\begin{itemize}
\item The \texttt{MeshClassifier} class identifies how the active cells and faces are located relative to the zero contour of the level set function, as illustrated in Figure~\ref{fig:location-to-level-set}. Its member function \texttt{location\_to\_level\_set()} takes a cell or face and
returns an enum, \texttt{LocationToLevelSet}, with values \{\texttt{inside}, \texttt{outside}, \texttt{intersected}\}.
- This information is typically needed when choosing what element a cell of the \texttt{DoFHandler} should use.
+ This information is typically needed when choosing what element (e.g., \texttt{FE\_Q} or \texttt{FE\_Nothing}) and/or what quadrature (defined over the
+ complete cell or over a part---see below) a cell
+ should use.
\item The \texttt{QuadratureGenerator} class, which implements the algorithm in \cite{saye2015}, generates high-order quadrature rules for the three different regions of a \texttt{BoundingBox}, $B$, defined by the sign of the level set function:
\begin{align}\label{eq:boundingbox}
The Computational Geometry Algorithms Library (CGAL, \url{https://www.cgal.org/}) is a widely used
library to describe geometries and meshes \cite{cgal}. \dealii now
has wrappers for CGAL classes and functions, provided in the new
-namespace \texttt{CGALWrappers}, and implementing functionality
-spanning from mesh generation to boolean operations between \dealii
+namespace \texttt{CGALWrappers}: they implement functionality
+spanning from mesh generation to boolean operations between
triangulations and cells. These wrappers are enabled only if \dealii
is compiled with \texttt{C++17}. \textit{Note: This feature is still experimental and
interfaces might change during the next release cycle.}
A related function is
\texttt{GridGenerator::surface\_mesh\_to\_volumetric\_mesh()}, which
-computes a tetrahedral volume triangulation \texttt{Triangulation<3>}, based on a
-given surface triangulation \texttt{Triangulation<2,3>} that bounds
+computes a tetrahedral volume mesh \texttt{Triangulation<3>}, based on a
+given surface mesh \texttt{Triangulation<2,3>} that bounds
the three dimensional shape.
-CGAL also provides Boolean operations on meshes, as available in the
-utility function
+CGAL also provides Boolean operations on meshes: they can be accessed via
+the utility function
\texttt{CGALWrappers::\allowbreak{}compute\_boolean\_operation()}. The
available operations are \textit{co-refinement}, \textit{intersection},
-\textit{union}, and \textit{difference}. Oftentimes, boolean
+\textit{union}, and \textit{difference}. Oftentimes, Boolean
operations and co-refinement around the intersection produces
-badly shaped mesh cells. To overcome this issue, one can use \texttt{CGALWrappers::\allowbreak{}remesh\_surface()}. Figure~\ref{fig:corefinement_remeshed} shows a graphical example.
-A possible workflow is the following:
+badly shaped mesh cells. To overcome this issue, one can use \texttt{CGALWrappers::\allowbreak{}remesh\_surface()}.
+A possible workflow to create a good-quality mesh of the union of a cube and
+a sphere mesh is given in the following listing:
\begin{c++}
// 1) create deal.II triangulations, e.g., cube and sphere (not shown)
Triangulation<spacedim> tria0, tria1;
// 5) convert surface to volume mesh via surface_mesh_to_volumetric_mesh()
\end{c++}
-The output of the boolean operation can be seen in Fig.~\ref{fig:corefinement}, while in Fig.~\ref{fig:corefinement_remeshed}
-the same mesh has been remeshed.
+The output of the Boolean operation is shown in Fig.~\ref{fig:corefinement};
+Fig.~\ref{fig:corefinement_remeshed} shows the mesh after remeshing.
\begin{figure}
\centering
\begin{subfigure}[b]{0.28\textwidth}
\caption{\it (a) Triangulation created by filling a heart-shaped surface implicitly described by a function $f$. (b) Union of a cube with a sphere with badly shaped cells at the intersection. (c) Remeshed version of the same triangulation.}
\end{figure}
-\texttt{CGALWrappers::compute\_quadrature\_on\_boolean\_operation()} returns a \texttt{Quadrature<3>} that allows exact integration of polynomials on polyhedral elements coming out of a \texttt{BooleanOperation} between \dealii cells.
-The quadrature rule is built by meshing the polyhedral region with tetrahedra, computing on each tetrahedron a \texttt{QGaussSimplex<3>} quadrature rule by using \texttt{QSimplex<3>::\allowbreak{}compute\_affine\_transformation()}, and finally
-collecting all of the rules together, giving a \texttt{Quadrature<3>} formula on the \emph{physical} element.
-
-
-These utility functions will be the building blocks for functions in the \texttt{NonMatching} namespace that will, e.g., assemble coupling terms like $(u,v)_{\Omega}$, with $\Omega$ a domain immersed in a fixed background mesh $B$ and $u,v$ finite element functions on $V_h(B)$, as needed, e.g., in
-the context of CutFEM (see Section~\ref{sec:cut}) or Nitsche's method to weakly impose boundary conditions at an interface. The same applies to coupling terms of the form $(u,q)_{\Omega}$ in formulations using Lagrange multipliers, where now $q \in Q_h(\Omega)$, with $Q_h(\Omega)$ the space of the multiplier variable.
-Note that the most relevant difference between this and the \texttt{QuadratureGenerator} in Section~\ref{sec:cut} is that the \texttt{Quadrature} objects are created directly from two overlapping grids, one
-spanning over $B$ and the other one over $\Omega$, and not from a level set function.
+The function \texttt{CGALWrappers::compute\_quadrature\_on\_boolean\_operation()} returns a quadrature that allows exact integration of polynomials on polyhedral elements created by a \texttt{Boolean\allowbreak Operation} between \dealii cells.
+The quadrature rule is built by 1) subdividing the polyhedral region in tetrahedra, 2) computing on each a \texttt{QGaussSimplex<3>} quadrature rule via \texttt{QSimplex<3>::\allowbreak{}compute\_affine\_transformation()}, and 3)
+collecting all in a single \texttt{Quadrature<3>} object on the \emph{physical} cell.
+
+The new utility functions will be the building blocks for functions in
+the \texttt{NonMatching} namespace in future releases. While our current
+CutFEM implementation (see Section~\ref{sec:cut}) relies on a level set description of the domain to determine
+the boundary and the quadrature rules, one could do that also based on non-matching overlapping
+grids~\cite{massing2013efficient}, for which the newly introduced, CGAL-based Boolean operations and
+quadrature-generation functionality might be helpful. We would like to point out
+that the functionalities are not limited to CutFEM but are also applicable
+for weakly imposing boundary conditions on the interface of an immersed boundary
+using Nitsche's method and coupling terms using Lagrange multipliers.
+
+%These utility functions will be the building blocks for functions in the \texttt{NonMatching} namespace that will, e.g., assemble coupling terms like $(u,v)_{\Omega}$, with $\Omega$ a domain immersed in a fixed background mesh $B$ and $u,v$ finite element functions on $V_h(B)$, as needed, e.g., in
+%the context of CutFEM (see Section~\ref{sec:cut}) or Nitsche's method to weakly impose boundary conditions at an interface. The same applies to coupling terms of the form $(u,q)_{\Omega}$ in formulations using Lagrange multipliers, where now $q \in Q_h(\Omega)$, with $Q_h(\Omega)$ the space of the multiplier variable.
+%Note that the most relevant difference between this and the \texttt{QuadratureGenerator} in Section~\ref{sec:cut} is that the \texttt{Quadrature} objects are created directly from two overlapping grids, one
+%spanning over $B$ and the other one over $\Omega$, and not from a level set function.
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