def implementationOne():
# constants
u0 = 1.0
C_f = 1.0
C_s = 1.0
# create the blocks, with no parameters or parameter functions
b0 = b.Block(
'0', # name
{'u': u0}) # boundary state, this is fixed
b1 = b.Block(
'1', # name
{
'u': 0,
'v': 0
}) # states, and their initial guesses
b2 = b.Block(
'2', # name
{'u': 0}) # states, and their initial guesses
# create the fluxes
# flux from 0 to 1
F_P = {'C_f': C_f} # dictionary of parameters for the flux
# flux is from b0, function is F_u
f_u01 = f.Flux(b0, F_u, F_P, 'f_u01')
# flux is from b1, function is F_u
f_u12 = f.Flux(b1, F_u, F_P, 'f_u12')
# flux is from b2, function is F_v
# parameter argument is optional, if not used
# we don't technically need a name, they really exist
# for debugging, and if we want to do something that
# requires us to remove the flux, here it is defaulted to ''
f_v21 = f.Flux(b2, F_v)
# add these fluxes to the right blocks
b1.addFlux(f_u01)
b1.addFlux(f_v21)
b2.addFlux(f_u12)
# create the sources
S_uP = {'C_s': C_s}
S_vP = {'g': g}
s_u1 = s.Source(S_u, S_uP, 's_u1')
s_u2 = s.Source(S_u, S_uP, 's_u2')
# again name argument is optional
s_v1 = s.Source(S_v, S_vP)
b1.addSource(s_u1)
b2.addSource(s_u2)
# create and solve the problem
problem = p.Problem([b1, b2])
problem.solve()
# the blocks now contain their own values,
# which are the correct values
print "Running first implementation of example code"
print b1
print b2
def implementationTwo():
# constants
u0 = 1.0
C_f = 1.0
C_s = 1.0
# create the blocks, with no parameters or parameter functions
b1 = b.Block(
'1', # name
{
'u': 0,
'v': 0
}) # states, and their initial guesses
b2 = b.Block(
'2', # name
{'u': 0}) # states, and their initial guesses
# create the fluxes
# flux from 0 to 1
F_P = {'C_f': C_f} # dictionary of parameters for the flux
# flux is from b1, function is F_u
f_u12 = f.Flux(b1, F_u, F_P, 'f_u12')
# flux is from b2, function is F_v
# name argument is optional
# parameter argument is optional too, if not used
f_v21 = f.Flux(b2, F_v)
# add these fluxes to the right blocks
b1.addFlux(f_v21)
b2.addFlux(f_u12)
# create the sources
# these are parameters for the new boundary source
S_0P = {'C_f': C_f, 'u_0': u0}
S_uP = {'C_s': C_s}
S_vP = {'g': g}
s_u0 = s.Source(S_0, S_0P, 'boundary')
s_u1 = s.Source(S_u, S_uP, 's_u1')
s_u2 = s.Source(S_u, S_uP, 's_u2')
# again name argument is optional
s_v1 = s.Source(S_v, S_vP)
# add the boundary source in with the other ones
b1.addSource(s_u0)
b1.addSource(s_u1)
b2.addSource(s_u2)
# create and solve the problem
problem = p.Problem([b1, b2])
problem.solve()
# the blocks now contain their own values,
# which are the correct values
print "Running second implementation of example code"
print b1
print b2
def diffusion2D(N):
d = 2./float(N) # spacing, delta
# initialize with exact solution at t = 0
B = []
for i in range(-1,N+1):
for j in range(-1,N+1):
x = (i*d+d/2)
y = (j*d+d/2)
# block has a simple name
name = '('+str(i)+','+str(j)+')'
B.append(b.Block(name,
# initialize to exact solution
{'u':math.sin(u_a*math.pi*x)*math.sin(u_b*math.pi*y),
'v':math.sin(v_a*math.pi*x)*math.sin(v_b*math.pi*y)},
None, # no parameter functions
{'x':x,'y':y}))
# Flux geometry
P = {'d':d*d}
n = N+2 # add two for the boundaries
for i in range(1,n-1):
for j in range(1,n-1):
# Add fluxes, looping around the edges
for k in [(i-1)*n+j, i*n+j-1,(i+1)*n+j, i*n+j+1]:
B[i*n+j].addFlux(f.Flux(B[k],difference,P))
interiorBlocks = [B[i*n+j] for i in range(1,n-1) for j in range(1,n-1)]
# sort out the boundary blocks
boundaryBlocks = []
bcRange = [j for j in range(1,n-1)] + \
[(n-1)*n+j for j in range(1,n-1)] + \
[i*n for i in range(0,n)] + \
[i*n+n-1 for i in range(0,n)]
for k in bcRange:
B[k].addSource(s.Source(time,None,B[k].name))
boundaryBlocks.append(B[k])
# solve the problem on the interior blocks,
# pass in the boundary blocks so they can be updated
# at each needed time.
# solve the unstead problem at the following timesteps
P = p.Problem(interiorBlocks,boundaryBlocks)
P.solveUnst(np.linspace(0,tf,10))
# calculate the error for accuracy checking
Eu = 0
Ev = 0
for bb in interiorBlocks:
(x,y) = (bb.p['x'],bb.p['y'])
ue = math.exp(-nu_u*math.pi*math.pi*(u_a*u_a+u_b*u_b)*tf)*math.sin(u_a*math.pi*x)*math.sin(u_b*math.pi*y)
ve = math.exp(-nu_v*math.pi*math.pi*(v_a*v_a+v_b*v_b)*tf)*math.sin(v_a*math.pi*x)*math.sin(v_b*math.pi*y)
Eu += (bb['u']-ue)**2
Ev += (bb['v']-ve)**2
return (math.sqrt(Eu/(n-2)/(n-2)),math.sqrt(Ev)/(n-2)/(n-2))
def poisson2D(N):
"""
lets define a uniform square mesh on [-1, 1] x [1, 1]
and create boundary blocks as we go,
initializing based on the exact solution, and naming the block by its coordinates
"""
d = 2. / float(N) # spacing, delta X
B = [
b.Block('(' + str(i * d - d / 2 - 1) + ',' + str(j * d - d / 2 - 1) +
')',
None,
u=math.exp((i * d - d / 2 - 1) * (j * d - d / 2 - 1)),
v=math.exp((i * d - d / 2 - 1)**2 + (j * d - d / 2 - 1)**2))
for i in range(0, N + 2) for j in range(0, N + 2)
]
# interior sources
# use the name to get the source values
for block in B:
(x, y) = eval(block.name)
block.addSource(
s.Source('const',
u=-(x * x + y * y) * math.exp(x * y),
v=-4.0 * (x * x + y * y + 1.0) * math.exp(x * x + y * y)))
# Flux geometry
G = {'type': 'edge', 'd': d * d, 'm': []}
n = N + 2 # add two for the boundaries
for i in range(1, n - 1):
for j in range(1, n - 1):
# Add fluxes, no "normals" so we cheat and define them cleverly
for k in [(i - 1) * n + j, i * n + j - 1, (i + 1) * n + j,
i * n + j + 1]:
B[i * n + j].addFlux(f.Flux(B[k], 'difference', G))
B[i * n + j].state['u'] = 0.
B[i * n + j].state['v'] = 0.
interiorBlocks = [
B[i * n + j] for i in range(1, n - 1) for j in range(1, n - 1)
]
# solve the problem on the interior blocks
P = p.Problem(interiorBlocks)
P.solve()
Eu = math.sqrt(
sum([(math.exp(eval(block.name)[0] * eval(block.name)[1]) -
block.state['u'])**2
for block in interiorBlocks]) / (n - 2) / (n - 2))
Ev = math.sqrt(
sum([(math.exp(eval(block.name)[0]**2 + eval(block.name)[1]**2) -
block.state['v'])**2
for block in interiorBlocks]) / (n - 2) / (n - 2))
return (Eu, Ev)
def diff2D(N):
"""
lets define a uniform square mesh on [-1, 1] x [1, 1]
and create boundary blocks as we go,
initializing based on the exact solution, and naming the block by its coordinates
"""
u_a = 1
u_b = 1
nu_u = 1
v_a = 1
v_b = 1
nu_v = 0.5
tf = 1
d = 2. / float(N) # spacing, delta
# initialize with exact solution at t = 0
B = [b.Block('('+str(i*d-d/2-1)+','+str(j*d-d/2-1)+')',None,
u = math.sin(u_a*math.pi*(i*d-d/2-1))*math.sin(u_b*math.pi*(j*d-d/2-1)),
v = math.sin(v_a*math.pi*(i*d-d/2-1))*math.sin(v_b*math.pi*(j*d-d/2-1))) \
for i in range(0,N+2) for j in range(0,N+2)]
# Flux geometry
G = {'type': 'edge', 'd': d * d, 'm': []}
n = N + 2 # add two for the boundaries
for i in range(1, n - 1):
for j in range(1, n - 1):
# Add fluxes, no "normals" so we cheat and define them cleverly
for k in [(i - 1) * n + j, i * n + j - 1, (i + 1) * n + j,
i * n + j + 1]:
B[i * n + j].addFlux(f.Flux(B[k], 'difference', G))
B[i * n + j].state['u'] = 0.
B[i * n + j].state['v'] = 0.
interiorBlocks = [
B[i * n + j] for i in range(1, n - 1) for j in range(1, n - 1)
]
boundaryBlocks = []
bcRange = [j for j in range(1,n-1)] + \
[(n-1)*n+j for j in range(1,n-1)] + \
[i*n for i in range(0,n)] + \
[i*n+n-1 for i in range(0,n)]
for k in bcRange:
(x, y) = eval(B[k].name)
B[k].addSource(s.Source('time', \
u = lambda t: math.exp(-nu_u*math.pi*math.pi*(u_a*u_a+u_b*u_b)*t)*math.sin(u_a*math.pi*x)*math.sin(u_b*math.pi*y), \
v = lambda t: math.exp(-nu_v*math.pi*math.pi*(v_a*v_a+v_b*v_b)*t)*math.sin(v_a*math.pi*x)*math.sin(v_b*math.pi*y)))
boundaryBlocks.append(B[k])
# solve the problem on the interior blocks
P = p.Problem(interiorBlocks, boundaryBlocks)
P.solveUnst(0, tf, 10)
# P.printSolution()
Eu = 0
Ev = 0
t = tf
for bb in interiorBlocks:
(x, y) = eval(bb.name)
ue = math.exp(-nu_u * math.pi * math.pi *
(u_a * u_a + u_b * u_b) * t) * math.sin(
u_a * math.pi * x) * math.sin(u_b * math.pi * y)
ve = math.exp(-nu_v * math.pi * math.pi *
(v_a * v_a + v_b * v_b) * t) * math.sin(
v_a * math.pi * x) * math.sin(v_b * math.pi * y)
Eu += (bb.state['u'] - ue)**2
Ev += (bb.state['v'] - ve)**2
# Eu = math.sqrt(sum([(math.exp(-nu_u*math.pi*math.pi*(u_a*u_a+u_b*u_b)*tf)*math.sin(u_a*math.pi*block.name[0])*math.sin(u_b*math.pi*block.name[1]) \
# -block.state['u'])**2 for block in interiorBlocks])/(n-2)/(n-2))
# Ev = math.sqrt(sum([(math.exp(-nu_v*math.pi*math.pi*(v_a*v_a+v_b*v_b)*tf)*math.sin(v_a*math.pi*block.name[0])*math.sin(v_b*math.pi*block.name[1]) \
# -block.state['v'])**2 for block in interiorBlocks])/(n-2)/(n-2))
return (math.sqrt(Eu / (n - 2) / (n - 2)),
math.sqrt(Ev) / (n - 2) / (n - 2))
""" Block Initialization """
w1 = b.Block('water1', 'constWater', T=13)
w2 = b.Block('water2', 'constWater', T=13)
a1 = b.Block('air1', 'constAir', T=20)
a2 = b.Block('air2', 'constAir', T=20)
w1.mdot = w0.mdot
w2.mdot = w1.mdot
a1.mdot = a0.mdot
a2.mdot = a1.mdot
a2.addSource(Sa)
w2.addSource(Sw)
a1.addFlux(f.Flux(aExt, 'heatCondSimple', extGeom))
a1.addFlux(f.Flux(aInt, 'heatCondSimple', intGeom))
a1.addFlux(f.Flux(w1, 'heatCondSimple', tubeGeom))
a1.addFlux(f.Flux(a0, 'heatConvection'))
w1.addFlux(f.Flux(a1, 'heatCondSimple', tubeGeom))
w1.addFlux(f.Flux(w0, 'heatConvection'))
a2.addFlux(f.Flux(a1, 'heatConvection'))
w2.addFlux(f.Flux(w1, 'heatConvection'))
""" Problem Initialization """
# Start the problem with solvable blocks, which
# are all the blocks except the first two
ICSolar = p.Problem([a1, a2, w1, w2])
ICSolar.solve()
ICSolar.printSolution()
def implementationThree():
# constants
u0 = 1.0
C_f = 1.0
C_s = 1.0
# create the blocks, with no parameters or parameter functions
b0 = b.Block(
'0', # name
{'u': u0}) # boundary state, this is fixed
b1 = b.Block(
'1', # name
{
'u': 0,
'v': 0
}, # states, and their initial guesses
{'g': g}, # parameter functions
{
'C_f': C_f,
'C_s': C_s
}) # parameter constants
b2 = b.Block(
'2', # name
{'u': 0}, # states, and their initial guesses
None, # no parameter functions are needed in this block
{
'C_f': C_f,
'C_s': C_s
}) # parameter constants
# create the fluxes
# flux from 0 to 1
# flux is from b0, function is F_u
# no parameter dictionaries needed
f_u01 = f.Flux(b0, F_u3)
# flux is from b1, function is F_u
f_u12 = f.Flux(b1, F_u3)
# flux is from b2, function is F_v
# name argument is optional
# parameter argument is optional too, if not used
f_v21 = f.Flux(b2, F_v)
# add these fluxes to the right blocks
b1.addFlux(f_u01)
b1.addFlux(f_v21)
b2.addFlux(f_u12)
# create the sources
# no parameters or names needed,
# they come from the plots
s_u1 = s.Source(S_u3)
s_u2 = s.Source(S_u3)
s_v1 = s.Source(S_v3)
b1.addSource(s_u1)
b2.addSource(s_u2)
# create and solve the problem
problem = p.Problem([b1, b2])
problem.solve()
# the blocks now contain their own values,
# which are the correct values
print "Running third implementation of example code"
print b1
print b2
def poisson2D(N):
"""
lets define a uniform square mesh on [-1, 1] x [1, 1]
and create boundary blocks as we go,
initializing based on the exact solution, and naming the block by its coordinates
"""
d = 2. / float(N) # spacing, Delta x = Delta y
# For N blocks in each direction, N+2 blocks since we have a constant block on each end surrounding the domain
# a ghost cell approach, there are no parameters in this case, or parameter functions needed
# the center of the cell is at (i*d+d/2,j*d+d/2)
B = []
for i in range(-1, N + 1):
for j in range(-1, N + 1):
x = (i * d + d / 2)
y = (j * d + d / 2)
# block has a simple name
name = '(' + str(i) + ',' + str(j) + ')'
B.append(
b.Block(
name,
{
'u': math.exp(x * y),
'v': math.exp(x**2 + y**2)
}, # initialize to exact solution
None, # no parameter functions
{
'x': x,
'y': y
})) # parameter of coordinates
# interior sources, -f['u'] and -f['v']
for block in B:
(x, y) = (block.p['x'], block.p['y'])
block.addSource(
s.Source(
s.constant, # standard function defined in source.py
{
'u': -(x * x + y * y) * math.exp(x * y),
'v': -4.0 * (x * x + y * y + 1.0) * math.exp(x * x + y * y)
}, # parameters
'constant')) # name
# Flux geometry
P = {'d': d * d} # divide by delta x^2 in the end
n = N + 2 # add two for the boundaries
for i in range(1, N + 1):
for j in range(1, N + 1):
# Add fluxes, figuring out which neighbors to connect to
for k in [(i - 1) * n + j, i * n + j - 1, (i + 1) * n + j,
i * n + j + 1]:
B[i * n + j].addFlux(f.Flux(B[k], difference, P))
interiorBlocks = [
B[i * n + j] for i in range(1, N + 1) for j in range(1, N + 1)
]
# solve the problem on the interior blocks
P = p.Problem(interiorBlocks)
P.solve()
# compute the L-2 error against the exact solution for both variables
Eu = math.sqrt(
sum([(math.exp(block.p['x'] * block.p['y']) - block['u'])**2
for block in interiorBlocks]) / (n - 2) / (n - 2))
Ev = math.sqrt(
sum([(math.exp(block.p['x']**2 + block.p['y']**2) - block['v'])**2
for block in interiorBlocks]) / (n - 2) / (n - 2))
return (Eu, Ev)
# add in the inflow block to make it easy to connect blocks
# These blocks are not used in the solve
water.append(w0)
air.append(a0)
#### Initialize the blocks we will solve on
for i in range(1, 2 * n + 2):
# Every block is named for its material in this case
water.append(b.Block('water' + str(i), 'water'))
air.append(b.Block('air' + str(i), 'air'))
if (i % 2 == 1): # odd regions are "tube" regions
# Water tube has one flux for heat conduction
water[i].addFlux(
f.Flux(water[i], air[i], 'heatConduction', tubeGeom))
# Air has three, corresponding to the windows and the water-tube
air[i].addFlux(f.Flux(water[i], air[i], 'heatConduction',
tubeGeom))
air[i].addFlux(f.Flux(aInt, air[i], 'heatConduction', IGUGeom))
air[i].addFlux(f.Flux(aExt, air[i], 'heatConduction', windowGeom))
else: # These are "module" region
water[i].addSource(Sw)
air[i].addSource(Sa)
# These are the connectivity between regions, each block takes heat
# from the block "below" it
air[i].addFlux(f.Flux(air[i - 1], air[i], 'heatConvection'))
water[i].addFlux(f.Flux(water[i - 1], water[i], 'heatConvection'))
# Initialize the state of every block to be the same for now