--- title: "TetGenTetrahedralize" language: "en" type: "Symbol" summary: "TetGenTetrahedralize[expr, settings] tetrahedralizes a TetGen expression using settings and returns the result in a new TetGen expression." keywords: - Tetrahedralization - mesh - mesh generation - surface mesh - volume mesh - boundary mesh - refinement - mesh refinement - 3D mesh - tetrahedra - grid - convex hull - delaunay - delaunay mesh canonical_url: "https://reference.wolfram.com/language/TetGenLink/ref/TetGenTetrahedralize.html" source: "Wolfram Language Documentation" related_guides: - title: "TetGenLink" link: "https://reference.wolfram.com/language/TetGenLink/guide/TetGenLink.en.md" related_tutorials: - title: "Introduction in TetGenLink User Guide" link: "https://reference.wolfram.com/language/TetGenLink/tutorial/Introduction.en.md" - title: "TetGenLink User Guide" link: "https://reference.wolfram.com/language/TetGenLink/tutorial/Overview.en.md" --- ## TetGenLink\` # TetGenTetrahedralize TetGenTetrahedralize[expr, settings] tetrahedralizes a TetGen expression using settings and returns the result in a new TetGen expression. ## Details and Options * To use ``TetGenTetrahedralize``, you first need to load it using ``Needs["TetGenLink`"]``. * The settings given to ``TetGenTetrahedralize`` are a string of different commands: | | | | ---- | ----------------------------------------------------------------------------------- | | "p" | tetrahedralize a piecewise linear complex (PLC) | | "q" | quality mesh generation; a minimum radius-edge ratio may be specified (default 2.0) | | "a" | apply a maximum tetrahedron volume constraint | | "A" | assign attributes to identify tetrahedra in certain regions | | "r" | reconstruct/refine a previously generated mesh | | "Y" | suppress splitting of boundary facets/segments | | "YY" | suppress splitting of exterior and interior boundary facets/segments | | "i" | insert a list of additional points into mesh | | "M" | do not merge coplanar facets | | "T" | set a tolerance for coplanar test (default $10^-8$) | | "d" | detect intersections of PLC facets | | "o2" | generate second-order subparametric elements | | "C" | check the consistency of the final mesh | | "Q" | quiet: no terminal output except error | * An empty string ``" "`` given to ``TetGenTetrahedralize`` creates a Delaunay tetrahedralization. * ``TetGenTetrahedralize`` has the following options: "TetrahedronRefinement" [`None`](https://reference.wolfram.com/language/ref/None.en.md) function to refine a tetrahedron --- ## Examples (3) ### Basic Examples (1) First, load the package: ```wl In[1]:= Needs["TetGenLink`"] ``` This creates an instance of a TetGen expression: ```wl In[2]:= inst = TetGenCreate[] Out[2]= TetGenExpression[1] ``` This sets up points and facets to use: ```wl In[3]:= pts = {{0., 0., 0.}, {2., 0., 0.}, {2., 0., 2.}, {0., 0., 2.}, {0., 12., 0.}, {2., 12., 0.}, {2., 12., 2.}, {0., 12., 2.}}; facets = {{{1, 2, 3, 4}}, {{5, 6, 7, 8}}, {{1, 5, 6, 2}}, {{2, 6, 7, 3}}, {{3, 7, 8, 4}}, {{4, 8, 5, 1}}}; ``` This sets the points and facets in the TetGen instance: ```wl In[4]:= TetGenSetPoints[inst, pts]; TetGenSetFacets[inst, facets]; ``` This carries out the tetrahedralization, returning a new TetGen instance: ```wl In[5]:= inst1 = TetGenTetrahedralize[ inst, "pq1.414a0.1"] Out[5]= TetGenExpression[2] ``` This extracts the points and faces from the tetrahedralization: ```wl In[6]:= elemPts = TetGenGetPoints[inst1]; elemFaces = TetGenGetFaces[inst1]; ``` This plots the faces: ```wl In[7]:= Graphics3D[GraphicsComplex[elemPts, Polygon[elemFaces]]] Out[7]= [image] ``` Set the "r" string in place of "p" to refine an existing TetGen instance: ```wl In[8]:= inst2 = TetGenTetrahedralize[ inst1, "rq1.414a0.01"] Out[8]= TetGenExpression[3] ``` This extracts the points and faces from the tetrahedralization: ```wl In[9]:= elemPts = TetGenGetPoints[inst2]; elemFaces = TetGenGetFaces[inst2]; ``` This plots the faces: ```wl In[10]:= Graphics3D[GraphicsComplex[elemPts, Polygon[elemFaces]]] Out[10]= [image] ``` ### Options (1) #### "TetrahedronRefinement" (1) First, load the package: ```wl In[1]:= Needs["TetGenLink`"] ``` This creates an instance of a TetGen expression: ```wl In[2]:= inst = TetGenCreate[] Out[2]= TetGenExpression[1] ``` This sets up points and facets to use: ```wl In[3]:= pts = {{0., 0., 0.}, {2., 0., 0.}, {2., 0., 2.}, {0., 0., 2.}, {0., 12., 0.}, {2., 12., 0.}, {2., 12., 2.}, {0., 12., 2.}}; facets = {{{1, 2, 3, 4}}, {{5, 6, 7, 8}}, {{1, 5, 6, 2}}, {{2, 6, 7, 3}}, {{3, 7, 8, 4}}, {{4, 8, 5, 1}}}; ``` This sets the points and facets in the TetGen instance: ```wl In[4]:= TetGenSetPoints[inst, pts]; TetGenSetFacets[inst, facets]; ``` This carries out the tetrahedralization, returning a new TetGen instance: ```wl In[5]:= inst1 = TetGenTetrahedralize[ inst, "pq1.414"] Out[5]= TetGenExpression[2] ``` This extracts the points and faces from the tetrahedralization: ```wl In[6]:= elemPts = TetGenGetPoints[inst1]; elemFaces = TetGenGetFaces[inst1]; ``` This plots the faces: ```wl In[7]:= Graphics3D[GraphicsComplex[elemPts, Polygon[elemFaces]]] Out[7]= [image] ``` This sets up a compiled function that returns ``True`` if a tetrahedron should be refined and ``False`` otherwise: ```wl In[8]:= cf = Compile[{{coordinates, _Real, 2}, {volume, _Real, 0}}, If[volume > 0.1, True, False] ]; ``` This carries out the tetrahedralization with the refinement function, returning a new TetGen instance: ```wl In[9]:= inst2 = TetGenTetrahedralize[inst, "pq1.414", "TetrahedronRefinement" -> cf] Out[9]= TetGenExpression[3] ``` In this case the "p" string is used as the refinement operates on the initial TetGen instance. This extracts the points and faces from the tetrahedralization: ```wl In[10]:= elemPts = TetGenGetPoints[inst2]; elemFaces = TetGenGetFaces[inst2]; ``` This plots the faces: ```wl In[11]:= Graphics3D[GraphicsComplex[elemPts, Polygon[elemFaces]]] Out[11]= [image] ``` Refine the already tetrahedralized TetGen instance: ```wl In[12]:= inst2 = TetGenTetrahedralize[inst1, "rq1.414", "TetrahedronRefinement" -> cf] Out[12]= TetGenExpression[4] ``` In this case the "r" string is used as the refinement operates on an already tetrahedralized TetGen instance. This extracts the points and faces from the tetrahedralization: ```wl In[13]:= elemPts = TetGenGetPoints[inst2]; elemFaces = TetGenGetFaces[inst2]; ``` This plots the faces: ```wl In[14]:= Graphics3D[GraphicsComplex[elemPts, Polygon[elemFaces]]] Out[14]= [image] ``` ### Possible Issues (1) Refinement of an already tetrahedralized TetGen instance needs to be done with the "r" string. First, load the package: ```wl In[1]:= Needs["TetGenLink`"] ``` Set up and tetrahedralize an initial TetGen instance: ```wl In[2]:= coordinates = {{0., 0., 0.}, {2., 0., 0.}, {2., 0., 2.}, {0., 0., 2.}, {0., 12., 0.}, {2., 12., 0.}, {2., 12., 2.}, {0., 12., 2.}}; facets = {{{1, 2, 3, 4}}, {{5, 6, 7, 8}}, {{1, 5, 6, 2}}, {{2, 6, 7, 3}}, {{3, 7, 8, 4}}, {{4, 8, 5, 1}}}; In[3]:= geometry = TetGenCreate[]; TetGenSetPoints[geometry, coordinates]; TetGenSetFacets[geometry, facets]; discretization = TetGenTetrahedralize[geometry, "pq"]; ``` Inspect the number of tetrahedra in the tetrahedralized instance: ```wl In[4]:= TetGenGetElements[discretization]//Length Out[4]= 12 ``` Tetrahedralization of an already tetrahedralized TetGen instance as a piecewise linear complex (PLC) does not work. No new elements are generated: ```wl In[5]:= refinement = TetGenTetrahedralize[discretization, "pqa.01"]; TetGenGetElements[refinement]//Length Out[5]= 0 ``` Refinement of an already tetrahedralized TetGen instance needs to be done with the "r" string: ```wl In[6]:= discretization = TetGenTetrahedralize[geometry, "pq"]; refinement = TetGenTetrahedralize[discretization, "rqa.01"]; In[7]:= TetGenGetElements[refinement]//Length Out[7]= 9414 ``` ## Tech Notes * [Introduction in *TetGenLink* User Guide](https://reference.wolfram.com/language/TetGenLink/tutorial/Introduction.en.md) * [*TetGenLink* User Guide](https://reference.wolfram.com/language/TetGenLink/tutorial/Overview.en.md) ## Related Guides * [TetGenLink](https://reference.wolfram.com/language/TetGenLink/guide/TetGenLink.en.md)