#!/usr/bin/env python # -*- coding: utf-8 -*- """ *********************************************************************************** tutorial_cv_10.py DAE Tools: pyDAE module, www.daetools.com Copyright (C) Dragan Nikolic *********************************************************************************** DAE Tools is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License version 3 as published by the Free Software Foundation. DAE Tools is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with the DAE Tools software; if not, see <http://www.gnu.org/licenses/>. ************************************************************************************ """ __doc__ = """ Code verification using the Method of Exact Solutions (Rotating Gaussian Hill problem). Reference (section 4.4.6.3 Convection-Diffusion): - D. Kuzmin (2010). A Guide to Numerical Methods for Transport Equations. `PDF <http://www.mathematik.uni-dortmund.de/~kuzmin/Transport.pdf>`_ Here, a 2D transient convection-diffusion problem in a rectangular (-1,1)x(-1,1) domain is solved using the FE method: .. code-block:: none dc/dt + div(u*c) - eps*nabla(c) = 0, in Omega = (-1,1)x(-1,1) The exact solution is given by the following function: .. code-block:: none (x0, y0) = (0.0, 0.5) x_bar(t) = x0*cos(t) - y0*sin(t) y_bar(t) = -x0*sin(t) + y0*cos(t) r2(x,y,t) = (x-x_bar(t))**2 + (y-y_bar(t))**2 c_exact(x,y,t) = 1.0 / (4*pi*eps*t) * exp(-r2(x,y,t) / (4*eps*t)) The initial conditions define a Gaussian hill which is rotated counterclockwise around the point (0.0, 0.0) using the velocity field u = (-y, x). Since at t = 0 the value of c_exact is the Dirac delta function it is better to start the simulation at t > 0. Therefore, the simulation is started and t = pi/2 by reinitialising variable c to: .. code-block:: none c(x,y,pi/2) = c_exact(x,y,pi/2) At t = 5/2 pi the peak smeared by the diffusion should arrive at the starting position. Homogeneous Dirichlet boundary conditions are prescribed at all four edges: .. code-block:: none c(x,y,t) = 0.0 The mesh is a rectangle (-1,1)x(-1,1): .. image:: _static/square(-1,1)x(-1,1)-64x64.png :width: 300 px The solution plots at t = pi/2 (the initial peak) and t = 5/2pi (96x96 grid): .. image:: _static/tutorial_cv_10-results1.png :height: 400 px .. image:: _static/tutorial_cv_10-results2.png :height: 400 px Animations for 32x32 and 96x96 grids: .. image:: _static/tutorial_cv_10-animation-32x32.gif :height: 400 px .. image:: _static/tutorial_cv_10-animation-96x96.gif :height: 400 px Again, some low-magnitude oscillations in the solution appear, which are more pronounced for coarser grids. In the original example this problem was resolved using the flux linearisation technique. The normalised global errors and the order of accuracy plots (no. elements = [32x32, 64x64, 96x96, 128x128], t = 5/2pi): .. image:: _static/tutorial_cv_10-results3.png :width: 800 px """ import os, sys, numpy, json, tempfile from time import localtime, strftime import matplotlib.pyplot as plt from daetools.pyDAE import * from daetools.solvers.deal_II import * from daetools.solvers.superlu import pySuperLU # Standard variable types are defined in variable_types.py from pyUnits import m, kg, s, K, Pa, mol, J, W eps = 1E-3 (x0, y0) = (0.0, 0.5) x_bar = lambda t: x0*numpy.cos(t) - y0*numpy.sin(t) y_bar = lambda t: -x0*numpy.sin(t) + y0*numpy.cos(t) r2 = lambda x,y,t: (x-x_bar(t))**2 + (y-y_bar(t))**2 ct = lambda x,y,t: 1.0 / (4*numpy.pi*eps*t) * numpy.exp(-r2(x,y,t) / (4*eps*t)) class VelocityFunction_2D(Function_2D): def __init__(self, n_components = 1): Function_2D.__init__(self, n_components) self.m_velocity = Tensor_1_2D() def gradient(self, point, component = 0): self.m_velocity[0] = -point.y self.m_velocity[1] = point.x return self.m_velocity def vector_gradient(self, point): return [self.value(point, c) for c in range(self.n_components)] c_t = daeVariableType("c_t", unit(), 0.0, 1E20, 0, 1e-07) class modTutorial(daeModel): def __init__(self, Name, Nx, Parent = None, Description = ""): daeModel.__init__(self, Name, Parent, Description) dofs = [dealiiFiniteElementDOF_2D(name='c', description='Something', fe = FE_Q_2D(1), multiplicity=1, variableType=c_t)] self.n_components = int(numpy.sum([dof.Multiplicity for dof in dofs])) meshes_dir = os.path.join(os.path.dirname(os.path.abspath(__file__)), 'meshes') mesh_file = os.path.join(meshes_dir, 'square(-1,1)x(-1,1)-%dx%d.msh' % (Nx, Nx)) # Store the object so it does not go out of scope while still in use by daetools self.fe_system = dealiiFiniteElementSystem_2D(meshFilename = mesh_file, quadrature = QGauss_2D(3), faceQuadrature = QGauss_1D(3), dofs = dofs) self.fe_model = daeFiniteElementModel('GaussianHill', self, 'GaussianHill problem', self.fe_system) def DeclareEquations(self): daeModel.DeclareEquations(self) # Create some auxiliary objects for readability phi_i = phi_2D('c', fe_i, fe_q) phi_j = phi_2D('c', fe_j, fe_q) dphi_i = dphi_2D('c', fe_i, fe_q) dphi_j = dphi_2D('c', fe_j, fe_q) xyz = xyz_2D(fe_q) JxW = JxW_2D(fe_q) # The counterclockwise velocity field (0.5-y, x-0.5) Function<dim>::gradient wrapper. self.fun_u = VelocityFunction_2D() u_grad = function_gradient_2D("u", self.fun_u, xyz) # Boundary IDs left_edge = 0 top_edge = 1 right_edge = 2 bottom_edge = 3 dirichletBC = {} dirichletBC[left_edge] = [ ('c', adoubleConstantFunction_2D(adouble(0.0), self.n_components)), ] dirichletBC[top_edge] = [ ('c', adoubleConstantFunction_2D(adouble(0.0), self.n_components)), ] dirichletBC[right_edge] = [ ('c', adoubleConstantFunction_2D(adouble(0.0), self.n_components)), ] dirichletBC[bottom_edge] = [ ('c', adoubleConstantFunction_2D(adouble(0.0), self.n_components)), ] # FE weak form terms accumulation = (phi_i * phi_j) * JxW diffusion = (dphi_i * dphi_j) * eps * JxW convection = phi_i * (u_grad * dphi_j) * JxW source = 0.0 * JxW weakForm = dealiiFiniteElementWeakForm_2D(Aij = diffusion + convection, Mij = accumulation, Fi = source, functionsDirichletBC = dirichletBC) # Setting the weak form of the FE system will declare a set of equations: # [Mij]{dx/dt} + [Aij]{x} = {Fi} and boundary integral equations self.fe_system.WeakForm = weakForm class simTutorial(daeSimulation): def __init__(self, Nx): daeSimulation.__init__(self) self.m = modTutorial("tutorial_cv_10", Nx) self.m.Description = __doc__ self.m.fe_model.Description = __doc__ def SetUpParametersAndDomains(self): pass def SetUpVariables(self): setFEInitialConditions(self.m.fe_model, self.m.fe_system, 'c', 0.0) def Run(self): # Get coordinates for every DOF sp = self.m.fe_system.GetDOFSupportPoints() # Define a peak def ic(internal_index, overall_index): p = sp[overall_index] return ct(p.x, p.y, numpy.pi/2) # Integrate for pi/2, c(x,y) = 0 everywhere self.Log.Message("Integrating for pi/2 seconds ... ", 0) time = self.IntegrateForTimeInterval(numpy.pi/2, eDoNotStopAtDiscontinuity) self.ReportData(self.CurrentTime) self.Log.SetProgress(int(100.0 * self.CurrentTime/self.TimeHorizon)); # Set the initial peak at t = pi/2 self.Log.Message("Setting an initial peak at t = pi/2 seconds ... ", 0) setFEInitialConditions(self.m.fe_model, self.m.fe_system, 'c', ic) self.Reinitialize() self.ReportData(self.CurrentTime) daeSimulation.Run(self) # Setup everything manually and run in a console def simulate(Nx): # Create Log, Solver, DataReporter and Simulation object log = daePythonStdOutLog() daesolver = daeIDAS() datareporter = daeDelegateDataReporter() simulation = simTutorial(Nx) daesolver.RelativeTolerance = 1E-6 # Do no print progress log.PrintProgress = False lasolver = pySuperLU.daeCreateSuperLUSolver() daesolver.SetLASolver(lasolver) simName = simulation.m.Name + 'Nx=%d'%Nx + strftime(" [%d.%m.%Y %H:%M:%S]", localtime()) results_folder = os.path.join(os.path.dirname(os.path.abspath(__file__)), 'tutorial_cv_10-results(Nx=%d)' % Nx) # Create three data reporters: # 1. DealII feDataReporter = simulation.m.fe_system.CreateDataReporter() datareporter.AddDataReporter(feDataReporter) if not feDataReporter.Connect(results_folder, simName): sys.exit() # 2. TCP/IP tcpipDataReporter = daeTCPIPDataReporter() datareporter.AddDataReporter(tcpipDataReporter) if not tcpipDataReporter.Connect("", simName): sys.exit() # 3. Data dr = daeNoOpDataReporter() datareporter.AddDataReporter(dr) # Enable reporting of all variables simulation.m.SetReportingOn(True) # Set the time horizon and the reporting interval simulation.ReportingInterval = 5.0/2.0*numpy.pi / 100 simulation.TimeHorizon = 5.0/2.0*numpy.pi # Initialize the simulation simulation.Initialize(daesolver, datareporter, log) # Save the model report and the runtime model report #simulation.m.fe_model.SaveModelReport(simulation.m.Name + ".xml") #simulation.m.fe_model.SaveRuntimeModelReport(simulation.m.Name + "-rt.xml") # Solve at time=0 (initialization) simulation.SolveInitial() # Run simulation.Run() simulation.Finalize() ########################################### # Plots and data # ########################################### results = dr.Process.dictVariables cvar = results[simulation.m.Name + '.GaussianHill.c'] points = cvar.Domains[0].Points c = cvar.Values[-1,:] # 2D array [t,omega] sp = simulation.m.fe_system.GetDOFSupportPoints() Nsp = len(sp) c_exact = numpy.zeros(Nsp) for i, p in enumerate(sp): c_exact[i] = ct(p.x, p.y, 5.0/2.0*numpy.pi) return points, c, c_exact def run(**kwargs): Nxs = numpy.array([32, 64, 96, 128]) n = len(Nxs) L = 1.0 hs = L / Nxs E = numpy.zeros(n) C = numpy.zeros(n) p = numpy.zeros(n) E2 = numpy.zeros(n) # The normalised global errors for i,Nx in enumerate(Nxs): points, numerical_sol, manufactured_sol = simulate(int(Nx)) E[i] = numpy.sqrt((1.0/Nx) * numpy.sum((numerical_sol-manufactured_sol)**2)) # Order of accuracy for i,Nx in enumerate(Nxs): p[i] = numpy.log(E[i]/E[i-1]) / numpy.log(hs[i]/hs[i-1]) C[i] = E[i] / hs[i]**p[i] C2 = 400.0 # constant for the second order slope line (to get close to the actual line) E2 = C2 * hs**2 # E for the second order slope fontsize = 14 fontsize_legend = 11 fig = plt.figure(figsize=(10,4), facecolor='white') fig.canvas.set_window_title('The Normalised global errors and the Orders of accuracy (Nelems = %s) (cv_10)' % Nxs.tolist()) ax = plt.subplot(121) plt.figure(1, facecolor='white') plt.loglog(hs, E, 'ro', label='E(h)') plt.loglog(hs, E2, 'b-', label='2nd order slope') plt.xlabel('h', fontsize=fontsize) plt.ylabel('||E||', fontsize=fontsize) plt.legend(fontsize=fontsize_legend) #plt.xlim((0.04, 0.11)) ax = plt.subplot(122) plt.figure(1, facecolor='white') plt.semilogx(hs[1:], p[1:], 'rs-', label='Order of Accuracy (p)') plt.xlabel('h', fontsize=fontsize) plt.ylabel('p', fontsize=fontsize) plt.legend(fontsize=fontsize_legend) #plt.xlim((0.04, 0.075)) #plt.ylim((2.0, 2.04)) plt.tight_layout() plt.show() if __name__ == "__main__": run()