def eva(self, num):
# construct basis functions, Set num points corresponding to num basis functions
basPoints = np.linspace(0, 1, num)
dx = basPoints[1] - basPoints[0]
aa, bb, cc = -dx, 0.0, dx
for x_p in basPoints:
self.theta.append(
fe.interpolate(
fe.Expression(
'x[0] < a || x[0] > c ? 0 : (x[0] >=a && x[0] <= b ? (x[0]-a)/(b-a) : 1-(x[0]-b)/(c-b))',
degree=2,
a=aa,
b=bb,
c=cc), self.Vc))
aa, bb, cc = aa + dx, bb + dx, cc + dx
u_trial, u_test = fe.TrialFunction(self.Vu), fe.TestFunction(self.Vu)
left = fe.inner(self.al * fe.nabla_grad(u_trial),
fe.nabla_grad(u_test)) * fe.dx
right = self.f * u_test * fe.dx
def boundaryD(x, on_boundary):
return on_boundary and fe.near(x[1], 1.0)
for i in range(num):
uH = fe.Function(self.Vu)
bcD = fe.DirichletBC(self.Vu, self.theta[i], boundaryD)
left_m, right_m = fe.assemble_system(left, right, bcD)
fe.solve(left_m, uH.vector(), right_m)
self.sol.append(uH)
def geneForwardMatrix(self, q_fun=fe.Constant(0.0), fR=fe.Constant(0.0), \
fI=fe.Constant(0.0)):
if self.haveFunctionSpace == False:
self.geneFunctionSpace()
xx, yy, dPML, sig0_, p_ = self.domain.xx, self.domain.yy, self.domain.dPML,\
self.domain.sig0, self.domain.p
# define the coefficents induced by PML
sig1 = fe.Expression('x[0] > x1 && x[0] < x1 + dd ? sig0*pow((x[0]-x1)/dd, p) : (x[0] < 0 && x[0] > -dd ? sig0*pow((-x[0])/dd, p) : 0)',
degree=3, x1=xx, dd=dPML, sig0=sig0_, p=p_)
sig2 = fe.Expression('x[1] > x2 && x[1] < x2 + dd ? sig0*pow((x[1]-x2)/dd, p) : (x[1] < 0 && x[1] > -dd ? sig0*pow((-x[1])/dd, p) : 0)',
degree=3, x2=yy, dd=dPML, sig0=sig0_, p=p_)
sR = fe.as_matrix([[(1+sig1*sig2)/(1+sig1*sig1), 0.0], [0.0, (1+sig1*sig2)/(1+sig2*sig2)]])
sI = fe.as_matrix([[(sig2-sig1)/(1+sig1*sig1), 0.0], [0.0, (sig1-sig2)/(1+sig2*sig2)]])
cR = 1 - sig1*sig2
cI = sig1 + sig2
# define the coefficients with physical meaning
angl_fre = self.kappa*np.pi
angl_fre2 = fe.Constant(angl_fre*angl_fre)
# define equations
u_ = fe.TestFunction(self.V)
du = fe.TrialFunction(self.V)
u_R, u_I = fe.split(u_)
duR, duI = fe.split(du)
def sigR(v):
return fe.dot(sR, fe.nabla_grad(v))
def sigI(v):
return fe.dot(sI, fe.nabla_grad(v))
F1 = - fe.inner(sigR(duR)-sigI(duI), fe.nabla_grad(u_R))*(fe.dx) \
- fe.inner(sigR(duI)+sigI(duR), fe.nabla_grad(u_I))*(fe.dx) \
- fR*u_R*(fe.dx) - fI*u_I*(fe.dx)
a2 = fe.inner(angl_fre2*q_fun*(cR*duR-cI*duI), u_R)*(fe.dx) \
+ fe.inner(angl_fre2*q_fun*(cR*duI+cI*duR), u_I)*(fe.dx) \
# define boundary conditions
def boundary(x, on_boundary):
return on_boundary
bc = [fe.DirichletBC(self.V.sub(0), fe.Constant(0.0), boundary), \
fe.DirichletBC(self.V.sub(1), fe.Constant(0.0), boundary)]
a1, L1 = fe.lhs(F1), fe.rhs(F1)
self.u = fe.Function(self.V)
self.A1 = fe.assemble(a1)
self.b1 = fe.assemble(L1)
self.A2 = fe.assemble(a2)
bc[0].apply(self.A1, self.b1)
bc[1].apply(self.A1, self.b1)
bc[0].apply(self.A2)
bc[1].apply(self.A2)
self.A = self.A1 + self.A2
def calTrueSol(para):
nx, ny = para['mesh_N'][0], para['mesh_N'][1]
mesh = fe.UnitSquareMesh(nx, ny)
Vu = fe.FunctionSpace(mesh, 'P', para['P'])
Vc = fe.FunctionSpace(mesh, 'P', para['P'])
al = fe.Constant(para['alpha'])
f = fe.Expression(para['f'], degree=5)
q1 = fe.interpolate(fe.Expression(para['q1'], degree=5), Vc)
q2 = fe.interpolate(fe.Expression(para['q2'], degree=5), Vc)
q3 = fe.interpolate(fe.Expression(para['q3'], degree=5), Vc)
theta = fe.interpolate(fe.Expression(para['q4'], degree=5), Vc)
class BoundaryX0(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[0], 0.0)
class BoundaryX1(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[0], 1.0)
class BoundaryY0(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[1], 0.0)
class BoundaryY1(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[1], 1.0)
boundaries = fe.MeshFunction('size_t', mesh, mesh.topology().dim() - 1)
boundaries.set_all(0)
bc0, bc1, bc2, bc3 = BoundaryX0(), BoundaryX1(), BoundaryY0(), BoundaryY1()
bc0.mark(boundaries, 1)
bc1.mark(boundaries, 2)
bc2.mark(boundaries, 3)
bc3.mark(boundaries, 4)
domains = fe.MeshFunction("size_t", mesh, mesh.topology().dim())
domains.set_all(0)
bcD = fe.DirichletBC(Vu, theta, boundaries, 4)
dx = fe.Measure('dx', domain=mesh, subdomain_data=domains)
ds = fe.Measure('ds', domain=mesh, subdomain_data=boundaries)
u_trial, u_test = fe.TrialFunction(Vu), fe.TestFunction(Vu)
u = fe.Function(Vu)
left = fe.inner(al * fe.nabla_grad(u_trial), fe.nabla_grad(u_test)) * dx
right = f * u_test * dx + (q1 * u_test * ds(1) + q2 * u_test * ds(2) +
q3 * u_test * ds(3))
left_m, right_m = fe.assemble_system(left, right, bcD)
fe.solve(left_m, u.vector(), right_m)
return u
def eva(self):
# construct solutions corresponding to the basis functions
class BoundaryX0(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[0], 0.0)
class BoundaryX1(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[0], 1.0)
class BoundaryY0(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[1], 0.0)
class BoundaryY1(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[1], 1.0)
boundaries = fe.MeshFunction('size_t', self.mesh,
self.mesh.topology().dim() - 1)
boundaries.set_all(0)
bc0, bc1, bc2, bc3 = BoundaryX0(), BoundaryX1(), BoundaryY0(
), BoundaryY1()
bc0.mark(boundaries, 1)
bc1.mark(boundaries, 2)
bc2.mark(boundaries, 3)
bc3.mark(boundaries, 4)
domains = fe.MeshFunction("size_t", self.mesh,
self.mesh.topology().dim())
domains.set_all(0)
dx = fe.Measure('dx', domain=self.mesh, subdomain_data=domains)
ds = fe.Measure('ds', domain=self.mesh, subdomain_data=boundaries)
u_trial, u_test = fe.TrialFunction(self.Vu), fe.TestFunction(self.Vu)
left = fe.inner(fe.nabla_grad(u_trial), fe.nabla_grad(u_test)) * dx
right = self.f * u_test * dx + (self.q1 * u_test * ds(1) +
self.q2 * u_test * ds(2) +
self.q3 * u_test * ds(3))
bcD = fe.DirichletBC(self.Vu, self.theta, boundaries, 4)
left_m, right_m = fe.assemble_system(left, right, bcD)
fe.solve(left_m, self.u.vector(), right_m)
f = fs.Constant((0, 0))
k = fs.Constant(dt)
mu = fs.Constant(mu)
rho = fs.Constant(rho)
# Define strain-rate tensor
def epsilon(u):
return fs.sym(fs.nabla_grad(u))
# Define stress tensor
def sigma(u, p):
return 2*mu*epsilon(u) - p*fs.Identity(len(u))
# Define variational problem for step 1 (Tentative velocity step)
F1 = rho*fs.dot((u - u_n) / k, v)*fs.dx + \
rho*fs.dot(fs.dot(u_n, fs.nabla_grad(u_n)), v)*fs.dx \
+ fs.inner(sigma(U, p_n), epsilon(v))*fs.dx \
+ fs.dot(p_n*n, v)*fs.ds - fs.dot(mu*fs.nabla_grad(U)*n, v)*fs.ds \
- fs.dot(f, v)*fs.dx
a1 = fs.lhs(F1)
L1 = fs.rhs(F1)
# Define variational problem for step 2 (Pressure correction step)
a2 = fs.dot(fs.nabla_grad(p), fs.nabla_grad(q))*fs.dx
L2 = fs.dot(fs.nabla_grad(p_n), fs.nabla_grad(q))*fs.dx \
- (1/k)*fs.div(u_)*q*fs.dx
# Define variational problem for step 3 (Velocity correction step)
a3 = fs.dot(u, v)*fs.dx
L3 = fs.dot(u_, v)*fs.dx - k*fs.dot(fs.nabla_grad(p_ - p_n), v)*fs.dx
def b(u, v, w):
return 0.5 * (fn.inner(fn.dot(u, fn.nabla_grad(v)), w) -
fn.inner(fn.dot(u, fn.nabla_grad(w)), v))
def contract(u, v):
return fn.inner(fn.nabla_grad(u), fn.nabla_grad(v))
def solve(self):
# TODO: when FEniCS ported to Python3, this should be exist_ok
try:
os.makedirs('results')
except OSError:
pass
z, w = (self.z, self.w)
u0 = d.Constant(0.0)
# Define the linear and bilinear forms
L = u0 * w * dx
# Define useful functions
cond = d.Function(self.DV)
U = d.Function(self.V)
# Initialize the max_e vector, that will store the cumulative max e values
max_e = d.Function(self.V)
max_e.vector()[:] = 0.0
max_e.rename("max_E", "Maximum energy deposition by location")
max_e_file = d.File("results/%s-max_e.pvd" % input_mesh)
max_e_per_step = d.Function(self.V)
max_e_per_step_file = d.File("results/%s-max_e_per_step.pvd" % input_mesh)
self.es = {}
self.max_es = {}
fi = d.File("results/%s-cond.pvd" % input_mesh)
potential_file = d.File("results/%s-potential.pvd" % input_mesh)
# Loop through the voltages and electrode combinations
for i, (anode, cathode, voltage) in enumerate(v.electrode_triples):
print("Electrodes %d (%lf) -> %d (0)" % (anode, voltage, cathode))
cond = d.project(self.sigma_start, V=self.DV)
# Define the Dirichlet boundary conditions on the active needles
uV = d.Constant(voltage)
term1_bc = d.DirichletBC(self.V, uV, self.patches, v.needles[anode])
term2_bc = d.DirichletBC(self.V, u0, self.patches, v.needles[cathode])
e = d.Function(self.V)
e.vector()[:] = max_e.vector()
# Re-evaluate conductivity
self.increase_conductivity(cond, e)
for j in range(v.max_restarts):
# Update the bilinear form
a = d.inner(d.nabla_grad(z), cond * d.nabla_grad(w)) * dx
# Solve again
print(" [solving...")
d.solve(a == L, U, bcs=[term1_bc, term2_bc])
print(" ....solved]")
# Extract electric field norm
for k in range(len(U.vector())):
if N.isnan(U.vector()[k]):
U.vector()[k] = 1e5
e_new = d.project(d.sqrt(d.dot(d.grad(U), d.grad(U))), self.V)
# Take the max of the new field and the established electric field
e.vector()[:] = N.array([max(*X) for X in zip(e.vector(), e_new.vector())])
# Re-evaluate conductivity
fi << cond
self.increase_conductivity(cond, e)
potential_file << U
# Save the max e function to a VTU
max_e_per_step.vector()[:] = e.vector()[:]
max_e_per_step_file << max_e_per_step
# Store this electric field norm, for this triple, for later reference
self.es[i] = e
# Store the max of this electric field norm and that for all previous triples
max_e_array = N.array([max(*X) for X in zip(max_e.vector(), e.vector())])
max_e.vector()[:] = max_e_array
# Create a new max_e function for storage, or it will be overwritten by the next iteration
max_e_new = d.Function(self.V)
max_e_new.vector()[:] = max_e_array
# Store this max e function for the cumulative coverage curve calculation later
self.max_es[i] = max_e_new
# Save the max e function to a VTU
max_e_file << max_e
self.max_e_count = i
def epsilon(u):
return 0.5 * (fs.nabla_grad(u) + fs.nabla_grad(u).T)
def sigI(v):
return fe.dot(sI, fe.nabla_grad(v))
def sigR(v):
return fe.dot(sR, fe.nabla_grad(v))
def nabla_grad(self, arg):
return FEN.nabla_grad(arg)
def epsilon(u):
return 0.5 * (fenics.nabla_grad(u) + fenics.nabla_grad(u).T
) #return sym(nabla_grad(u))
def xest_implement_2d_myosin(self):
#Parameters
total_time = 1.0
number_of_time_steps = 100
delta_t = total_time / number_of_time_steps
nx = ny = 100
domain_size = 1.0
lambda_ = 5.0
mu = 2.0
gamma = 1.0
eta_b = 0.0
eta_s = 1.0
k_b = 1.0
k_u = 1.0
# zeta_1 = -0.5
zeta_1 = 0.0
zeta_2 = 1.0
mu_a = 1.0
K_0 = 1.0
K_1 = 0.0
K_2 = 0.0
K_3 = 0.0
D = 0.25
alpha = 3
c = 0.1
# Sub domain for Periodic boundary condition
class PeriodicBoundary(fenics.SubDomain):
# Left boundary is "target domain" G
def inside(self, x, on_boundary):
# return True if on left or bottom boundary AND NOT on one of the two corners (0, 1) and (1, 0)
return bool(
(fenics.near(x[0], 0) or fenics.near(x[1], 0)) and
(not ((fenics.near(x[0], 0) and fenics.near(x[1], 1)) or
(fenics.near(x[0], 1) and fenics.near(x[1], 0))))
and on_boundary)
def map(self, x, y):
if fenics.near(x[0], 1) and fenics.near(x[1], 1):
y[0] = x[0] - 1.
y[1] = x[1] - 1.
elif fenics.near(x[0], 1):
y[0] = x[0] - 1.
y[1] = x[1]
else: # near(x[1], 1)
y[0] = x[0]
y[1] = x[1] - 1.
periodic_boundary_condition = PeriodicBoundary()
#Set up finite elements
mesh = fenics.RectangleMesh(fenics.Point(0, 0),
fenics.Point(domain_size, domain_size), nx,
ny)
vector_element = fenics.VectorElement('P', fenics.triangle, 2, dim=2)
single_element = fenics.FiniteElement('P', fenics.triangle, 2)
mixed_element = fenics.MixedElement(vector_element, single_element)
V = fenics.FunctionSpace(
mesh,
mixed_element,
constrained_domain=periodic_boundary_condition)
v, r = fenics.TestFunctions(V)
full_trial_function = fenics.Function(V)
u, rho = fenics.split(full_trial_function)
full_trial_function_n = fenics.Function(V)
u_n, rho_n = fenics.split(full_trial_function_n)
#Define non-linear weak formulation
def epsilon(u):
return 0.5 * (fenics.nabla_grad(u) + fenics.nabla_grad(u).T
) #return sym(nabla_grad(u))
def sigma_e(u):
return lambda_ * ufl.nabla_div(u) * fenics.Identity(
2) + 2 * mu * epsilon(u)
def sigma_d(u):
return eta_b * ufl.nabla_div(u) * fenics.Identity(
2) + 2 * eta_s * epsilon(u)
# def sigma_a(u,rho):
# return ( -zeta_1*rho/(1+zeta_2*rho)*mu_a*fenics.Identity(2)*(K_0+K_1*ufl.nabla_div(u)+
# K_2*ufl.nabla_div(u)*ufl.nabla_div(u)+K_3*ufl.nabla_div(u)*ufl.nabla_div(u)*ufl.nabla_div(u)))
def sigma_a(u, rho):
return -zeta_1 * rho / (
1 + zeta_2 * rho) * mu_a * fenics.Identity(2) * (K_0)
F = (gamma * fenics.dot(u, v) * fenics.dx -
gamma * fenics.dot(u_n, v) * fenics.dx +
fenics.inner(sigma_d(u), fenics.nabla_grad(v)) * fenics.dx -
fenics.inner(sigma_d(u_n), fenics.nabla_grad(v)) * fenics.dx -
delta_t *
fenics.inner(sigma_e(u) + sigma_a(u, rho), fenics.nabla_grad(v)) *
fenics.dx + rho * r * fenics.dx - rho_n * r * fenics.dx +
ufl.nabla_div(rho * u) * r * fenics.dx -
ufl.nabla_div(rho * u_n) * r * fenics.dx - D * delta_t *
fenics.dot(fenics.nabla_grad(rho), fenics.nabla_grad(r)) *
fenics.dx + delta_t *
(-k_u * rho * fenics.exp(alpha * ufl.nabla_div(u)) + k_b *
(1 - c * ufl.nabla_div(u))) * r * fenics.dx)
# F = ( gamma*fenics.dot(u,v)*fenics.dx - gamma*fenics.dot(u_n,v)*fenics.dx + fenics.inner(sigma_d(u),fenics.nabla_grad(v))*fenics.dx -
# fenics.inner(sigma_d(u_n),fenics.nabla_grad(v))*fenics.dx - delta_t*fenics.inner(sigma_e(u)+sigma_a(u,rho),fenics.nabla_grad(v))*fenics.dx
# +rho*r*fenics.dx-rho_n*r*fenics.dx + ufl.nabla_div(rho*u)*r*fenics.dx - ufl.nabla_div(rho*u_n)*r*fenics.dx -
# D*delta_t*fenics.dot(fenics.nabla_grad(rho),fenics.nabla_grad(r))*fenics.dx +delta_t*(-k_u*rho*fenics.exp(alpha*ufl.nabla_div(u))+k_b*(1-c*ufl.nabla_div(u))))
vtkfile_rho = fenics.File(
os.path.join(os.path.dirname(__file__), 'output', 'myosin_2d',
'solution_rho.pvd'))
vtkfile_u = fenics.File(
os.path.join(os.path.dirname(__file__), 'output', 'myosin_2d',
'solution_u.pvd'))
# rho_0 = fenics.Expression(((('0.0'),('0.0'),('0.0')),('sin(x[0])')), degree=1 )
# full_trial_function_n = fenics.project(rho_0, V)
time = 0.0
for time_index in range(number_of_time_steps):
# Update current time
time += delta_t
# Compute solution
fenics.solve(F == 0, full_trial_function)
# Save to file and plot solution
vis_u, vis_rho = full_trial_function.split()
vtkfile_rho << (vis_rho, time)
vtkfile_u << (vis_u, time)
full_trial_function_n.assign(full_trial_function)
def epsilon(u):
return fs.sym(fs.nabla_grad(u))
def sigma(u):
return lambda_ * fs.nabla_grad(u) * fs.Identity(d) + 2 * mu * epsilon(u)
def epsilon(u):
return 0.5 * (nabla_grad(u) + nabla_grad(u).T)
def solve(self):
z, w = (self.z, self.w)
u0 = d.Constant(0.0)
# Define the linear and bilinear forms
L = u0 * w * dx
# Define useful functions
cond = d.Function(self.DV)
U = d.Function(self.V)
# Initialize the max_e vector, that will store the cumulative max e values
max_e = d.Function(self.V)
max_e.vector()[:] = 0.0
max_e.rename("max_E", "Maximum energy deposition by location")
max_e_file = d.File("../%s-max_e.pvd" % input_mesh)
max_e_per_step = d.Function(self.V)
max_e_per_step_file = d.File("../%s-max_e_per_step.pvd" % input_mesh)
self.es = {}
self.max_es = {}
fi = d.File("../%s-cond.pvd" % input_mesh)
potential_file = d.File("../%s-potential.pvd" % input_mesh)
# Loop through the voltages and electrode combinations
for i, (anode, cathode, voltage) in enumerate(v.electrode_triples):
print("Electrodes %d (%lf) -> %d (0)" % (anode, voltage, cathode))
cond = d.project(self.sigma_start, V=self.DV)
# Define the Dirichlet boundary conditions on the active needles
uV = d.Constant(voltage)
term1_bc = d.DirichletBC(self.V, uV, self.patches, anode)
term2_bc = d.DirichletBC(self.V, u0, self.patches, cathode)
e = d.Function(self.V)
e.vector()[:] = max_e.vector()
# Re-evaluate conductivity
self.increase_conductivity(cond, e)
for j in range(v.max_restarts):
# Update the bilinear form
a = d.inner(d.nabla_grad(z), cond * d.nabla_grad(w)) * dx
# Solve again
print(" [solving...")
d.solve(a == L, U, bcs=[term1_bc, term2_bc])
print(" ....solved]")
# Extract electric field norm
for k in range(len(U.vector())):
if N.isnan(U.vector()[k]):
U.vector()[k] = 1e5
e_new = d.project(d.sqrt(d.dot(d.grad(U), d.grad(U))), self.V)
# Take the max of the new field and the established electric field
e.vector()[:] = N.array(
[max(*X) for X in zip(e.vector(), e_new.vector())])
# Re-evaluate conductivity
fi << cond
self.increase_conductivity(cond, e)
potential_file << U
# Save the max e function to a VTU
max_e_per_step.vector()[:] = e.vector()[:]
max_e_per_step_file << max_e_per_step
# Store this electric field norm, for this triple, for later reference
self.es[i] = e
# Store the max of this electric field norm and that for all previous triples
max_e_array = N.array(
[max(*X) for X in zip(max_e.vector(), e.vector())])
max_e.vector()[:] = max_e_array
# Create a new max_e function for storage, or it will be overwritten by the next iteration
max_e_new = d.Function(self.V)
max_e_new.vector()[:] = max_e_array
# Store this max e function for the cumulative coverage curve calculation later
self.max_es[i] = max_e_new
# Save the max e function to a VTU
max_e_file << max_e
self.max_e_count = i
