def create_function_marker(PHI, W, xlimits, ylimits):
x, y, z = fd.SpatialCoordinate(PHI.ufl_domain())
x_func, y_func, z_func = (
fd.Function(PHI),
fd.Function(PHI),
fd.Function(PHI),
)
with fda.stop_annotating():
x_func.interpolate(x)
y_func.interpolate(y)
z_func.interpolate(z)
domain = "{[i, j]: 0 <= i < f.dofs and 0<= j <= 3}"
instruction = f"""
f[i, j] = 1.0 if (x[i, 0] < {xlimits[1]} and x[i, 0] > {xlimits[0]}) and (y[i, 0] < {ylimits[0]} or y[i, 0] > {ylimits[1]}) and z[i, 0] < 1e-7 else 0.0
"""
I_BC = fd.Function(W)
fd.par_loop(
(domain, instruction),
dx,
{
"f": (I_BC, fd.RW),
"x": (x_func, fd.READ),
"y": (y_func, fd.READ),
"z": (z_func, fd.READ),
},
is_loopy_kernel=True,
)
return I_BC
def test_line_search():
mesh = fd.UnitSquareMesh(50, 50)
# Shape derivative
S = fd.VectorFunctionSpace(mesh, "CG", 1)
s = fd.Function(S, name="deform")
mesh.coordinates.assign(mesh.coordinates + s)
# Level set
PHI = fd.FunctionSpace(mesh, "CG", 1)
x, y = fd.SpatialCoordinate(mesh)
with fda.stop_annotating():
phi = fd.interpolate(-(x - 0.5), PHI)
phi.rename("original")
solver_parameters = {
"ts_atol": 1e-4,
"ts_rtol": 1e-4,
"ts_dt": 1e-2,
"ts_exact_final_time": "matchstep",
"ts_monitor": None,
}
## Search direction
with fda.stop_annotating():
delta_x = fd.interpolate(fd.as_vector([-100 * x, 0.0]), S)
delta_x.rename("velocity")
# Cost function
J = fd.assemble(hs(-phi) * dx)
# Reduced Functional
c = fda.Control(s)
Jhat = LevelSetFunctional(J, c, phi)
# InfDim Problem
beta_param = 0.08
reg_solver = RegularizationSolver(S,
mesh,
beta=beta_param,
gamma=1e5,
dx=dx)
solver_parameters = {"hj_solver": solver_parameters}
problem = InfDimProblem(Jhat,
reg_solver,
solver_parameters=solver_parameters)
new_phi = fd.Function(PHI, name="new_ls")
orig_phi = fd.Function(PHI)
with fda.stop_annotating():
orig_phi.assign(phi)
problem.delta_x.assign(delta_x)
AJ, AC = 1.0, 1.0
C = np.array([])
merit = merit_eval_new(AJ, J, AC, C)
rtol = 1e-4
new_phi, newJ, newG, newH = line_search(
problem,
orig_phi,
new_phi,
merit_eval_new,
merit,
AJ,
AC,
dt=1.0,
tol_merit=rtol,
maxtrials=20,
)
assert_allclose(newJ, J, rtol=rtol)
def heat_exchanger_optimization(mu=0.03, n_iters=1000):
output_dir = "2D/"
path = os.path.abspath(__file__)
dir_path = os.path.dirname(path)
mesh = fd.Mesh(f"{dir_path}/2D_mesh.msh")
# Perturb the mesh coordinates. Necessary to calculate shape derivatives
S = fd.VectorFunctionSpace(mesh, "CG", 1)
s = fd.Function(S, name="deform")
mesh.coordinates.assign(mesh.coordinates + s)
# Initial level set function
x, y = fd.SpatialCoordinate(mesh)
PHI = fd.FunctionSpace(mesh, "CG", 1)
phi_expr = sin(y * pi / 0.2) * cos(x * pi / 0.2) - fd.Constant(0.8)
# Avoid recording the operation interpolate into the tape.
# Otherwise, the shape derivatives will not be correct
with fda.stop_annotating():
phi = fd.interpolate(phi_expr, PHI)
phi.rename("LevelSet")
fd.File(output_dir + "phi_initial.pvd").write(phi)
# Physics
mu = fd.Constant(mu) # viscosity
alphamin = 1e-12
alphamax = 2.5 / (2e-4)
parameters = {
"mat_type": "aij",
"ksp_type": "preonly",
"ksp_converged_reason": None,
"pc_type": "lu",
"pc_factor_mat_solver_type": "mumps",
}
stokes_parameters = parameters
temperature_parameters = parameters
u_inflow = 2e-3
tin1 = fd.Constant(10.0)
tin2 = fd.Constant(100.0)
P2 = fd.VectorElement("CG", mesh.ufl_cell(), 2)
P1 = fd.FiniteElement("CG", mesh.ufl_cell(), 1)
TH = P2 * P1
W = fd.FunctionSpace(mesh, TH)
U = fd.TrialFunction(W)
u, p = fd.split(U)
V = fd.TestFunction(W)
v, q = fd.split(V)
epsilon = fd.Constant(10000.0)
def hs(phi, epsilon):
return fd.Constant(alphamax) * fd.Constant(1.0) / (
fd.Constant(1.0) + exp(-epsilon * phi)) + fd.Constant(alphamin)
def stokes(phi, BLOCK_INLET_MOUTH, BLOCK_OUTLET_MOUTH):
a_fluid = mu * inner(grad(u), grad(v)) - div(v) * p - q * div(u)
darcy_term = inner(u, v)
return (a_fluid * dx + hs(phi, epsilon) * darcy_term * dx(0) +
alphamax * darcy_term *
(dx(BLOCK_INLET_MOUTH) + dx(BLOCK_OUTLET_MOUTH)))
# Dirichlet boundary conditions
inflow1 = fd.as_vector([
u_inflow * sin(
((y - (line_sep -
(dist_center + inlet_width))) * pi) / inlet_width),
0.0,
])
inflow2 = fd.as_vector([
u_inflow * sin(((y - (line_sep + dist_center)) * pi) / inlet_width),
0.0,
])
noslip = fd.Constant((0.0, 0.0))
# Stokes 1
bcs1_1 = fd.DirichletBC(W.sub(0), noslip, WALLS)
bcs1_2 = fd.DirichletBC(W.sub(0), inflow1, INLET1)
bcs1_3 = fd.DirichletBC(W.sub(1), fd.Constant(0.0), OUTLET1)
bcs1_4 = fd.DirichletBC(W.sub(0), noslip, INLET2)
bcs1_5 = fd.DirichletBC(W.sub(0), noslip, OUTLET2)
bcs1 = [bcs1_1, bcs1_2, bcs1_3, bcs1_4, bcs1_5]
# Stokes 2
bcs2_1 = fd.DirichletBC(W.sub(0), noslip, WALLS)
bcs2_2 = fd.DirichletBC(W.sub(0), inflow2, INLET2)
bcs2_3 = fd.DirichletBC(W.sub(1), fd.Constant(0.0), OUTLET2)
bcs2_4 = fd.DirichletBC(W.sub(0), noslip, INLET1)
bcs2_5 = fd.DirichletBC(W.sub(0), noslip, OUTLET1)
bcs2 = [bcs2_1, bcs2_2, bcs2_3, bcs2_4, bcs2_5]
# Forward problems
U1, U2 = fd.Function(W), fd.Function(W)
L = inner(fd.Constant((0.0, 0.0, 0.0)), V) * dx
problem = fd.LinearVariationalProblem(stokes(-phi, INMOUTH2, OUTMOUTH2),
L,
U1,
bcs=bcs1)
solver_stokes1 = fd.LinearVariationalSolver(
problem,
solver_parameters=stokes_parameters,
options_prefix="stokes_1")
solver_stokes1.solve()
problem = fd.LinearVariationalProblem(stokes(phi, INMOUTH1, OUTMOUTH1),
L,
U2,
bcs=bcs2)
solver_stokes2 = fd.LinearVariationalSolver(
problem,
solver_parameters=stokes_parameters,
options_prefix="stokes_2")
solver_stokes2.solve()
# Convection difussion equation
ks = fd.Constant(1e0)
cp_value = 5.0e5
cp = fd.Constant(cp_value)
T = fd.FunctionSpace(mesh, "DG", 1)
t = fd.Function(T, name="Temperature")
w = fd.TestFunction(T)
# Mesh-related functions
n = fd.FacetNormal(mesh)
h = fd.CellDiameter(mesh)
u1, p1 = fd.split(U1)
u2, p2 = fd.split(U2)
def upwind(u):
return (dot(u, n) + abs(dot(u, n))) / 2.0
u1n = upwind(u1)
u2n = upwind(u2)
# Penalty term
alpha = fd.Constant(500.0)
# Bilinear form
a_int = dot(grad(w), ks * grad(t) - cp * (u1 + u2) * t) * dx
a_fac = (fd.Constant(-1.0) * ks * dot(avg(grad(w)), jump(t, n)) * dS +
fd.Constant(-1.0) * ks * dot(jump(w, n), avg(grad(t))) * dS +
ks("+") *
(alpha("+") / avg(h)) * dot(jump(w, n), jump(t, n)) * dS)
a_vel = (dot(
jump(w),
cp * (u1n("+") + u2n("+")) * t("+") - cp *
(u1n("-") + u2n("-")) * t("-"),
) * dS + dot(w,
cp * (u1n + u2n) * t) * ds)
a_bnd = (dot(w,
cp * dot(u1 + u2, n) * t) * (ds(INLET1) + ds(INLET2)) +
w * t * (ds(INLET1) + ds(INLET2)) - w * tin1 * ds(INLET1) -
w * tin2 * ds(INLET2) + alpha / h * ks * w * t *
(ds(INLET1) + ds(INLET2)) - ks * dot(grad(w), t * n) *
(ds(INLET1) + ds(INLET2)) - ks * dot(grad(t), w * n) *
(ds(INLET1) + ds(INLET2)))
aT = a_int + a_fac + a_vel + a_bnd
LT_bnd = (alpha / h * ks * tin1 * w * ds(INLET1) +
alpha / h * ks * tin2 * w * ds(INLET2) -
tin1 * ks * dot(grad(w), n) * ds(INLET1) -
tin2 * ks * dot(grad(w), n) * ds(INLET2))
problem = fd.LinearVariationalProblem(derivative(aT, t), LT_bnd, t)
solver_temp = fd.LinearVariationalSolver(
problem,
solver_parameters=temperature_parameters,
options_prefix="temperature",
)
solver_temp.solve()
# fd.solve(eT == 0, t, solver_parameters=temperature_parameters)
# Cost function: Flux at the cold outlet
scale_factor = 4e-4
Jform = fd.assemble(
fd.Constant(-scale_factor * cp_value) * inner(t * u1, n) * ds(OUTLET1))
# Constraints: Pressure drop on each fluid
power_drop = 1e-2
Power1 = fd.assemble(p1 / power_drop * ds(INLET1))
Power2 = fd.assemble(p2 / power_drop * ds(INLET2))
phi_pvd = fd.File("phi_evolution.pvd")
def deriv_cb(phi):
with stop_annotating():
phi_pvd.write(phi[0])
c = fda.Control(s)
# Reduced Functionals
Jhat = LevelSetFunctional(Jform, c, phi, derivative_cb_pre=deriv_cb)
P1hat = LevelSetFunctional(Power1, c, phi)
P1control = fda.Control(Power1)
P2hat = LevelSetFunctional(Power2, c, phi)
P2control = fda.Control(Power2)
Jhat_v = Jhat(phi)
print("Initial cost function value {:.5f}".format(Jhat_v), flush=True)
print("Power drop 1 {:.5f}".format(Power1), flush=True)
print("Power drop 2 {:.5f}".format(Power2), flush=True)
beta_param = 0.08
# Regularize the shape derivatives only in the domain marked with 0
reg_solver = RegularizationSolver(S,
mesh,
beta=beta_param,
gamma=1e5,
dx=dx,
design_domain=0)
tol = 1e-5
dt = 0.05
params = {
"alphaC": 1.0,
"debug": 5,
"alphaJ": 1.0,
"dt": dt,
"K": 1e-3,
"maxit": n_iters,
"maxtrials": 5,
"itnormalisation": 10,
"tol_merit":
5e-3, # new merit can be within 0.5% of the previous merit
# "normalize_tol" : -1,
"tol": tol,
}
solver_parameters = {
"reinit_solver": {
"h_factor": 2.0,
}
}
# Optimization problem
problem = InfDimProblem(
Jhat,
reg_solver,
ineqconstraints=[
Constraint(P1hat, 1.0, P1control),
Constraint(P2hat, 1.0, P2control),
],
solver_parameters=solver_parameters,
)
results = nlspace_solve(problem, params)
return results
}
def update_forcings(t):
with timed_stage('update forcings'):
print_output("Updating tidal field at t={}".format(t))
elev = prepare.myboundary.set_tidal_field(
Function(bathymetry2d.function_space()), t, dt)
tidal_elev.project(elev)
v = prepare.myboundary.set_velocity_field(
Function(VectorFunctionSpace(mesh2d, "CG", 1)), t, dt)
tidal_v.project(v)
print_output("Done updating tidal field")
# run as normal (this run will be annotated by firedrake_adjoint)
#solver_obj.load_state(670,outputdir='./outputs/redata_5min_normaldepth')
solver_obj.assign_initial_conditions(uv=as_vector((1e-7, 0.0)),
elev=Constant(0.0))
#place detectors code
with stop_annotating():
locations, names = prepare.detectors.get_detectors(mesh2d)
cb = DetectorsCallback(solver_obj,
locations, ['elev_2d', 'uv_2d'],
name='detectors',
detector_names=names)
solver_obj.add_callback(cb, 'timestep')
solver_obj.iterate(update_forcings=update_forcings)
def gradient_test_acoustic(model,
mesh,
V,
comm,
vp_exact,
vp_guess,
mask=None): #{{{
import firedrake_adjoint as fire_adj
with fire_adj.stop_annotating():
if comm.comm.rank == 0:
print('######## Starting gradient test ########', flush=True)
sources = spyro.Sources(model, mesh, V, comm)
receivers = spyro.Receivers(model, mesh, V, comm)
wavelet = spyro.full_ricker_wavelet(
model["timeaxis"]["dt"],
model["timeaxis"]["tf"],
model["acquisition"]["frequency"],
)
point_cloud = receivers.set_point_cloud(comm)
# simulate the exact model
if comm.comm.rank == 0:
print('######## Running the exact model ########', flush=True)
p_exact_recv = forward(model, mesh, comm, vp_exact, sources, wavelet,
point_cloud)
# simulate the guess model
if comm.comm.rank == 0:
print('######## Running the guess model ########', flush=True)
p_guess_recv, Jm = forward(model,
mesh,
comm,
vp_guess,
sources,
wavelet,
point_cloud,
fwi=True,
true_rec=p_exact_recv)
if comm.comm.rank == 0:
print("\n Cost functional at fixed point : " + str(Jm) + " \n ",
flush=True)
# compute the gradient of the control (to be verified)
if comm.comm.rank == 0:
print(
'######## Computing the gradient by automatic differentiation ########',
flush=True)
control = fire_adj.Control(vp_guess)
dJ = fire_adj.compute_gradient(Jm, control)
if mask:
dJ *= mask
# File("gradient.pvd").write(dJ)
#steps = [1e-3, 1e-4, 1e-5, 1e-6, 1e-7] # step length
#steps = [1e-4, 1e-5, 1e-6, 1e-7] # step length
steps = [1e-5, 1e-6, 1e-7, 1e-8] # step length
with fire_adj.stop_annotating():
delta_m = Function(V) # model direction (random)
delta_m.assign(dJ)
Jhat = fire_adj.ReducedFunctional(Jm, control)
derivative = enlisting.Enlist(Jhat.derivative())
hs = enlisting.Enlist(delta_m)
projnorm = sum(hi._ad_dot(di) for hi, di in zip(hs, derivative))
# this deepcopy is important otherwise pertubations accumulate
vp_original = vp_guess.copy(deepcopy=True)
if comm.comm.rank == 0:
print(
'######## Computing the gradient by finite diferences ########',
flush=True)
errors = []
for step in steps: # range(3):
# steps.append(step)
# perturb the model and calculate the functional (again)
# J(m + delta_m*h)
vp_guess = vp_original + step * delta_m
p_guess_recv, Jp = forward(model,
mesh,
comm,
vp_guess,
sources,
wavelet,
point_cloud,
fwi=True,
true_rec=p_exact_recv)
fd_grad = (Jp - Jm) / step
if comm.comm.rank == 0:
print("\n Cost functional for step " + str(step) + " : " +
str(Jp) + ", fd approx.: " + str(fd_grad) +
", grad'*dir : " + str(projnorm) + " \n ",
flush=True)
errors.append(100 * ((fd_grad - projnorm) / projnorm))
fire_adj.get_working_tape().clear_tape()
# all errors less than 1 %
errors = np.array(errors)
assert (np.abs(errors) < 5.0).all()
def compliance():
parser = argparse.ArgumentParser(description="Compliance problem with MMA")
parser.add_argument(
"--nref",
action="store",
dest="nref",
type=int,
help="Number of mesh refinements",
default=2,
)
parser.add_argument(
"--uniform",
action="store",
dest="uniform",
type=int,
help="Use uniform mesh",
default=0,
)
parser.add_argument(
"--inner_product",
action="store",
dest="inner_product",
type=str,
help="Inner product, euclidean or L2",
default="L2",
)
parser.add_argument(
"--output_dir",
action="store",
dest="output_dir",
type=str,
help="Directory for all the output",
default="./",
)
args = parser.parse_args()
nref = args.nref
inner_product = args.inner_product
output_dir = args.output_dir
assert inner_product == "L2" or inner_product == "euclidean"
mesh = fd.Mesh("./beam_uniform.msh")
#mh = fd.MeshHierarchy(mesh, 2)
#mesh = mh[-1]
if nref > 0:
mh = fd.MeshHierarchy(mesh, nref)
mesh = mh[-1]
elif nref < 0:
raise RuntimeError("Non valid mesh argument")
V = fd.VectorFunctionSpace(mesh, "CG", 1)
u, v = fd.TrialFunction(V), fd.TestFunction(V)
# Elasticity parameters
E, nu = 1e0, 0.3
mu, lmbda = fd.Constant(E / (2 * (1 + nu))), fd.Constant(
E * nu / ((1 + nu) * (1 - 2 * nu))
)
# Helmholtz solver
RHO = fd.FunctionSpace(mesh, "CG", 1)
rho = fd.interpolate(fd.Constant(0.1), RHO)
af, b = fd.TrialFunction(RHO), fd.TestFunction(RHO)
#filter_radius = fd.Constant(0.2)
#x, y = fd.SpatialCoordinate(mesh)
#x_ = fd.interpolate(x, RHO)
#y_ = fd.interpolate(y, RHO)
#aH = filter_radius * inner(grad(af), grad(b)) * dx + af * b * dx
#LH = rho * b * dx
rhof = fd.Function(RHO)
solver_params = {
"ksp_type": "preonly",
"pc_type": "lu",
"pc_factor_mat_solver_type": "mumps",
"mat_mumps_icntl_14": 200,
"mat_mumps_icntl_24": 1,
}
#fd.solve(aH == LH, rhof, solver_parameters=solver_params)
rhof.assign(rho)
rhofControl = fda.Control(rhof)
eps = fd.Constant(1e-5)
p = fd.Constant(3.0)
def simp(rho):
return eps + (fd.Constant(1.0) - eps) * rho ** p
def epsilon(v):
return sym(nabla_grad(v))
def sigma(v):
return 2.0 * mu * epsilon(v) + lmbda * tr(epsilon(v)) * Identity(2)
DIRICHLET = 3
NEUMANN = 4
a = inner(simp(rhof) * sigma(u), epsilon(v)) * dx
load = fd.Constant((0.0, -1.0))
L = inner(load, v) * ds(NEUMANN)
u_sol = fd.Function(V)
bcs = fd.DirichletBC(V, fd.Constant((0.0, 0.0)), DIRICHLET)
fd.solve(a == L, u_sol, bcs=bcs, solver_parameters=solver_params)
c = fda.Control(rho)
J = fd.assemble(fd.Constant(1e-4) * inner(u_sol, load) * ds(NEUMANN))
Vol = fd.assemble(rhof * dx)
VolControl = fda.Control(Vol)
with fda.stop_annotating():
Vlimit = fd.assemble(fd.Constant(1.0) * dx(domain=mesh)) * 0.5
rho_viz_f = fd.Function(RHO, name="rho")
plot_file = f"{output_dir}/design_{inner_product}.pvd"
controls_f = fd.File(plot_file)
def deriv_cb(j, dj, rho):
with fda.stop_annotating():
rho_viz_f.assign(rhofControl.tape_value())
controls_f.write(rho_viz_f)
Jhat = fda.ReducedFunctional(J, c, derivative_cb_post=deriv_cb)
Volhat = fda.ReducedFunctional(Vol, c)
class VolumeConstraint(fda.InequalityConstraint):
def __init__(self, Vhat, Vlimit, VolControl):
self.Vhat = Vhat
self.Vlimit = float(Vlimit)
self.VolControl = VolControl
def function(self, m):
# Compute the integral of the control over the domain
integral = self.VolControl.tape_value()
with fda.stop_annotating():
value = -integral / self.Vlimit + 1.0
return [value]
def jacobian(self, m):
with fda.stop_annotating():
gradients = self.Vhat.derivative()
with gradients.dat.vec as v:
v.scale(-1.0 / self.Vlimit)
return [gradients]
def output_workspace(self):
return [0.0]
def length(self):
"""Return the number of components in the constraint vector (here, one)."""
return 1
lb = 1e-5
ub = 1.0
problem = fda.MinimizationProblem(
Jhat,
bounds=(lb, ub),
constraints=[VolumeConstraint(Volhat, Vlimit, VolControl)],
)
parameters_mma = {
"move": 0.2,
"maximum_iterations": 200,
"m": 1,
"IP": 0,
"tol": 1e-6,
"accepted_tol": 1e-4,
"norm": inner_product,
#"norm": "euclidean",
"gcmma": False,
}
solver = MMASolver(problem, parameters=parameters_mma)
rho_opt = solver.solve()
with open(f"{output_dir}/finished_{inner_product}.txt", "w") as f:
f.write("Done")
def jacobian(self, m):
with fda.stop_annotating():
gradients = self.Vhat.derivative()
with gradients.dat.vec as v:
v.scale(-1.0 / self.Vlimit)
return [gradients]
def function(self, m):
# Compute the integral of the control over the domain
integral = self.VolControl.tape_value()
with fda.stop_annotating():
value = -integral / self.Vlimit + 1.0
return [value]
def deriv_cb(j, dj, rho):
with fda.stop_annotating():
rho_viz_f.assign(rhofControl.tape_value())
controls_f.write(rho_viz_f)
def compliance_optimization(n_iters=200):
output_dir = "cantilever/"
path = os.path.abspath(__file__)
dir_path = os.path.dirname(path)
m = fd.Mesh(f"{dir_path}/mesh_cantilever.msh")
mesh = fd.MeshHierarchy(m, 0)[-1]
# Perturb the mesh coordinates. Necessary to calculate shape derivatives
S = fd.VectorFunctionSpace(mesh, "CG", 1)
s = fd.Function(S, name="deform")
mesh.coordinates.assign(mesh.coordinates + s)
# Initial level set function
x, y = fd.SpatialCoordinate(mesh)
PHI = fd.FunctionSpace(mesh, "CG", 1)
lx = 2.0
ly = 1.0
phi_expr = (
-cos(6.0 / lx * pi * x) * cos(4.0 * pi * y)
- 0.6
+ max_value(200.0 * (0.01 - x ** 2 - (y - ly / 2) ** 2), 0.0)
+ max_value(100.0 * (x + y - lx - ly + 0.1), 0.0)
+ max_value(100.0 * (x - y - lx + 0.1), 0.0)
)
# Avoid recording the operation interpolate into the tape.
# Otherwise, the shape derivatives will not be correct
with fda.stop_annotating():
phi = fd.interpolate(phi_expr, PHI)
phi.rename("LevelSet")
fd.File(output_dir + "phi_initial.pvd").write(phi)
# Physics. Elasticity
rho_min = 1e-5
beta = fd.Constant(200.0)
def hs(phi, beta):
return fd.Constant(1.0) / (
fd.Constant(1.0) + exp(-beta * phi)
) + fd.Constant(rho_min)
H1_elem = fd.VectorElement("CG", mesh.ufl_cell(), 1)
W = fd.FunctionSpace(mesh, H1_elem)
u = fd.TrialFunction(W)
v = fd.TestFunction(W)
# Elasticity parameters
E, nu = 1.0, 0.3
mu, lmbda = fd.Constant(E / (2 * (1 + nu))), fd.Constant(
E * nu / ((1 + nu) * (1 - 2 * nu))
)
def epsilon(u):
return sym(nabla_grad(u))
def sigma(v):
return 2.0 * mu * epsilon(v) + lmbda * tr(epsilon(v)) * Identity(2)
a = inner(hs(-phi, beta) * sigma(u), nabla_grad(v)) * dx
t = fd.Constant((0.0, -75.0))
L = inner(t, v) * ds(2)
bc = fd.DirichletBC(W, fd.Constant((0.0, 0.0)), 1)
parameters = {
"ksp_type": "preonly",
"pc_type": "lu",
"mat_type": "aij",
"ksp_converged_reason": None,
"pc_factor_mat_solver_type": "mumps",
}
u_sol = fd.Function(W)
F = fd.action(a, u_sol) - L
problem = fd.NonlinearVariationalProblem(F, u_sol, bcs=bc)
solver = fd.NonlinearVariationalSolver(
problem, solver_parameters=parameters
)
solver.solve()
# fd.solve(
# a == L, u_sol, bcs=[bc], solver_parameters=parameters
# ) # , nullspace=nullspace)
with fda.stop_annotating():
fd.File("u_sol.pvd").write(u_sol)
# Cost function: Compliance
J = fd.assemble(
fd.Constant(1e-2)
* inner(hs(-phi, beta) * sigma(u_sol), epsilon(u_sol))
* dx
)
# Constraint: Volume
with fda.stop_annotating():
total_volume = fd.assemble(fd.Constant(1.0) * dx(domain=mesh))
VolPen = fd.assemble(hs(-phi, beta) * dx)
# Needed to track the value of the volume
VolControl = fda.Control(VolPen)
Vval = total_volume / 2.0
phi_pvd = fd.File("phi_evolution.pvd", target_continuity=fd.H1)
def deriv_cb(phi):
with fda.stop_annotating():
phi_pvd.write(phi[0])
c = fda.Control(s)
Jhat = LevelSetFunctional(J, c, phi, derivative_cb_pre=deriv_cb)
Vhat = LevelSetFunctional(VolPen, c, phi)
beta_param = 0.1
# Boundary conditions for the shape derivatives.
# They must be zero at the boundary conditions.
bcs_vel = fd.DirichletBC(S, fd.Constant((0.0, 0.0)), (1, 2))
# Regularize the shape derivatives
reg_solver = RegularizationSolver(
S,
mesh,
beta=beta_param,
gamma=1.0e5,
dx=dx,
bcs=bcs_vel,
output_dir=None,
)
# Hamilton-Jacobi equation to advect the level set
dt = 0.05
tol = 1e-5
# Optimization problem
vol_constraint = Constraint(Vhat, Vval, VolControl)
problem = InfDimProblem(Jhat, reg_solver, ineqconstraints=vol_constraint)
parameters = {
"ksp_type": "preonly",
"pc_type": "lu",
"mat_type": "aij",
"ksp_converged_reason": None,
"pc_factor_mat_solver_type": "mumps",
}
params = {
"alphaC": 3.0,
"K": 0.1,
"debug": 5,
"alphaJ": 1.0,
"dt": dt,
"maxtrials": 10,
"maxit": n_iters,
"itnormalisation": 50,
"tol": tol,
}
results = nlspace_solve(problem, params)
return results
def deriv_cb(phi):
with fda.stop_annotating():
phi_pvd.write(phi[0])
def compliance_bridge():
parser = argparse.ArgumentParser(description="Heat exchanger")
parser.add_argument(
"--n_iters",
dest="n_iters",
type=int,
action="store",
default=1000,
help="Number of optimization iterations",
)
parser.add_argument(
"--output_dir",
dest="output_dir",
type=str,
action="store",
default="./",
help="Output directory",
)
opts = parser.parse_args()
output_dir = opts.output_dir
# Elasticity parameters
E, nu = 1.0, 0.3
rho_min = fd.Constant(1e-4) # Min Vol fraction
eps = fd.Constant(100.0) # Heaviside parameter
mu, lmbda = fd.Constant(E / (2 * (1 + nu))), fd.Constant(
E * nu / ((1 + nu) * (1 - 2 * nu)))
mesh = fd.RectangleMesh(20, 40, 0.5, 1, quadrilateral=True)
mh = fd.MeshHierarchy(mesh, 1)
m = fd.ExtrudedMeshHierarchy(mh, height=1, base_layer=40)
mesh = m[-1]
S = fd.VectorFunctionSpace(mesh, "CG", 1)
s = fd.Function(S, name="deform")
mesh.coordinates.assign(mesh.coordinates + s)
x, y, z = fd.SpatialCoordinate(mesh)
PHI = fd.FunctionSpace(mesh, "CG", 1)
lx = 1.0
ly = 2.0
lz = ly
phi_expr = (-cos(4.0 / lx * pi * x) * cos(4.0 * pi / ly * y) *
cos(4.0 / lz * pi * z) - 0.6)
with fda.stop_annotating():
phi = fd.interpolate(-phi_expr, PHI)
phi.rename("LevelSet")
H1 = fd.VectorElement("CG", mesh.ufl_cell(), 1)
W = fd.FunctionSpace(mesh, H1)
print(f"DOFS: {W.dim()}")
modes = [fd.Function(W) for _ in range(6)]
modes[0].interpolate(fd.Constant([1, 0, 0]))
modes[1].interpolate(fd.Constant([0, 1, 0]))
modes[2].interpolate(fd.Constant([0, 0, 1]))
modes[3].interpolate(fd.as_vector([0, z, -y]))
modes[4].interpolate(fd.as_vector([-z, 0, x]))
modes[5].interpolate(fd.as_vector([y, -x, 0]))
nullmodes = fd.VectorSpaceBasis(modes)
# Make sure they're orthonormal.
nullmodes.orthonormalize()
u = fd.TrialFunction(W)
v = fd.TestFunction(W)
def epsilon(u):
return sym(nabla_grad(u))
def sigma(v):
return 2.0 * mu * epsilon(v) + lmbda * tr(epsilon(v)) * Identity(3)
# Variational forms
a = inner(hs(phi, eps, min_value=rho_min) * sigma(u),
nabla_grad(v)) * dx(degree=2)
t = fd.Constant((0.0, 0.0, -1.0e-1))
L = inner(t, v) * ds_t
# Dirichlet BCs
ylimits = (0.2, 1.8)
xlimits = (0.4, 0.6)
I_BC = create_function_marker(PHI, W, xlimits, ylimits)
bc1 = MyBC(W, 0, I_BC)
bc2 = fd.DirichletBC(W.sub(0), fd.Constant(0.0), 2)
bc3 = fd.DirichletBC(W.sub(1), fd.Constant(0.0), 4)
u_sol = fd.Function(W)
fd.solve(
a == L,
u_sol,
bcs=[bc1, bc2, bc3],
solver_parameters=gamg_parameters,
near_nullspace=nullmodes,
)
# Cost function
Jform = fd.assemble(
inner(hs(phi, eps, min_value=rho_min) * sigma(u_sol), epsilon(u_sol)) *
dx(degree=2))
# Constraint
VolPen = fd.assemble(hs(phi, eps, min_value=rho_min) * dx(degree=2))
total_vol = fd.assemble(fd.Constant(1.0) * dx(domain=mesh), annotate=False)
VolControl = fda.Control(VolPen)
Vval = 0.15 * total_vol
# Plotting
global_counter1 = itertools.count()
phi_pvd = fd.File(f"{output_dir}/level_set_evolution.pvd")
def deriv_cb(phi):
iter = next(global_counter1)
if iter % 10 == 0:
phi_pvd.write(phi[0])
c = fda.Control(s)
Jhat = LevelSetFunctional(Jform, c, phi, derivative_cb_pre=deriv_cb)
Vhat = LevelSetFunctional(VolPen, c, phi)
# Regularization solver. Zero on the BCs boundaries
beta_param = 0.005
bcs_vel_1 = MyBC(S, 0, I_BC)
bcs_vel_2 = fd.DirichletBC(S, fd.Constant((0.0, 0.0, 0.0)), "top")
bcs_vel = [bcs_vel_1, bcs_vel_2]
reg_solver = RegularizationSolver(
S,
mesh,
beta=beta_param,
gamma=1.0e4,
dx=dx,
bcs=bcs_vel,
output_dir=None,
solver_parameters=gamg_parameters,
)
dt = 0.05
tol = 1e-5
params = {
"alphaC": 1.0,
"K": 0.0001,
"debug": 5,
"maxit": opts.n_iters,
"alphaJ": 2.0,
"dt": dt,
"maxtrials": 500,
"tol_merit":
5e-2, # new merit can be within 0.5% of the previous merit
"itnormalisation": 50,
"tol": tol,
}
hj_solver_parameters["ts_dt"] = dt / 50.0
solver_parameters = {
"hj_solver": hj_solver_parameters,
"reinit_solver": reinit_solver_parameters,
}
vol_constraint = Constraint(Vhat, Vval, VolControl)
problem = InfDimProblem(
Jhat,
reg_solver,
ineqconstraints=vol_constraint,
solver_parameters=solver_parameters,
)
_ = nlspace_solve(problem, params)