PDMT, acryonm for Parallel Dual Meshing Tool, is a polyhedral meshing/remsehing tool. It harnesses the power of the finite element framework (FreeFEM) to facilitate the seamless transformation of a triangular or tetrahedral mesh into a more versatile and efficient polyhedral mesh. PDMT adeptly identifies the dual structure of the original triangular mesh, thereby establishing a solid foundation for the subsequent creation of the polyhedral mesh. The underlying Voronoi frame is utilized to form this new polyhedral mesh, ensuring optimal utilization of computational resources and enhancing the mesh's adaptability for a diverse range of engineering simulations and scientific analyses.

3D mesh example

3D surface mesh example

2D mesh example

Dependencies

To compile and use PDMT, you will need the following dependencies:

• FreeFEM
• MedCoupling (optional for .med mesh support)

Compilation

Compilation with precompiled MedCoupling (install procedure 1)

Below, we present a step-by-step guide on how to compile and install PDMT with precompiled MedCoupling support for FreeFEM:

Prepare the Build Configuration:

Run the following command to prepare the build configuration:

autoreconf -i

Configure the Build:

Now, let's configure the build by specifying the necessary options using the configure script:

./configure \
--prefix=/home/Work/tmp/pdmt \
--with-medcoupling=/home/Install/TarPackages/SALOME-9.15.0-native-UB24.04/BINARIES-UB24.04/MEDCOUPLING \
--with-medfile=/home/Install/TarPackages/SALOME-9.15.0-native-UB24.04/BINARIES-UB24.04/medfile         \
--with-hdf5=/home/Install/TarPackages/SALOME-9.15.0-native-UB24.04/BINARIES-UB24.04/hdf5

In this configuration:

• PDMT will be installed in the /home/Work/tmp/pdmt directory. You can choose a directory that you wish to install PDMT.
• The root directories for medcoupling , medfile, and hdf5 which come as precompiled with SALOME are provided /home/Install/TarPackages/SALOME-9.15.0-native-UB24.04/BINARIES-UB24.04/. To get your precompiled SALOME click here.
• To proceed with the build, ensure that FreeFEM is already installed and available in your $PATH. If not use flag --with-FreeFEM to configure with FreeFEM installed elsewhere.

Please note you will need to adapt each flag to your specific system.

Compile PDMT:

With the configuration set, it's time to compile PDMT and make sure it is ready for use:

make

Install PDMT:

Once the compilation process is successful, proceed to install PDMT using the following command:

make install

By following these steps, you should have successfully compiled and installed PDMT linked with precompiled MedCoupling support for FreeFEM.

Compilation with MedCoupling (install procedure 2)

Here is a step-by-step guide for the typical compilation process:

Prepare the Build Configuration:

Begin by running the following command to set up the build configuration:

autoreconf -i

Configure the Build:

Next, use the configure script to configure the build with the required options:

./configure \
--prefix=/home/Work/tmp/pdmt \
--with-dependencies

In this configuration:

• PDMT will be installed in the /home/Work/tmp/pdmt directory. You can choose a directory that you wish to install PDMT.
• After successfull install, the root directories for medcoupling, medfile, and hdf5 will be automatically located in ext/MEDCOUPLING-9.15.0/INSTALL.
• Ensure that FreeFEM is already installed and available in your $PATH. If not use flag --with-FreeFEM to configure with FreeFEM installed elsewhere.

Compile PDMT:

With the configuration set, it's time to compile PDMT and make sure it is ready for use:

make

Install PDMT:

Once the compilation process is successful, proceed to install PDMT using the following command:

make install

By following these steps, you should have successfully compiled and installed PDMT linked with MedCoupling support for FreeFEM.

Check the compilation

To ensure a successful compilation, you can run the following command to perform checks:

make check

Note

to use and run PDMT with med support please make sure that your $LD_LIBRARY_PATH variable contains medcoupling, medfile, and hdf5 paths. For example for the install above

export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:/home/Install/TarPackages/SALOME-9.15.0-native-UB24.04/BINARIES-UB24.04/MEDCOUPLING/lib:/home/Install/TarPackages/SALOME-9.15.0-native-UB24.04/BINARIES-UB24.04/medfile/lib:/home/Install/TarPackages/SALOME-9.15.0-native-UB24.04/BINARIES-UB24.04/hdf5/lib

By following these steps, you will have successfully compiled and installed PDMT with MedCoupling support for FreeFEM. Enjoy the poly meshes for your computational simulations!

Usage

After installation is done you can simply launch the PDMT mesh conversion via a TUI in any folder of choice. The list of command line flags it accepts

• --debug : to print some verbos info about the meshing process
• --mesh : to provide mesh for conversion, it accepts .mesh, .msh, .vtk, .med(conditional) formats. Also accepts ("square" or "circle").
• --dimension: input mesh type (2 by default, 3 for tetrahedra, or 3S for a triangular surface embedded in 3D).
• --feature_angle: preserve 3D boundary edges sharper than this angle (45 degrees by default).
• --conserve_edge: comma-separated Gmsh/MED edge-group names that must remain as feature-edge chains in 3D/3S output, or ALL to conserve every available edge group.
• --mode: dual construction for every dimension, either subdivided_dual or smooth_dual. The defaults are smooth_dual for 2D/3D and subdivided_dual for 3S.
• --smooth_iterations: number of boundary-aware dual-area (2D/3S) or dual-volume (3D) balancing passes (0 by default).
• --smooth_relaxation: strength of each 2D/3D/3S balancing pass, in (0,1] (0.3 by default).

Example 1:

PDMT --debug --mesh /your/mesh/file.mesh

2D polygonal meshes

Use --dimension 2 to convert a planar triangular mesh into a polygonal dual mesh. Each input vertex produces one output polygon.

The default --mode smooth_dual connects neighboring triangle barycentres directly. This is the original PDMT 2D construction and produces the smallest number of polygon edges:

PDMT --dimension 2 \
  --mesh square \
  --square_mesh_size 4 \
  --mode smooth_dual \
  --out_mesh square_smooth.vtu

With --mode subdivided_dual, every dual line passes through the corresponding primal-edge midpoint. Around an interior primal vertex, the polygon therefore alternates between triangle barycentres and edge midpoints. Boundary polygons also retain the original boundary vertex, so the output remains conforming and has the same exterior boundary as the input mesh:

PDMT --dimension 2 \
  --mesh square \
  --square_mesh_size 4 \
  --mode subdivided_dual \
  --out_mesh square_subdivided.vtu

The two modes support the same 2D input and output formats. Use subdivided_dual when the dual should explicitly follow the barycentric subdivision, and smooth_dual when straighter, less subdivided polygon boundaries are preferred.

Both modes support boundary-aware polygon-area regularization:

PDMT --dimension 2 \
  --mesh square \
  --square_mesh_size 20 \
  --mode smooth_dual \
  --smooth_iterations 3 \
  --smooth_relaxation 0.3 \
  --out_mesh square_regularized.vtu

Each pass measures the actual output polygon areas. Boundary polygons are compared directly with adjacent interior polygons, then weighted triangle centres and edge points are moved to redistribute area. Original boundary vertices remain fixed, and every weighted boundary point remains on its original primal edge. A relaxation of 0.3 applies thirty percent of the multiplicative area correction in each pass. With verbose output, PDMT reports the polygon-area coefficient of variation and mean boundary/interior area ratio before and after regularization.

3D Polyhedral meshes

PDMT can convert a tetrahedral mesh into a conforming polyhedral mesh. Every input vertex is treated as a seed/generator for polytopal mesh hence produces one polyhedron. Triangle fans around each primal edge are merged into a single polygonal face, matching the normal dual topology instead of exposing the barycentric sub-tetrahedra. Boundary labels and sharp feature edges are retained.

Use --dimension 3 and write legacy VTK (.vtk), XML VTK (.vtu), or—when PDMT is built with MEDCoupling—MED (.med):

PDMT --dimension 3 \
	--mesh /your/mesh/tetrahedra.mesh \
  	--out_mesh polyhedra.vtu

The default --feature_angle 45 keeps sharp corners while merging smooth surface regions. Use a larger value for more aggressive merging, for example--feature_angle 80; use a smaller value to preserve more boundary edges. Edges separating different boundary labels are always preserved.

Tetrahedral 3D mode supports both dual representations. smooth_dual is the default and connects neighboring tetrahedron barycentres directly across ordinary internal subdivisions. subdivided_dual retains the intervening tetra-face barycentres and smooth boundary-edge midpoints:

PDMT --dimension 3 \
  --mesh tetrahedra.msh \
  --mode subdivided_dual \
  --out_mesh subdivided_polyhedra.vtu

In either mode, domain-boundary anchors, sharp features, region interfaces, and curves selected by --conserve_edge take priority and remain in the polyhedron connectivity. This keeps every polyhedron closed and conforming.

Boundary-aware 3D regularization

Either 3D dual representation can be regularized by redistributing volume between neighbouring dual cells. Each pass measures the current polyhedral volumes. A boundary cell is compared directly with adjacent interior cells, rather than mostly with other truncated boundary cells. PDMT then moves the shared dual points inside their original primal edges, faces, and tetrahedra. The tetrahedral seed vertices themselves do not move:

PDMT --dimension 3 \
  --mesh tetrahedra.med \
  --med_mesh_name TetrahedralMesh \
  --mode smooth_dual \
  --smooth_iterations 3 \
  --smooth_relaxation 0.3 \
  --out_mesh regularized_polyhedra.vtu

Two to five iterations with a relaxation between 0.2 and 0.4 are a reasonable starting point. The operation does not modify the input file, move the primal boundary, or remove explicitly conserved geometry. Weighted dual edge points remain on their original primal edge, weighted face points remain inside their original face, and weighted tetrahedron points remain inside their original tetrahedron. With verbose output, PDMT reports both the cell-volume coefficient of variation and the mean boundary/interior cell-volume ratio before and after regularization.

Named physical curve groups in an ASCII Gmsh 2.x input can be protected independently of the angle criterion. For example, to retain the group bla from cylinder_edge.msh:

PDMT --dimension 3 \
  --mesh msh/cylinder_edge.msh \
  --conserve_edge bla \
  --out_mesh cylinder_poly.vtu

Multiple group names are comma-separated, for example --conserve_edge inlet_rim,outlet_rim. Each original curve segment remains on the output boundary as a geometrically identical chain split at the dual edge midpoint.

Use --conserve_edge ALL to retain every populated physical curve group in a Gmsh file, or every group on the one-dimensional edge level of a MED mesh:

PDMT --dimension 3 \
  --mesh tetrahedra.med \
  --med_mesh_name TetrahedralMesh \
  --conserve_edge ALL \
  --out_mesh all_curves_conserved.vtu

The 3D loader accepts tetrahedral .mesh, .msh, .vtk, and—when PDMT is built with MED support—.med files. Select a non-default MED input mesh with --med_mesh_name. MED output uses native NORM_POLYHED cells and stores exterior polygonal faces at level -1, including boundary family labels. For example:

PDMT --dimension 3 --mesh tetrahedra.med \
  --med_mesh_name TetrahedralMesh --out_mesh polyhedra.med

Three-dimensional .typ2 output is not supported. The VTK files use cell type 42 (VTK_POLYHEDRON). In VTU output, pdmt_face_connectivity,pdmt_face_offsets, and pdmt_face_labels are stored as field data so polygon topology and boundary tags remain available to downstream tools.

The plugin API is also available directly in FreeFEM:

real[int,int] nodes(0,0);
int[int][int] faces, cells;
int[int] cellLabels, faceLabels;

PdmtBuildDual3D(Th3,
  nodes=nodes, faces=faces, cells=cells,
  labels=cellLabels, faceLabels=faceLabels,
  featureAngle=45.0,
  meshFile="tetrahedra.msh", conserveEdge="ridge,corner",
  mode="smooth_dual",
  smoothIterations=3, smoothRelaxation=0.3);

PdmtPolyMeshWrite("polyhedra.vtu",
  nodes=nodes, cells=cells, faces=faces,
  labels=cellLabels, faceLabels=faceLabels);

Cell face references use signed, one-based face IDs: the sign records whether the face orientation agrees with the global face connectivity.

3D surface polytopal meshes

Use --dimension 3S for a triangular surface embedded in 3D. Each input vertex produces one dual polygon on a smooth part of the surface. A sharp, boundary, region-interface, or explicitly conserved edge splits the polygon fan so the feature remains part of the polygon connectivity.

PDMT --dimension 3S \
  --mesh surface.msh \
  --feature_angle 45 \
  --conserve_edge ridge,rim \
  --out_mesh surface_dual.vtu

The default --mode subdivided_dual connects triangle barycentres through primal-edge midpoints and therefore follows the piecewise-triangular surface. With --mode smooth_dual, ordinary midpoint vertices are removed and adjacent triangle barycentres are connected by straight polygon edges:

PDMT --dimension 3S \
  --mesh surface.msh \
  --mode smooth_dual \
  --conserve_edge ridge,rim \
  --out_mesh smooth_surface_dual.vtu

Protected geometry has priority in both modes. Boundary edges, edges selected by --feature_angle, region interfaces, and --conserve_edge curves retain their primal vertices and edge midpoints so those feature segments remain in the output connectivity.

Both 3S modes also support boundary-aware surface-cell area regularization:

PDMT --dimension 3S \
  --mesh ./surface.msh \
  --mode smooth_dual \
  --smooth_iterations 3 \
  --smooth_relaxation 0.3 \
  --out_mesh regularized-surface.vtu

Each pass measures the areas of the actual output polygons, including the straight connections used by smooth_dual, and adjusts triangle points and unprotected edge points inside their original input simplices. Original surface vertices and all boundary, feature, region-interface, and conserved edge midpoints remain fixed. When a feature edge splits the fan of one input vertex, each resulting output polygon receives its own balancing weight; this prevents a small feature-side cell from being hidden by the total area around the seed. With verbose output, PDMT reports the surface-cell area coefficient of variation and mean boundary/interior area ratio before and after regularization.

The 3S loader accepts .mesh, .msh, .vtk, and—when MED support is enabled—.med triangular surfaces. It writes .vtk, .vtu, or native MED polygon cells. Use --med_mesh_name for the surface mesh name inside an input MED file:

PDMT --dimension 3S --mesh surface.med \
  --med_mesh_name TriangularMesh --out_mesh surface_dual.med

For Gmsh inputs, named edge groups require an ASCII Gmsh 2.x file. For MED inputs, PDMT reads the groups from the one-dimensional edge level.

The Gmsh file must contain type-2 triangle elements; exporting only the physical curves is not a surface mesh. From a .geo file, an ASCII 2.x surface export can be generated with:

gmsh -2 model.geo -format msh2 -o surface.msh

For generating meshes that are presented above you can use the mesh files provided in msh folder. From top left clockwise:

• for the disk mesh with five holes mesh is provided in .vtk format

PDMT --debug --mesh /msh/disk5holes.vtk

• for the pentagon mesh is provided in .mesh fromat

PDMT --debug --mesh /msh/pentagon.mesh

• for the triangle with one hole at the center mesh is provided in .med format

PDMT --debug --mesh /msh/triangle1hole.med --med_mesh_name Mesh_1

• for the disk with large decentered hole mesh is provided in .msh format

PDMT --debug --mesh /msh/disk1hole.msh