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) m, h = s # Assign parameters E_h = self._parameters["E_h"] E_m = self._parameters["E_m"] delta_h = self._parameters["delta_h"] k_h = self._parameters["k_h"] k_m = self._parameters["k_m"] tau_h0 = self._parameters["tau_h0"] tau_m = self._parameters["tau_m"] # Init return args F_expressions = [ufl.zero()]*2 # Expressions for the m gate component m_inf = 1.0/(1.0 + ufl.exp((E_m - V)/k_m)) F_expressions[0] = (-m + m_inf)/tau_m # Expressions for the h gate component h_inf = 1.0/(1.0 + ufl.exp((-E_h + V)/k_h)) tau_h = 2*tau_h0*ufl.exp(delta_h*(-E_h + V)/k_h)/(1 + ufl.exp((-E_h +\ V)/k_h)) F_expressions[1] = (-h + h_inf)/tau_h # Return results return dolfin.as_vector(F_expressions)
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 I(self, v, s, time=None): """ Transmembrane current """ # Imports # No imports for now time = time if time else Constant(0.0) # Assign states _is_vector = False # Assign parameters a = self._parameters["a"] stim_start = self._parameters["stim_start"] stim_amplitude = self._parameters["stim_amplitude"] c_1 = self._parameters["c_1"] c_2 = self._parameters["c_2"] v_rest = self._parameters["v_rest"] stim_duration = self._parameters["stim_duration"] v_peak = self._parameters["v_peak"] current = -(v - v_rest)*(v_peak - v)*(-(v_peak - v_rest)*a + v -\ v_rest)*c_1/((v_peak - v_rest)*(v_peak - v_rest)) + (v -\ v_rest)*c_2*s/(v_peak - v_rest) - (1.0 - 1.0/(1.0 +\ ufl.exp(-5.0*stim_start + 5.0*time)))*stim_amplitude/(1.0 +\ ufl.exp(-5.0*stim_start + 5.0*time - 5.0*stim_duration)) return current
def eval(self, values, x): u = self.omega*self.t - 3*pow(6,0.5) if abs(x[0] - self.center) < 0.25: Tp = ((0.25*pow(u,2) - 0.5)*exp(-0.25*pow(u,2)) - 13*exp(-13.5))/(0.5 + 13*exp(-13.5)) values[0] = 0.0 values[1] = 5000*Tp # [KPa] else: values[0] = 0.0 values[1] = 0.0 #5000*Tp
def eval(self, values, x): # Ricker pulse Parameters r = ((x[0] - xc)**2 + (x[1] - yc)**2)**0.5 u = self.omega*self.t - 3*pow(6,0.5) Sp = (1.0 - (r / rd) ** 2) ** 3 #if self.t <= 6 * pow(6, 0.5) / self.omega: Tp = ((0.25 * pow(u, 2) - 0.5) * exp(-0.25 * pow(u, 2)) \ -13 * exp(-13.5)) / (0.5 + 13 * exp(-13.5)) #else: # Tp = 0 values[0] = (5e4)*physical_parameters.amplitude * Tp * Sp * (x[0] - xc) / r values[1] = (5e4)*physical_parameters.amplitude * Tp * Sp * (x[1] - yc) / r
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 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 eval(self, values, x): if abs(x[0] - self.center) < 0.25: values[0] = 0 values[1] = 5000 * (1 - 0.5 * self.omega ** 2 * self.t ** 2) * exp( - 0.25 * self.omega ** 2 * self.t ** 2) else: values[0] = 0.0 values[1] = 0.0 #5000*Tp
def hs( phi: fd.Function, epsilon: fd.Constant = fd.Constant(10000.0), width_h: float = None, shift: float = 0.0, min_value: fd.Function = fd.Constant(0.0), ): """Heaviside approximation Args: phi (fd.Function): Level set epsilon ([type], optional): Parameter to approximate the Heaviside. Defaults to Constant(10000.0). width_h (float): Width of the Heaviside approximation transition in terms of multiple of the mesh element size shift: (float): Shift the level set value to define the interface. Returns: [type]: [description] """ if width_h: if epsilon: fd.warning( "Epsilon and width_h are both defined, pick one or the other. \ Overriding epsilon choice") mesh = phi.ufl_domain() hmin = min_mesh_size(mesh) epsilon = fd.Constant(math.log(0.99**2 / 0.01**2) / (width_h * hmin)) return (fd.Constant(1.0) / (fd.Constant(1.0) + ufl.exp(-epsilon * (phi - fd.Constant(shift)))) + min_value)
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 eta_dash(eta_dash0, dt, temp, hum): """Ageing viscosity of dashpot element Parameters ---------- eta_dash0: Previous viscosity [MPa day] dt: Timestep [days] temp: Temperature [K] hum: Rel. humidity [-] Returns ------- Ageing viscosity [MPa day] Note ---- The Book, page 470, 471, 477, formulas (10.33, 10.36, 10.39, 10.60) """ beta_sT = ufl.exp(3000 * (1.0 / room_temp - 1.0 / temp)) alpha_s = 0.1 beta_sh = alpha_s + (1 - alpha_s) * hum**2 psi_s = beta_sT * beta_sh return eta_dash0 + dt * psi_s / MPS_q4 * 1.0e-6
def guccione_dev(self, params, f0, s0, C): n0 = cross(f0, s0) # anisotropic invariants - keep in mind that for growth, self.E is the elastic part of E E_ff = dot(dot(self.E, f0), f0) # fiber GL strain E_ss = dot(dot(self.E, s0), s0) # cross-fiber GL strain E_nn = dot(dot(self.E, n0), n0) # radial GL strain E_fs = dot(dot(self.E, f0), s0) E_fn = dot(dot(self.E, f0), n0) E_sn = dot(dot(self.E, s0), n0) c_0 = params['c_0'] b_f = params['b_f'] b_t = params['b_t'] b_fs = params['b_fs'] Q = b_f * E_ff**2. + b_t * (E_ss**2. + E_nn**2. + 2. * E_sn**2. ) + b_fs * (2. * E_fs**2. + 2. * E_fn**2.) psi_dev = 0.5 * c_0 * (exp(Q) - 1.) S = 2. * diff(psi_dev, C) return S
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 f4(temp): """Arrhenius activation term Note ---- Saetta 1993, page 763, formula inlined """ return A * ufl.exp(- E0 / (R * temp))
def test_dolfin_expression_compilation_of_math_functions(dolfin): # Define some PyDOLFIN coefficients mesh = dolfin.UnitSquareMesh(3, 3) # Using quadratic element deliberately for accuracy V = dolfin.FunctionSpace(mesh, "CG", 2) u = dolfin.Function(V) u.interpolate(dolfin.Expression("x[0]*x[1]")) w0 = u # Define ufl expression with math functions v = abs(ufl.cos(u))/2 + 0.02 uexpr = ufl.sin(u) + ufl.tan(v) + ufl.exp(u) + ufl.ln(v) + ufl.atan(v) + ufl.acos(v) + ufl.asin(v) #print dolfin.assemble(uexpr**2*dolfin.dx, mesh=mesh) # 11.7846508409 # Define expected output from compilation ucode = 'v_w0[0]' vcode = '0.02 + fabs(cos(v_w0[0])) / 2' funcs = 'asin(%(v)s) + (acos(%(v)s) + (atan(%(v)s) + (log(%(v)s) + (exp(%(u)s) + (sin(%(u)s) + tan(%(v)s))))))' oneliner = funcs % {'u':ucode, 'v':vcode} # Oneliner version (ignoring reuse): expected_lines = ['double s[1];', 'Array<double> v_w0(1);', 'w0->eval(v_w0, x);', 's[0] = %s;' % oneliner, 'values[0] = s[0];'] #cppcode = format_dolfin_expression(classname="DebugExpression", shape=(), eval_body=expected_lines) #print '-'*100 #print cppcode #print '-'*100 #dolfin.plot(dolfin.Expression(cppcode=cppcode, mesh=mesh)) #dolfin.interactive() # Split version (handles reuse of v, no other reuse): expected_lines = ['double s[2];', 'Array<double> v_w0(1);', 'w0->eval(v_w0, x);', 's[0] = %s;' % (vcode,), 's[1] = %s;' % (funcs % {'u':ucode,'v':'s[0]'},), 'values[0] = s[1];'] # Define expected evaluation values: [(x,value), (x,value), ...] import math x, y = 0.6, 0.7 u = x*y v = abs(math.cos(u))/2 + 0.02 v0 = .52 expected0 = math.tan(v0) + 1 + math.log(v0) + math.atan(v0) + math.acos(v0) + math.asin(v0) expected = math.sin(u) + math.tan(v) + math.exp(u) + math.log(v) + math.atan(v) + math.acos(v) + math.asin(v) expected_values = [((0.0, 0.0), (expected0,)), ((x, y), (expected,)), ] # Execute all tests check_dolfin_expression_compilation(uexpr, expected_lines, expected_values, members={'w0':w0})
def assembleWmm(self, x): """ Assemble the derivative of the parameter equation with respect to the parameter (Newton method) """ trial = dl.TrialFunction(self.Vh[PARAMETER]) test = dl.TestFunction(self.Vh[PARAMETER]) u = vector2Function(x[STATE], self.Vh[STATE]) m = vector2Function(x[PARAMETER], self.Vh[PARAMETER]) p = vector2Function(x[ADJOINT], self.Vh[ADJOINT]) varf = ufl.inner(ufl.exp(m)*sigma(u),strain(p))*test*trial*ufl.dx return dl.assemble(varf)
def test_complex_algebra(self): z1 = ComplexValue(1j) z2 = ComplexValue(1+1j) # Remember that ufl.algebra functions return ComplexValues, but ufl.mathfunctions return complex Python scalar # Any operations with a ComplexValue and a complex Python scalar promote to ComplexValue assert z1*z2 == ComplexValue(-1+1j) assert z2/z1 == ComplexValue(1-1j) assert pow(z2, z1) == ComplexValue((1+1j)**1j) assert sqrt(z2) * as_ufl(1) == ComplexValue(cmath.sqrt(1+1j)) assert ((sin(z2) + cosh(z2) - atan(z2)) * z1) == ComplexValue((cmath.sin(1+1j) + cmath.cosh(1+1j) - cmath.atan(1+1j))*1j) assert (abs(z2) - ln(z2))/exp(z1) == ComplexValue((abs(1+1j) - cmath.log(1+1j))/cmath.exp(1j))
def test_latex_formatting_of_cmath(): x = ufl.SpatialCoordinate(ufl.triangle)[0] assert expr2latex(ufl.exp(x)) == r"e^{x_0}" assert expr2latex(ufl.ln(x)) == r"\ln(x_0)" assert expr2latex(ufl.sqrt(x)) == r"\sqrt{x_0}" assert expr2latex(abs(x)) == r"\|x_0\|" assert expr2latex(ufl.sin(x)) == r"\sin(x_0)" assert expr2latex(ufl.cos(x)) == r"\cos(x_0)" assert expr2latex(ufl.tan(x)) == r"\tan(x_0)" assert expr2latex(ufl.asin(x)) == r"\arcsin(x_0)" assert expr2latex(ufl.acos(x)) == r"\arccos(x_0)" assert expr2latex(ufl.atan(x)) == r"\arctan(x_0)"
def nn(eps, u, p, v, q): #return inner(grad(p), grad(q)) * dx def sigma_(vec, func=ufl.tanh): v = [func(vec[i]) for i in range(vec.ufl_shape[0])] return ufl.as_vector(v) relu = lambda vec: conditional(ufl.gt(vec, 0), vec, (ufl.exp(vec) - 1)) sigma = lambda vec: sigma_(vec, func=relu) nn = dot(W_3, sigma(ufl.transpose(as_vector([W_1, W_2])) * eps * grad(p) + b_1)) + b_2 return inner(nn, grad(q))*dx, inner(nn, nn)*dx
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) == 3) m, h, n = s # Assign parameters # Init return args F_expressions = [ufl.zero()] * 3 # Expressions for the m gate component alpha_m = (-5.0 - 0.1 * V) / (-1.0 + ufl.exp(-5.0 - V / 10.0)) beta_m = 4 * ufl.exp(-25.0 / 6.0 - V / 18.0) F_expressions[0] = (1 - m) * alpha_m - beta_m * m # Expressions for the h gate component alpha_h = 0.07 * ufl.exp(-15.0 / 4.0 - V / 20.0) beta_h = 1.0 / (1 + ufl.exp(-9.0 / 2.0 - V / 10.0)) F_expressions[1] = (1 - h) * alpha_h - beta_h * h # Expressions for the n gate component alpha_n = (-0.65 - 0.01 * V) / (-1.0 + ufl.exp(-13.0 / 2.0 - V / 10.0)) beta_n = 0.125 * ufl.exp(-15.0 / 16.0 - V / 80.0) F_expressions[2] = (1 - n) * alpha_n - beta_n * n # Return results return dolfin.as_vector(F_expressions)
def test_cpp_formatting_of_cmath(): x, y = ufl.SpatialCoordinate(ufl.triangle) # Test cmath functions assert expr2cpp(ufl.exp(x)) == "exp(x[0])" assert expr2cpp(ufl.ln(x)) == "log(x[0])" assert expr2cpp(ufl.sqrt(x)) == "sqrt(x[0])" assert expr2cpp(abs(x)) == "fabs(x[0])" assert expr2cpp(ufl.sin(x)) == "sin(x[0])" assert expr2cpp(ufl.cos(x)) == "cos(x[0])" assert expr2cpp(ufl.tan(x)) == "tan(x[0])" assert expr2cpp(ufl.asin(x)) == "asin(x[0])" assert expr2cpp(ufl.acos(x)) == "acos(x[0])" assert expr2cpp(ufl.atan(x)) == "atan(x[0])"
def assembleC(self, x): """ Assemble the derivative of the forward problem with respect to the parameter """ trial = dl.TrialFunction(self.Vh[PARAMETER]) test = dl.TestFunction(self.Vh[STATE]) u = vector2Function(x[STATE], Vh[STATE]) m = vector2Function(x[PARAMETER], Vh[PARAMETER]) Cvarf = ufl.inner(ufl.exp(m)*trial*sigma(u), strain(test))*ufl.dx C = dl.assemble(Cvarf) # print ( "||m||", x[PARAMETER].norm("l2"), "||u||", x[STATE].norm("l2"), "||C||", C.norm("linf") ) self.bc0.zero(C) return C
def assembleWmu(self, x): """ Assemble the derivative of the parameter equation with respect to the state """ trial = dl.TrialFunction(self.Vh[STATE]) test = dl.TestFunction(self.Vh[PARAMETER]) p = vector2Function(x[ADJOINT], self.Vh[ADJOINT]) m = vector2Function(x[PARAMETER], self.Vh[PARAMETER]) varf = ufl.inner(ufl.exp(m)*sigma(trial), strain(p))*test*ufl.dx Wmu = dl.assemble(varf) Wmu_t = Transpose(Wmu) self.bc0.zero(Wmu_t) Wmu = Transpose(Wmu_t) return Wmu
def nn(u, p, v, q): # return inner(grad(p), grad(q)) * dx, None, None def sigma_(vec, func=ufl.tanh): v = [func(vec[i]) for i in range(vec.ufl_shape[0])] return ufl.as_vector(v) relu = lambda vec: conditional(ufl.gt(vec, 0), vec, (ufl.exp(vec) - 1)) sigma = lambda vec: sigma_(vec, func=relu)#lambda x:x) nn_p = dot(W_3, sigma(ufl.transpose(as_vector([W_1, W_2])) * u + b_1)) + b_2 #nn_q = dot(W_3, sigma(ufl.transpose(as_vector([W_1, W_2])) * grad(q) + b_1)) + b_2 return inner(nn_p, v)*dx, inner(nn_p, nn_p)*dx, nn_p
def dirac_delta(phi: fd.Function, epsilon=fd.Constant(10000.0), width_h=None): """Dirac delta approximation Args: phi (fd.Function): Level set epsilon ([type], optional): Parameter to approximate the Heaviside. Defaults to Constant(10000.0). width_h (float): Width of the Heaviside approximation transition in terms of multiple of the mesh element size Returns: [type]: [description] """ if width_h: if epsilon: fd.warning( "Epsilon and width_h are both defined, pick one or the other. \ Overriding epsilon choice") mesh = phi.ufl_domain() hmin = min_mesh_size(mesh) epsilon = fd.Constant(math.log(0.95**2 / 0.05**2) / (width_h * hmin)) return (fd.Constant(epsilon) * ufl.exp(-epsilon * phi) / (fd.Constant(1.0) + ufl.exp(-epsilon * phi))**2)
def nn(u, v): inp = as_vector( [avg(u), jump(u), *grad(avg(u)), *grad(jump(u)), *n('+')]) def sigma_(vec, func=ufl.tanh): v = [func(vec[i]) for i in range(vec.ufl_shape[0])] return ufl.as_vector(v) relu = lambda vec: conditional(ufl.gt(vec, 0), vec, (ufl.exp(vec) - 1)) sigma = lambda vec: sigma_(vec, func=relu) nn = dot(W_2, sigma(ufl.transpose(as_vector(W_1)) * inp + b_1)) + b_2 return inner(nn, jump(v) + avg(v)) * dS, inner(nn, nn) * dS
def pfuncs(clargs, params0, t0=0.0): """Create functions for parameters Required arguments: clargs: Command line arguments. params0: a mappable of initial numerical values of parameters Optional parameter: t0=0.0: initial time (time at which params have valuve params0 Returns a dict mapping param names (from params0) to functions. Each function has the call signature func(t, params={}) and returns the value of the parameter at time t. """ import ufl from KSDG import ParameterList, KSDGException decays = ParameterList() decays.decode(clargs.decay, allow_new=True) slopes = ParameterList() slopes.decode(clargs.slope, allow_new=True) keys = set(decays.keys()) | set(slopes.keys()) extras = keys - set(params0.keys()) if extras: raise KSDGException(', '.join([k for k in extras]) + ': no such parameter') funcs = {} for k in params0.keys(): d = decays[k] if k in decays else 0.0 s = slopes[k] if k in slopes else 0.0 pt0 = params0[k] if k in params0 else 1.0 if d == 0 and s == 0: def func(t, params={}, p0=pt0): return p0 elif d == 0: p0 = pt0 - s * t0 def func(t, params={}, p0=pt0, s=s): return p0 + s * t else: a = (ufl.exp(d * t0) * (d * pt0 - s) - s) / d pinf = s / d def func(t, params={}, d=d, pinf=pinf, a=a): return pinf + a * ufl.exp(-d * t) funcs[k] = func return funcs
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 nn(u, v): #return inner(grad(p), grad(q)) * dx def sigma_(vec, func=ufl.tanh): v = [func(vec[i]) for i in range(vec.ufl_shape[0])] return ufl.as_vector(v) relu = lambda vec: conditional(ufl.gt(vec, 0), vec, (ufl.exp(vec) - 1)) sigma = lambda vec: sigma_(vec, func=relu) #lambda x:x) #from IPython import embed #embed() n1 = dot(W_2, sigma(W_1 * u) + b_1) n2 = dot(W_3_1, sigma(W_3_2 * u.dx(0)) + b_2) return inner(n1, v) * dx + inner(n2, v.dx(0)) * dx, inner( n1, n1) * dx + inner(n2, n2) * dx, n1
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 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 assembleA(self,x, assemble_adjoint = False, assemble_rhs = False): """ Assemble the matrices and rhs for the forward/adjoint problems """ trial = dl.TrialFunction(self.Vh[STATE]) test = dl.TestFunction(self.Vh[STATE]) m = vector2Function(x[PARAMETER], self.Vh[PARAMETER]) Avarf = ufl.inner(ufl.exp(m)*sigma(trial), strain(test))*ufl.dx if not assemble_adjoint: bform = ufl.inner(self.f, test)*ufl.dx Matrix, rhs = dl.assemble_system(Avarf, bform, self.bc) else: # Assemble the adjoint of A (i.e. the transpose of A) u = vector2Function(x[STATE], self.Vh[STATE]) obs = vector2Function(self.u_o, self.Vh[STATE]) bform = ufl.inner(obs - u, test)*ufl.dx Matrix, rhs = dl.assemble_system(dl.adjoint(Avarf), bform, self.bc0) if assemble_rhs: return Matrix, rhs else: return Matrix
def rhs(states, time, parameters, dy=None): """ Compute right hand side """ # Imports import ufl import dolfin # Assign states assert(isinstance(states, dolfin.Function)) assert(states.function_space().depth() == 1) assert(states.function_space().num_sub_spaces() == 17) Xr1, Xr2, Xs, m, h, j, d, f, fCa, s, r, Ca_SR, Ca_i, g, Na_i, V, K_i =\ dolfin.split(states) # Assign parameters assert(isinstance(parameters, (dolfin.Function, dolfin.Constant))) if isinstance(parameters, dolfin.Function): assert(parameters.function_space().depth() == 1) assert(parameters.function_space().num_sub_spaces() == 45) else: assert(parameters.value_size() == 45) P_kna, g_K1, g_Kr, g_Ks, g_Na, g_bna, g_CaL, g_bca, g_to, K_mNa, K_mk,\ P_NaK, K_NaCa, K_sat, Km_Ca, Km_Nai, alpha, gamma, K_pCa, g_pCa,\ g_pK, Buf_c, Buf_sr, Ca_o, K_buf_c, K_buf_sr, K_up, V_leak, V_sr,\ Vmax_up, a_rel, b_rel, c_rel, tau_g, Na_o, Cm, F, R, T, V_c,\ stim_amplitude, stim_duration, stim_period, stim_start, K_o =\ dolfin.split(parameters) # Reversal potentials E_Na = R*T*ufl.ln(Na_o/Na_i)/F E_K = R*T*ufl.ln(K_o/K_i)/F E_Ks = R*T*ufl.ln((Na_o*P_kna + K_o)/(Na_i*P_kna + K_i))/F E_Ca = 0.5*R*T*ufl.ln(Ca_o/Ca_i)/F # Inward rectifier potassium current alpha_K1 = 0.1/(1.0 + 6.14421235332821e-6*ufl.exp(0.06*V - 0.06*E_K)) beta_K1 = (3.06060402008027*ufl.exp(0.0002*V - 0.0002*E_K) +\ 0.367879441171442*ufl.exp(0.1*V - 0.1*E_K))/(1.0 + ufl.exp(0.5*E_K -\ 0.5*V)) xK1_inf = alpha_K1/(alpha_K1 + beta_K1) i_K1 = 0.430331482911935*ufl.sqrt(K_o)*(-E_K + V)*g_K1*xK1_inf # Rapid time dependent potassium current i_Kr = 0.430331482911935*ufl.sqrt(K_o)*(-E_K + V)*Xr1*Xr2*g_Kr # Rapid time dependent potassium current xr1 gate xr1_inf = 1.0/(1.0 + 0.0243728440732796*ufl.exp(-0.142857142857143*V)) alpha_xr1 = 450.0/(1.0 + ufl.exp(-9/2 - V/10.0)) beta_xr1 = 6.0/(1.0 + 13.5813245225782*ufl.exp(0.0869565217391304*V)) tau_xr1 = alpha_xr1*beta_xr1 # Rapid time dependent potassium current xr2 gate xr2_inf = 1.0/(1.0 + 39.1212839981532*ufl.exp(0.0416666666666667*V)) alpha_xr2 = 3.0/(1.0 + 0.0497870683678639*ufl.exp(-0.05*V)) beta_xr2 = 1.12/(1.0 + 0.0497870683678639*ufl.exp(0.05*V)) tau_xr2 = alpha_xr2*beta_xr2 # Slow time dependent potassium current i_Ks = (Xs*Xs)*(V - E_Ks)*g_Ks # Slow time dependent potassium current xs gate xs_inf = 1.0/(1.0 + 0.69967253737513*ufl.exp(-0.0714285714285714*V)) alpha_xs = 1100.0/ufl.sqrt(1.0 +\ 0.188875602837562*ufl.exp(-0.166666666666667*V)) beta_xs = 1.0/(1.0 + 0.0497870683678639*ufl.exp(0.05*V)) tau_xs = alpha_xs*beta_xs # Fast sodium current i_Na = (m*m*m)*(-E_Na + V)*g_Na*h*j # Fast sodium current m gate m_inf = 1.0/((1.0 +\ 0.00184221158116513*ufl.exp(-0.110741971207087*V))*(1.0 +\ 0.00184221158116513*ufl.exp(-0.110741971207087*V))) alpha_m = 1.0/(1.0 + ufl.exp(-12.0 - V/5.0)) beta_m = 0.1/(1.0 + 0.778800783071405*ufl.exp(0.005*V)) + 0.1/(1.0 +\ ufl.exp(7.0 + V/5.0)) tau_m = alpha_m*beta_m # Fast sodium current h gate h_inf = 1.0/((1.0 + 15212.5932856544*ufl.exp(0.134589502018843*V))*(1.0 +\ 15212.5932856544*ufl.exp(0.134589502018843*V))) alpha_h = 4.43126792958051e-7*ufl.exp(-0.147058823529412*V)/(1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V)) beta_h = (310000.0*ufl.exp(0.3485*V) + 2.7*ufl.exp(0.079*V))/(1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V)) + 0.77*(1.0 - 1.0/(1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V)))/(0.13 +\ 0.0497581410839387*ufl.exp(-0.0900900900900901*V)) tau_h = 1.0/(alpha_h + beta_h) # Fast sodium current j gate j_inf = 1.0/((1.0 + 15212.5932856544*ufl.exp(0.134589502018843*V))*(1.0 +\ 15212.5932856544*ufl.exp(0.134589502018843*V))) alpha_j = (37.78 + V)*(-6.948e-6*ufl.exp(-0.04391*V) -\ 25428.0*ufl.exp(0.2444*V))/((1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V))*(1.0 +\ 50262745825.954*ufl.exp(0.311*V))) beta_j = 0.6*(1.0 - 1.0/(1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V)))*ufl.exp(0.057*V)/(1.0 +\ 0.0407622039783662*ufl.exp(-0.1*V)) +\ 0.02424*ufl.exp(-0.01052*V)/((1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V))*(1.0 +\ 0.00396086833990426*ufl.exp(-0.1378*V))) tau_j = 1.0/(alpha_j + beta_j) # Sodium background current i_b_Na = (-E_Na + V)*g_bna # L type ca current i_CaL = 4.0*(F*F)*(-0.341*Ca_o +\ Ca_i*ufl.exp(2.0*F*V/(R*T)))*V*d*f*fCa*g_CaL/((-1.0 +\ ufl.exp(2.0*F*V/(R*T)))*R*T) # L type ca current d gate d_inf = 1.0/(1.0 + 0.513417119032592*ufl.exp(-0.133333333333333*V)) alpha_d = 0.25 + 1.4/(1.0 +\ 0.0677244716592409*ufl.exp(-0.0769230769230769*V)) beta_d = 1.4/(1.0 + ufl.exp(1.0 + V/5.0)) gamma_d = 1.0/(1.0 + 12.1824939607035*ufl.exp(-0.05*V)) tau_d = gamma_d + alpha_d*beta_d # L type ca current f gate f_inf = 1.0/(1.0 + 17.4117080633276*ufl.exp(0.142857142857143*V)) tau_f = 80.0 + 165.0/(1.0 + ufl.exp(5/2 - V/10.0)) +\ 1125.0*ufl.exp(-0.00416666666666667*((27.0 + V)*(27.0 + V))) # L type ca current fca gate alpha_fCa = 1.0/(1.0 + 8.03402376701711e+27*ufl.elem_pow(Ca_i, 8.0)) beta_fCa = 0.1/(1.0 + 0.00673794699908547*ufl.exp(10000.0*Ca_i)) gama_fCa = 0.2/(1.0 + 0.391605626676799*ufl.exp(1250.0*Ca_i)) fCa_inf = 0.157534246575342 + 0.684931506849315*gama_fCa +\ 0.684931506849315*beta_fCa + 0.684931506849315*alpha_fCa tau_fCa = 2.0 d_fCa = (-fCa + fCa_inf)/tau_fCa # Calcium background current i_b_Ca = (V - E_Ca)*g_bca # Transient outward current i_to = (-E_K + V)*g_to*r*s # Transient outward current s gate s_inf = 1.0/(1.0 + ufl.exp(4.0 + V/5.0)) tau_s = 3.0 + 85.0*ufl.exp(-0.003125*((45.0 + V)*(45.0 + V))) + 5.0/(1.0 +\ ufl.exp(-4.0 + V/5.0)) # Transient outward current r gate r_inf = 1.0/(1.0 + 28.0316248945261*ufl.exp(-0.166666666666667*V)) tau_r = 0.8 + 9.5*ufl.exp(-0.000555555555555556*((40.0 + V)*(40.0 + V))) # Sodium potassium pump current i_NaK = K_o*Na_i*P_NaK/((K_mk + K_o)*(Na_i + K_mNa)*(1.0 +\ 0.0353*ufl.exp(-F*V/(R*T)) + 0.1245*ufl.exp(-0.1*F*V/(R*T)))) # Sodium calcium exchanger current i_NaCa = (-(Na_o*Na_o*Na_o)*Ca_i*alpha*ufl.exp((-1.0 + gamma)*F*V/(R*T))\ + (Na_i*Na_i*Na_i)*Ca_o*ufl.exp(F*V*gamma/(R*T)))*K_NaCa/((1.0 +\ K_sat*ufl.exp((-1.0 + gamma)*F*V/(R*T)))*((Na_o*Na_o*Na_o) +\ (Km_Nai*Km_Nai*Km_Nai))*(Km_Ca + Ca_o)) # Calcium pump current i_p_Ca = Ca_i*g_pCa/(K_pCa + Ca_i) # Potassium pump current i_p_K = (-E_K + V)*g_pK/(1.0 +\ 65.4052157419383*ufl.exp(-0.167224080267559*V)) # Calcium dynamics i_rel = ((Ca_SR*Ca_SR)*a_rel/((Ca_SR*Ca_SR) + (b_rel*b_rel)) + c_rel)*d*g i_up = Vmax_up/(1.0 + (K_up*K_up)/(Ca_i*Ca_i)) i_leak = (-Ca_i + Ca_SR)*V_leak g_inf = (1.0 - 1.0/(1.0 + 0.0301973834223185*ufl.exp(10000.0*Ca_i)))/(1.0 +\ 1.97201988740492e+55*ufl.elem_pow(Ca_i, 16.0)) + 1.0/((1.0 +\ 0.0301973834223185*ufl.exp(10000.0*Ca_i))*(1.0 +\ 5.43991024148102e+20*ufl.elem_pow(Ca_i, 6.0))) d_g = (-g + g_inf)/tau_g Ca_i_bufc = 1.0/(1.0 + Buf_c*K_buf_c/((K_buf_c + Ca_i)*(K_buf_c + Ca_i))) Ca_sr_bufsr = 1.0/(1.0 + Buf_sr*K_buf_sr/((K_buf_sr + Ca_SR)*(K_buf_sr +\ Ca_SR))) # Sodium dynamics # Membrane i_Stim = -(1.0 - 1.0/(1.0 + ufl.exp(-5.0*stim_start +\ 5.0*time)))*stim_amplitude/(1.0 + ufl.exp(-5.0*stim_start + 5.0*time\ - 5.0*stim_duration)) # Potassium dynamics # The ODE system: 17 states # Init test function _v = dolfin.TestFunction(states.function_space()) # Derivative for state Xr1 dy = ((-Xr1 + xr1_inf)/tau_xr1)*_v[0] # Derivative for state Xr2 dy += ((-Xr2 + xr2_inf)/tau_xr2)*_v[1] # Derivative for state Xs dy += ((-Xs + xs_inf)/tau_xs)*_v[2] # Derivative for state m dy += ((-m + m_inf)/tau_m)*_v[3] # Derivative for state h dy += ((-h + h_inf)/tau_h)*_v[4] # Derivative for state j dy += ((j_inf - j)/tau_j)*_v[5] # Derivative for state d dy += ((d_inf - d)/tau_d)*_v[6] # Derivative for state f dy += ((-f + f_inf)/tau_f)*_v[7] # Derivative for state fCa dy += ((1.0 - 1.0/((1.0 + ufl.exp(60.0 + V))*(1.0 + ufl.exp(-10.0*fCa +\ 10.0*fCa_inf))))*d_fCa)*_v[8] # Derivative for state s dy += ((-s + s_inf)/tau_s)*_v[9] # Derivative for state r dy += ((-r + r_inf)/tau_r)*_v[10] # Derivative for state Ca_SR dy += ((-i_leak + i_up - i_rel)*Ca_sr_bufsr*V_c/V_sr)*_v[11] # Derivative for state Ca_i dy += ((-i_up - (i_CaL + i_p_Ca + i_b_Ca - 2.0*i_NaCa)*Cm/(2.0*F*V_c) +\ i_leak + i_rel)*Ca_i_bufc)*_v[12] # Derivative for state g dy += ((1.0 - 1.0/((1.0 + ufl.exp(60.0 + V))*(1.0 + ufl.exp(-10.0*g +\ 10.0*g_inf))))*d_g)*_v[13] # Derivative for state Na_i dy += ((-3.0*i_NaK - 3.0*i_NaCa - i_Na - i_b_Na)*Cm/(F*V_c))*_v[14] # Derivative for state V dy += (-i_Ks - i_to - i_Kr - i_p_K - i_NaK - i_NaCa - i_Na - i_p_Ca -\ i_b_Na - i_CaL - i_Stim - i_K1 - i_b_Ca)*_v[15] # Derivative for state K_i dy += ((-i_Ks - i_to - i_Kr - i_p_K - i_Stim - i_K1 +\ 2.0*i_NaK)*Cm/(F*V_c))*_v[16] # Return dy return dy