def _construct_eigenproblem(self, u: ufl.Argument, v: ufl.Argument) \ -> Tuple[ufl.algebra.Operator, ufl.algebra.Operator]: """Construct left- and right-hand sides of eigenvalue problem. Parameters ---------- u : ufl.Argument A function belonging to the function space under consideration. v : ufl.Argument A function belonging to the function space under consideration. Returns ------- a : ufl.algebra.Operator Left hand side form. b : ufl.algebra.Operator Right hand side form. """ g = self.metric_tensor sqrt_g = fenics.sqrt(fenics.det(g)) inv_g = fenics.inv(g) # $a(u, v) = \int_M \nabla u \cdot g^{-1} \nabla v \, \sqrt{\det(g)} \, d x$. a = fenics.dot(fenics.grad(u), inv_g * fenics.grad(v)) * sqrt_g # $b(u, v) = \int_M u \, v \, \sqrt{\det(g)} \, d x$. b = fenics.dot(u, v) * sqrt_g return a, b
def set_target_velocity(self, u=None, v=None, U=None): """ Set target velocity. Accepts a list of surface velocity data, and generates a dolfin expression from these. Then projects this onto the velocity function space. The sum square error between this velocity and modelled surface velocity is the objective function. :param u : Surface velocity :param v : Surface velocity perpendicular to :attr:`u` :param U : 2-D surface velocity data """ model = self.model S = model.S Q = model.Q if u != None and v != None: model.u_o = project(u, Q) model.v_o = project(v, Q) elif U != None: Smag = project(sqrt(S.dx(0)**2 + S.dx(1)**2 + 1e-10), Q) model.U_o.interpolate(U) model.u_o = project(-model.U_o * S.dx(0) / Smag, Q) model.v_o = project(-model.U_o * S.dx(1) / Smag, Q)
def __setup_decrease_computation(self): """Initializes attributes and solver for the frobenius norm check. Returns ------- None """ if not self.angle_change > 0: raise ConfigError('MeshQuality', 'angle_change', 'This parameter has to be positive.') options = [[['ksp_type', 'preonly'], ['pc_type', 'jacobi'], ['pc_jacobi_type', 'diagonal'], ['ksp_rtol', 1e-16], ['ksp_atol', 1e-20], ['ksp_max_it', 1000]]] self.ksp_frobenius = PETSc.KSP().create() _setup_petsc_options([self.ksp_frobenius], options) self.trial_dg0 = fenics.TrialFunction(self.form_handler.DG0) self.test_dg0 = fenics.TestFunction(self.form_handler.DG0) if not (self.angle_change == float('inf')): self.search_direction_container = fenics.Function( self.form_handler.deformation_space) self.a_frobenius = self.trial_dg0 * self.test_dg0 * self.dx self.L_frobenius = fenics.sqrt( fenics.inner(fenics.grad(self.search_direction_container), fenics.grad(self.search_direction_container)) ) * self.test_dg0 * self.dx
def CompressibleOgden(Lambda, Mu, Alpha, C, Ic, J): # Invariant of Right Cauchy-Green deformation tensor def I1(C): return fe.tr(C) def I2(C): c1 = C[0,0]*C[1,1] + C[0,0]*C[2,2] + C[1,1]*C[2,2] c2 = C[0,1]*C[0,1] + C[0,2]*C[0,2] + C[1,2]*C[1,2] return c1 - c2 def I3(C): return fe.det(C) # Define function necessary for eigenvalues computation def v_inv(C): return (I1(C)/3.)**2 - I2(C)/3. def s_inv(C): return (I1(C)/3.)**3 - I1(C)*I2(C)/6. + I3(C)/2. def phi_inv(C): arg = s_inv(C)/v_inv(C)*fe.sqrt(1./v_inv(C)) # numerical issues if arg~0 # https://fenicsproject.org/qa/12299 # /nan-values-when-computing-arccos-1-0-bug/ arg_cond = fe.conditional( fe.ge(arg, 1-fe.DOLFIN_EPS), 1-fe.DOLFIN_EPS,fe.conditional( fe.le(arg, -1+fe.DOLFIN_EPS), -1+fe.DOLFIN_EPS, arg )) return fe.acos(arg_cond)/3. # Eigenvalues of the strech tensor C Lambda1 = Ic/3. + 2*fe.sqrt(v_inv(C))*fe.cos(phi_inv(C)) Lambda2 = Ic/3. - 2*fe.sqrt(v_inv(C))*fe.cos(fe.pi/3. + phi_inv(C)) Lambda3 = Ic/3. - 2*fe.sqrt(v_inv(C))*fe.cos(fe.pi/3. - phi_inv(C)) D1 = Lambda/2 Lambda1b = J**(-1/3) * Lambda1 Lambda2b = J**(-1/3) * Lambda2 Lambda3b = J**(-1/3) * Lambda3 # Constitutive model Psi = 2 * Mu * (Lambda1b**(Alpha/2) + Lambda2b**(Alpha/2) + Lambda3b**(Alpha/2) - 3) / Alpha**2 + D1 * (J-1)**2 return Psi
def phi_inv(C): arg = s_inv(C)/v_inv(C)*fe.sqrt(1./v_inv(C)) # numerical issues if arg~0 # https://fenicsproject.org/qa/12299 # /nan-values-when-computing-arccos-1-0-bug/ arg_cond = fe.conditional( fe.ge(arg, 1-fe.DOLFIN_EPS), 1-fe.DOLFIN_EPS,fe.conditional( fe.le(arg, -1+fe.DOLFIN_EPS), -1+fe.DOLFIN_EPS, arg )) return fe.acos(arg_cond)/3.
def local_refine(mesh, center, r): xc, yc = center cell_markers = MeshFunction("bool", mesh, mesh.topology().dim()) for c in cells(mesh): mp = c.midpoint() cell_markers[c] = sqrt((mp[0] - xc) * (mp[0] - xc) + (mp[1] - yc) * (mp[1] - yc)) < r mesh = refine(mesh, cell_markers) return mesh
def avg_condition_number(mesh): """Computes average mesh quality based on the condition number of the reference mapping. This quality criterion uses the condition number (in the Frobenius norm) of the (linear) mapping from the elements of the mesh to the reference element. Computes the average of the condition number over all elements. Parameters ---------- mesh : dolfin.cpp.mesh.Mesh The mesh, whose quality shall be computed. Returns ------- float The average mesh quality based on the condition number. """ DG0 = fenics.FunctionSpace(mesh, 'DG', 0) jac = Jacobian(mesh) inv = JacobianInverse(mesh) options = [ ['ksp_type', 'preonly'], ['pc_type', 'jacobi'], ['pc_jacobi_type', 'diagonal'], ['ksp_rtol', 1e-16], ['ksp_atol', 1e-20], ['ksp_max_it', 1000] ] ksp = PETSc.KSP().create() _setup_petsc_options([ksp], [options]) dx = fenics.Measure('dx', mesh) a = fenics.TrialFunction(DG0)*fenics.TestFunction(DG0)*dx L = fenics.sqrt(fenics.inner(jac, jac))*fenics.sqrt(fenics.inner(inv, inv))*fenics.TestFunction(DG0)*dx cond = fenics.Function(DG0) A, b = _assemble_petsc_system(a, L) _solve_linear_problem(ksp, A, b, cond.vector().vec(), options) cond.vector().apply('') return np.average(np.sqrt(mesh.geometric_dimension()) / cond.vector()[:])
def distance_function_line_segement_ufl(P, A=[-1, 0], B=[1, 0]): AB = [None, None] AB[0] = B[0] - A[0] AB[1] = B[1] - A[1] BP = [None, None] BP[0] = P[0] - B[0] BP[1] = P[1] - B[1] AP = [None, None] AP[0] = P[0] - A[0] AP[1] = P[1] - A[1] AB_BP = AB[0] * BP[0] + AB[1] * BP[1] AB_AP = AB[0] * AP[0] + AB[1] * AP[1] y = P[1] - B[1] x = P[0] - B[0] df1 = fe.sqrt(x**2 + y**2) xi1 = 1 y = P[1] - A[1] x = P[0] - A[0] df2 = fe.sqrt(x**2 + y**2) xi2 = 0 x1 = AB[0] y1 = AB[1] x2 = AP[0] y2 = AP[1] mod = fe.sqrt(x1**2 + y1**2) df3 = np.absolute(x1 * y2 - y1 * x2) / mod xi3 = fe.conditional(fe.gt(x2**2 + y2**2 - df3**2, 0), fe.sqrt(x2**2 + y2**2 - df3**2) / mod, 0) df = fe.conditional(fe.gt(AB_BP, 0), df1, fe.conditional(fe.lt(AB_AP, 0), df2, df3)) xi = fe.conditional(fe.gt(AB_BP, 0), xi1, fe.conditional(fe.lt(AB_AP, 0), xi2, xi3)) return df, xi
def compute_errors(u_approx, u_ref, V, total_error_tol=10**-4): error_normalized = (u_ref - u_approx) / u_ref # compute pointwise L2 error error_pointwise = project(abs(error_normalized), V) # project onto function space error_total = sqrt( assemble(inner(error_pointwise, error_pointwise) * dx)) # determine L2 norm to estimate total error error_pointwise.rename("error", " ") assert (error_total < total_error_tol) return error_total, error_pointwise
def psi_minus_linear_elasticity_model_C(epsilon, lamda, mu): sqrt_delta = fe.conditional( fe.gt(fe.tr(epsilon)**2 - 4 * fe.det(epsilon), 0), fe.sqrt(fe.tr(epsilon)**2 - 4 * fe.det(epsilon)), 0) eigen_value_1 = (fe.tr(epsilon) + sqrt_delta) / 2 eigen_value_2 = (fe.tr(epsilon) - sqrt_delta) / 2 tr_epsilon_minus = fe.conditional(fe.lt(fe.tr(epsilon), 0.), fe.tr(epsilon), 0.) eigen_value_1_minus = fe.conditional(fe.lt(eigen_value_1, 0.), eigen_value_1, 0.) eigen_value_2_minus = fe.conditional(fe.lt(eigen_value_2, 0.), eigen_value_2, 0.) return lamda / 2 * tr_epsilon_minus**2 + mu * (eigen_value_1_minus**2 + eigen_value_2_minus**2)
def _handle_function( obj, **kwargs, ): data = [] scatter = kwargs.get("scatter", False) norm = kwargs.get("norm", False) component = kwargs.get("component", None) if len(obj.ufl_shape) == 0: if scatter: surface = _scatter_plot_function(obj, **kwargs) else: surface = _surface_plot_function(obj, **kwargs) data.append(surface) elif len(obj.ufl_shape) == 1: if norm or component == "magnitude": V = obj.function_space().split()[0].collapse() magnitude = fe.project(fe.sqrt(fe.inner(obj, obj)), V) else: magnitude = None if component is None: if norm: surface = _surface_plot_function(magnitude, **kwargs) data.append(surface) cones = _cone_plot(obj, **kwargs) data.append(cones) else: if component == "magnitude": surface = _surface_plot_function(magnitude, **kwargs) data.append(surface) else: for i, comp in enumerate(["x", "y", "z"]): if component not in [comp, comp.upper()]: continue surface = _surface_plot_function( obj.sub(i, deepcopy=True), **kwargs ) data.append(surface) if kwargs.get("wireframe", True): lines = _wireframe_plot_mesh(obj.function_space().mesh()) data.append(lines) return data
bc_p = fe.DirichletBC(W.sub(1), bc_p, dbc_top) def Max(a, b): return (a + b + abs(a - b)) / 2. def Min(a, b): return (a + b - abs(a - b)) / 2. ns_conv = fe.inner(v, fe.grad(u) * u) * fe.dx ns_press = p * fe.div(v) * fe.dx #s = fe.grad(u) + fe.grad(u).T sij = 0.5 * (fe.grad(u) + fe.grad(u).T) S = fe.sqrt(EPSILON + 2. * fe.inner(0.5 * (fe.grad(u) + fe.grad(u).T), 0.5 * (fe.grad(u) + fe.grad(u).T))) lmx = 0.01 nu_tv = lmx**2. * S #nu_tv = 0.5 * fe.inner(sij, sij) ** 0.5 #nu_tv = lmx * (2 * fe.inner(sij, sij)) ** (0.5) #ns_tv = fe.inner((nu_tv) * fe.grad(v), fe.grad(u)) * fe.dx ns_visc = (nu + nu_tv) * fe.inner(fe.grad(v), fe.grad(u)) * fe.dx #ns_visc = nu * fe.inner(fe.grad(v), fe.grad(u)) * fe.dx ns_conti = q * fe.div(u) * fe.dx ns_forcing = fe.dot(v, b) * fe.dx NS = ns_conv + ns_press + ns_visc - ns_conti + ns_forcing #NS = ns_conv + ns_press + ns_tv + ns_visc + ns_conti + ns_forcing N = 5
def differentiated_apparent_viscosity(self, II): return self.ty / II**1.5 \ * ((1 + sqrt(II)/self.eps) * exp(-sqrt(II) / self.eps) - 1)
def eval(self, values, x): values[0] = min(M_max, S_b * (R_el - sqrt(x[0]**2 + x[1]**2)))
f = fs.Constant((0, 0, -rho * g)) T = fs.Constant((0, 0, 0)) a = fs.inner(sigma(u), epsilon(v)) * fs.dx L = fs.dot(f, v) * fs.dx + fs.dot(T, v) * fs.ds # Compute solution u = fs.Function(V) fs.solve(a == L, u, bc) # Plot solution plt.figure() fs.plot(u, title='Displacement', mode='displacement') # Plot stress s = sigma(u) - (1. / 3) * fs.tr(sigma(u)) * fs.Identity(d) # deviatoric stress von_Mises = fs.sqrt(3. / 2 * fs.inner(s, s)) V = fs.FunctionSpace(mesh, 'P', 1) von_Mises = fs.project(von_Mises, V) plt.figure() fs.plot(von_Mises, title='Stress intensity') # Compute magnitude of displacement u_magnitude = fs.sqrt(fs.dot(u, u)) u_magnitude = fs.project(u_magnitude, V) plt.figure() fs.plot(u_magnitude, title='Displacement magnitude') print('min/max u:', u_magnitude.vector().min(), u_magnitude.vector().max()) # Save solution to file in VTK format fs.File('results/displacement.pvd') << u fs.File('results/von_mises.pvd') << von_Mises
def solve(self, **kwargs): """ Solves the variational form of the electrostatics as defined in the End of Master thesis from Ximo Gallud Cidoncha: A comprehensive numerical procedure for solving the Taylor-Melcher leaky dielectric model with charge evaporation. Parameters ---------- **kwargs : dict Accepted kwargs are: - electrostatics_solver_settings: The user may define its own solver parameters. They must be defined as follows: solver_parameters = {"snes_solver": {"linear_solver": "mumps", "maximum_iterations": 50, "report": True, "error_on_nonconvergence": True, 'line_search': 'bt', 'relative_tolerance': 1e-4}} where: - snes_solver is the type of solver to be used. In this case, it is compulsory to use snes, since it's the solver accepted by multiphenics. However, one may try other options if only FEniCS is used. These are: krylov_solver and lu_solver. - linear_solver is the type of linear solver to be used. - maximum_iterations is the maximum number of iterations the solver will try to solve the problem. In case no convergence is achieved, the variable error_on_nonconvergence will raise an error in case this is True. If the user preferes not to raise an error when no convergence, the script will continue with the last results obtained in the iteration process. - line_search is the type of line search technique to be used for solving the problem. It is stronly recommended to use the backtracking (bt) method, since it has been proven to be the most robust one, specially in cases where sqrt are defined, where NaNs may appear due to a bad initial guess or a bad step in the iteration process. - relative_tolerance will tell the solver the parameter to consider convergence on the solution. All this options, as well as all the other options available can be consulted by calling the method Poisson.check_solver_options(). - initial_potential: Dolfin/FEniCS function which will be used as an initial guess on the iterative process. This must be introduced along with kwarg initial_surface_charge_density. Optional. - initial_surface_charge_density: Dolfin/FEniCS function which will be used as an initial guess on the iterative process. This must be introduced along with kwarg initial_potential. Optional. Raises ------ TypeError This error will raise when the convection charge has not one of the following types: - Dolfin Function. - FEniCS UserExpression. - FEniCS Constant. - Integer or float number, which will be converted to a FEniCS Constant. Returns ------- phi : dolfin.function.function.Function Dolfin function containing the potential solution. surface_charge_density : dolfin.function.function.Function Dolfin function conataining the surface charge density solution. """ # -------------------------------------------------------------------- # EXTRACT THE INPUTS # # -------------------------------------------------------------------- # Check if the type of j_conv is the proper one. if not isinstance(self.j_conv, (int, float)) \ and not Poisson.isDolfinFunction(self.j_conv) \ and not Poisson.isfenicsexpression(self.j_conv) \ and not Poisson.isfenicsconstant(self.j_conv): conv_type = type(self.j_conv) raise TypeError( f'Convection charge must be an integer, float, Dolfin function, FEniCS UserExpression or FEniCS constant, not {conv_type}.' ) else: if isinstance(self.j_conv, (int, float)): self.j_conv = fn.Constant(float(self.j_conv)) # Extract the solver parameters. solver_parameters = kwargs.get('electrostatics_solver_settings') # -------------------------------------------------------------------- # FUNCTION SPACES # # -------------------------------------------------------------------- # Extract the restrictions to create the function spaces. """ This variable will be used by multiphenics when creating function spaces. It will create function spaces on the introduced restrictions. """ restrictions_block = [ self.restrictions_dict['domain_rtc'], self.restrictions_dict['interface_rtc'] ] # Base Function Space. V = fn.FunctionSpace(self.mesh, 'Lagrange', 2) # Block Function Space. """ Block Function Spaces are similar to FEniCS function spaces. However, since we are creating function spaces based on the block of restrictions, we need to create a 'block of function spaces' for each of the restrictions. That block of functions is the list [V, V] from the line of code below this comment. They are assigned in the same order in which the block of restrictions has been created, that is: - V -> domain_rtc - V -> interface_rtc """ W = mp.BlockFunctionSpace([V, V], restrict=restrictions_block) # Check the dimensions of the created block function spaces. for ix, _ in enumerate(restrictions_block): assert W.extract_block_sub_space( (ix, )).dim() > 0., f'Subdomain {ix} has dimension 0.' # -------------------------------------------------------------------- # TRIAL/TEST FUNCTIONS # # -------------------------------------------------------------------- # Trial Functions. dphisigma = mp.BlockTrialFunction(W) # Test functions. vl = mp.BlockTestFunction(W) (v, l) = mp.block_split(vl) phisigma = mp.BlockFunction(W) (phi, sigma) = mp.block_split(phisigma) # -------------------------------------------------------------------- # MEASURES # # -------------------------------------------------------------------- self.get_measures() self.dS = self.dS( self.boundaries_ids['Interface']) # Restrict to the interface. # Check proper marking of the interface. assert fn.assemble( 1 * self.dS(domain=self.mesh) ) > 0., "The length of the interface is zero, wrong marking. Check the files in Paraview." # -------------------------------------------------------------------- # DEFINE THE F TERM # # -------------------------------------------------------------------- n = fn.FacetNormal(self.mesh) t = fn.as_vector((n[1], -n[0])) # Define auxiliary terms. r = fn.SpatialCoordinate(self.mesh)[0] K = 1 + self.Lambda * (self.T_h - 1) E_v_n_aux = fn.dot(-fn.grad(phi("-")), n("-")) def expFun(): sqrterm = E_v_n_aux expterm = (self.Phi / self.T_h) * (1 - pow(self.B, 0.25) * fn.sqrt(sqrterm)) return fn.exp(expterm) def sigma_fun(): num = K * E_v_n_aux + self.eps_r * self.j_conv den = K + (self.T_h / self.Chi) * expFun() return r * num / den # Define the relative permittivity. class relative_perm(fn.UserExpression): def __init__(self, markers, subdomain_ids, relative, **kwargs): super().__init__(**kwargs) self.markers = markers self.subdomain_ids = subdomain_ids self.relative = relative def eval_cell(self, values, x, cell): if self.markers[cell.index] == self.subdomain_ids['Vacuum']: values[0] = 1. else: values[0] = self.relative rel_perm = relative_perm(self.subdomains, self.subdomains_ids, relative=self.eps_r, degree=0) # Define the variational form. # vacuum_int = r*fn.inner(fn.grad(phi), fn.grad(v))*self.dx(self.subdomains_ids['Vacuum']) # liquid_int = self.eps_r*r*fn.inner(fn.grad(phi), fn.grad(v))*self.dx(self.subdomains_ids['Liquid']) F = [ r * rel_perm * fn.inner(fn.grad(phi), fn.grad(v)) * self.dx - r * sigma("-") * v("-") * self.dS, r * sigma_fun() * l("-") * self.dS - r * sigma("-") * l("-") * self.dS ] J = mp.block_derivative(F, phisigma, dphisigma) # -------------------------------------------------------------------- # BOUNDARY CONDITIONS # # -------------------------------------------------------------------- bcs_block = [] for i in self.boundary_conditions: if 'Dirichlet' in self.boundary_conditions[i]: bc_val = self.boundary_conditions[i]['Dirichlet'][0] bc = mp.DirichletBC(W.sub(0), bc_val, self.boundaries, self.boundaries_ids[i]) # Check the created boundary condition. assert len(bc.get_boundary_values() ) > 0., f'Wrongly defined boundary {i}' bcs_block.append(bc) bcs_block = mp.BlockDirichletBC([bcs_block]) # -------------------------------------------------------------------- # SOLVE # # -------------------------------------------------------------------- # Define and assign the initial guesses. if kwargs.get('initial_potential') is None: """ Check if the user is introducing a potential from a previous iteration. """ phiv, phil, sigma_init = self.solve_initial_problem() # phi_init = self.solve_initial_problem_v2() phi.assign(phiv) sigma.assign(sigma_init) else: phi.assign(kwargs.get('initial_potential')) sigma.assign(kwargs.get('initial_surface_charge_density')) # Apply the initial guesses to the main function. phisigma.apply('from subfunctions') # Solve the problem with the solver options (either default or user). problem = mp.BlockNonlinearProblem(F, phisigma, bcs_block, J) solver = mp.BlockPETScSNESSolver(problem) solver_type = [i for i in solver_parameters.keys()][0] solver.parameters.update(solver_parameters[solver_type]) solver.solve() # Extract the solutions. (phi, _) = phisigma.block_split() self.phi = phi # -------------------------------------------------------------------- # Compute the electric field at vacuum and correct the surface charge density. self.E_v = self.get_electric_field('Vacuum') self.E_v_n = self.get_normal_field(n("-"), self.E_v) self.E_t = self.get_tangential_component(t("+"), self.E_v) C = self.Phi / self.T_h * (1 - self.B**0.25 * fn.sqrt(self.E_v_n)) self.sigma = (K * self.E_v_n) / (K + self.T_h / self.Chi * fn.exp(-C))
bc_p = fe.DirichletBC(W.sub(1), bc_p, dbc_top) def Max(a, b): return (a + b + abs(a - b)) / 2. def Min(a, b): return (a + b - abs(a - b)) / 2. ns_conv = fe.inner(v, fe.grad(u) * u) * fe.dx ns_press = p * fe.div(v) * fe.dx #s = fe.grad(u) + fe.grad(u).T sij = 0.5 * (fe.grad(u) + fe.grad(u).T) S = fe.sqrt(EPSILON + 2. * fe.inner(0.5 * (fe.grad(u) + fe.grad(u).T), 0.5 * (fe.grad(u) + fe.grad(u).T))) lmx = 0.3 nu_tv = lmx**2. * S #nu_tv = 0.5 * fe.inner(sij, sij) ** 0.5 #nu_tv = lmx * (2 * fe.inner(sij, sij)) ** (0.5) #ns_tv = fe.inner((nu_tv) * fe.grad(v), fe.grad(u)) * fe.dx ns_visc = (nu + nu_tv) * fe.inner(fe.grad(v), fe.grad(u)) * fe.dx #ns_visc = nu * fe.inner(fe.grad(v), fe.grad(u)) * fe.dx ns_conti = q * fe.div(u) * fe.dx ns_forcing = fe.dot(v, b) * fe.dx NS = ns_conv + ns_press + ns_visc + ns_conti + ns_forcing #NS = ns_conv + ns_press + ns_tv + ns_visc + ns_conti + ns_forcing N = 5
def evaporated_charge(): return (self.sigma * self.T_h) / (self.eps_r * self.Chi) * fn.exp( -self.Phi / self.T_h * (1 - self.B**0.25 * fn.sqrt(self.E_v_n)))
def sqrt(self, f): return FEN.sqrt(f)
def distance_function_point_ufl(P, A=[0, 0]): x = P[0] - A[0] y = P[1] - A[1] return fe.sqrt(x * x + y * y)
prm['newton_solver']['preconditioner'] = 'ilu' # Invoke the solver #cprint("\nSOLUTION OF THE NONLINEAR PROBLEM", 'blue', attrs=['bold']) #cprint("The solution of the nonlinear system is in progress...", 'red') solver.solve() ############################# POST-PROCESSING ################################ #cprint("\nSOLUTION POST-PROCESSING", 'blue', attrs=['bold']) # Save solution to file in VTK format #cprint("Saving displacement solution to file...", 'green') uViewer = fe.File('paraview/displacement.pvd') uViewer << u # Maximum and minimum displacement u_magnitude = fe.sqrt(fe.dot(u, u)) u_magnitude = fe.project(u_magnitude, W) print('Min/Max displacement:', u_magnitude.vector().array().min(), u_magnitude.vector().array().max()) # Computation of the large deformation strains #cprint("Computing the deformation tensor and saving to file...", 'green') epsilon_u = largeKinematics(u) epsilon_u_project = fe.project(epsilon_u, Z) epsilonViewer = fe.File('paraview/strain.pvd') epsilonViewer << epsilon_u_project # Computation of the stresses #cprint("Stress derivation and saving to file...", 'green') S = fe.diff(psi, E)
def map_function_ufl(x_hat, control_points, impact_radii, map_type, boundary_info=None): if len(control_points) == 0: return x_hat x_hat = fe.variable(x_hat) df, rho = distance_function_segments_ufl(x_hat, control_points, impact_radii) grad_x_hat = fe.diff(df, x_hat) delta_x_hat = fe.conditional( fe.gt(df, rho), fe.Constant((0., 0.)), grad_x_hat * (rho * ratio_function_ufl(df / rho, map_type) - df)) if boundary_info is None: return delta_x_hat + x_hat else: last_control_point = control_points[-1] points, directions, rho_default = boundary_info mid_point, mid_point1, mid_point2 = points direct_vec, rotated_vec = directions aux_control_point1 = last_control_point + rho_default * rotated_vec aux_control_point2 = last_control_point - rho_default * rotated_vec w1 = np.linalg.norm(mid_point1 - aux_control_point1) w2 = np.linalg.norm(mid_point2 - aux_control_point2) w0 = np.linalg.norm(mid_point - last_control_point) assert np.absolute(2 * w0 - w1 - w2) < 1e-5 AB = mid_point - last_control_point AP = x_hat - last_control_point x1 = AB[0] y1 = AB[1] x2 = AP[0] y2 = AP[1] mod = fe.sqrt(x1**2 + y1**2) df_to_direct = (x1 * y2 - y1 * x2) / mod # AB x AP df_to_rotated = (x1 * x2 + y1 * y2) / mod k1 = rho_default * (w1 + w2) / (rho_default * (w1 + w2) + df_to_direct * (w1 - w2)) new_df_to_direct = rho_default * ratio_function_ufl( df_to_direct / rho_default, map_type) k2 = rho_default * (w1 + w2) / (rho_default * (w1 + w2) + new_df_to_direct * (w1 - w2)) new_df_to_rotated = df_to_rotated * k1 / k2 x = fe.as_vector(last_control_point + direct_vec * new_df_to_rotated + rotated_vec * new_df_to_direct) return fe.conditional( fe.gt(df_to_rotated, 0), fe.conditional(fe.gt(np.absolute(df_to_direct), rho), x_hat, x), delta_x_hat + x_hat)
def extract_all_info(self, general_inputs, liquid_properties): """ Extract all the important data generated from the electrostatics simulation. Args: general_inputs: Object containing the SimulationGeneralParameters class. liquid_properties: Object obtained from the Liquid_Parameters class, which is located at Liquids.py Returns: """ self.potential = self.class_caller.phi self.vacuum_electric_field = self.class_caller.E_v # Vacuum electric field. self.normal_component_vacuum = self.class_caller.E_v_n # Normal component of the electric field @interface self.tangential_component = self.class_caller.E_t # @interface self.surface_charge_density = self.class_caller.sigma # Surface charge density at interface. # Get coordinates of the nodes and midpoints. r_nodes, z_nodes = general_inputs.get_nodepoints(self.mesh, self.boundaries, self.boundaries_ids['Interface']) self.coords_nodes = [r_nodes, z_nodes] r_mids, z_mids = general_inputs.get_midpoints(self.boundaries, self.boundaries_ids['Interface']) self.coords_mids = [r_mids, z_mids] # Split the electric field into radial and axial components. self.radial_component_vacuum, self.axial_component_vacuum = \ PostProcessing.extract_from_function(self.vacuum_electric_field, self.coords_nodes) # E_v_n_array = PostProcessing.extract_from_function(Electrostatics.normal_component_vacuum, coords_mids) E_t_array = PostProcessing.extract_from_function(self.tangential_component, self.coords_mids) # Define an auxiliary term for the computations. K = 1 + general_inputs.Lambda * (general_inputs.T_h - 1) self.normal_component_liquid = (self.normal_component_vacuum - self.surface_charge_density) / \ liquid_properties.eps_r E_l_n_array = PostProcessing.extract_from_function(self.normal_component_liquid, self.coords_mids) # Get components of the liquid field. self.radial_component_liquid, self.axial_component_liquid = \ Poisson.get_liquid_electric_field(mesh=self.mesh, subdomain_data=self.boundaries, boundary_id=self.boundaries_ids['Interface'], normal_liquid=E_l_n_array, tangential_liquid=E_t_array) self.radial_component_liquid.append(self.radial_component_liquid[-1]) self.axial_component_liquid.append(self.axial_component_liquid[-1]) # Calculate the non-dimensional evaporated charge and current. self.evaporated_charge = (self.surface_charge_density * general_inputs.T_h) / (liquid_properties.eps_r * general_inputs.Chi) \ * fn.exp(-general_inputs.Phi / general_inputs.T_h * (1 - pow(general_inputs.B, 1 / 4) * fn.sqrt( self.normal_component_vacuum)) ) self.conducted_charge = K * self.normal_component_liquid # Calculate the emitted current through the interface. self.emitted_current = self.class_caller.get_nd_current(self.evaporated_charge) # Compute the normal component of the electric stress at the meniscus (electric pressure). self.normal_electric_stress = (self.normal_component_vacuum ** 2 - liquid_properties.eps_r * self.normal_component_liquid ** 2) + \ (liquid_properties.eps_r - 1) * self.tangential_component ** 2
K = 1+Lambda*(T_h - 1) E_l_n = (Electrostatics.E_v_n-Electrostatics.sigma)/LiquidInps.eps_r E_l_n_array = PostProcessing.extract_from_function(E_l_n, coords_mids) sigma_arr = PostProcessing.extract_from_function(Electrostatics.sigma, coords_mids) # Get components of the liquid field. E_l_r, E_l_z = Poisson.get_liquid_electric_field(mesh=mesh, subdomain_data=boundaries, boundary_id=boundaries_ids['Interface'], normal_liquid=E_l_n_array, tangential_liquid=E_t_array) E_l_r.append(E_l_r[-1]) E_l_z.append(E_l_z[-1]) # Calculate the non-dimensional evaporated charge and current. j_ev = (Electrostatics.sigma*T_h)/(LiquidInps.eps_r*Chi) * fn.exp(-Phi/T_h * ( 1-pow(B, 1/4)*fn.sqrt(Electrostatics.E_v_n))) j_cond = K*E_l_n I_h = Electrostatics.get_nd_current(j_ev) j_ev_arr = PostProcessing.extract_from_function(j_ev, coords_mids) j_cond_arr = PostProcessing.extract_from_function(j_cond, coords_mids) # Compute the normal component of the electric stress at the meniscus (electric pressure). n_taue_n = (Electrostatics.E_v_n**2-LiquidInps.eps_r*E_l_n**2) + (LiquidInps.eps_r-1)*Electrostatics.E_t**2 n_taue_n_arr = PostProcessing.extract_from_function(n_taue_n, coords_mids) # %% DATA POSTPROCESSING. # Check charge conservation. charge_check = abs(j_ev_arr-j_cond_arr)/j_ev_arr print(f'Maximum relative difference between evaporated and conducted charge is {max(charge_check)}')
def differentiated_apparent_viscosity(self, II): return self.ty / (pi * II) \ * (self.eps / (self.eps**2 + II) - atan(sqrt(II) / self.eps) / sqrt(II))
#d2 = fe.Expression('1 - x[0]', degree=1) #d3 = fe.Expression('1 - x[1]', degree=1) d4 = fe.Expression('x[1] - 0', degree=1) d = fe.Constant(10) #d = Min(d1, d4) xi = nu_trial / nu fv1 = fe.elem_pow(xi, 3) / (fe.elem_pow(xi, 3) + Cv1 * Cv1 * Cv1) #fv2 = fe.Constant(1) fv2 = 1 - xi / (1 + xi * fv1) #ft2 = fe.Constant(1) ft2 = Ct3 * fe.exp(-Ct4 * xi * xi) Omega = fe.Constant(0.5) * (fe.grad(u) - fe.grad(u).T) #S = fe.Constant(2) * fe.inner(Omega, Omega) S = fe.sqrt(fe.Constant(2) * fe.inner(Omega, Omega)) #Stilde = fe.Constant(1) #Stilde = nu_trial /(kappa * kappa * d * d) * fv2 #Stilde = nu_trial * fe.Constant(1000) Stilde = S + nu_trial / (kappa * kappa * d * d) * fv2 ft2 = Ct3 * fe.exp(fe.Constant(-1) * Ct4 * xi * xi) #ft2 = fe.Constant(0) #r =fe.Constant(1) #r = nu_trial / (Stilde * kappa * kappa * d * d) r = Max(Min(nu_trial / (Stilde * kappa * kappa * d * d), fe.Constant(10)), -10) #r = fe.Constant(0.1) #g = fe.Constant(0.01) g = r + Cw2 * (r * r * r * r * r * r - r) #fw = g * ((1 + Cw3 * Cw3 * Cw3 * Cw3 * Cw3 * Cw3) / (g * g * g * g * g * g + Cw3 * Cw3 * Cw3 * Cw3 * Cw3 * Cw3)) fw = fe.elem_pow(
def apparent_viscosity(self, II): return 2 * self.mu + 2 * self.ty / sqrt(II) * ( 1 - exp(-sqrt(II) / self.eps))
def expFun(): sqrterm = E_v_n_aux expterm = (self.Phi / self.T_h) * (1 - pow(self.B, 0.25) * fn.sqrt(sqrterm)) return fn.exp(expterm)
d1 = fe.Expression('x[0] - 0', degree=1) #d2 = fe.Expression('1 - x[0]', degree=1) #d3 = fe.Expression('1 - x[1]', degree=1) d = fe.Expression('x[1] - 0', degree=1) #d = fe.Constant(10) #d = Min(d1, d4) xi = nu_trial / nu fv1 = xi**3 / (xi**3 + Cv1 * Cv1 * Cv1) #fv1 = fe.elem_pow(xi, 3) / (fe.elem_pow(xi, 3) + Cv1 * Cv1 * Cv1) fv2 = 1 - xi / (1 + xi * fv1) ft2 = Ct3 * fe.exp(-Ct4 * xi * xi) Omega = fe.Constant(0.5) * (fe.grad(u) - fe.grad(u).T) S = fe.sqrt(EPSILON + fe.Constant(2) * fe.inner(Omega, Omega)) Stilde = S + nu_trial / (kappa * kappa * d * d) * fv2 ft2 = Ct3 * fe.exp(fe.Constant(-1) * Ct4 * xi * xi) r = Min(nu_trial / (Stilde * kappa * kappa * d * d), 10) g = r + Cw2 * (r * r * r * r * r * r - r) fw = fe.elem_pow( EPSILON + abs(g * ((1 + Cw3 * Cw3 * Cw3 * Cw3 * Cw3 * Cw3) / (g * g * g * g * g * g + Cw3 * Cw3 * Cw3 * Cw3 * Cw3 * Cw3))), fe.Constant(1. / 6.)) ns_conv = fe.inner(v, fe.grad(u) * u) * fe.dx ns_press = p * fe.div(v) * fe.dx s = fe.grad(u) + fe.grad(u).T if MODEL: ns_tv = fe.inner((fv1 * nu_trial) * fe.grad(v), fe.grad(u)) * fe.dx
def apparent_viscosity(self, II): return 2 * self.mu + 2 * self.ty / (pi * sqrt(II)) * atan( sqrt(II) / self.eps)
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