vx(i,j+1)
add_to_sparse(Vx(k[i,j]),Vx(k[i,j-1]), 2*eta_n[i+1,j]/dx2 ,row,col,data) #Coefficient for Vx(i,j-1)
add_to_sparse(Vx(k[i,j]),Vx(k[i,j]), -2*eta_n[i+1,j+1]/dx2-2*eta_n[i+1,j]/dx2 ,row,col,data) #Coefficient for Vx(i,j)
#dSxy/dy=etas(i+1,j)*((vx(i+1,j)-vx(i,j))/dy^2+(vy(i+1,j)-vy(i+1,j-1))/dx/dy)-
# -etas(i,j)*((vx(i,j)-vx(i-1,j))/dy^2+(vy(i,j)-vy(i,j-1))/dx/dy)-
add_to_sparse(Vx(k[i,j]),Vx(k[i+1,j]), eta_s[i+1,j]/dy2 ,row,col,data) #Coefficient for Vx(i+1,j)
add_to_sparse(Vx(k[i,j]),Vx(k[i-1,j]), eta_s[i,j]/dy2 ,row,col,data) #Coefficient for Vx(i-1,j)
add_to_sparse(Vx(k[i,j]),Vx(k[i,j]), -eta_s[i+1,j]/dy2-eta_s[i,j]/dy2 ,row,col,data) #Coefficient for Vx(i,j)
add_to_sparse(Vx(k[i,j]),Vy(k[i+1,j]), eta_s[i+1,j]/dx/dy ,row,col,data) #Coefficient for Vy(i+1,j)
add_to_sparse(Vx(k[i,j]),Vy(k[i+1,j-1]),-eta_s[i+1,j]/dx/dy ,row,col,data) #Coefficient for Vy(i+1,j-1)
add_to_sparse(Vx(k[i,j]),Vy(k[i,j]), -eta_s[i,j]/dx/dy ,row,col,data) #Coefficient for Vy(i,j)
add_to_sparse(Vx(k[i,j]),Vy(k[i,j-1]), eta_s[i,j]/dx/dy ,row,col,data) #Coefficient for Vy(i,j-1)
# -dP/dx=(P(i+1,j)-P(i+1,j+1))/dx
add_to_sparse(Vx(k[i][j]),P(k[i+1][j]), kcont/dx ,row,col,data) # Coefficient for P(i+1,j)
add_to_sparse(Vx(k[i][j]),P(k[i+1][j+1]),-kcont/dx ,row,col,data) # Coefficient for P(i+1,j+1)
# Right-hand-side part:0
vector[Vx(k[i][j])] = 0 # Right-hand-side part
# y-Stokes equation
# Ghost vy unknowns (j=j_res) and boundary nodes (i=0, i=i_res-1, j=0, j=j_res-1)
# Ghost vy unknowns (j=jres: vy(i,j)=0
if j==j_res-1:
add_to_sparse(Vy(k[i][j]),Vy(k[i][j]) ,kbond ,row,col,data) # Coefficient for Vy(i,j)
vector[Vy(k[i][j])] = 0 # Right-hand-side part
# Upper and lower boundaries (i=0, i=i_res)
elif (i==0 or i==i_res-1) and j<j_res:
# Free slip, No slip: vy(i,j)=0
add_to_sparse(Vy(k[i][j]),Vy(k[i][j]) , kbond ,row,col,data) # Coefficient for Vy(i,j)
vector[Vy(k[i][j])] = 0 # Right-hand-side part
# Left boundary, iner points (j=0, 0<i<i_res)
elif j==0 and 0 < i < i_res-1:
# Free slip dvy/dx=0: vy(i,j)-vy(i,j+1)=0
if left_boundary == "sleep":
add_to_sparse(Vy(k[i][j]),Vy(k[i][j]) , kbond ,row,col,data) # Coefficient for Vy(i,j)
add_to_sparse(Vy(k[i][j]),Vy(k[i][j+1]) ,-kbond ,row,col,data) # Coefficient for Vy(i,j+1)
vector[Vy(k[i][j])] = 0 # Right-hand-side part
## No slip vy=0: vy(i,j)-1/3*vy(i,j+1)=0
else:
add_to_sparse(Vy(k[i][j]),Vy(k[i][j]) , kbond ,row,col,data) # Coefficient for Vy(i,j)
add_to_sparse(Vy(k[i][j]),Vy(k[i][j+1]) ,-(1.0/3)*kbond ,row,col,data) # Coefficient for Vy(i,j+1)
vector[Vy(k[i][j])] = 0 # Right-hand-side part
# Right boundary, iner points (j=j_res-1, 0<i<i_res)
elif j==j_res-2 and 0 < i <i_res-1:
# Free slip dvy/dx=0: vy(i,j)-vy(i,j-1)=0
if right_boundary=="sleep":
add_to_sparse(Vy(k[i][j]),Vy(k[i][j]) , kbond ,row,col,data) # Coefficient for Vy(i,j)
add_to_sparse(Vy(k[i][j]),Vy(k[i][j-1]) ,-kbond ,row,col,data) # Coefficient for Vy(i,j-1)
vector[Vy(k[i][j])] = 0 # Right-hand-side part
## No slip vy=0: vy(i,j)-1/3*vy(i,j-1)=0
else:
add_to_sparse(Vy(k[i][j]),Vy(k[i][j]) , kbond ,row,col,data) # Coefficient for Vy(i,j)
add_to_sparse(Vy(k[i][j]),Vy(k[i][j-1]) ,-(1.0/3)*kbond ,row,col,data) # Coefficient for Vy(i,j-1)
vector[Vy(k[i][j])] = 0 # Right-hand-side part
else:
# Internal nodes: dSyy/dy+dSxy/dx-dP/dy=-gy*RHO
#dSyy/dy=2*etan(i+1,j+1)*(vy(i+1,j)-vy(i,j))/dy^2-2*etan(i,j+1)*(vy(i,j)-vy(i-1,j))/dy^2
add_to_sparse(Vy(k[i,j]),Vy(k[i+1,j]), 2*eta_n[i+1,j+1]/dy2 ,row,col,data) #Coefficient for Vy(i+1,j)
add_to_sparse(Vy(k[i,j]),Vy(k[i-1,j]), 2*eta_n[i,j+1]/dy2 ,row,col,data) #Coefficient for Vy(i-1,j)
add_to_sparse(Vy(k[i,j]),Vy(k[i,j]), -2*eta_n[i+1,j+1]/dy2-2*eta_n[i,j+1]/dy2 ,row,col,data) #Coefficient for Vy(i,j)
#dSxy/dx=etas(i,j+1)*((vy(i,j+1)-vy(i,j))/dx^2+(vx(i,j+1)-vx(i-1,j+1))/dx/dy)-
# -etas(i,j)*((vy(i,j)-vy(i,j-1))/dx^2+(vx(i,j)-vx(i-1,j))/dx/dy)-
add_to_sparse(Vy(k[i,j]),Vy(k[i,j+1]), eta_s[i,j+1]/dx2 ,row,col,data) #Coefficient for Vy(i+1,j)
add_to_sparse(Vy(k[i,j]),Vy(k[i,j-1]), eta_s[i,j]/dx2 ,row,col,data) #Coefficient for Vy(i-1,j)
add_to_sparse(Vy(k[i,j]),Vy(k[i,j]), -eta_s[i,j+1]/dx2-eta_s[i,j]/dx2 ,row,col,data) #Coefficient for Vy(i,j)
add_to_sparse(Vy(k[i,j]),Vx(k[i,j+1]), eta_s[i,j+1]/dx/dy ,row,col,data) #Coefficient for Vx(i+1,j)
add_to_sparse(Vy(k[i,j]),Vx(k[i-1,j+1]),-eta_s[i,j+1]/dx/dy ,row,col,data) #Coefficient for Vx(i+1,j-1)
add_to_sparse(Vy(k[i,j]),Vx(k[i,j]), -eta_s[i,j]/dx/dy ,row,col,data) #Coefficient for Vx(i,j)
add_to_sparse(Vy(k[i,j]),Vx(k[i-1,j]), eta_s[i,j]/dx/dy ,row,col,data) #Coefficient for Vx(i,j-1)
# -dP/dy=(P(i,j+1)-P(i+1,j+1))/dx
add_to_sparse(Vy(k[i][j]),P(k[i][j+1]), kcont/dy ,row,col,data) # Coefficient for P(i,j+1)
add_to_sparse(Vy(k[i][j]),P(k[i+1][j+1]),-kcont/dy ,row,col,data) # Coefficient for P(i+1,j+1)
# Right part: -RHO*gy
vector[Vy(k[i][j])] = -gy_0 * (rho[i,j] + rho[i,j+1])/2.0 # Right-hand-side part
mtx = sparse.coo_matrix((data, (row, col)), shape=(3*j_res*i_res, 3*j_res*i_res))
mtx = mtx.tocsr()
return mtx, vector