Add a monolithic incompressible Navier-Stokes example and a fully implicit mass sweeper

pySDC/projects/StroemungsRaum/problem_classes/NavierStokes_2D_monolithic_FEniCS.py

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+import logging
+import os
+
+import dolfin as df
+import numpy as np
+
+from pySDC.core.problem import Problem
+from pySDC.implementations.datatype_classes.fenics_mesh import fenics_mesh
+
+
+class fenics_NSE_2D_Monolithic(Problem):
+    r"""
+    Example implementing the forced two-dimensional incompressible Navier-Stokes equations with
+    time-dependent Dirichlet boundary conditions
+
+    .. math::
+        \frac{\partial u}{\partial t} = - u \cdot \nabla u  + \nu \nabla u - \nabla p + f
+                      0 = \nabla \cdot u
+
+    for :math:`x \in \Omega`, where the forcing term :math:`f` is defined by
+
+    .. math::
+        f(x, t) = (0, 0).
+
+    This implementation follows a monolithic approach, where velocity and pressure are
+    solved simultaneously in a coupled system using a mixed finite element formulation.
+
+    Boundary conditions are applied on subsets of the boundary:
+    - no-slip on channel walls and cylinder surface,
+    - a time-dependent inflow profile at the inlet,
+    - pressure condition at the outflow.
+
+    In this class the problem is implemented in the way that the spatial part is solved using ``FEniCS`` [1]_. Hence, the problem
+    is reformulated to the *weak formulation*
+
+    .. math:
+        \int_\Omega u_t v\,dx = - \int_\Omega u \cdot \nabla u v\,dx -  \nu \int_\Omega \nabla u \nabla v\,dx + \int_\Omega p \nabla \cdot v\,dx + \int_\Omega f v\,dx
+        \int_\Omega \nabla \cdot u q\,dx = 0
+
+    Parameters
+    ----------
+    t0 : float, optional
+        Starting time.
+    order : int, optional
+        Defines the order of the elements in the function space.
+    nu : float, optional
+        Diffusion coefficient :math:`\nu`.
+
+    Attributes
+    ----------
+    V : FunctionSpace
+        Defines the function space of the trial and test functions for velocity.
+    Q : FunctionSpace
+        Defines the function space of the trial and test functions for pressure.
+    W : FunctionSpace
+        Defines the mixed function space for the coupled velocity-pressure system.
+    M : scalar, vector, matrix or higher rank tensor
+        Denotes the expression :math:`\int_\Omega u_t v\,dx`.
+    Mf : scalar, vector, matrix or higher rank tensor
+        Denotes the expression :math:`\int_\Omega u v\,dx + \int_\Omega p q\,dx`.
+    g : Expression
+        The forcing term :math:`f` in the heat equation.
+    bc : DirichletBC
+        Denotes the time-dependent Dirichlet boundary conditions.
+    bc_hom : DirichletBC
+        Denotes the homogeneous Dirichlet boundary conditions, potentially required for fixing the residual
+    fix_bc_for_residual: boolean
+        flag to indicate that the residual requires special treatment due to boundary conditions
+
+    References
+    ----------
+    .. [1] The FEniCS Project Version 1.5. M. S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg,
+        C. Richardson, J. Ring, M. E. Rognes, G. N. Wells. Archive of Numerical Software (2015).
+    .. [2] Automated Solution of Differential Equations by the Finite Element Method. A. Logg, K.-A. Mardal, G. N.
+        Wells and others. Springer (2012).
+    """
+
+    dtype_u = fenics_mesh
+    dtype_f = fenics_mesh
+
+    df.set_log_active(False)
+
+    def __init__(self, t0=0.0, order=2, nu=0.001):
+
+        # set logger level for FFC and dolfin
+        logging.getLogger('FFC').setLevel(logging.WARNING)
+        logging.getLogger('UFL').setLevel(logging.WARNING)
+
+        # set solver and form parameters
+        df.parameters["form_compiler"]["optimize"] = True
+        df.parameters["form_compiler"]["cpp_optimize"] = True
+        df.parameters['allow_extrapolation'] = True
+
+        # load mesh
+        path = f"{os.path.dirname(__file__)}/../meshs/"
+        mesh = df.Mesh(path + '/cylinder.xml')
+
+        # define function spaces for future reference (Taylor-Hood)
+        P2 = df.VectorElement("P", mesh.ufl_cell(), order)
+        P1 = df.FiniteElement("P", mesh.ufl_cell(), order - 1)
+        TH = df.MixedElement([P2, P1])
+        self.W = df.FunctionSpace(mesh, TH)
+        self.V = df.FunctionSpace(mesh, P2)
+        self.Q = df.FunctionSpace(mesh, P1)
+
+        # print the number of DoFs for debugging purposes
+        tmp = df.Function(self.W)
+        self.logger.debug('DoFs on this level:', len(tmp.vector()[:]))
+
+        super().__init__(self.W)
+        self._makeAttributeAndRegister('t0', 'order', 'nu', localVars=locals(), readOnly=True)
+
+        # Trial and test function for the Mixed FE space
+        self.u, self.p = df.TrialFunctions(self.W)
+        self.v, self.q = df.TestFunctions(self.W)
+
+        # velocity mass matrix
+        a_M = df.inner(self.u, self.v) * df.dx
+        self.M = df.assemble(a_M)
+
+        # full mass matrix
+        a_Mf = df.inner(self.u, self.v) * df.dx + df.inner(self.p, self.q) * df.dx
+        self.Mf = df.assemble(a_Mf)
+
+        # define the time-dependent inflow profile as an Expression
+        Uin = '4.0*1.5*sin(pi*t/8)*x[1]*(0.41 - x[1]) / pow(0.41, 2)'
+        self.u_in = df.Expression((Uin, '0'), pi=np.pi, t=t0, degree=self.order)
+
+        # define boundaries
+        inflow = 'near(x[0], 0)'
+        outflow = 'near(x[0], 2.2)'
+        walls = 'near(x[1], 0) || near(x[1], 0.41)'
+        cylinder = 'on_boundary && x[0]>0.1 && x[0]<0.3 && x[1]>0.1 && x[1]<0.3'
+
+        # define boundary conditions
+        bc_in = df.DirichletBC(self.W.sub(0), self.u_in, inflow)
+        bc_out = df.DirichletBC(self.W.sub(1), 0, outflow)
+        bc_walls = df.DirichletBC(self.W.sub(0), (0, 0), walls)
+        bc_cylinder = df.DirichletBC(self.W.sub(0), (0, 0), cylinder)
+        self.bc = [bc_cylinder, bc_walls, bc_out, bc_in]
+
+        # homogeneous boundary conditions for fixing the residual
+        bc_hom_u = df.DirichletBC(self.W.sub(0), df.Constant((0, 0)), 'on_boundary')
+        bc_hom_p = df.DirichletBC(self.W.sub(1), df.Constant(0), 'on_boundary')
+        self.bc_hom = [bc_hom_u, bc_hom_p]
+        self.fix_bc_for_residual = True
+
+        # define measure for drag and lift computation
+        Cylinder = df.CompiledSubDomain(cylinder)
+        CylinderBoundary = df.MeshFunction("size_t", mesh, mesh.topology().dim() - 1, 0)
+        Cylinder.mark(CylinderBoundary, 1)
+        self.dsc = df.Measure("ds", domain=mesh, subdomain_data=CylinderBoundary, subdomain_id=1)
+
+        # set forcing term as expression
+        self.g = df.Expression(('0', '0'), a=np.pi, b=self.nu, t=self.t0, degree=self.order)
+
+        # initialize XDMF files for velocity and pressure if needed
+        self.xdmffile_p = None
+        self.xdmffile_u = None
+
+    def solve_system(self, rhs, factor, u0, t):
+        r"""
+        Dolfin's nonlinear solver for :
+
+        (u,v) + factor * (u \cdot \nabla u, v) + factor * \nu (\nabla u, \nabla v) - factor * (p, \nabla \cdot v) - factor * (g, v) - factor * (div(u), q) = (rhs_u, v) + (rhs_p, q)
+
+        Parameters
+        ----------
+        rhs : dtype_f
+            Right-hand side for the nonlinear system.
+        factor : float
+            Abbrev. for the node-to-node stepsize (or any other factor required).
+        u0 : dtype_u
+            Initial guess for the iterative solver (not used here so far).
+        t : float
+            Current time.
+
+        Returns
+        -------
+        w : dtype_u
+            Solution.
+        """
+        # introduce the coupled solution vector for velocity and pressure
+        w = self.dtype_u(u0)
+        u, p = df.split(w.values)
+
+        # get the SDC right-hand side
+        rhs = self.__invert_mass_matrix(rhs)
+        rhs_u, rhs_p = df.split(rhs.values)
+
+        # update time in Boundary conditions
+        self.u_in.t = t
+
+        # get the forcing term
+        self.g.t = t
+        g = df.interpolate(self.g, self.V)
+
+        # build the variational form for the coupled system
+        F = df.dot(u, self.v) * df.dx
+        F += factor * df.dot(df.dot(u, df.nabla_grad(u)), self.v) * df.dx
+        F += factor * self.nu * df.inner(df.nabla_grad(u), df.nabla_grad(self.v)) * df.dx
+        F -= factor * df.dot(p, df.div(self.v)) * df.dx
+        F -= factor * df.dot(g, self.v) * df.dx
+        F -= factor * df.dot(df.div(u), self.q) * df.dx
+        F -= df.dot(rhs_u, self.v) * df.dx
+        F -= df.dot(rhs_p, self.q) * df.dx
+
+        # solve the nonlinear system using Newton's method
+        df.solve(F == 0, w.values, self.bc, solver_parameters={"newton_solver": {"absolute_tolerance": 1e-15}})
+
+        return w
+
+    def eval_f(self, w, t):
+        """
+        Routine to evaluate both parts of the right-hand side of the problem.
+
+        Parameters
+        ----------
+        w : dtype_u
+            Current values of the numerical solution.
+        t : float
+            Current time at which the numerical solution is computed.
+
+        Returns
+        -------
+        f : dtype_f
+            The right-hand side.
+        """
+
+        f = self.dtype_f(self.W)
+
+        u, p = df.split(w.values)
+
+        # get the forcing term
+        self.g.t = t
+        g = self.dtype_f(df.interpolate(self.g, self.V), val=self.V)
+
+        F = -1.0 * df.dot(df.dot(u, df.nabla_grad(u)), self.v) * df.dx
+        F -= self.nu * df.inner(df.nabla_grad(u), df.nabla_grad(self.v)) * df.dx
+        F += df.dot(p, df.div(self.v)) * df.dx
+        F += df.dot(g.values, self.v) * df.dx
+        F += df.dot(df.div(u), self.q) * df.dx
+
+        f.values.vector()[:] = df.assemble(F)
+
+        return f
+
+    def apply_mass_matrix(self, w):
+        r"""
+        Routine to apply velocity mass matrix.
+
+        Parameters
+        ----------
+        w : dtype_u
+            Current values of the numerical solution.
+
+        Returns
+        -------
+        me : dtype_u
+            The product :math:`M \vec{w}`.
+        """
+
+        me = self.dtype_u(self.W)
+        self.M.mult(w.values.vector(), me.values.vector())
+
+        return me
+
+    def __invert_mass_matrix(self, w):
+        r"""
+        Helper routine to invert the full mass matrix Mf.
+
+        Parameters
+        ----------
+        w : dtype_u
+            Current values of the numerical solution.
+
+        Returns
+        -------
+        me : dtype_u
+            The product :math:`Mf^{-1} \vec{w}`.
+        """
+
+        me = self.dtype_u(self.W)
+        df.solve(self.Mf, me.values.vector(), w.values.vector())
+        return me
+
+    def u_exact(self, t):
+        r"""
+        Routine to compute the exact solution at time :math:`t`.
+
+        Parameters
+        ----------
+        t : float
+            Time of the exact solution.
+
+        Returns
+        -------
+        me : dtype_u
+            Exact solution.
+        """
+        # define the exact solution
+        w = df.Function(self.W)
+
+        # assign the exact solution as an Expression
+        df.assign(w.sub(0), df.interpolate(df.Expression(('0.0', '0.0'), degree=self.order), self.V))
+        df.assign(w.sub(1), df.interpolate(df.Expression('0.0', degree=self.order - 1), self.Q))
+
+        # update time in Boundary conditions
+        self.u_in.t = t
+        [bc.apply(w.vector()) for bc in self.bc]
+
+        me = self.dtype_u(w, val=self.W)
+
+        return me
+
+    def fix_residual(self, res):
+        """
+        Applies homogeneous Dirichlet boundary conditions to the residual
+
+        Parameters
+        ----------
+        res : dtype_u
+              Residual
+        """
+        [bc.apply(res.values.vector()) for bc in self.bc_hom]
+
+        return None