def holzapfelogden_dev(self, params, f0, s0, C):
# anisotropic invariants - keep in mind that for growth, self.C is the elastic part of C (hence != to function input variable C)
I4 = dot(dot(self.C,f0), f0)
I6 = dot(dot(self.C,s0), s0)
I8 = dot(dot(self.C,s0), f0)
# to guarantee initial configuration is stress-free (in case of initially non-orthogonal fibers f0 and s0)
I8 -= dot(f0,s0)
a_0, b_0 = params['a_0'], params['b_0']
a_f, b_f = params['a_f'], params['b_f']
a_s, b_s = params['a_s'], params['b_s']
a_fs, b_fs = params['a_fs'], params['b_fs']
try: fiber_comp = params['fiber_comp']
except: fiber_comp = False
# conditional parameters: fibers are only active in tension if fiber_comp is False
if not fiber_comp:
a_f_c = conditional(ge(I4,1.), a_f, 0.)
a_s_c = conditional(ge(I6,1.), a_s, 0.)
else:
a_f_c = a_f
a_s_c = a_s
# Holzapfel-Ogden (Holzapfel and Ogden 2009) material w/o split applied to invariants I4, I6, I8 (Sansour 2008)
psi_dev = a_0/(2.*b_0)*(exp(b_0*(self.Ic_bar-3.)) - 1.) + \
a_f_c/(2.*b_f)*(exp(b_f*(I4-1.)**2.) - 1.) + a_s_c/(2.*b_s)*(exp(b_s*(I6-1.)**2.) - 1.) + \
a_fs/(2.*b_fs)*(exp(b_fs*I8**2.) - 1.)
S = 2.*diff(psi_dev,C)
return S
def g(self, lam):
amp_min = self.params['amp_min']
amp_max = self.params['amp_max']
lam_threslo = self.params['lam_threslo']
lam_maxlo = self.params['lam_maxlo']
lam_threshi = self.params['lam_threshi']
lam_maxhi = self.params['lam_maxhi']
# Diss Hirschvogel eq. 2.107
# TeX: g(\lambda_{\mathrm{myo}}) = \begin{cases} a_{\mathrm{min}}, & \lambda_{\mathrm{myo}} \leq \hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,lo}}, \\ a_{\mathrm{min}}+\frac{1}{2}\left(a_{\mathrm{max}}-a_{\mathrm{min}}\right)\left(1-\cos \frac{\pi(\lambda_{\mathrm{myo}}-\hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,lo}})}{\hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,lo}}-\hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,lo}}}\right), & \hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,lo}} \leq \lambda_{\mathrm{myo}} \leq \hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,lo}}, \\ a_{\mathrm{max}}, & \hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,lo}} \leq \lambda_{\mathrm{myo}} \leq \hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,hi}}, \\ a_{\mathrm{min}}+\frac{1}{2}\left(a_{\mathrm{max}}-a_{\mathrm{min}}\right)\left(1-\cos \frac{\pi(\lambda_{\mathrm{myo}}-\hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,hi}})}{\hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,hi}}-\hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,hi}}}\right), & \hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,hi}} \leq \lambda_{\mathrm{myo}} \leq \hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,hi}}, \\ a_{\mathrm{min}}, & \lambda_{\mathrm{myo}} \geq \hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,hi}} \end{cases}
return conditional(
le(lam, lam_threslo), amp_min,
conditional(
And(ge(lam, lam_threslo), le(lam, lam_maxlo)), amp_min + 0.5 *
(amp_max - amp_min) * (1. - cos(pi * (lam - lam_threslo) /
(lam_maxlo - lam_threslo))),
conditional(
And(ge(lam, lam_maxlo), le(lam, lam_threshi)), amp_max,
conditional(
And(ge(lam, lam_threshi), le(lam, lam_maxhi)),
amp_min + 0.5 * (amp_max - amp_min) *
(1. - cos(pi * (lam - lam_maxhi) /
(lam_maxhi - lam_threshi))),
conditional(ge(lam, lam_maxhi), amp_min, as_ufl(0))))))
def __init__(self, functionSpace, value, xL, xR, eps=1e-10):
cond = 1
x = ufl.SpatialCoordinate(functionSpace)
for l, r, c in zip(xL, xR, x):
if l is not None:
cond *= ufl.conditional(c > l - eps, 1, 0)
if r is not None:
cond *= ufl.conditional(c < r + eps, 1, 0)
DirichletBC.__init__(self, functionSpace, value, cond)
def increase_conductivity(self, cond, e):
# Set up the three way choice function
intermediate = e * self.k + self.h
not_less_than = ufl.conditional(ufl.gt(e, self.threshold_irreversible), self.sigma_end, intermediate)
cond_expression = ufl.conditional(ufl.lt(e, self.threshold_reversible), self.sigma_start, not_less_than)
# Project this onto the function space
cond_function = d.project(ufl.Max(cond_expression, cond), cond.function_space())
cond.assign(cond_function)
def test_conditional(mode, compile_args):
cell = ufl.triangle
element = ufl.FiniteElement("Lagrange", cell, 1)
u, v = ufl.TrialFunction(element), ufl.TestFunction(element)
x = ufl.SpatialCoordinate(cell)
condition = ufl.Or(ufl.ge(ufl.real(x[0] + x[1]), 0.1),
ufl.ge(ufl.real(x[1] + x[1]**2), 0.1))
c1 = ufl.conditional(condition, 2.0, 1.0)
a = c1 * ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx
x1x2 = ufl.real(x[0] + ufl.as_ufl(2) * x[1])
c2 = ufl.conditional(ufl.ge(x1x2, 0), 6.0, 0.0)
b = c2 * ufl.conj(v) * ufl.dx
forms = [a, b]
compiled_forms, module = ffcx.codegeneration.jit.compile_forms(
forms,
parameters={'scalar_type': mode},
cffi_extra_compile_args=compile_args)
form0 = compiled_forms[0][0].create_cell_integral(-1)
form1 = compiled_forms[1][0].create_cell_integral(-1)
ffi = cffi.FFI()
c_type, np_type = float_to_type(mode)
A1 = np.zeros((3, 3), dtype=np_type)
w1 = np.array([1.0, 1.0, 1.0], dtype=np_type)
c = np.array([], dtype=np.float64)
coords = np.array([0.0, 0.0, 1.0, 0.0, 0.0, 1.0], dtype=np.float64)
form0.tabulate_tensor(
ffi.cast('{type} *'.format(type=c_type), A1.ctypes.data),
ffi.cast('{type} *'.format(type=c_type), w1.ctypes.data),
ffi.cast('{type} *'.format(type=c_type), c.ctypes.data),
ffi.cast('double *', coords.ctypes.data), ffi.NULL, ffi.NULL, 0)
expected_result = np.array([[2, -1, -1], [-1, 1, 0], [-1, 0, 1]],
dtype=np_type)
assert np.allclose(A1, expected_result)
A2 = np.zeros(3, dtype=np_type)
w2 = np.array([1.0, 1.0, 1.0], dtype=np_type)
coords = np.array([0.0, 0.0, 1.0, 0.0, 0.0, 1.0], dtype=np.float64)
form1.tabulate_tensor(
ffi.cast('{type} *'.format(type=c_type), A2.ctypes.data),
ffi.cast('{type} *'.format(type=c_type), w2.ctypes.data),
ffi.cast('{type} *'.format(type=c_type), c.ctypes.data),
ffi.cast('double *', coords.ctypes.data), ffi.NULL, ffi.NULL, 0)
expected_result = np.ones(3, dtype=np_type)
assert np.allclose(A2, expected_result)
def test_cpp_formatting_of_conditionals():
x, y = ufl.SpatialCoordinate(ufl.triangle)
# Test conditional expressions
assert expr2cpp(ufl.conditional(ufl.lt(x, 2), y, 3)) \
== "x[0] < 2 ? x[1]: 3"
assert expr2cpp(ufl.conditional(ufl.gt(x, 2), 4 + y, 3)) \
== "x[0] > 2 ? 4 + x[1]: 3"
assert expr2cpp(ufl.conditional(ufl.And(ufl.le(x, 2), ufl.ge(y, 4)), 7, 8)) \
== "x[0] <= 2 && x[1] >= 4 ? 7: 8"
assert expr2cpp(ufl.conditional(ufl.Or(ufl.eq(x, 2), ufl.ne(y, 4)), 7, 8)) \
== "x[0] == 2 || x[1] != 4 ? 7: 8"
def increase_conductivity(self, cond, e):
# Set up the three way choice function
intermediate = e * self.k + self.h
not_less_than = ufl.conditional(ufl.gt(e, self.threshold_irreversible),
self.sigma_end, intermediate)
cond_expression = ufl.conditional(ufl.lt(e, self.threshold_reversible),
self.sigma_start, not_less_than)
# Project this onto the function space
cond_function = d.project(ufl.Max(cond_expression, cond),
cond.function_space())
cond.assign(cond_function)
def handle_conditional(v, si, deps, SV_factors, FV, sv2fv, e2fi):
fac0 = SV_factors[deps[0]]
fac1 = SV_factors[deps[1]]
fac2 = SV_factors[deps[2]]
assert not fac0, "Cannot have argument in condition."
if not (fac1 or fac2): # non-arg ? non-arg : non-arg
# Record non-argument subexpression
sv2fv[si] = add_to_fv(v, FV, e2fi)
factors = noargs
else:
f0 = FV[sv2fv[deps[0]]]
f1 = FV[sv2fv[deps[1]]]
f2 = FV[sv2fv[deps[2]]]
# Term conditional(c, argument, non-argument) is not legal unless non-argument is 0.0
assert fac1 or isinstance(f1, Zero)
assert fac2 or isinstance(f2, Zero)
assert () not in fac1
assert () not in fac2
z = as_ufl(0.0)
# In general, can decompose like this:
# conditional(c, sum_i fi*ui, sum_j fj*uj) -> sum_i conditional(c, fi, 0)*ui + sum_j conditional(c, 0, fj)*uj
mas = sorted(set(fac1.keys()) | set(fac2.keys()))
factors = {}
for k in mas:
fi1 = fac1.get(k)
fi2 = fac2.get(k)
f1 = z if fi1 is None else FV[fi1]
f2 = z if fi2 is None else FV[fi2]
factors[k] = add_to_fv(conditional(f0, f1, f2), FV, e2fi)
return factors
def compute_welding_size(evo, V, threshold_temp, x, ds):
'''Computes either the welding depth or the welding radius.
Parameters:
evo: ndarray
The coefficients of the solution to the corresponding forward
problem in the basis of the space V (see solve_forward).
V: dolfin.FunctionSpace
The FEM space of the problem being solved.
threshold_temp: float
The threshold temperature used as a criterion for successful welding.
ds: dolfin.Measure
The measure on either the symmetry axis (for welding depth) or
the top boundary (for welding radius).
Returns:
evo_size: ndarray
Contains size of either the welding depth or the welding radius over
all time steps.
'''
theta_k = dolfin.Function(V)
Nt = len(evo) - 1
evo_size = np.zeros(Nt+1)
for k in range(Nt+1):
theta_k.vector().set_local(evo[k])
evo_size[k] = dolfin.assemble(
conditional(ge(theta_k, threshold_temp), 1., 0.) * ds
)
return evo_size
def p_sat(temp):
"""Water vapour saturation pressure
Parameters
----------
temp0: Previous temp function [K]
Note
----
Kunzel 1995, page 40, formula (50).
"""
a = conditional(ge(temp, 273.15), 17.08, 22.44)
theta0 = conditional(ge(temp, 273.15), 234.18, 272.44)
return 611. * exp(a * (temp - 273.15) / (theta0 + (temp - 273.15)))
def bessi0(x):
"""
Modified Bessel function of the first kind.
Code taken from
[Flannery et al. 1992] B.P. Flannery,
W.H. Press, S.A. Teukolsky, W. Vetterling,
"Numerical recipes in C", Press Syndicate
of the University of Cambridge, New York
(1992).
"""
ax = abs(x)
y1 = x / 3.75
y1 *= y1
expr1 = 1.0 + y1 * (3.5156229 + y1 * (3.0899424 + y1 *
(1.2067492 + y1 *
(0.2659732 + y1 *
(0.360768e-1 + y1 * 0.45813e-2)))))
y2 = 3.75 / ax
expr2 = (ufl.exp(ax) / ufl.sqrt(ax) *
(0.39894228 + y2 *
(0.1328592e-1 + y2 *
(0.225319e-2 + y2 *
(-0.157565e-2 + y2 *
(0.916281e-2 + y2 *
(-0.2057706e-1 + y2 *
(0.2635537e-1 + y2 *
(-0.1647633e-1 + y2 * 0.392377e-2)))))))))
return ufl.conditional(ax < 3.75, expr1, expr2)
def sliding_law(self, alpha, U):
constants = self.params.constants
bed = self.bed
H = self.H
rhoi = constants.rhoi
rhow = constants.rhow
g = constants.g
sl = self.params.ice_dynamics.sliding_law
vel_rp = constants.vel_rp
fl_ex = self.float_conditional(H)
C = alpha * alpha
u, v = split(U)
if sl == 'linear':
B2 = C
elif sl == 'weertman':
N = (1 - fl_ex) * (H * rhoi * g + ufl.Min(bed, 0.0) * rhow * g)
U_mag = sqrt(U[0]**2 + U[1]**2 + vel_rp**2)
# Need to catch N <= 0.0 here, as it's raised to
# 1/3 (forward) and -2/3 (adjoint)
N_term = ufl.conditional(N > 0.0, N**(1.0 / 3.0), 0)
B2 = (1 - fl_ex) * (C * N_term * U_mag**(-2.0 / 3.0))
return B2
def liquidity(theta_k, theta_kp1, solidus, implicitness=1.):
theta_avg = avg(theta_k, theta_kp1, implicitness)
form = theta_avg - Constant(solidus)
# filter to liquid parts
form *= conditional(ge(form, 0.), 1., 0.)
return form
def bessk0(x):
"""
Modified Bessel function of the second kind.
Code taken from
[Flannery et al. 1992] B.P. Flannery,
W.H. Press, S.A. Teukolsky, W. Vetterling,
"Numerical recipes in C", Press Syndicate
of the University of Cambridge, New York
(1992).
"""
y1 = x * x / 4.0
expr1 = -ufl.ln(x / 2.0) * bessi0(x) + (
-0.57721566 + y1 * (0.42278420 + y1 *
(0.23069756 + y1 *
(0.3488590e-1 + y1 *
(0.262698e-2 + y1 *
(0.10750e-3 + y1 * 0.74e-5))))))
y2 = 2.0 / x
expr2 = (ufl.exp(-x) / ufl.sqrt(x) *
(1.25331414 + y2 * (-0.7832358e-1 + y2 *
(0.2189568e-1 + y2 *
(-0.1062446e-1 + y2 *
(0.587872e-2 + y2 *
(-0.251540e-2 + y2 * 0.53208e-3)))))))
return ufl.conditional(x > 2, expr2, expr1)
def _rush_larsen_step(rhs_exprs, diff_rhs_exprs, linear_terms,
system_size, y0, stage_solution, dt, time, a, c,
v, DX, time_dep_expressions):
# If we need to replace the original solution with stage solution
repl = None
if stage_solution is not None:
if system_size > 1:
repl = {y0: stage_solution}
else:
repl = {y0[0]: stage_solution}
# If we have time dependent expressions
if time_dep_expressions and abs(float(c)) > DOLFIN_EPS:
time_ = time
time = time + dt * float(c)
repl.update(_replace_dict_time_dependent_expression(time_dep_expressions,
time_, dt, float(c)))
repl[time_] = time
# If all terms are linear (using generalized=True) we add a safe
# guard to the linearized term. See below
safe_guard = sum(linear_terms) == system_size
# Add componentwise contribution to rl form
rl_ufl_form = ufl.zero()
num_rl_steps = 0
for ind in range(system_size):
# forward euler step
fe_du_i = rhs_exprs[ind] * dt * float(a)
# If exact integration
if linear_terms[ind]:
num_rl_steps += 1
# Rush Larsen step Safeguard the divisor: let's hope
# diff_rhs_exprs[ind] is never 1.0e-16! Let's get rid of
# this when the conditional fixes land properly in UFL.
eps = Constant(1.0e-16)
rl_du_i = rhs_exprs[ind] / (diff_rhs_exprs[ind] + eps) * (
ufl.exp(diff_rhs_exprs[ind] * dt) - 1.0)
# If safe guard
if safe_guard:
du_i = ufl.conditional(ufl.lt(abs(diff_rhs_exprs[ind]), 1e-8), fe_du_i, rl_du_i)
else:
du_i = rl_du_i
else:
du_i = fe_du_i
# If we should replace solution in form with stage solution
if repl:
du_i = ufl.replace(du_i, repl)
rl_ufl_form += (y0[ind] + du_i) * v[ind]
return rl_ufl_form * DX
def handle_conditional(v, fac, sf, F):
fac0 = fac[0]
fac1 = fac[1]
fac2 = fac[2]
assert not fac0, "Cannot have argument in condition."
if not (fac1 or fac2): # non-arg ? non-arg : non-arg
raise RuntimeError("No arguments")
else:
f0 = sf[0]
f1 = sf[1]
f2 = sf[2]
# Term conditional(c, argument, non-argument) is not legal unless non-argument is 0.0
assert fac1 or isinstance(f1, Zero)
assert fac2 or isinstance(f2, Zero)
assert () not in fac1
assert () not in fac2
z = as_ufl(0.0)
# In general, can decompose like this:
# conditional(c, sum_i fi*ui, sum_j fj*uj) -> sum_i conditional(c, fi, 0)*ui + sum_j conditional(c, 0, fj)*uj
mas = sorted(set(fac1.keys()) | set(fac2.keys()))
factors = {}
for k in mas:
fi1 = fac1.get(k)
fi2 = fac2.get(k)
f1 = z if fi1 is None else F.nodes[fi1]['expression']
f2 = z if fi2 is None else F.nodes[fi2]['expression']
factors[k] = graph_insert(F, conditional(f0, f1, f2))
return factors
def damage(eps_eqv, mesh, E, f_t, G_f):
h_cb = ufl.CellVolume(mesh) ** (1 / 3)
eps0 = f_t / E
eps_f = G_f / 1.0e+6 / (f_t * h_cb) + eps0 / 2
dmg = ufl.Min(1.0 - eps0 / eps_eqv * ufl.exp(- (eps_eqv - eps0) / (eps_f - eps0)), 1.0 - dmg_eps)
return ufl.conditional(eps_eqv <= eps0, 0.0, 0.0 if args.damage_off else dmg)
def res_dtheta_growth(self, u_, p_, ivar, theta_old_, thres, dt, rquant):
theta_ = ivar["theta"]
grfnc = growthfunction(theta_, self.I)
try:
omega = self.gandrparams['thres_tol']
except:
omega = 0
try:
reduc = self.gandrparams['trigger_reduction']
except:
reduc = 1
assert (reduc <= 1 and reduc > 0)
# threshold should not be lower than specified (only relevant for multiscale analysis, where threshold is set element-wise)
threshold = conditional(gt(thres, self.gandrparams['growth_thres']),
(1. + omega) * thres,
self.gandrparams['growth_thres'])
# trace of elastic Mandel stress
if self.growth_trig == 'volstress':
trigger = reduc * tr(self.M_e(u_, p_, self.kin.C(u_), ivar))
# elastic fiber stretch
elif self.growth_trig == 'fibstretch':
trigger = reduc * self.fibstretch_e(self.kin.C(u_), theta_,
self.kin.fib_funcs[0])
else:
raise NameError("Unknown growth_trig!")
# growth function
ktheta = grfnc.grfnc1(trigger, threshold, self.gandrparams)
# growth residual
r_growth = theta_ - theta_old_ - ktheta * (trigger - threshold) * dt
# tangent
K_growth = diff(r_growth, theta_)
# increment
del_theta = -r_growth / K_growth
if rquant == 'res_del':
return r_growth, del_theta
elif rquant == 'ktheta':
return ktheta
elif rquant == 'tang':
return K_growth
else:
raise NameError("Unknown return quantity!")
def grfnc1(self, trigger, thres, params):
thetamax, thetamin = params['thetamax'], params['thetamin']
tau_gr, tau_gr_rev = params['tau_gr'], params['tau_gr_rev']
gamma_gr, gamma_gr_rev = params['gamma_gr'], params['gamma_gr_rev']
k_plus = (1./tau_gr) * ((thetamax-self.theta)/(thetamax-thetamin))**(gamma_gr)
k_minus = (1./tau_gr_rev) * ((self.theta-thetamin)/(thetamax-thetamin))**(gamma_gr_rev)
k = conditional(ge(trigger,thres), k_plus, k_minus)
return k
def F(self, v, s, time=None):
"""
Right hand side for ODE system
"""
time = time if time else Constant(0.0)
# Assign states
V = v
assert (len(s) == 2)
v, w = s
# Assign parameters
u_c = self._parameters["u_c"]
u_v = self._parameters["u_v"]
tau_v1_minus = self._parameters["tau_v1_minus"]
tau_v2_minus = self._parameters["tau_v2_minus"]
tau_v_plus = self._parameters["tau_v_plus"]
tau_w_minus = self._parameters["tau_w_minus"]
tau_w_plus = self._parameters["tau_w_plus"]
V_0 = self._parameters["V_0"]
V_fi = self._parameters["V_fi"]
# Init return args
F_expressions = [ufl.zero()] * 2
# Expressions for the p component
p = ufl.conditional(ufl.lt((-V_0 + V) / (V_fi - V_0), u_c), 0, 1)
# Expressions for the q component
q = ufl.conditional(ufl.lt((-V_0 + V) / (V_fi - V_0), u_v), 0, 1)
# Expressions for the v gate component
tau_v_minus = tau_v1_minus * q + tau_v2_minus * (1 - q)
F_expressions[0] = (1 - p) * (1 - v) / tau_v_minus - p * v / tau_v_plus
# Expressions for the w gate component
F_expressions[1] = (1 - p) * (1 - w) / tau_w_minus - p * w / tau_w_plus
# Return results
return dolfin.as_vector(F_expressions)
def smooth(up_u_adv, up_u, o):
# Calculate averaged velocities
# Define the indicator function of the turbine support
chi = ufl.conditional(ufl.gt(tf, 0), 1, 0)
c_diff = Constant(1e6)
F1 = chi * (inner(up_u - up_u_adv, o) + c_diff * inner(grad(up_u), grad(o))) * dx
invchi = 1 - chi
F2 = inner(invchi * up_u, o) * dx
F = F1 + F2
return F
def eig(A):
"""Eigenvalues of 3x3 tensor"""
eps = 1.0e-12
q = ufl.tr(A) / 3.0
p1 = 0.5 * (A[0, 1]**2 + A[1, 0]**2 + A[0, 2]**2 + A[2, 0]**2 +
A[1, 2]**2 + A[2, 1]**2)
p2 = (A[0, 0] - q)**2 + (A[1, 1] - q)**2 + (A[2, 2] - q)**2 + 2 * p1
p = ufl.sqrt(p2 / 6)
B = (A - q * ufl.Identity(3))
r = ufl.det(B) / (2 * p**3)
r = ufl.Max(ufl.Min(r, 1.0 - eps), -1.0 + eps)
phi = ufl.acos(r) / 3.0
eig0 = ufl.conditional(p2 < eps, q, q + 2 * p * ufl.cos(phi))
eig2 = ufl.conditional(p2 < eps, q,
q + 2 * p * ufl.cos(phi + (2 * numpy.pi / 3)))
eig1 = ufl.conditional(p2 < eps, q, 3 * q - eig0 -
eig2) # since trace(A) = eig1 + eig2 + eig3
return eig0, eig1, eig2
def __init__(self, spline, ufl_element):
self._ufl_coef = ufl.Coefficient(ufl_element)
t = self._ufl_coef
knots = spline.knots
coef_array = spline.coef_array
# assigning left polynomial
form = ufl.conditional(lt(t, knots[0]), 1., 0.)\
* Polynomial(coef_array[0])(t)
# assigning internal polynomials
for knot, knot_, coef in\
zip(knots[:-1], knots[1:], coef_array[1:-1]):
form += ufl.conditional(And(ge(t, knot), lt(t, knot_)), 1., 0.)\
* Polynomial(coef)(t)
# assigning right polynomial
form += ufl.conditional(ge(t, knots[-1]), 1., 0.)\
* Polynomial(coef_array[-1])(t)
self._ufl_form = form
def _I(self, v, s, time):
"""
Original gotran transmembrane current dV/dt
"""
time = time if time else Constant(0.0)
# Assign states
V = v
assert(len(s) == 7)
m, h, j, Cai, d, f, x1 = s
# Assign parameters
E_Na = self._parameters["E_Na"]
g_Na = self._parameters["g_Na"]
g_Nac = self._parameters["g_Nac"]
g_s = self._parameters["g_s"]
IstimAmplitude = self._parameters["IstimAmplitude"]
IstimPulseDuration = self._parameters["IstimPulseDuration"]
IstimStart = self._parameters["IstimStart"]
C = self._parameters["C"]
# Init return args
current = [ufl.zero()]*1
# Expressions for the Sodium current component
i_Na = (g_Nac + g_Na*(m*m*m)*h*j)*(-E_Na + V)
# Expressions for the Slow inward current component
E_s = -82.3 - 13.0287*ufl.ln(0.001*Cai)
i_s = g_s*(-E_s + V)*d*f
# Expressions for the Time dependent outward current component
i_x1 = 0.00197277571153*(-1 +\
21.7584023962*ufl.exp(0.04*V))*ufl.exp(-0.04*V)*x1
# Expressions for the Time independent outward current component
i_K1 = 0.0035*(-4 +\
119.85640019*ufl.exp(0.04*V))/(8.33113748769*ufl.exp(0.04*V) +\
69.4078518388*ufl.exp(0.08*V)) + 0.0035*(4.6 + 0.2*V)/(1 -\
0.398519041085*ufl.exp(-0.04*V))
# Expressions for the Stimulus protocol component
Istim = ufl.conditional(ufl.And(ufl.ge(time, IstimStart),\
ufl.le(time, IstimPulseDuration + IstimStart)), IstimAmplitude,\
0)
# Expressions for the Membrane component
current[0] = (-i_K1 + Istim - i_Na - i_x1 - i_s)/C
# Return results
return current[0]
def _I(self, v, s, time):
"""
Original gotran transmembrane current dV/dt
"""
time = time if time else Constant(0.0)
# Assign states
V = v
# Assign parameters
g_leak = self._parameters["g_L"]
Cm = self._parameters["Cm"]
E_leak = self._parameters["E_L"]
stim_type = self._parameters["stim_type"]
g_s = self._parameters["g_S"]
alpha = self._parameters["alpha"]
v_eq = self._parameters["v_eq"]
t0 = self._parameters["t_start"]
t1 = self._parameters["t_stop"]
# Init return args
current = [ufl.zero()] * 1
# Expressions for the Membrane component
if stim_type == 0:
i_stim = g_s * (-v_eq + V) * ufl.conditional(
ufl.ge(time, t0), 1, 0) * ufl.exp((t0 - time) / alpha)
elif stim_type == 1:
i_stim = -g_s * ufl.conditional(ufl.ge(time, t0), 1, 0)
elif stim_type == 2:
i_stim = -g_s * ufl.conditional(
ufl.And(ufl.ge(time, t0), ufl.le(time, t1)), 1, 0)
i_leak = g_leak * (-E_leak + V)
current[0] = (-i_leak - i_stim) / Cm
# Return results
return current[0]
def smooth(u, up_u, o):
# Calculate averaged velocities
# Define the indicator function of the turbine support
chi = ufl.conditional(ufl.gt(tf, 0), 1, 0)
# Solve the Helmholtz equation in each turbine area to obtain averaged velocity values
c_diff = Constant(1e6)
F1 = chi * (inner(up_u - norm_approx(u), o) + Constant(distance_to_upstream) / norm_approx(u) * (inner(dot(grad(norm_approx(u)), u), o) + c_diff * inner(grad(up_u), grad(o)))) * dx
invchi = 1 - chi
F2 = inner(invchi * up_u, o) * dx
F = F1 + F2
return F
def smooth(up_u_adv, up_u, o):
# Calculate averaged velocities
# Define the indicator function of the turbine support
chi = ufl.conditional(ufl.gt(tf, 0), 1, 0)
c_diff = Constant(1e6)
F1 = chi * (inner(up_u - up_u_adv, o) +
c_diff * inner(grad(up_u), grad(o))) * dx
invchi = 1 - chi
F2 = inner(invchi * up_u, o) * dx
F = F1 + F2
return F
def _I(self, v, s, time):
"""
Original gotran transmembrane current dV/dt
"""
time = time if time else Constant(0.0)
# Assign states
V = v
assert (len(s) == 2)
v, w = s
# Assign parameters
u_c = self._parameters["u_c"]
g_fi_max = self._parameters["g_fi_max"]
tau_0 = self._parameters["tau_0"]
tau_r = self._parameters["tau_r"]
k = self._parameters["k"]
tau_si = self._parameters["tau_si"]
u_csi = self._parameters["u_csi"]
Cm = self._parameters["Cm"]
V_0 = self._parameters["V_0"]
V_fi = self._parameters["V_fi"]
# Init return args
current = [ufl.zero()] * 1
# Expressions for the p component
p = ufl.conditional(ufl.lt((-V_0 + V) / (V_fi - V_0), u_c), 0, 1)
# Expressions for the Fast inward current component
tau_d = Cm / g_fi_max
J_fi = -(1 - (-V_0 + V)/(V_fi - V_0))*(-u_c + (-V_0 + V)/(V_fi -\
V_0))*p*v/tau_d
# Expressions for the Slow outward current component
J_so = p / tau_r + (1 - p) * (-V_0 + V) / (tau_0 * (V_fi - V_0))
# Expressions for the Slow inward current component
J_si = -(1 + ufl.tanh(k*(-u_csi + (-V_0 + V)/(V_fi -\
V_0))))*w/(2*tau_si)
# Expressions for the Stimulus protocol component
J_stim = 0
# Expressions for the Membrane component
current[0] = (V_0 - V_fi) * (J_stim + J_fi + J_si + J_so)
# Return results
return current[0]
def velocity(
theta_k, theta_kp1,
dt,
liquidus, solidus,
velocity_max=0.,
filter_solidification=True,
mean=False,
filter_negative=True,
inverse_sign=True,
x=None,
):
'''Provides the UFL form of the velocity function.
Parameters:
theta_k: dolfin.Function(V)
Function representing the state from the current time slice.
theta_kp1: dolfin.Function(V)
Function representing the state from the next time slice.
velocity_max: float
The maximal allowed velocity which will not be penalized.
implicitness: float
The weight of theta_kp1 in the expression. Normally, should stay
default.
implicitness: float
Returns:
form: UFL form
The UFL form of theta_k and theta_kp1.
'''
velocity_ = velocity_expression(theta_k, theta_kp1, dt) \
+ dolfin.Constant(velocity_max)
if filter_solidification:
ind = solidification_indicator(theta_k, theta_kp1, solidus, liquidus)
velocity_ *= ind
if mean:
area = dolfin.assemble(integral(ind, x))
if area > 0:
velocity_ /= area
if filter_negative:
velocity_ *= conditional(le(velocity_, 0.), 1., 0.)
if inverse_sign:
velocity_ *= -1
return velocity_
def smooth(u, up_u, o):
# Calculate averaged velocities
# Define the indicator function of the turbine support
chi = ufl.conditional(ufl.gt(tf, 0), 1, 0)
# Solve the Helmholtz equation in each turbine area to obtain averaged velocity values
c_diff = Constant(1e6)
F1 = chi * (inner(up_u - norm_approx(u), o) +
Constant(distance_to_upstream) / norm_approx(u) *
(inner(dot(grad(norm_approx(u)), u), o) +
c_diff * inner(grad(up_u), grad(o)))) * dx
invchi = 1 - chi
F2 = inner(invchi * up_u, o) * dx
F = F1 + F2
return F
def test_latex_formatting_of_variables():
x, y = ufl.SpatialCoordinate(ufl.triangle)
# Test user-provided C variables for subexpressions
# we can use variables for x[0], and sum, and power
assert expr2latex(x**2 + y**2, variables={x**2: 'x2', y**2: 'y2'}) == "x2 + y2"
assert expr2latex(x**2 + y**2, variables={x: 'z', y**2: 'y2'}) == r"{z}^{2} + y2"
assert expr2latex(x**2 + y**2, variables={x**2 + y**2: 'q'}) == "q"
# we can use variables in conditionals
if 0:
assert expr2latex(ufl.conditional(ufl.Or(ufl.eq(x, 2), ufl.ne(y, 4)), 7, 8), variables={ufl.eq(x, 2): 'c1', ufl.ne(y, 4): 'c2'}) == "c1 || c2 ? 7: 8"
# we can replace coefficients (formatted by user provided code)
V = ufl.FiniteElement("CG", ufl.triangle, 1)
f = ufl.Coefficient(V, count=0)
assert expr2latex(f, variables={f: 'f'}) == "f"
assert expr2latex(f**3, variables={f: 'f'}) == r"{f}^{3}"
# variables do not replace derivatives of variable expressions
assert expr2latex(f.dx(0), variables={f: 'f'}) == r"\overset{0}{w}_{, 0}"
# variables do replace variable expressions that are themselves derivatives
assert expr2latex(f.dx(0), variables={f.dx(0): 'df', ufl.grad(f)[0]: 'df'}) == "df"
assert expr2latex(ufl.grad(f)[0], variables={f.dx(0): 'df', ufl.grad(f)[0]: 'df'}) == "df"
def _I(self, v, s, time):
"""
Original gotran transmembrane current dV/dt
"""
time = time if time else Constant(0.0)
# Assign states
V = v
assert (len(s) == 2)
s, m = s
# Assign parameters
Cm = self._parameters["Cm"]
E_L = self._parameters["E_L"]
g_L = self._parameters["g_L"]
# synapse components
alpha = self._parameters["alpha"]
g_S = self._parameters["g_S"]
t0 = self._parameters["t0"]
v_eq = self._parameters["v_eq"]
# Init return args
current = [ufl.zero()] * 1
# Expressions for the Membrane component
# FIXME: base on stim_type + add to Hodgkin
# if
# i_Stim = g_S*(-v_eq + V)*ufl.conditional(ufl.ge(time, t0), 1, 0)*ufl.exp((t0 - time)/alpha)
# elif sss:
# i_Stim = g_S*ufl.conditional(ufl.ge(time, t0), 1, 0)
# else:
# i_Stim = g_S*ufl.conditional(ufl.And(ufl.ge(time, t0),
# ufl.le(time, t1), 1, 0))
i_Stim = g_S * (-v_eq + V) * ufl.conditional(ufl.ge(
time, t0), 1, 0) * ufl.exp((t0 - time) / alpha)
i_L = g_L * (-E_L + V)
current[0] = (-i_L - i_Stim) / Cm
# Return results
return current[0]
def beta_cr(dt):
"""Beta creep coefficients
Parameters
----------
dt: Timestep size [day]
Returns
-------
NumPy array of beta creep coefficients [-]
Note
----
The Book, page 160, formula (5.36)
"""
beta = []
for ta in tau:
beta.append(ufl.conditional(dt / ta > 30.0, 0.0, ufl.exp(-dt / ta)))
return beta
def test_cpp_formatting_with_variables():
x, y = ufl.SpatialCoordinate(ufl.triangle)
# Test user-provided C variables for subexpressions
# we can use variables for x[0], and sum, and power
assert expr2cpp(x**2 + y**2, variables={x**2: 'x2', y**2: 'y2'}) == "x2 + y2"
assert expr2cpp(x**2 + y**2, variables={x: 'z', y**2: 'y2'}) == "pow(z, 2) + y2"
assert expr2cpp(x**2 + y**2, variables={x**2 + y**2: 'q'}) == "q"
# we can use variables in conditionals
assert expr2cpp(ufl.conditional(ufl.Or(ufl.eq(x, 2), ufl.ne(y, 4)), 7, 8),
variables={ufl.eq(x, 2): 'c1', ufl.ne(y, 4): 'c2'}) == "c1 || c2 ? 7: 8"
# we can replace coefficients (formatted by user provided code)
V = ufl.FiniteElement("CG", ufl.triangle, 1)
f = ufl.Coefficient(V, count=0)
assert expr2cpp(f, variables={f: 'f'}) == "f"
assert expr2cpp(f**3, variables={f: 'f'}) == "pow(f, 3)"
# variables do not replace derivatives of variable expressions
assert expr2cpp(f.dx(0), variables={f: 'f'}) == "d1_w0[0]"
# This test depends on which differentiation algorithm is in use
# in UFL, representing derivatives as SpatialDerivative or Grad:
# variables do replace variable expressions that are themselves derivatives
assert expr2cpp(f.dx(0), variables={f.dx(0): 'df', ufl.grad(f)[0]: 'df'}) == "df"
assert expr2cpp(ufl.grad(f)[0], variables={f.dx(0): 'df', ufl.grad(f)[0]: 'df'}) == "df"
def _I(self, v, s, time):
"""
Original gotran transmembrane current dV/dt
"""
time = time if time else Constant(0.0)
# Assign states
V = v
assert(len(s) == 2)
m, h = s
# Assign parameters
amp = self._parameters["amp"]
duration = self._parameters["duration"]
stimStart = self._parameters["stimStart"]
E_Na = self._parameters["E_Na"]
g_Na = self._parameters["g_Na"]
E_K = self._parameters["E_K"]
g_K = self._parameters["g_K"]
k_r = self._parameters["k_r"]
# Init return args
current = [ufl.zero()]*1
# Expressions for the Fast sodium current component
i_Na = g_Na*ufl.elem_pow(m, 3.0)*(-E_Na + V)*h
# Expressions for the Potassium current component
i_K = g_K*(-E_K + V)*ufl.exp((E_K - V)/k_r)
# Expressions for the Membrane component
i_stim = ufl.conditional(ufl.And(ufl.gt(time, stimStart),\
ufl.le(time, duration + stimStart)), amp, 0)
current[0] = -i_K - i_Na - i_stim
# Return results
return current[0]
def SubDomainData(geometric_expr):
"""Creates a subdomain data object from a boolean-valued UFL expression.
The result can be attached as the subdomain_data field of a
:class:`ufl.Measure`. For example:
x = mesh.coordinates
sd = SubDomainData(x[0] < 0.5)
assemble(f*dx(subdomain_data=sd))
"""
import firedrake.functionspace as functionspace
import firedrake.projection as projection
# Find domain from expression
m = geometric_expr.ufl_domain()
# Find selected cells
fs = functionspace.FunctionSpace(m, 'DG', 0)
f = projection.project(ufl.conditional(geometric_expr, 1, 0), fs)
# Create cell subset
indices, = np.nonzero(f.dat.data_ro_with_halos > 0.5)
return op2.Subset(m.cell_set, indices)
def __define_boundary_terms(self):
if self.verb > 2: print0("Defining boundary terms")
self.bnd_vector_form = numpy.empty(self.DD.G, dtype=object)
self.bnd_matrix_form = numpy.empty(self.DD.G, dtype=object)
n = FacetNormal(self.DD.mesh)
i,p,q = ufl.indices(3)
natural_boundaries = self.BC.vacuum_boundaries.union(self.BC.incoming_fluxes.keys())
nonzero = lambda x: numpy.all(x > 0)
nonzero_inc_flux = any(map(nonzero, self.BC.incoming_fluxes.values()))
if nonzero_inc_flux and not self.fixed_source_problem:
coupled_solver_error(__file__,
"define boundary terms",
"Incoming flux specified for an eigenvalue problem"+\
"(Q must be given whenever phi_inc is; it may possibly be zero everywhere)")
for g in range(self.DD.G):
self.bnd_vector_form[g] = ufl.zero()
self.bnd_matrix_form[g] = ufl.zero()
if natural_boundaries:
try:
ds = Measure("ds")[self.DD.boundaries]
except TypeError:
coupled_solver_error(__file__,
"define boundary terms",
"File assigning boundary indices to facets required if vacuum or incoming flux boundaries "
"are specified.")
for bnd_idx in natural_boundaries:
# NOTE: The following doesn't work because ufl.abs requires a function
#
#for g in range(self.DD.G):
# self.bnd_matrix_form[g] += \
# abs(self.tensors.G[p,q,i]*n[i])*self.u[g][q]*self.v[g][p]*ds(bnd_idx)
#
# NOTE: Instead, the following explicit loop has to be used; this makes tensors.G unneccessary
#
for pp in range(self.DD.M):
omega_p_dot_n = self.DD.ordinates_matrix[i,pp]*n[i]
for g in range(self.DD.G):
self.bnd_matrix_form[g] += \
abs(omega_p_dot_n)*self.tensors.Wp[pp]*self.u[g][pp]*self.v[g][pp]*ds(bnd_idx)
if nonzero_inc_flux:
for pp in range(self.DD.M):
omega_p_dot_n = self.DD.ordinates_matrix[i,pp]*n[i]
for bnd_idx, psi_inc in self.BC.incoming_fluxes.iteritems():
if psi_inc.shape != (self.DD.M, self.DD.G):
coupled_solver_error(__file__,
"define boundary terms",
"Incoming flux with incorrect number of groups and directions specified: "+
"{}, expected ({}, {})".format(psi_inc.shape, self.DD.M, self.DD.G))
for g in range(self.DD.G):
self.bnd_vector_form[g] += \
ufl.conditional(omega_p_dot_n < 0,
omega_p_dot_n*self.tensors.Wp[pp]*psi_inc[pp,g]*self.v[g][pp]*ds(bnd_idx),
ufl.zero()) # FIXME: This assumes zero adjoint outgoing flux
else: # Apply vacuum b.c. everywhere
ds = Measure("ds")
for pp in range(self.DD.M):
omega_p_dot_n = self.DD.ordinates_matrix[i,pp]*n[i]
for g in range(self.DD.G):
self.bnd_matrix_form[g] += abs(omega_p_dot_n)*self.tensors.Wp[pp]*self.u[g][pp]*self.v[g][pp]*ds()
def xtest_latex_formatting_of_conditionals():
# Test conditional expressions
assert expr2latex(ufl.conditional(ufl.lt(x, 2), y, 3)) == "x_0 < 2 ? x_1: 3"
assert expr2latex(ufl.conditional(ufl.gt(x, 2), 4 + y, 3)) == "x_0 > 2 ? 4 + x_1: 3"
assert expr2latex(ufl.conditional(ufl.And(ufl.le(x, 2), ufl.ge(y, 4)), 7, 8)) == "x_0 <= 2 && x_1 >= 4 ? 7: 8"
assert expr2latex(ufl.conditional(ufl.Or(ufl.eq(x, 2), ufl.ne(y, 4)), 7, 8)) == "x_0 == 2 || x_1 != 4 ? 7: 8"
tf = project(Expression("sin(2*pi*x[0])"), V)
def norm_approx(u, alpha=1e-4):
# A smooth approximation to ||u||
return sqrt(inner(u, u)+alpha**2)
# Advect
shift = 0.1
theta = 0.5
uhalf = Constant(1.-theta)*u0 + Constant(theta)*u
F = (inner(u-u0, q) + Constant(shift)*inner(uhalf.dx(0)*1, q) + Constant(1e-8)*inner(grad(u), grad(q)))*dx
solve(F == 0, u)
plot(u, interactive=True, title="Computed after shifting")
u_shift = project(Expression("1+sin(2*pi*(x[0]-shift))", shift=shift), V)
plot(u_shift, interactive=True, title="Expected after shifting")
# Diffuse
a = Function(V)
chi = ufl.conditional(ufl.ge(tf, 0.0), 0, 1)
F1 = chi*(inner(a-u0, q) + Constant(1e3)*inner(grad(a), grad(q)))*dx
invchi = 1-chi
F2 = inner(invchi*a, q)*dx
F = F1 + F2
solve(F == 0, a)
adg = interpolate(a, Vdg)
f = File("averaged.pvd")
f << adg
plot(adg, interactive=True, title="Averaged")