def backward_step():
''' Poiseuille flow with a boundary asymmetry '''
m = 22
n = 67
step_height = 12
step_width = 10
obstacle = gds.utils.flatten([[(j, i) for i in range(step_height)]
for j in range(step_width)])
G, (l, r, t, b) = gds.triangular_lattice(m, n, with_boundaries=True)
for g in [G, l, r, t, b]:
g.remove_nodes_from(obstacle)
for i in range(step_width + 1):
b.add_node((i, step_height))
if i > 0:
b.add_edge((i - 1, step_height), (i, step_height))
for j in range(step_height + 1):
b.add_node((step_width, j))
if i > 0:
b.add_edge((step_width, j - 1), (step_width, j))
G.remove_edges_from(list(nx.edge_boundary(G, l, l)))
G.remove_edges_from(
list(
nx.edge_boundary(G, [(0, 2 * i + 1) for i in range(m // 2)],
[(1, 2 * i) for i in range(m // 2 + 1)])))
G.remove_edges_from(list(nx.edge_boundary(G, r, r)))
G.remove_edges_from(
list(
nx.edge_boundary(G, [(n // 2, 2 * i + 1) for i in range(m // 2)],
[(n // 2, 2 * i) for i in range(m // 2 + 1)])))
weight = 1.
nx.set_edge_attributes(G, weight, name='w')
inlet_v = 1.0
# outlet_v=2*(m - step_height - 2)*inlet_v / (m - 2)
outlet_p = 0.0
# ref_p=0.0
# grad_p=100.0
velocity, pressure = navier_stokes(G,
viscosity=100.,
density=1.0,
inlets=l.nodes,
outlets=r.nodes,
w_key='w')
pressure.set_constraints(dirichlet=gds.combine_bcs(
# {n: grad_p/2 for n in l.nodes},
# {n: -grad_p/2 for n in r.nodes if n[1] > step_height//2}
{(n // 2 + 1, j): outlet_p
for j in range(n)}
# {(n//2+1,m): ref_p}
))
velocity.set_constraints(dirichlet=gds.combine_bcs(
{((0, i), (1, i)): inlet_v
for i in range(step_height + 1, m)},
# {((n//2, i), (n//2+1, i)): outlet_v for i in range(1, m)},
# {((n//2-1, 2*i+1), (n//2, 2*i+1)): outlet_v for i in range(0, m//2)},
gds.zero_edge_bc(t),
gds.zero_edge_bc(b),
))
return velocity, pressure
def voronoi_poiseuille():
np.random.seed(401)
gradP = 1.0
n_boundary = 10
n_interior = 100
G, (l, r, t, b) = gds.voronoi_lattice(n_boundary,
n_interior,
with_boundaries=True,
eps=0.07)
# tb = set(t.nodes()) | set(b.nodes())
v_free_l, v_free_r = set(l.nodes()), set(r.nodes())
v_free = v_free_l | v_free_r
v_bd_l = set()
# for v in v_free_l:
# u = v - 1j
# G.add_node(u)
# G.add_edge(u, v)
# G.nodes[u]['pos'] = (G.nodes[v]['pos'][0] - 0.1, G.nodes[v]['pos'][1])
# v_bd_l.add(u)
v_bd_r = set()
# for v in v_free_r:
# u = v - 1j
# G.add_node(u)
# G.add_edge(v, u)
# G.nodes[u]['pos'] = (G.nodes[v]['pos'][0] + 0.1, G.nodes[v]['pos'][1])
# v_bd_r.add(u)
v_bd = v_bd_l | v_bd_r
pressure, velocity = navier_stokes(G,
viscosity=1.,
density=1e-2,
v_free=v_bd | v_free)
pressure.set_constraints(dirichlet=gds.combine_bcs(
{n: gradP / 2
for n in v_free_l | v_bd_l},
{n: -gradP / 2
for n in v_free_r | v_bd_r},
))
# e_free_mask = np.array([1 if len(set(velocity.iX[i]) - v_bd)==2 else 0 for i in range(velocity.ndim)])
# local_state = {'t': None, 'div': None}
# def free_boundaries(t, e):
# if local_state['t'] != t:
# local_state['div'] = velocity.div(velocity.y*e_free_mask)
# local_state['t'] = t
# if e[1] in v_bd_l:
# return -np.clip(local_state['div'][pressure.X[e[0]]], 0, None)
# elif e[1] in v_bd_r:
# return -np.clip(local_state['div'][pressure.X[e[0]]], None, 0)
velocity.set_constraints(dirichlet=gds.combine_bcs(
# free_boundaries,
gds.zero_edge_bc(t),
gds.zero_edge_bc(b),
))
return pressure, velocity
def sq_poiseuille():
m = 14
n = 28
gradP = 6.0
G, (l, r, t, b) = gds.square_lattice(m, n, with_boundaries=True)
v_free_l = set(l.nodes()) - (set(t.nodes()) | set(b.nodes()))
v_free_r = set(r.nodes()) - (set(t.nodes()) | set(b.nodes()))
v_free = v_free_l | v_free_r
pressure, velocity = navier_stokes(G,
viscosity=1.,
density=1e-2,
v_free=v_free)
e_free = np.array([velocity.X[(n, (n[0] + 1, n[1]))] for n in v_free_l] +
[velocity.X[((n[0] - 1, n[1]), n)] for n in v_free_r],
dtype=np.intp)
e_free_mask = np.ones(velocity.ndim)
e_free_mask[e_free] = 0
pressure, velocity = navier_stokes(G,
viscosity=1.,
density=1e-2,
v_free=v_free)
pressure.set_constraints(dirichlet=gds.combine_bcs(
{n: gradP / 2
for n in l.nodes},
{(n[0] + 1, n[1]): gradP / 2
for n in l.nodes},
{n: -gradP / 2
for n in r.nodes},
{(n[0] - 1, n[1]): -gradP / 2
for n in r.nodes},
))
local_state = {'t': None, 'div': None}
def free_boundaries(t, e):
if local_state['t'] != t:
local_state['div'] = velocity.div(velocity.y * e_free_mask)
if e[0][1] == e[1][1] and e[0][0] + 1 == e[1][0]:
if e[0] in v_free:
return local_state['div'][pressure.X[e[1]]]
elif e[1] in v_free:
return -local_state['div'][pressure.X[e[0]]]
velocity.set_constraints(dirichlet=gds.combine_bcs(gds.zero_edge_bc(
t), gds.zero_edge_bc(b), free_boundaries))
return pressure, velocity
def von_karman_projected():
m = 20
n = 40
gradP = 100.0
inlet_v = 5.0
outlet_p = 0.0
G, (l, r, t, b) = gds.triangular_lattice(m, n, with_boundaries=True)
# G, (l, r, t, b) = gds.square_lattice(m, n, with_boundaries=True)
# G, (l, r, t, b) = gds.hexagonal_lattice(m, n, with_boundaries=True)
# G = gds.triangular_cylinder(m, n)
# G = gds.square_cylinder(m, n)
# l, r = G.l_boundary, G.r_boundary
j, k = 3, m // 2
# Introduce occlusion
obstacle = [
(j, k),
# (j+1, k),
# (j+1, k+1),
# (j+1, k-1),
# (j-1, k+1),
# (j, k+2),
]
G.remove_nodes_from(obstacle)
velocity, pressure = fluid_projected.navier_stokes(G, viscosity=0.0001)
pressure.set_constraints(dirichlet=gds.combine_bcs(
{n: gradP / 2
for n in l.nodes}, {n: -gradP / 2
for n in r.nodes}))
# velocity.set_constraints(dirichlet=gds.combine_bcs(
# {((0, i), (1, i)): inlet_v for i in range(1, m)},
# {((n//2, i), (n//2+1, i)): inlet_v for i in range(1, m)},
# {((n//2-1, 2*i+1), (n//2, 2*i+1)): inlet_v for i in range(0, m//2)},
# # gds.utils.bidict({e: 0 for e in obstacle_boundary}),
# gds.utils.bidict({e: 0 for e in t.edges}),
# gds.utils.bidict({e: 0 for e in b.edges})
# ))
sys = gds.couple({
'velocity':
velocity,
# 'divergence': velocity.project(gds.GraphDomain.nodes, lambda v: v.div()),
'vorticity':
velocity.project(gds.GraphDomain.faces, lambda v: v.curl()),
'pressure':
pressure,
# 'tracer': lagrangian_tracer(velocity),
# 'advective': velocity.project(gds.GraphDomain.edges, lambda v: -advector(v)),
# 'L2': velocity.project(PointObservable, lambda v: np.sqrt(np.dot(v.y, v.y))),
# 'dK/dt': velocity.project(PointObservable, lambda v: np.dot(v.y, v.leray_project(-advector(v)))),
})
gds.render(sys,
canvas=gds.grid_canvas(sys.observables.values(), 3),
edge_max=0.6,
dynamic_ranges=True,
edge_colors=True,
edge_palette=cc.bgy)
def heat_free_lr():
temp = heat_grid(10)
temp.set_constraints(dirichlet=gds.combine_bcs(
lambda x: 0. if x[1] == 9 else None,
lambda x: 1. if x[1] == 0 else None,
))
gds.render(temp)
def tri_poiseuille():
m = 14
n = 58
gradP = 1.0
G, (l, r, t, b) = gds.triangular_lattice(m, n, with_boundaries=True)
# G.remove_edges_from(l.edges())
# G.remove_edges_from(r.edges())
# G.remove_edges_from(G.subgraph([(0, 2*i+1) for i in range(m//2)] + [(1, 2*i) for i in range(m//2+1)]).edges())
# G.remove_edges_from(G.subgraph([(n//2-1, 2*i+1) for i in range(m//2)] + [(n//2, 2*i) for i in range(m//2+1)]).edges())
v_free_l = set(l.nodes()) - (set(t.nodes()) | set(b.nodes()))
# v_free_l_int = set((n[0]+1, n[1]) for n in v_free_l)
v_free_r = set(r.nodes()) - (set(t.nodes()) | set(b.nodes()))
# v_free_r_int = set((n[0]-1, n[1]) for n in v_free_r)
v_free = v_free_l | v_free_r
pressure, velocity = navier_stokes(G,
viscosity=1.,
density=1e-2,
v_free=v_free)
e_free = np.array([velocity.X[(n, (n[0] + 1, n[1]))] for n in v_free_l] +
[velocity.X[((n[0] - 1, n[1]), n)] for n in v_free_r],
dtype=np.intp)
e_free_mask = np.ones(velocity.ndim)
e_free_mask[e_free] = 0
pressure.set_constraints(dirichlet=gds.combine_bcs(
{n: gradP / 2
for n in l.nodes},
{(n[0] + 1, n[1]): gradP / 2
for n in l.nodes},
{n: -gradP / 2
for n in r.nodes},
{(n[0] - 1, n[1]): -gradP / 2
for n in r.nodes},
))
local_state = {'t': None, 'div': None}
def free_boundaries(t, e):
if local_state['t'] != t:
local_state['div'] = velocity.div(velocity.y * e_free_mask)
if e[0][1] == e[1][1] and e[0][0] + 1 == e[1][0]:
if e[0] in v_free:
return local_state['div'][pressure.X[e[1]]]
elif e[1] in v_free:
return -local_state['div'][pressure.X[e[0]]]
velocity.set_constraints(dirichlet=gds.combine_bcs(gds.zero_edge_bc(
t), gds.zero_edge_bc(b), free_boundaries))
return pressure, velocity
def grid_tv_boundary():
G = gds.square_lattice(10, 10)
temperature = gds.node_gds(G)
temperature.set_evolution(dydt=lambda t, y: temperature.laplacian())
temperature.set_constraints(dirichlet=gds.combine_bcs(
lambda x: 0 if x[0] == 9 else None,
lambda t, x: np.sin(t + x[1] / 4)**2 if x[0] == 0 else None))
gds.render(temperature, title='Heat equation with time-varying boundary')
def poiseuille(G, dG, viscosity=50, density=1, gradP=10):
(l, r, t, b) = dG
tb = set(t.nodes()) | set(b.nodes())
lr = set(l.nodes()) | set(r.nodes())
v_free = lr - tb
e_free = set(gds.edge_domain(G, lr))
e_normal = set(nx.edge_boundary(G, lr))
velocity, pressure = navier_stokes(G, viscosity=viscosity, density=density, v_free=v_free, e_free=e_free, e_normal=e_normal)
pressure.set_constraints(dirichlet=gds.combine_bcs(
{n: gradP/2 for n in l.nodes},
{n: -gradP/2 for n in r.nodes},
))
velocity.set_constraints(dirichlet=gds.combine_bcs(
gds.zero_edge_bc(t),
gds.zero_edge_bc(b),
))
return velocity, pressure
def lid_driven_cavity(G, dG, v=10, viscosity=10):
(l, r, t, b) = dG
v_free = (set(l.nodes()) & set(t.nodes())) | (set(r.nodes())
& set(t.nodes()))
print('v_free:', v_free)
velocity, pressure = navier_stokes(G, viscosity=viscosity, v_free=v_free)
velocity.set_constraints(dirichlet=gds.combine_bcs(
gds.const_edge_bc(t, v),
gds.zero_edge_bc(b),
gds.zero_edge_bc(l),
gds.zero_edge_bc(r),
))
# pressure.set_constraints(dirichlet={(0, 0): 0.}) # Pressure reference
return velocity, pressure
def sq_poiseuille_projected(viscosity, density): m=14 n=28 gradP=6.0 G, (l, r, t, b) = gds.square_lattice(m, n, with_boundaries=True) v_free_l = set(l.nodes()) - (set(t.nodes()) | set(b.nodes())) v_free_r = set(r.nodes()) - (set(t.nodes()) | set(b.nodes())) v_free = v_free_l | v_free_r velocity = navier_stokes_projected(G, viscosity=viscosity, density=density, body_force=lambda t, y: gradP*np.ones_like(y), v_free=v_free) velocity.set_constraints(dirichlet=gds.combine_bcs( gds.zero_edge_bc(t), gds.zero_edge_bc(b), )) return velocity
def von_karman():
m = 20
n = 50
gradP = 80.0
inlet_v = 20.0
outlet_p = 0.0
G, (l, r, t, b) = gds.triangular_lattice(m, n, with_boundaries=True)
tb = set(t.nodes()) | set(b.nodes())
lr = set(l.nodes()) | set(r.nodes())
v_free = lr - tb
e_free = set(gds.edge_domain(G, lr))
e_normal = set(nx.edge_boundary(G, lr))
j, k = 8, m // 2
# Introduce occlusion
obstacle = [
(j, k),
(j + 1, k),
(j + 2, k),
# (j, k+1),
# (j, k-1),
# (j-1, k),
# (j+1, k+1),
# (j+1, k-1),
# (j, k+2),
# (j, k-2),
]
obstacle_boundary = gds.utils.flatten([G.neighbors(n) for n in obstacle])
obstacle_boundary = gds.edge_domain(G, obstacle_boundary)
G.remove_nodes_from(obstacle)
# G.remove_edges_from(list(nx.edge_boundary(G, l, l)))
# G.remove_edges_from(list(nx.edge_boundary(G, [(0, 2*i+1) for i in range(m//2)], [(1, 2*i) for i in range(m//2+1)])))
# G.remove_edges_from(list(nx.edge_boundary(G, r, r)))
# G.remove_edges_from(list(nx.edge_boundary(G, [(n//2, 2*i+1) for i in range(m//2)], [(n//2, 2*i) for i in range(m//2+1)])))
velocity, pressure = fluid.navier_stokes(G,
viscosity=10,
v_free=v_free,
e_free=e_free,
e_normal=e_normal)
pressure.set_constraints(dirichlet=gds.combine_bcs(
{n: gradP / 2
for n in l.nodes}, {n: -gradP / 2
for n in r.nodes}
# {n: 0 for n in l.nodes}
# {(n//2+1, j): outlet_p for j in range(n)}
))
gradation = np.linspace(-0.1, 0.1, m + 1)
velocity.set_constraints(dirichlet=gds.combine_bcs(
# {((0, i), (1, i)): inlet_v + gradation[i] for i in range(1, m)},
# {((n//2, i), (n//2+1, i)): inlet_v - gradation[i] for i in range(1, m)},
# {((n//2-1, 2*i+1), (n//2, 2*i+1)): inlet_v - gradation[2*i+1] for i in range(0, m//2)},
{e: 0
for e in obstacle_boundary},
gds.zero_edge_bc(t),
gds.zero_edge_bc(b),
))
sys = gds.couple({
'velocity': velocity,
# 'divergence': velocity.project(gds.GraphDomain.nodes, lambda v: v.div()),
# 'vorticity': velocity.project(gds.GraphDomain.faces, lambda v: v.curl()),
'pressure': pressure,
})
gds.render(sys,
canvas=gds.grid_canvas(sys.observables.values(), 3),
edge_max=0.6,
dynamic_ranges=True,
edge_palette=cc.bgy)
def hex_poiseuille():
m = 14
n = 29
gradP = 1.0
G, (l, r, t, b) = gds.hexagonal_lattice(m, n, with_boundaries=True)
v_free_l = set(l.nodes()) - (set(t.nodes()) | set(b.nodes()))
# v_free_l.remove(min(v_free_l, key=lambda x: x[1]))
# v_free_l.remove(max(v_free_l, key=lambda x: x[1]))
v_free_r = set(r.nodes()) - (set(t.nodes()) | set(b.nodes()))
# v_free_r.remove(min(v_free_r, key=lambda x: x[1]))
# v_free_r.remove(max(v_free_r, key=lambda x: x[1]))
v_free = v_free_l | v_free_r
v_bd_l = set()
for v in v_free_l:
u = (-1, v[1])
G.add_node(u)
G.add_edge(u, v)
G.nodes[u]['pos'] = (G.nodes[v]['pos'][0] - 1.0, G.nodes[v]['pos'][1])
v_bd_l.add(u)
v_bd_r = set()
for v in v_free_r:
u = (n + 1, v[1])
G.add_node(u)
G.add_edge(v, u)
G.nodes[u]['pos'] = (G.nodes[v]['pos'][0] + 1.0, G.nodes[v]['pos'][1])
v_bd_r.add(u)
v_bd = v_bd_l | v_bd_r
pressure, velocity = navier_stokes(G,
viscosity=1.,
density=1e-2,
v_free=v_bd | v_free)
e_free_mask = np.array([
1 if len(set(velocity.iX[i]) - v_bd) == 2 else 0
for i in range(velocity.ndim)
])
pressure.set_constraints(dirichlet=gds.combine_bcs(
{n: gradP / 2
for n in v_free_l | v_bd_l},
{n: -gradP / 2
for n in v_free_r | v_bd_r},
))
local_state = {'t': None, 'div': None}
def free_boundaries(t, e):
if local_state['t'] != t:
local_state['div'] = velocity.div(velocity.y * e_free_mask)
local_state['t'] = t
if e[1] in v_bd_l:
return -local_state['div'][pressure.X[e[0]]]
elif e[1] in v_bd_r:
return -local_state['div'][pressure.X[e[0]]]
# elif (e[0][1] == e[1][1] == 0) and (e[0][0] == e[1][0] - 1):
# return 0
# elif (e[0][1] == e[1][1] == n) and (e[0][0] == e[1][0] - 1):
# return 0
velocity.set_constraints(dirichlet=gds.combine_bcs(
free_boundaries,
gds.zero_edge_bc(t),
gds.zero_edge_bc(b),
))
return pressure, velocity
