From: Marco Feder Date: Wed, 1 Jun 2022 23:44:54 +0000 (+0200) Subject: Add CGAL section X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=a84160ee72bc8ce27b71b65996b9714e6d776577;p=release-papers.git Add CGAL section --- diff --git a/9.4/paper.bib b/9.4/paper.bib index 8f75e8f..f7f40ea 100644 --- a/9.4/paper.bib +++ b/9.4/paper.bib @@ -1342,3 +1342,12 @@ doi = {10.1504/IJCSE.2009.029164} year = {2022}, volume = {in press} } + +@book{ cgal +, title = "{CGAL} User and Reference Manual" +, author = "{The CGAL Project}" +, publisher = "{CGAL Editorial Board}" +, edition = "{5.4.1}" +, year = 2022 +, url = "https://doc.cgal.org/5.4.1/Manual/packages.html" +} \ No newline at end of file diff --git a/9.4/paper.tex b/9.4/paper.tex index f78841a..a5df767 100644 --- a/9.4/paper.tex +++ b/9.4/paper.tex @@ -200,6 +200,7 @@ The major changes of this release are: \item Advances in matrix-free infrastructure (see Section~\ref{sec:mf}); \item Advances in multigrid infrastructure (see Section~\ref{sec:multigrid}); \item CutFEM support (see Section~\ref{sec:cut}); + \item Integration with the Computational Geometry Algorithms Library (CGAL) (see Section~\ref{sec:cgalwrappers}); \item Performance improvement of particle infrastructure (see Section~\ref{sec:particles}); \item Two new tutorial programs and a new code gallery program (see Section~\ref{subsec:steps}). \end{itemize} @@ -678,12 +679,88 @@ The new \texttt{step-85} tutorial shows how many of these classes go together. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Integration with the Computational Geometry Algorithms - Library (CGAL)}\label{sec:cgal} - + Library (CGAL)}\label{sec:cgalwrappers} + If you have CGAL (\href{https://www.cgal.org/}{{https://www.cgal.org/}}) installed \cite{cgal}, you can use wrappers around CGAL classes and functions contained in the new namespace \texttt{CGALWrappers}, allowing capabilities spanning from mesh generation to boolean operations between \dealii{} triangulations and cells. CGAL wrappers are enabled only if \dealii{} is compiled with \texttt{C++17}. We have introduced the following utilities to the \dealii's \texttt{CGALWrappers} namespace. \begin{itemize} -\item new classes +\item Refinement criteria for tetrahedra or surface facets can be stored and specified in a struct named \texttt{AdditionalData}. +Depending on the template parameter \texttt{dim}, different criteria can be chosen. See the documentation of this struct for a list and an explanation of the possible parameters. +\item \texttt{GridGenerator::implicit\_function()} creates a \texttt{Triangulation} out of the zero level set of an implicit function $f$. +Apart from the final \texttt{Triangulation} and the implicit funcion $f$ itself, also an interior point for which $f<0$ must be supplied by the user, along with the radius of a ball that will contain the generated triangulation. +Mesh-related entities must be specified with \texttt{AdditionalData}. +When \texttt{dim} is \texttt{3}, the mesh is filled with tetrahedra. A prototypical use case is the following, where the surface is the zero level set of Taubin's heart function $f=\bigl ( x^2 + \frac{9y^2}{4} +z^2 -1 \bigr ) -x^2 z^3 - \frac{9y^2z^3}{80}$. The resulting \texttt{Triangulation<3>} can be appreciated in Figure~\ref{fig:heart_tria}. +\begin{c++} +// Empty triangulation. +Triangulation<3, 3> tria; +//Taubin's heart surface +ImplicitFunction implicit_function; +CGALWrappers::AdditionalData<3> data; +data.cell_size = .05; +GridGenerator::implicit_function( + tria, implicit_function, data, Point<3>(0, 0, 0), 10.0); +\end{c++} + +\begin{figure}[h!] + \centering + \includegraphics[width=.6\paperwidth]{png/heart_implicit.png} + \caption{\texttt{Triangulation<3>} of a heart surface starting from an implicit function $f$. \label{fig:heart_tria}} +\end{figure} + +\item \texttt{GridGenerator::surface\_mesh\_to\_volumetric\_mesh()} fills a \texttt{Triangulation<3>} starting from a \texttt{Triangulation<2,3>}. +The surface triangulation bounds the three dimensional volume, which is meshed inside with tetrahedra. +\item Corefinement and boolean operations between \texttt{CGAL::Surface\_mesh} types are exploited to perform boolean operations between \dealii{} +triangulations, and are available in the utility \texttt{CGALWrappers::compute\_boolean\_operation()}. The available operations are \texttt{corefinement}, \texttt{intersection}, \texttt{union} and \texttt{difference}, and they are +described in the enum class \texttt{BooleanOperation}. Also, several wrappers allow to go back and forth from a \dealii{} Triangulation type to a CGAL Triangulation type. In general, after a \texttt{BooleanOperation} the corefinement around the intersection gives most of the times +badly shaped elements. To overcome this issue, one can use \texttt{CGALWrappers::remesh\_surface()}. See Figure~\ref{fig:corefinement_remeshed} for a graphical example. +A possible workflow is the following: +\begin{c++} + Triangulation tria0, tria1; + GridGenerator::hyper_cube(tria0,-1.,1.); + GridGenerator::hyper_ball(tria1,{1.,1.,1.},0.6); + tria0.refine_global(3); + tria1.refine_global(3); + + // Move to CGAL surfaces, assume Kernel is already defined. + CGAL::Surface_mesh surface_mesh0, surface_mesh1; + CGALWrappers::dealii_tria_to_cgal_surface_mesh(tria0, surface_mesh0); + CGALWrappers::dealii_tria_to_cgal_surface_mesh(tria1, surface_mesh1); + + // Compute the union of the two meshes + CGALWrappers::compute_boolean_operation(surface_mesh0, + surface_mesh1, + BooleanOperation::compute_union, + out_mesh); + // Now back to deal.II + Triangulation<2, 3> tria_out; + CGALWrappers::cgal_surface_mesh_to_dealii_triangulation(out_mesh, tria_out); + // tria_out is now a valid deal.II triangulation + \end{c++} + The output of the boolean operation can be appreciated in Figure~\ref{fig:corefinement}, while in Figure~\ref{fig:corefinement_remeshed} + the same mesh has been remeshed. + \begin{figure}[h] + \centering + \begin{subfigure}[b]{0.4\textwidth} + \centering + \includegraphics[height=.2\paperheight]{png/intersection_cube_sphere_mesh.png} + \caption{\label{fig:corefinement}} + \end{subfigure} + \qquad + \begin{subfigure}[b]{0.5\textwidth} + \centering + \includegraphics[height=.2\paperheight]{png/cube_sphere_remeshed.png} + \caption{ \label{fig:corefinement_remeshed}} + \end{subfigure} + \caption{a) Union of the unit cube with an hyper ball. Notice how the corefinement around the intersection gives badly shaped elements. b) The remeshed \dealii{} Triangulation.} + \end{figure} + +\item \texttt{CGALWrappers::compute\_quadrature\_on\_boolean\_operation()} returns a \texttt{Quadrature<3>} that allows to integrate exactly 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>::compute\_affine\_transformation()}, and finally +collecting all of the rules together, giving a \texttt{Quadrature<3>} formula on the \emph{physical} element. \end{itemize} + +These utilities will be the building blocks for adding functions to the \texttt{NonMatching} namespace that will assemble coupling terms like $(u,v)_{B}$, with $B$ a domain immersed in a fixed background mesh $\Omega$ and $u,v$ finite element functions on $V_h(\Omega)$, as usually happens using Nitsche's method to weakly impose boundary conditions at an interface. The same applies to coupling terms of the form $(u,q)_B$ in formulations using Lagrange multipliers, where now $q \in Q_h(B)$, with $Q_h(B)$ the space of the multiplier variable. + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Performance improvement of particle infrastructure}\label{sec:particles} @@ -820,7 +897,6 @@ following: \end{multicols} \dealii{} can interface with many other libraries: -\todo[inline]{Need to update, in particular add CGAL.} \begin{multicols}{3} \begin{itemize} \item ADOL-C \cite{Griewank1996a,adol-c} @@ -828,6 +904,7 @@ following: \item ARPACK \cite{arpack} \item Assimp \cite{assimp} \item BLAS and LAPACK \cite{lapack} + \item CGAL \cite{cgal} \item cuSOLVER \cite{cusolver} \item cuSPARSE \cite{cusparse} \item Gmsh \cite{geuzaine2009gmsh} diff --git a/9.4/png/cube_sphere_remeshed.png b/9.4/png/cube_sphere_remeshed.png new file mode 100644 index 0000000..e3b66e2 Binary files /dev/null and b/9.4/png/cube_sphere_remeshed.png differ diff --git a/9.4/png/heart_implicit.png b/9.4/png/heart_implicit.png new file mode 100644 index 0000000..19aa077 Binary files /dev/null and b/9.4/png/heart_implicit.png differ diff --git a/9.4/png/intersection_cube_sphere_mesh.png b/9.4/png/intersection_cube_sphere_mesh.png new file mode 100644 index 0000000..dd5307d Binary files /dev/null and b/9.4/png/intersection_cube_sphere_mesh.png differ