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url = {https://arxiv.org/abs/2106.13877}
}
+% ------------------------------------
+% Step 85
+% ------------------------------------
+
+@article{saye_2015,
+ title = {High-{Order} {Quadrature} {Methods} for {Implicitly} {Defined} {Surfaces} and {Volumes} in {Hyperrectangles}},
+ volume = {37},
+ issn = {1064-8275, 1095-7197},
+ url = {http://epubs.siam.org/doi/10.1137/140966290},
+ doi = {10.1137/140966290},
+ language = {en},
+ number = {2},
+ urldate = {2016-01-27},
+ journal = {SIAM Journal on Scientific Computing},
+ author = {Saye, R. I.},
+ month = jan,
+ year = {2015},
+ pages = {A993--A1019}
+}
+
+@article{burman_hansbo_2012,
+ title = {Fictitious domain finite element methods using cut elements: {II}. {A} stabilized {Nitsche} method},
+ volume = {62},
+ issn = {01689274},
+ url = {http://linkinghub.elsevier.com/retrieve/pii/S0168927411000298},
+ doi = {10.1016/j.apnum.2011.01.008},
+ language = {en},
+ number = {4},
+ urldate = {2015-12-17},
+ journal = {Applied Numerical Mathematics},
+ author = {Burman, Erik and Hansbo, Peter},
+ month = apr,
+ year = {2012},
+ pages = {328--341}
+}
+
+@article{cutfem_2015,
+ title = {{CutFEM}: {Discretizing} geometry and partial differential equations},
+ volume = {104},
+ issn = {00295981},
+ url = {http://doi.wiley.com/10.1002/nme.4823},
+ doi = {10.1002/nme.4823},
+ language = {en},
+ number = {7},
+ urldate = {2016-01-27},
+ journal = {International Journal for Numerical Methods in Engineering},
+ author = {Burman, Erik and Claus, Susanne and Hansbo, Peter and Larson, Mats G. and Massing, André},
+ month = nov,
+ year = {2015},
+ pages = {472--501}
+}
+
+
% ------------------------------------
% References used elsewhere
% ------------------------------------
* <td> Solving the fourth-order biharmonic equation using a lifting operator approach.
* </td></tr>
*
+ * <tr valign="top">
+ * <td>step-85</td>
+ * <td> Solving the Poisson equation using the cut finite element method.
+ * </td></tr>
+ *
* </table>
*
*
* </td>
* <td>
* step-60,
- * step-70
+ * step-70,
+ * step-85
* </td>
* </tr>
*
--- /dev/null
+##
+# CMake script for the step-85 tutorial program:
+##
+
+# Set the name of the project and target:
+SET(TARGET "step-85")
+
+# Declare all source files the target consists of. Here, this is only
+# the one step-X.cc file, but as you expand your project you may wish
+# to add other source files as well. If your project becomes much larger,
+# you may want to either replace the following statement by something like
+# FILE(GLOB_RECURSE TARGET_SRC "source/*.cc")
+# FILE(GLOB_RECURSE TARGET_INC "include/*.h")
+# SET(TARGET_SRC ${TARGET_SRC} ${TARGET_INC})
+# or switch altogether to the large project CMakeLists.txt file discussed
+# in the "CMake in user projects" page accessible from the "User info"
+# page of the documentation.
+SET(TARGET_SRC
+ ${TARGET}.cc
+ )
+
+# Usually, you will not need to modify anything beyond this point...
+
+CMAKE_MINIMUM_REQUIRED(VERSION 3.1.0)
+
+FIND_PACKAGE(deal.II 10.0.0
+ HINTS ${deal.II_DIR} ${DEAL_II_DIR} ../ ../../ $ENV{DEAL_II_DIR}
+ )
+IF(NOT ${deal.II_FOUND})
+ MESSAGE(FATAL_ERROR "\n"
+ "*** Could not locate a (sufficiently recent) version of deal.II. ***\n\n"
+ "You may want to either pass a flag -DDEAL_II_DIR=/path/to/deal.II to cmake\n"
+ "or set an environment variable \"DEAL_II_DIR\" that contains this path."
+ )
+ENDIF()
+
+DEAL_II_INITIALIZE_CACHED_VARIABLES()
+PROJECT(${TARGET})
+DEAL_II_INVOKE_AUTOPILOT()
--- /dev/null
+step-12 step-46
--- /dev/null
+<i>
+This program was contributed by Simon Sticko.
+
+The material is based upon work partially supported by
+eSSENCE of e-Science and the Swedish Research Council
+under grants 2014-6088 (Kreiss) and 2017-05038 (Massing).
+</i>
+
+<a name="Intro"></a>
+<h1>Introduction</h1>
+
+<h3>The Cut Finite Element Method</h3>
+
+In this example, we show how to use the cut finite element method (CutFEM) in deal.II.
+For illustration, we want to solve the simplest possible problem,
+so we again consider Poisson's equation:
+@f{align*}
+ -\Delta u &= f \qquad && \text{in }\, \Omega,
+ \\
+ u &= u_D \qquad && \text{on }\, \Gamma = \partial \Omega,
+@f}
+where we choose $f(x) = 4$ and $u_D(x) = 1$.
+CutFEM is an immersed method.
+In this context,
+"immersed" means that the mesh is unfitted to the geometry of the domain, $\Omega$.
+Instead, $\Omega$ floats freely on top of a uniform background mesh, $\mathcal{T}^h$.
+@image html step-85-background-mesh.svg
+Since we no longer use the mesh to describe the geometry of the domain,
+we need some other way to represent it.
+This can be done in several ways but here we assume that $\Omega$ is described by a level set function,
+$\psi : \mathbb{R}^{\text{dim}} \to \mathbb{R}$ such that
+@f{align*}
+ \Omega &= \{x \in \mathbb{R}^{\text{dim}} : \psi(x) < 0 \}, \\
+ \Gamma &= \{x \in \mathbb{R}^{\text{dim}} : \psi(x) = 0 \}.
+@f}
+For simplicity, we choose $\Omega$ to be a unit disk, so that
+@f{equation*}
+ \psi(x) = \| x \| - 1.
+@f}
+As can be seen from the figure below,
+the level set function is negative for points in $\Omega$,
+zero on the boundary, and positive everywhere else.
+@image html step-85-level-set.png
+To solve this problem,
+we want to distribute degrees of freedom over the smallest submesh, $\mathcal{T}_\Omega^h$,
+that completely covers the domain:
+@f{equation*}
+ \mathcal{T}_\Omega^h = \{ T \in \mathcal{T}^{h} : T \cap \Omega \neq \emptyset \}.
+@f}
+This is usually referred to as the "active mesh".
+@image html step-85-active-mesh.svg
+The finite element space where we want to find our numerical solution, $u_h$, is now
+@f{equation*}
+ V_\Omega^h = \{ v \in C(\mathcal{N}_\Omega^h) : v \in Q_p(T), \, T \in \mathcal{T}_\Omega^h \},
+@f}
+where
+@f{equation*}
+ \mathcal{N}_\Omega^h = \bigcup_{T \in \mathcal{T}_\Omega^h} \overline{T},
+@f}
+and $\overline{T}$ denotes the closure of $T$.
+The set $\mathcal{N}_\Omega^h$ is sometimes referred to as the "fictitious domain".
+Since $\Omega \subset \mathcal{N}_\Omega^h$,
+we see that the numerical solution is defined over a slightly larger region than the analytical solution.
+
+In this type of immersed finite element method,
+the standard way to apply boundary conditions is using Nitsche's method.
+Multiplying the PDE with a test function, $v_h \in V_\Omega^h$,
+and integrating by parts over $\Omega$, as usual, gives us
+@f{equation*}
+ (\nabla u_h, \nabla v_h)_\Omega - (\partial_n u_h, v_h)_\Gamma = (f,v)_\Omega.
+@f}
+Let $\gamma_D > 0$ be a scalar penalty parameter and let $h$ be some measure of the local cell size.
+We now note that the following terms are consistent with the Dirichlet boundary condition:
+@f{align*}
+ -(u_h, \partial_n v_h)_\Gamma &= -(u_D, \partial_n v_h)_\Gamma, \\
+ \left (\frac{\gamma_D}{h} u_h, v_h \right )_\Gamma &= \left (\frac{\gamma_D}{h}u_D, v_h \right )_\Gamma.
+@f}
+Thus, we can add these to the weak formulation to enforce the boundary condition.
+This leads to the following weak formulation:
+Find $u_h \in V_\Omega^h$ such that
+@f{equation*}
+ a_h(u_h, v_h) = L_h(v_h), \quad \forall v_h \in V_\Omega^h,
+@f}
+where
+@f{align*}
+ a_h(u_h, v_h) &= (\nabla u_h, \nabla v_h)_\Omega
+ - (\partial_n u_h, v_h)_\Gamma
+ - (u_h, \partial_n v_h)_\Gamma
+ + \left (\frac{\gamma_D}{h} u_h, v_h \right )_\Gamma,
+ \\
+ L_h(v_h) &= (f,v)_\Omega
+ + \left (u_D, \frac{\gamma_D}{h} v_h -\partial_n v_h \right )_\Gamma.
+@f}
+In this formulation, there is one big difference,
+compared to a standard boundary-fitted finite element method.
+On each cell,
+we need to integrate over the part of the domain and the part of the boundary that falls within the cell.
+Thus, on each cell intersected by $\Gamma$,
+we need special quadrature rules that only integrate over these parts of the cell,
+that is, over $T \cap \Omega$ and $T \cap \Gamma$.
+@image html immersed_quadratures.svg
+Since $\Omega \cap T$ is the part of the cell that lies inside the domain,
+we shall refer to the following regions
+@f{align*}
+ \{x \in T : \psi(x) < 0 \}, \\
+ \{x \in T : \psi(x) > 0 \}, \\
+ \{x \in T : \psi(x) = 0 \},
+@f}
+as the "inside", "outside" and the "surface region" of the cell $T$.
+
+The above finite element method that uses the bilinear form $a_h(\cdot, \cdot)$
+is sometimes referred to as the "naive weak formulation"
+because it suffers from the so-called "small cut problem".
+Depending on how $\Omega$ is located relative to $\mathcal{T}_h$,
+a cut between a cell, $T \in \mathcal{T}_h$, and $\Omega$ can become arbitrarily small:
+$|\Omega \cap T | \rightarrow 0$.
+For Neumann boundary conditions,
+the consequence is that the stiffness matrix can become arbitrarily ill-conditioned
+as the cut-size approaches zero.
+For a Dirichlet condition, the situation is even worse.
+For any finite choice of Nitsche constant, $\gamma_D$,
+the bilinear form $a_h(\cdot,\cdot)$ loses coercivity as the size of a cell cut approaches zero.
+This makes the above weak formulation essentially useless
+because as we refine we typically can not control how the cells intersect $\Gamma$.
+One way to avoid this problem is to add a so-called ghost penalty term, $g_h$,
+to the weak formulation (see e.g. @cite burman_hansbo_2012 and @cite cutfem_2015).
+This leads to the stabilized cut finite element method,
+which reads: Find $u_h \in V_\Omega^h$ such that
+@f{equation*}
+ A_h(u_h, v_h) = L_h(v_h), \quad \forall v_h \in V_\Omega^h,
+@f}
+where
+@f{equation*}
+ A_h(u_h,v_h) = a_h(u_h,v_h) + g_h(u_h, v_h).
+@f}
+The point of this ghost penalty is that it makes the numerical method essentially independent
+of how $\Omega$ relates to the background mesh.
+In particular, $A_h$ can be shown to be continuous and coercive,
+with constants that do not depend on how $\Omega$ intersects $\mathcal{T}^h$.
+To define the ghost penalty, let $\mathcal{T}_\Gamma^h$ be the set of intersected cells:
+@f{equation*}
+ \mathcal{T}_{\Gamma}^h = \{ T \in \mathcal{T}_{\Omega}^{h} : T \cap \Gamma \neq \emptyset \},
+@f}
+and let $\mathcal{F}_h$ denote the interior faces of the intersected cells in the active mesh:
+@f{equation*}
+ \mathcal{F}_h = \{ F = \overline{T}_+ \cap \overline{T}_- : \,
+ T_+ \in \mathcal{T}_{\Gamma}^h, \,
+ T_- \in \mathcal{T}_{\Omega}^h
+ \}.
+@f}
+@image html step-85-ghost-faces.svg
+The ghost penalty acts on these faces and reads
+@f{equation*}
+ g_h(u_h,v_h) = \gamma_A \sum_{F \in \mathcal{F}_h} g_F(u_h, v_h),
+@f}
+where $g_F$ is the face-wise ghost penalty:
+@f{equation*}
+ g_F(u_h, v_h) = \gamma_A \sum_{k=0}^p \left(\frac{h_F^{2k-1}}{k!^2}[\partial_n^k u_h], [\partial_n^k v_h] \right)_F.
+@f}
+Here, $\gamma_A$ is a penalty parameter and $h_F$ is some measure of the face size.
+We see that $g_F$ penalizes the jumps in the face-normal derivatives, $\partial_n^k$,
+over $F = \overline{T}_+ \cap \overline{T}_-$.
+Since we include all normal derivatives up to the polynomial degree,
+we weakly force the piecewise polynomial to behave as a single polynomial over $\overline{T}_+ \cup \overline{T}_-$.
+Hand-wavingly speaking,
+this is the reason why we obtain a cut-independent method when we enforce $g_F(u_h, v_h) = 0$ over the faces in $\mathcal{F}_h$.
+Here, we shall use a continuous space of $Q_1$-elements,
+so the ghost penalty is reduced to
+@f{equation*}
+ g_h(u_h,v_h) = \gamma_A \sum_{F \in \mathcal{F}_h} (h_F [\partial_n u_h], [\partial_n v_h])_F.
+@f}
+
+<h3>Discrete Level Set Function</h3>
+A typical use case of a level set method is a problem where the domain is advected in a velocity field,
+such that the domain deforms with time.
+For such a problem,
+one would typically solve for an approximation of the level set function, $\psi_h \in V^h$,
+in a separate finite element space over the whole background mesh:
+@f{equation*}
+ V^h = \{ v \in C(\mathcal{N}^h) : v \in Q_p(T), \, T \in \mathcal{T}^h \},
+@f}
+where $\mathcal{N}^h = \bigcup_{T \in \mathcal{T}^h} \overline{T}$.
+Even if we solve a much simpler problem with a stationary domain in this tutorial,
+we shall, just to illustrate, still use a discrete level set function for the Poisson problem.
+Technically,
+this is a so-called "variational crime" because we are actually not using the bilinear form $a_h$ but instead
+@f{equation*}
+ a_h^\star(u_h, v_h) = (\nabla u_h, \nabla v_h)_{\Omega_h}
+ - (\partial_n u_h, v_h)_{\Gamma_h} + \ldots
+@f}
+This is an approximation of $a_h$ since we integrate over the approximations of the geometry that we get via the discrete level set function:
+@f{align*}
+ \Omega_h &= \{x \in \mathbb{R}^{\text{dim}} : \psi_h(x) < 0 \}, \\
+ \Gamma_h &= \{x \in \mathbb{R}^{\text{dim}} : \psi_h(x) = 0 \}.
+@f}
+Using $\Omega_h$ instead of $\Omega$ in the method will give rise to a larger error in the numerical solution.
+This is often referred to as the "geometrical error".
+However, when the same element order, $p$, is used in $V^h$ and $V_\Omega^h$,
+one can often show that the method gives the same order of convergence
+as if the exact domain would have been used.
+Furthermore, deal.II allows us to independently choose a more accurate geometry representation
+with a higher-order level set function, compared to the function space for solving the Poisson equation.
+
+<h3>The MeshClassifier Class</h3>
+Even if we have used $\mathcal{T}_\Omega^h$ to define the finite element space,
+we will not create this submesh in practice. As in step-46, we shall
+instead use the hp-framework. To create $V_\Omega^h$, we first add an FE_Q and an
+FE_Nothing element to an hp::FECollection. We then iterate over each cell,
+$T$, and depending on whether $T$ belongs to $\mathcal{T}_\Omega^h$ or not,
+we set the active_fe_index to either 0 or 1. To do so, we need to
+determine if a given cell is in $\mathcal{T}_\Omega^h$ or not.
+For this purpose, we will use the class NonMatching::MeshClassifier.
+The MeshClassifier takes the discrete level set function,
+described as a (DoFHandler, Vector)-pair, as arguments to its constructor:
+@code
+ MeshClassifier(const DoFHandler<dim> &level_set_dof_handler,
+ const VectorType & level_set);
+@endcode
+When we call the reclassify() function on an object of this class,
+each active cell and face is associated with one of the values
+{inside, outside, intersected} of the enum NonMatching::LocationToLevelSet.
+Here, "inside" means that the level set function is negative over the whole cell
+so that it lies completely inside the domain.
+Analogously, "outside" means that $\psi$ is positive over the whole cell,
+and "intersected" means that $\psi$ varies in sign over $T$
+so that the zero-contour of $\psi$ goes through $T$.
+
+| LocationToLevelSet | $\psi(x)$ for $x \in T$ | Relation to $\Omega$ |
+|:------------------:|:------------------------:|:------------------------------:|
+| inside | $\psi(x) < 0$ | $T \cap \Omega = T$ |
+| outside | $0 < \psi(x)$ | $T \cap \Omega = \emptyset$ |
+| intersected | $\psi(x)$ varies in sign | $T \cap \Gamma \neq \emptyset$ |
+
+Each active face is classified in the same way, according to how the sign of $\psi$ varies over the face.
+MeshClassifier lets you query this information for a given cell/face via its MeshClassifier::location_to_level_set() methods:
+@code
+ NonMatching::MeshClassifier<dim> mesh_classifier(dof_handler, level_set);
+ mesh_classifier.reclassify();
+
+ for (const auto &cell : triangulation.active_cell_iterators())
+ {
+ NonMatching::LocationToLevelSet cell_location =
+ mesh_classifier.location_to_level_set(cell);
+
+ for (const unsigned int f : cell->face_indices())
+ {
+ NonMatching::LocationToLevelSet face_location =
+ mesh_classifier.location_to_level_set(cell, f);
+ }
+ }
+@endcode
--- /dev/null
+techniques
--- /dev/null
+<h1>Results</h1>
+
+The numerical solution for one of the refinements is shown in the below figure.
+The zero-contour of the level set function is shown as a white line.
+On the intersected cells,
+we see that the numerical solution has a value also outside $\overline{\Omega}$.
+As mentioned earlier, this extension of the solution is artificial.
+
+The results of the convergence study is shown in the table below.
+We see that the $L^2$ error decreases as we refine and that the estimated
+order of convergence, EOC, is close to 2.
+
+@image html step-85-solution.png
+
+| Cycle | Mesh size | $L^2$-Error | EOC |
+|:-----:|:---------:|:-----------:|:----:|
+| 0 | 0.3025 | 8.0657e-02 | - |
+| 1 | 0.1513 | 1.8711e-02 | 2.11 |
+| 2 | 0.0756 | 4.1624e-03 | 2.17 |
+| 3 | 0.0378 | 9.3979e-04 | 2.15 |
--- /dev/null
+Solving Poisson's equation using the cut finite element method.
--- /dev/null
+/* ---------------------------------------------------------------------
+ *
+ * Copyright (C) 2021 - 2021 by the deal.II authors
+ *
+ * This file is part of the deal.II library.
+ *
+ * The deal.II library is free software; you can use it, redistribute
+ * it, and/or modify it under the terms of the GNU Lesser General
+ * Public License as published by the Free Software Foundation; either
+ * version 2.1 of the License, or (at your option) any later version.
+ * The full text of the license can be found in the file LICENSE.md at
+ * the top level directory of deal.II.
+ *
+ * ---------------------------------------------------------------------
+ */
+
+// @sect3{Include files}
+
+// The first include files have all been treated in previous examples.
+
+#include <deal.II/base/function.h>
+
+#include <deal.II/base/convergence_table.h>
+#include <deal.II/base/point.h>
+#include <deal.II/base/quadrature.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_interface_values.h>
+#include <deal.II/fe/fe_nothing.h>
+#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_update_flags.h>
+#include <deal.II/fe/fe_values.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/filtered_iterator.h>
+#include <deal.II/grid/tria.h>
+
+#include <deal.II/hp/fe_collection.h>
+#include <deal.II/hp/q_collection.h>
+
+#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/solver_control.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/sparsity_pattern.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <fstream>
+#include <vector>
+
+// The first new header contains some common level set functions.
+// For example, the spherical geometry that we use here.
+#include <deal.II/base/function_signed_distance.h>
+
+// We also need 3 new headers from the NonMatching namespace.
+#include <deal.II/non_matching/fe_immersed_values.h>
+#include <deal.II/non_matching/fe_values.h>
+#include <deal.II/non_matching/mesh_classifier.h>
+
+// @sect3{The LaplaceSolver class Template}
+// We then define the main class that solves the Laplace problem.
+
+namespace Step85
+{
+ using namespace dealii;
+
+ template <int dim>
+ class LaplaceSolver
+ {
+ public:
+ LaplaceSolver();
+
+ void run();
+
+ private:
+ void make_grid();
+
+ void setup_discrete_level_set();
+
+ void distribute_dofs();
+
+ void initialize_matrices();
+
+ void assemble_system();
+
+ void solve();
+
+ void output_results() const;
+
+ double compute_L2_error() const;
+
+ bool face_has_ghost_penalty(
+ const typename Triangulation<dim>::active_cell_iterator &cell,
+ const unsigned int face_index) const;
+
+ const unsigned int fe_degree;
+
+ const Functions::ConstantFunction<dim> rhs_function;
+ const Functions::ConstantFunction<dim> boundary_condition;
+
+ Triangulation<dim> triangulation;
+
+ // We need two separate DoFHandlers. The first manages the DoFs for the
+ // discrete level set function that describes the geometry of the domain.
+ const FE_Q<dim> fe_level_set;
+ DoFHandler<dim> level_set_dof_handler;
+ Vector<double> level_set;
+
+ // The second DoFHandler manages the DoFs for the solution of the Poisson
+ // equation.
+ hp::FECollection<dim> fe_collection;
+ DoFHandler<dim> dof_handler;
+ Vector<double> solution;
+
+ NonMatching::MeshClassifier<dim> mesh_classifier;
+
+ SparsityPattern sparsity_pattern;
+ SparseMatrix<double> stiffness_matrix;
+ Vector<double> rhs;
+ };
+
+
+
+ template <int dim>
+ LaplaceSolver<dim>::LaplaceSolver()
+ : fe_degree(1)
+ , rhs_function(4.0)
+ , boundary_condition(1.0)
+ , fe_level_set(fe_degree)
+ , level_set_dof_handler(triangulation)
+ , dof_handler(triangulation)
+ , mesh_classifier(level_set_dof_handler, level_set)
+ {}
+
+
+
+ // @sect3{Setting up the Background Mesh}
+ // We generate a background mesh with perfectly Cartesian cells. Our domain is
+ // a unit disc centered at the origin, so we need to make the background mesh
+ // a bit larger than $[-1, 1]^{\text{dim}}$ to completely cover $\Omega$.
+ template <int dim>
+ void LaplaceSolver<dim>::make_grid()
+ {
+ std::cout << "Creating background mesh" << std::endl;
+
+ GridGenerator::hyper_cube(triangulation, -1.21, 1.21);
+ triangulation.refine_global(2);
+ }
+
+
+
+ // @sect3{Setting up the Discrete Level Set Function}
+ // The discrete level set function is defined on the whole background mesh.
+ // Thus, to set up the DoFHandler for the level set function, we distribute
+ // DoFs over all elements in $\mathcal{T}_h$. We then set up the discrete
+ // level set function by interpolating onto this finite element space.
+ template <int dim>
+ void LaplaceSolver<dim>::setup_discrete_level_set()
+ {
+ std::cout << "Setting up discrete level set function" << std::endl;
+
+ level_set_dof_handler.distribute_dofs(fe_level_set);
+ level_set.reinit(level_set_dof_handler.n_dofs());
+
+ const Functions::SignedDistance::Sphere<dim> signed_distance_sphere;
+ VectorTools::interpolate(level_set_dof_handler,
+ signed_distance_sphere,
+ level_set);
+ }
+
+
+
+ // @sect3{Setting up the Finite Element Space}
+ // To set up the finite element space $V_\Omega^h$, we will use 2 different
+ // elements: FE_Q and FE_Nothing. For better readability we define an enum for
+ // the indices in the order we store them in the hp::FECollection.
+ enum ActiveFEIndex
+ {
+ lagrange = 0,
+ nothing = 1
+ };
+
+ // We then use the MeshClassifier to check LocationToLevelSet for each cell in
+ // the mesh and tell the DoFHandler to use FE_Q on elements that are inside or
+ // intersected, and FE_Nothing on the elements that are outside.
+ template <int dim>
+ void LaplaceSolver<dim>::distribute_dofs()
+ {
+ std::cout << "Distributing degrees of freedom" << std::endl;
+
+ fe_collection.push_back(FE_Q<dim>(fe_degree));
+ fe_collection.push_back(FE_Nothing<dim>());
+
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ {
+ const NonMatching::LocationToLevelSet cell_location =
+ mesh_classifier.location_to_level_set(cell);
+
+ if (cell_location == NonMatching::LocationToLevelSet::outside)
+ cell->set_active_fe_index(ActiveFEIndex::nothing);
+ else
+ cell->set_active_fe_index(ActiveFEIndex::lagrange);
+ }
+
+ dof_handler.distribute_dofs(fe_collection);
+ }
+
+
+
+ // @sect3{Sparsity Pattern}
+ // The added ghost penalty results in a sparsity pattern similar to a DG
+ // method with a symmetric-interior-penalty term. Thus, we can use the
+ // make_flux_sparsity_pattern() function to create it. However, since the
+ // ghost-penalty terms only act on the faces in $\mathcal{F}_h$, we can pass
+ // in a lambda function that tells make_flux_sparsity_pattern() over which
+ // faces the flux-terms appear. This gives us a sparsity pattern with minimal
+ // number of entries. When passing a lambda function,
+ // make_flux_sparsity_pattern requires us to also pass cell and face coupling
+ // tables to it. If the problem was vector-valued, these tables would allow us
+ // to couple only some of the vector components. This is discussed in step-46.
+ template <int dim>
+ void LaplaceSolver<dim>::initialize_matrices()
+ {
+ std::cout << "Initializing matrices" << std::endl;
+
+ const auto face_has_flux_coupling = [&](const auto & cell,
+ const unsigned int face_index) {
+ return this->face_has_ghost_penalty(cell, face_index);
+ };
+
+ DynamicSparsityPattern dsp(dof_handler.n_dofs(), dof_handler.n_dofs());
+
+ const unsigned int n_components = fe_collection.n_components();
+ Table<2, DoFTools::Coupling> cell_coupling(n_components, n_components);
+ Table<2, DoFTools::Coupling> face_coupling(n_components, n_components);
+ cell_coupling[0][0] = DoFTools::always;
+ face_coupling[0][0] = DoFTools::always;
+
+ const AffineConstraints<double> constraints;
+ const bool keep_constrained_dofs = true;
+
+ DoFTools::make_flux_sparsity_pattern(dof_handler,
+ dsp,
+ constraints,
+ keep_constrained_dofs,
+ cell_coupling,
+ face_coupling,
+ numbers::invalid_subdomain_id,
+ face_has_flux_coupling);
+ sparsity_pattern.copy_from(dsp);
+
+ stiffness_matrix.reinit(sparsity_pattern);
+ solution.reinit(dof_handler.n_dofs());
+ rhs.reinit(dof_handler.n_dofs());
+ }
+
+
+
+ // The following function describes which faces are part of the set
+ // $\mathcal{F}_h$. That is, it returns true if the face of the incoming cell
+ // belongs to the set $\mathcal{F}_h$.
+ template <int dim>
+ bool LaplaceSolver<dim>::face_has_ghost_penalty(
+ const typename Triangulation<dim>::active_cell_iterator &cell,
+ const unsigned int face_index) const
+ {
+ if (cell->at_boundary(face_index))
+ return false;
+
+ const NonMatching::LocationToLevelSet cell_location =
+ mesh_classifier.location_to_level_set(cell);
+
+ const NonMatching::LocationToLevelSet neighbor_location =
+ mesh_classifier.location_to_level_set(cell->neighbor(face_index));
+
+ if (cell_location == NonMatching::LocationToLevelSet::intersected &&
+ neighbor_location != NonMatching::LocationToLevelSet::outside)
+ return true;
+
+ if (neighbor_location == NonMatching::LocationToLevelSet::intersected &&
+ cell_location != NonMatching::LocationToLevelSet::outside)
+ return true;
+
+ return false;
+ }
+
+
+
+ // @sect3{Assembling the System}
+ template <int dim>
+ void LaplaceSolver<dim>::assemble_system()
+ {
+ std::cout << "Assembling" << std::endl;
+
+ const unsigned int n_dofs_per_cell = fe_collection[0].dofs_per_cell;
+ FullMatrix<double> local_stiffness(n_dofs_per_cell, n_dofs_per_cell);
+ Vector<double> local_rhs(n_dofs_per_cell);
+ std::vector<types::global_dof_index> local_dof_indices(n_dofs_per_cell);
+
+ const double ghost_parameter = 0.5;
+ const double nitsche_parameter = 5 * (fe_degree + 1) * fe_degree;
+
+ // Since the ghost penalty is similar to a DG flux term, the simplest way to
+ // assemble it is to use an FEInterfaceValues object.
+ const QGauss<dim - 1> face_quadrature(fe_degree + 1);
+ FEInterfaceValues<dim> fe_interface_values(fe_collection[0],
+ face_quadrature,
+ update_gradients |
+ update_JxW_values |
+ update_normal_vectors);
+
+
+ // As we iterate over the cells in the mesh, we would in principle have to
+ // do the following on each cell, $T$,
+ //
+ // 1. Construct one quadrature rule to integrate over the intersection with
+ // the domain, $T \cap \Omega$, and one quadrature rule to integrate over
+ // the intersection with the boundary, $T \cap \Gamma$.
+ // 2. Create FEValues-like objects with the new quadratures.
+ // 3. Assemble the local matrix using the created FEValues-objects.
+ //
+ // To make the assembly easier, we use the class NonMatching::FEValues,
+ // which does the above steps 1 and 2 for us. The algorithm @cite saye_2015
+ // that is used to generate the quadrature rules on the intersected cells
+ // uses a 1-dimensional quadrature rule as base. Thus, we pass a 1D
+ // Gauss--Legendre quadrature to the constructor of NonMatching::FEValues.
+ // On the non-intersected cells, a tensor product of this 1D-quadrature will
+ // be used.
+ //
+ // As stated in the introduction, each cell has 3 different regions: inside,
+ // surface, and outside, where the level set function in each region is
+ // negative, zero, and positive. We need an UpdateFlags variable for each
+ // such region. These are stored on an object of type
+ // NonMatching::RegionUpdateFlags, which we pass to NonMatching::FEValues.
+ const QGauss<1> quadrature_1D(fe_degree + 1);
+
+ NonMatching::RegionUpdateFlags region_update_flags;
+ region_update_flags.inside = update_values | update_gradients |
+ update_JxW_values | update_quadrature_points;
+ region_update_flags.surface = update_values | update_gradients |
+ update_JxW_values | update_quadrature_points |
+ update_normal_vectors;
+
+ NonMatching::FEValues<dim> non_matching_fe_values(fe_collection,
+ quadrature_1D,
+ region_update_flags,
+ mesh_classifier,
+ level_set_dof_handler,
+ level_set);
+
+ // As we iterate over the cells, we don't need to do anything on the cells
+ // that have FENothing elements. To disregard them we use an iterator
+ // filter.
+ for (const auto &cell :
+ dof_handler.active_cell_iterators() |
+ IteratorFilters::ActiveFEIndexEqualTo(ActiveFEIndex::lagrange))
+ {
+ local_stiffness = 0;
+ local_rhs = 0;
+
+ const double cell_side_length = cell->minimum_vertex_distance();
+
+ // First, we call the reinit function of our NonMatching::FEValues
+ // object. In the background, NonMatching::FEValues uses the
+ // MeshClassifier passed to its constructor to check if the incoming
+ // cell is intersected. If that is the case, NonMatching::FEValues calls
+ // the NonMatching::QuadratureGenerator in the background to create the
+ // immersed quadrature rules.
+ non_matching_fe_values.reinit(cell);
+
+ // After calling reinit, we can retrieve a dealii::FEValues object with
+ // quadrature points that corresponds to integrating over the inside
+ // region of the cell. This is the object we use to do the local
+ // assembly. This is similar to how hp::FEValues builds dealii::FEValues
+ // objects. However, one difference here is that the dealii::FEValues
+ // object is returned as an optional. This is a type that wraps an
+ // object that may or may not be present. This requires us to add an
+ // if-statement to check if the returned optional contains a value,
+ // before we use it. This might seem odd at first. Why does the function
+ // not just return a reference to a const FEValues<dim>? The reason is
+ // that in an immersed method, we have essentially no control of how the
+ // cuts occur. Even if the cell is formally intersected: $T \cap \Omega
+ // \neq \emptyset$, it might be that the cut is only of floating point
+ // size $|T \cap \Omega| \sim \epsilon$. When this is the case, we can
+ // not expect that the algorithm that generates the quadrature rule
+ // produces anything useful. It can happen that the algorithm produces 0
+ // quadrature points. When this happens, the returned optional will not
+ // contain a value, even if the cell is formally intersected.
+ const std_cxx17::optional<FEValues<dim>> &inside_fe_values =
+ non_matching_fe_values.get_inside_fe_values();
+
+ if (inside_fe_values)
+ for (const unsigned int q :
+ inside_fe_values->quadrature_point_indices())
+ {
+ const Point<dim> &point = inside_fe_values->quadrature_point(q);
+ for (const unsigned int i : inside_fe_values->dof_indices())
+ {
+ for (const unsigned int j : inside_fe_values->dof_indices())
+ {
+ local_stiffness(i, j) +=
+ inside_fe_values->shape_grad(i, q) *
+ inside_fe_values->shape_grad(j, q) *
+ inside_fe_values->JxW(q);
+ }
+ local_rhs(i) += rhs_function.value(point) *
+ inside_fe_values->shape_value(i, q) *
+ inside_fe_values->JxW(q);
+ }
+ }
+
+ // In the same way, we can use NonMatching::FEValues to retrieve an
+ // FEFaceValues-like object to integrate over $T \cap \Gamma$. The only
+ // thing that is new here is the type of the object. The transformation
+ // from quadrature weights to JxW-values is different for surfaces, so
+ // we need a new class: NonMatching::FEImmersedSurfaceValues. In
+ // addition to the ordinary functions shape_value(..), shape_grad(..),
+ // etc., one can use its normal_vector(..)-function to get an outward
+ // normal to the immersed surface, $\Gamma$. In terms of the level set
+ // function, this normal reads
+ // @f{equation*}
+ // n = \frac{\nabla \psi}{\| \nabla \psi \|}.
+ // @f}
+ // An additional benefit of std::optional is that we do not need any
+ // other check for whether we are on intersected cells: In case we are
+ // on an inside cell, we get an empty object here.
+ const std_cxx17::optional<NonMatching::FEImmersedSurfaceValues<dim>>
+ &surface_fe_values = non_matching_fe_values.get_surface_fe_values();
+
+ if (surface_fe_values)
+ {
+ for (const unsigned int q :
+ surface_fe_values->quadrature_point_indices())
+ {
+ const Point<dim> &point =
+ surface_fe_values->quadrature_point(q);
+ const Tensor<1, dim> &normal =
+ surface_fe_values->normal_vector(q);
+ for (const unsigned int i : surface_fe_values->dof_indices())
+ {
+ for (const unsigned int j :
+ surface_fe_values->dof_indices())
+ {
+ local_stiffness(i, j) +=
+ (-normal * surface_fe_values->shape_grad(i, q) *
+ surface_fe_values->shape_value(j, q) +
+ -normal * surface_fe_values->shape_grad(j, q) *
+ surface_fe_values->shape_value(i, q) +
+ nitsche_parameter / cell_side_length *
+ surface_fe_values->shape_value(i, q) *
+ surface_fe_values->shape_value(j, q)) *
+ surface_fe_values->JxW(q);
+ }
+ local_rhs(i) +=
+ boundary_condition.value(point) *
+ (nitsche_parameter / cell_side_length *
+ surface_fe_values->shape_value(i, q) -
+ normal * surface_fe_values->shape_grad(i, q)) *
+ surface_fe_values->JxW(q);
+ }
+ }
+ }
+
+ cell->get_dof_indices(local_dof_indices);
+
+ stiffness_matrix.add(local_dof_indices, local_stiffness);
+ rhs.add(local_dof_indices, local_rhs);
+
+ // The assembly of the ghost penalty term is straight forward. As we
+ // iterate over the local faces, we first check if the current face
+ // belongs to the set $\mathcal{F}_h$. The actual assembly is simple
+ // using FEInterfaceValues. Assembling in this we will traverse each
+ // internal face in the mesh twice, so in order to get the penalty
+ // constant we expect, we multiply the penalty term with a factor 1/2.
+ for (unsigned int f : cell->face_indices())
+ if (face_has_ghost_penalty(cell, f))
+ {
+ const unsigned int invalid_subface =
+ numbers::invalid_unsigned_int;
+
+ fe_interface_values.reinit(cell,
+ f,
+ invalid_subface,
+ cell->neighbor(f),
+ cell->neighbor_of_neighbor(f),
+ invalid_subface);
+
+ const unsigned int n_interface_dofs =
+ fe_interface_values.n_current_interface_dofs();
+ FullMatrix<double> local_stabilization(n_interface_dofs,
+ n_interface_dofs);
+ for (unsigned int q = 0;
+ q < fe_interface_values.n_quadrature_points;
+ ++q)
+ {
+ const Tensor<1, dim> normal = fe_interface_values.normal(q);
+ for (unsigned int i = 0; i < n_interface_dofs; ++i)
+ for (unsigned int j = 0; j < n_interface_dofs; ++j)
+ {
+ local_stabilization(i, j) +=
+ .5 * ghost_parameter * cell_side_length * normal *
+ fe_interface_values.jump_in_shape_gradients(i, q) *
+ normal *
+ fe_interface_values.jump_in_shape_gradients(j, q) *
+ fe_interface_values.JxW(q);
+ }
+ }
+
+ const std::vector<types::global_dof_index>
+ local_interface_dof_indices =
+ fe_interface_values.get_interface_dof_indices();
+
+ stiffness_matrix.add(local_interface_dof_indices,
+ local_stabilization);
+ }
+ }
+ }
+
+
+ // @sect3{Solving the System}
+ template <int dim>
+ void LaplaceSolver<dim>::solve()
+ {
+ std::cout << "Solving system" << std::endl;
+
+ const unsigned int max_iterations = solution.size();
+ SolverControl solver_control(max_iterations);
+ SolverCG<> solver(solver_control);
+ solver.solve(stiffness_matrix, solution, rhs, PreconditionIdentity());
+ }
+
+
+
+ // @sect3{Data Output}
+ // Since both DoFHandler instances use the same triangulation, we can add both
+ // the level set function and the solution to the same vtu-file. Further, we
+ // do not want to output the cells that have LocationToLevelSet value outside.
+ // To disregard them, we write a small lambda function and use the
+ // set_cell_selection function of the DataOut class.
+ template <int dim>
+ void LaplaceSolver<dim>::output_results() const
+ {
+ std::cout << "Writing vtu file" << std::endl;
+
+ DataOut<dim> data_out;
+ data_out.add_data_vector(dof_handler, solution, "solution");
+ data_out.add_data_vector(level_set_dof_handler, level_set, "level_set");
+
+ data_out.set_cell_selection(
+ [this](const typename Triangulation<dim>::cell_iterator &cell) {
+ return cell->is_active() &&
+ mesh_classifier.location_to_level_set(cell) !=
+ NonMatching::LocationToLevelSet::outside;
+ });
+
+ data_out.build_patches();
+ std::ofstream output("step-85.vtu");
+ data_out.write_vtu(output);
+ }
+
+
+
+ // @sect3{$L^2$-Error}
+ // To test that the implementation works as expected, we want to compute the
+ // error in the solution in the $L^2$-norm. The analytical solution to the
+ // Poisson problem stated in the introduction reads
+ // @f{align*}
+ // u(x) = 1 - \frac{2}{\text{dim}}(\| x \|^2 - 1) , \qquad x \in
+ // \overline{\Omega}.
+ // @f}
+ // We first create a function corresponding to the analytical solution:
+ template <int dim>
+ class AnalyticalSolution : public Function<dim>
+ {
+ public:
+ double value(const Point<dim> & point,
+ const unsigned int component = 0) const override;
+ };
+
+
+
+ template <int dim>
+ double AnalyticalSolution<dim>::value(const Point<dim> & point,
+ const unsigned int component) const
+ {
+ AssertIndexRange(component, this->n_components);
+ (void)component;
+
+ return 1. - 2. / dim * (point.norm_square() - 1.);
+ }
+
+
+
+ // Of course, the analytical solution, and thus also the error, is only
+ // defined in $\overline{\Omega}$. Thus, to compute the $L^2$-error we must
+ // proceed in the same way as when we assembled the linear system. We first
+ // create an NonMatching::FEValues object.
+ template <int dim>
+ double LaplaceSolver<dim>::compute_L2_error() const
+ {
+ std::cout << "Computing L2 error" << std::endl;
+
+ const QGauss<1> quadrature_1D(fe_degree + 1);
+
+ NonMatching::RegionUpdateFlags region_update_flags;
+ region_update_flags.inside =
+ update_values | update_JxW_values | update_quadrature_points;
+
+ NonMatching::FEValues<dim> non_matching_fe_values(fe_collection,
+ quadrature_1D,
+ region_update_flags,
+ mesh_classifier,
+ level_set_dof_handler,
+ level_set);
+
+ // We then iterate iterate over the cells that have LocationToLevelSetValue
+ // value inside or intersected again. For each quadrature point, we compute
+ // the pointwise error and use this to compute the integral.
+ const AnalyticalSolution<dim> analytical_solution;
+ double error_L2_squared = 0;
+
+ for (const auto &cell :
+ dof_handler.active_cell_iterators() |
+ IteratorFilters::ActiveFEIndexEqualTo(ActiveFEIndex::lagrange))
+ {
+ non_matching_fe_values.reinit(cell);
+
+ const std_cxx17::optional<FEValues<dim>> &fe_values =
+ non_matching_fe_values.get_inside_fe_values();
+
+ if (fe_values)
+ {
+ std::vector<double> solution_values(fe_values->n_quadrature_points);
+ fe_values->get_function_values(solution, solution_values);
+
+ for (const unsigned int q : fe_values->quadrature_point_indices())
+ {
+ const Point<dim> &point = fe_values->quadrature_point(q);
+ const double error_at_point =
+ solution_values.at(q) - analytical_solution.value(point);
+ error_L2_squared +=
+ std::pow(error_at_point, 2) * fe_values->JxW(q);
+ }
+ }
+ }
+
+ return std::sqrt(error_L2_squared);
+ }
+
+
+
+ // @sect3{A Convergence Study}
+ // Finally, we do a convergence study to check that the $L^2$-error decreases
+ // with the expected rate. We refine the background mesh a few times. In each
+ // refinement cycle, we solve the problem, compute the error, and add the
+ // $L^2$-error and the mesh size to a ConvergenceTable.
+ template <int dim>
+ void LaplaceSolver<dim>::run()
+ {
+ ConvergenceTable convergence_table;
+ const unsigned int n_refinements = 3;
+
+ make_grid();
+ for (unsigned int cycle = 0; cycle <= n_refinements; cycle++)
+ {
+ std::cout << "Refinement cycle " << cycle << std::endl;
+ triangulation.refine_global(1);
+ setup_discrete_level_set();
+ std::cout << "Classifying cells" << std::endl;
+ mesh_classifier.reclassify();
+ distribute_dofs();
+ initialize_matrices();
+ assemble_system();
+ solve();
+ if (cycle == 1)
+ output_results();
+ const double error_L2 = compute_L2_error();
+ const double cell_side_length =
+ triangulation.begin_active()->minimum_vertex_distance();
+
+ convergence_table.add_value("Cycle", cycle);
+ convergence_table.add_value("Mesh size", cell_side_length);
+ convergence_table.add_value("L2-Error", error_L2);
+
+ convergence_table.evaluate_convergence_rates(
+ "L2-Error", ConvergenceTable::reduction_rate_log2);
+ convergence_table.set_scientific("L2-Error", true);
+
+ std::cout << std::endl;
+ convergence_table.write_text(std::cout);
+ std::cout << std::endl;
+ }
+ }
+
+} // namespace Step85
+
+
+
+// @sect3{The main() function}
+int main()
+{
+ const int dim = 2;
+
+ Step85::LaplaceSolver<dim> laplace_solver;
+ laplace_solver.run();
+}
projects / dealii.git / commitdiff
