def render_all():
G, dG = gds.square_lattice(14, 28, with_boundaries=True)
v1, p1 = poiseuille(G, dG)
G, dG = gds.triangular_lattice(14, 58, with_boundaries=True)
v2, p2 = poiseuille(G, dG)
G, dG = gds.hexagonal_lattice(14, 29, with_boundaries=True)
v3, p3 = poiseuille(G, dG)
G, dG = gds.voronoi_lattice(10, 100, with_boundaries=True, eps=0.07)
v4, p4 = poiseuille(G, dG)
sys = gds.couple({
'velocity (sq)': v1,
'velocity (tri)': v2,
'velocity (hex)': v3,
'velocity (voronoi)': v4,
'pressure (sq)': p1,
'pressure (tri)': p2,
'pressure (hex)': p3,
'pressure (voronoi)': p4,
'vorticity (sq)': v1.project(gds.GraphDomain.faces, lambda v: v.curl()),
'vorticity (tri)': v2.project(gds.GraphDomain.faces, lambda v: v.curl()),
'vorticity (hex)': v3.project(gds.GraphDomain.faces, lambda v: v.curl()),
'vorticity (voronoi)': v4.project(gds.GraphDomain.faces, lambda v: v.curl()),
})
gds.render(sys, canvas=gds.grid_canvas(sys.observables.values(), 4), edge_max=0.6, dynamic_ranges=True, plot_width=900, node_size=0.04)
def square_edge_diffusion():
m, n = 10, 10
G, (l, r, t, b) = gds.square_lattice(n, m, with_boundaries=True)
faces, outer_face = gds.embedded_faces(G)
for j in range(m):
G.add_edge((n - 1, j), (0, j))
aux_faces = [((n - 1, j), (0, j), (0, j + 1), (n - 1, j + 1))
for j in range(m - 1)]
G.faces = faces + aux_faces # Hacky
G.rendered_faces = np.array(range(len(faces)), dtype=np.intp) # Hacky
v = gds.edge_gds(G)
v.set_evolution(dydt=lambda t, y: v.laplacian(y))
velocity = 1.0
def boundary(e):
if e in b.edges:
return velocity
elif e == ((0, 0), (m - 1, 0)):
return -velocity
elif e in t.edges or e == ((0, n - 1), (m - 1, n - 1)):
return 0.0
return None
v.set_constraints(dirichlet=boundary)
return v
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 euler_vortex_street():
'''
Vortex street translation in the inviscid model.
x-periodic hexagonal lattice.
'''
# m, n = 4, 20
# G = gds.hexagonal_lattice(m, n)
# speed = 1
# # G = gds.contract_pairs(G, [((0, j), (n, j)) for j in range(1, 2*m)])
# # G = gds.remove_pos(G)
# velocity, pressure = fluid_projected.euler(G)
# y0 = np.zeros(velocity.ndim)
# for j in range(n-1):
# if j == 5:
# e = ((j, m), (j, m+1))
# y_e = np.zeros(velocity.ndim)
# y_e[velocity.X[e]] = 1
# y_f = speed * velocity.curl_face@y_e
# y_e = velocity.curl_face.T@y_f
# y0 += y_e
m, n = 3, 20
G = gds.square_lattice(m, n)
speed = 1
velocity, pressure = fluid_projected.euler(G)
y0 = np.zeros(velocity.ndim)
for j in [m // 2]:
i = 5
e = ((i, j), (i + 1, j))
y_e = np.zeros(velocity.ndim)
y_e[velocity.X[e]] = 1
y_f = speed * velocity.curl_face @ y_e
y0 += velocity.curl_face.T @ y_f
velocity.set_initial(y0=lambda x: y0[velocity.X[x]])
velocity.advect()
sys = gds.couple({
'velocity':
velocity,
'vorticity':
velocity.project(gds.GraphDomain.faces, lambda v: v.curl()),
'advective':
velocity.project(gds.GraphDomain.edges, lambda v: -v.advect()),
'pressure':
pressure,
# 'divergence': velocity.project(gds.GraphDomain.nodes, lambda v: v.div()),
})
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,
n_spring_iters=2000)
def fluid_on_grid():
G = gds.square_lattice(10, 10)
i, o = (3, 3), (6, 6)
dG = nx.Graph()
dG.add_nodes_from([i, o])
velocity, pressure = navier_stokes(G, dG)
pressure.set_constraints(dirichlet={i: 10.0, o: -10.0})
tracer = lagrangian_tracer(velocity, [i])
return velocity, pressure, tracer
def heat_grid(n=10, steady_state=False) -> gds.node_gds:
G = gds.square_lattice(n, n)
temp = gds.node_gds(G)
if steady_state:
''' Steady-state version (for comparison) '''
temp.set_evolution(cost=lambda t, y: temp.laplacian(y))
else:
temp.set_evolution(dydt=lambda t, y: temp.laplacian(y))
return temp
def advection_on_grid(): n = 5 G = gds.square_lattice(n, n) def v_field(e): if e[1][0] >= e[0][0] and e[1][1] >= e[0][1]: return 1 else: return -1 conc, flow = advection(G, v_field) conc.set_initial(y0 = lambda x: 1.0 if x == (0, 0) else 0.) # delta initial condition return conc, flow
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 poiseuille(): m, n = 8, 10 G = gds.square_lattice(n, m) velocity, pressure = fluid_projected.navier_stokes(G) def pressure_values(x): if x[0] == 0: return 0.2 if x[0] == n-1: return -0.2 return None pressure.set_constraints(dirichlet=pressure_values) def no_slip(x): if x[0][1] == x[1][1] == 0 or x[0][1] == x[1][1] == m-1: return 0. return None velocity.set_constraints(dirichlet=no_slip) 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 sq_lid_driven_cavity_ivp():
m = 18
n = 21
v = 1.0
G, (l, r, t, b) = gds.square_lattice(m, n, with_boundaries=True)
velocity, pressure = navier_stokes(G, viscosity=200., density=0.1)
def walls(e):
if e in t.edges(): return v
elif e in b.edges(): return -v
elif e in r.edges(): return -v
elif (e[1], e[0]) in r.edges(): return v
elif e in l.edges(): return v
elif (e[1], e[0]) in l.edges(): return -v
return 0
velocity.set_initial(y0=walls)
pressure.set_constraints(dirichlet={(0, 0): 0.}) # Pressure reference
return velocity, pressure
def poiseuille_flow():
''' API example '''
G, (G_L, G_R, G_T, G_B) = gds.square_lattice(m, n, with_boundaries=True)
velocity = Field(GraphDomain.Edges, G)
velocity.set_dirichlet_boundary(set(G_T.edges()) | set(G_B.edges()), 0.)
velocity.set_free_boundary(set(G_L.edges()) | set(G_R.edges()))
pressure = Field(GraphDomain.Nodes, G)
pressure.set_dirichlet_boundary(set(G_L.edges()), 1.)
pressure.set_dirichlet_boundary(set(G_R.edges()), -1.)
pressure.set_free_boundary(set(G_T.edges()) | set(G_B.edges()))
dt = 1e-3
velocity.set_evolution(dydt=-advect(velocity, velocity) -
grad(pressure) / density +
laplacian(velocity) * viscosity / density)
pressure.set_evolution(
lhs=div(velocity / dt - advect(velocity, velocity)) -
laplacian(pressure) / density +
laplacian(div(velocity)) * viscosity / density)
def sq_couette(): m, n = 11, 10 G, (l, r, t, b) = gds.square_lattice(m, n, with_boundaries=True) faces, outer_face = gds.embedded_faces(G) for j in range(m): G.add_edge((n-1, j), (0, j)) aux_faces = [((n-1, j), (0, j), (0, j+1), (n-1, j+1)) for j in range(m-1)] G.faces = faces + aux_faces # Hacky G.rendered_faces = np.array(range(len(faces)), dtype=np.intp) # Hacky pressure, velocity = navier_stokes(G, viscosity=1., density=1e-2) vel = 1.0 def walls(e): if e in t.edges or e == ((0, m-1), (n-1, m-1)): return 0 elif e in b.edges: return vel elif e == ((0, 0), (n-1, 0)): return -vel return None velocity.set_constraints(dirichlet=walls) return pressure, velocity
def sq_couette_ivp(viscosity, density): m, n = 11, 10 G, (l, r, t, b) = gds.square_lattice(m, n, with_boundaries=True, with_lattice_components=True) faces, outer_face = gds.embedded_faces(G) for j in range(m): G.add_edge((n-1, j), (0, j)) # for i in range(n): # G.add_edge((i, 0), (i, m-1)) aux_faces = [((n-1, j), (0, j), (0, j+1), (n-1, j+1)) for j in range(m-1)] # aux_faces += [((i, 0), (i, m-1), (i+1, m-1), (i+1, 0)) for i in range(n-1)] G.faces = faces + aux_faces # Hacky G.rendered_faces = np.array(range(len(faces)), dtype=np.intp) # Hacky pressure, velocity = navier_stokes(G, viscosity=viscosity, density=density) vel = 1.0 def walls(e): if e[0][1] == e[1][1] == m//2: if e[0][0] == 0 and e[1][0] == n-1: return -vel return vel return 0 velocity.set_initial(y0=walls) return pressure, velocity
'diffusion_tri':
v2.project(gds.GraphDomain.edges, lambda v: v.laplacian()),
'diffusion_hex':
v3.project(gds.GraphDomain.edges, lambda v: v.laplacian()),
'advection_square':
v1.project(gds.GraphDomain.edges, lambda v: -v.advect()),
'advection_tri':
v2.project(gds.GraphDomain.edges, lambda v: -v.advect()),
'advection_hex':
v3.project(gds.GraphDomain.edges, lambda v: -v.advect()),
'divergence_square':
v1.project(gds.GraphDomain.nodes, lambda v: v.div()),
'divergence_tri':
v2.project(gds.GraphDomain.nodes, lambda v: v.div()),
'divergence_hex':
v3.project(gds.GraphDomain.nodes, lambda v: v.div()),
})
sys.solve_to_disk(5.0, 0.01, 'lid_driven_cavity')
if __name__ == '__main__':
# render()
# G, dG = gds.triangular_lattice(18, 42, with_boundaries=True)
G, dG = gds.square_lattice(16, 30, with_boundaries=True)
# G, dG = gds.hexagonal_lattice(18, 21, with_boundaries=True)
v, p = lid_driven_cavity(G, dG, viscosity=200)
fluid_test(v, p, columns=3, edge_palette=cc.bgy)