TetGen

TetGen
TetGen
A Quality Tetrahedral Mesh Generator and
3D Delaunay Triangulator
Version 1.5
User’s Manual
Also available as WIAS Technical Report No. 13, 2013
Hang Si
[email protected]
http://www.tetgen.org
c
2002
– 2013
ii
Abstract
TetGen is a software for tetrahedral mesh generation. Its goal is to
generate good quality tetrahedral meshes suitable for numerical methods and scientific computing. It can be used as either a standalone
program or a library component integrated in other software.
The purpose of this document is to give a brief explanation of
the kind of tetrahedralizations and meshing problems handled by TetGen and to give a fairly detailed documentation about the usage of the
program. Readers will learn how to create tetrahedral meshes using
input files from the command line. Furthermore, the programming
interface for calling TetGen from other programs is explained.
keywords: tetrahedral mesh generation, Delaunay tetrahedralization, weighted Delaunay triangulation, constrained Delaunay tetrahedralization, mesh quality, mesh refinement, mesh adaption, mesh
coarsening
AMS Classification: 65M50, 65N50
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Contents
Contents
v
1 Introduction
1.1 Triangulations of Point Sets . . . . . . . . . . . . . . . . . .
1.1.1 Delaunay Triangulations, Voronoi Diagrams . . . . .
1.1.2 Weighted Delaunay Triangulations, Power Diagrams .
1.1.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . .
1.2 Tetrahedral Meshes of 3d Spaces . . . . . . . . . . . . . . . .
1.2.1 Piecewise Linear Complexes (PLCs) . . . . . . . . . .
1.2.2 Steiner Points . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Boundary Conformity . . . . . . . . . . . . . . . . .
1.2.4 Constrained Delaunay Tetrahedralizations . . . . . .
1.2.5 Mesh Quality, Tetrahedron Shape Measures . . . . .
1.2.6 Mesh Adaptation, Mesh Sizing Functions . . . . . . .
1.2.7 Mesh Optimization . . . . . . . . . . . . . . . . . . .
1.2.8 Algorithms . . . . . . . . . . . . . . . . . . . . . . .
1.3 Description of the Meshing Process . . . . . . . . . . . . . .
2 General Information
2.1 Language, Platforms
2.2 Memory requirement
2.3 CPU time estimation
2.4 Performance . . . . .
2.5 Errors . . . . . . . .
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3 Getting Started
3.1 Compilation . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Using make . . . . . . . . . . . . . . . . . . . . .
3.1.2 Using cmake . . . . . . . . . . . . . . . . . . . . .
3.1.3 Remarks on Using Shewchuk’s Robust Predicates
3.1.4 Using CGAL’s Robust Predicates . . . . . . . . .
3.2 A Short Tutorial . . . . . . . . . . . . . . . . . . . . . .
3.3 Visualization . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 TetView . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Medit and Paraview . . . . . . . . . . . . . . . .
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4 Using TetGen
4.1 Command Line Syntax . . . . . . . . . . . . . . . . . . . . .
4.2 Command Line Switches . . . . . . . . . . . . . . . . . . . .
4.2.1 Delaunay and weighted Delaunay tetrahedralizations
4.2.2 Boundary conformity and recovery (-p, -Y) . . . . . .
4.2.3 Quality mesh generation (-q) . . . . . . . . . . . . . .
4.2.4 Adaptive mesh generation (-a, -m) . . . . . . . . . .
4.2.5 Reconstructing a tetrahedral mesh (-r) . . . . . . . .
4.2.6 Mesh optimization (-O) . . . . . . . . . . . . . . . .
4.2.7 Mesh coarsening (-R) . . . . . . . . . . . . . . . . . .
4.2.8 Inserting additional points (-i) . . . . . . . . . . . . .
4.2.9 Assigning region attributes (-A) . . . . . . . . . . . .
4.2.10 Mesh output switches (-f, -e, -n, -z, -o2) . . . . . . .
4.2.11 Mesh statistics (-V) . . . . . . . . . . . . . . . . . . .
4.2.12 Memory allocation (-x) . . . . . . . . . . . . . . . . .
4.2.13 Miscellaneous . . . . . . . . . . . . . . . . . . . . . .
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5 File Formats
5.1 Useful Things to Know . . . . . . . . . . . . . . .
5.1.1 A Boundary Description of PLCs . . . . .
5.1.2 Boundary Markers . . . . . . . . . . . . .
5.2 TetGen’s File Formats . . . . . . . . . . . . . . .
5.2.1 .node files . . . . . . . . . . . . . . . . . .
5.2.2 .poly files . . . . . . . . . . . . . . . . . .
5.2.3 .smesh files . . . . . . . . . . . . . . . . .
5.2.4 .ele files . . . . . . . . . . . . . . . . . . .
5.2.5 .face files . . . . . . . . . . . . . . . . . . .
5.2.6 .edge files . . . . . . . . . . . . . . . . . .
5.2.7 .vol files . . . . . . . . . . . . . . . . . . .
5.2.8 .mtr files . . . . . . . . . . . . . . . . . . .
5.2.9 .var files . . . . . . . . . . . . . . . . . . .
5.2.10 .neigh files . . . . . . . . . . . . . . . . . .
5.2.11 .v.node, .v.edge, .v.face, .v.cell . . . . . . .
5.3 Supported File Formats . . . . . . . . . . . . . .
5.3.1 .off files . . . . . . . . . . . . . . . . . . .
5.3.2 .ply files . . . . . . . . . . . . . . . . . . .
5.3.3 .stl files . . . . . . . . . . . . . . . . . . .
5.3.4 .mesh files . . . . . . . . . . . . . . . . . .
5.4 File Format Examples . . . . . . . . . . . . . . .
5.4.1 A PLC with Two Boundary Markers . . .
5.4.2 A PLC with Two Sub-regions (Materials)
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5.4.3
A PLC with Two Sub-regions and Two Holes . . . . . 75
6 Calling TetGen from Another Program
6.1 The Header File . . . . . . . . . . . . .
6.2 The Calling Convention . . . . . . . .
6.3 The tetgenio Data Type . . . . . . .
6.4 Description of Arrays . . . . . . . . . .
6.4.1 Memory Management . . . . . .
6.4.2 The facet Data Structure . . .
6.5 A Complete Example . . . . . . . . . .
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A Basic Definitions
88
A.1 Simplices, Simplicial Complexes . . . . . . . . . . . . . . . . . 88
A.2 Polyhedra and Faces . . . . . . . . . . . . . . . . . . . . . . . 89
A.3 CSG and B-Rep Models of 3d Domains . . . . . . . . . . . . . 89
B List of Error Codes and Messages
90
References
91
Index
95
vii
1
1
Introduction
TetGen is a robust, fast, and easy-to-use software for generating tetrahedral
meshes suitable in many applications.
For a set of 3d (weighted) points, TetGen generates the Delaunay and
weighted Delaunay tetrahedralization as well as their duals, the Voronoi diagram and power diagram. For a 3d polyhedral domain, TetGen generates the
constrained Delaunay tetrahedralization and an isotropic adaptive tetrahedral mesh of it. Domain boundaries (edges and faces) are respected and can
be preserved in the resulting mesh. The shapes of resulting tetrahedra can
be provably good for a large class of inputs. One of its main applications is to
simulate physical phenomena by numerical methods, such as finite element
and finite volume methods. A good quality mesh is essential to achieve high
accuracy and efficiency of the simulations.
The algorithms of TetGen are Delaunay-based. They can preserve arbitrary complex geometry and topology. TetGen uses a constrained Delaunay
refinement algorithm which guarantees termination and good mesh quality.
The robustness of TetGen is enhanced by using advanced technologies developed in computational geometry. A technical paper describing the algorithms
and technologies used in TetGen is available [24].
TetGen is an outcome of a long-term research project supported by Weierstrass Institute for Applied Analysis and Stochastics (WIAS). It is continuously developed and improved.
TetGen is written in C++. It uses only C standard library. It is easy to compile and runs on all major 32-bit and 64-bit computer systems. The source
code of TetGen is freely available at http://www.tetgen.org. It is distributed under the the terms of the GNU Affero General Public License
(AGPL) (v 3.0 or later) or a commercial license provided by WIAS.
The remainder of this section is to give a brief description of the triangulation and meshing problems considered in TetGen, and an overview of the
implemented algorithms. For basic usage of TetGen, most of the information
are not necessary to know, but Sections 1.2.1 and 1.2.5 contain some necessary guidelines to create correct inputs and to generate quality tetrahedral
meshes.
2
1 INTRODUCTION
Figure 1: The Delaunay triangulation (left) and its dual Voronoi diagram
(right) of a 2d point set.
1.1
Triangulations of Point Sets
Triangulations are basic geometric structures. A triangulation of a set V of
points is a simplicial complex S whose vertex set is a subset of or equal to V ,
and the underlying space of S is the convex hull of V . Given a point set, there
are many triangulations of it. Among them, the Delaunay triangulation is
of the most interested one. Its dual is the Voronoi diagram of the point set.
Delaunay triangulations and Voronoi diagrams have many nice mathematical
properties [1, 12, 7]. They are extensively used in many applications.
1.1.1
Delaunay Triangulations, Voronoi Diagrams
Let V be a set of points in Rd , σ be a k-simplex (0 ≤ k ≤ d) whose vertices
are in V . A circumsphere of σ is a sphere that passes through all vertices of
σ. If k = d, σ has a unique circumsphere, otherwise, there are infinitely many
circumspheres of σ. We say that σ is Delaunay if there exists a circumsphere
of σ such that no vertex of V lies inside it.
A Delaunay triangulation D of V is a simplicial complex such that all
simplices are Delaunay, and the underlying space of D is the convex hull of
V [6]. Figure 1 left illustrates a 2d Delaunay triangulation. A 3d Delaunay
triangulation is also called a Delaunay tetrahedralization.
A Delaunay triangulation of V is unique if V is in general position, i.e.,
no d + 2 points in V lie on a common sphere. Otherwise, we say that V
contains degeneracies, i.e., there are d+2 points in V lie on a common sphere.
Degeneracies can be removed by applying an arbitrary small perturbation
onto the coordinates of points in V .
The dual of the Delaunay triangulation is the Voronoi diagram defined on
the same vertex set (see Figure 1 right). For any vertex p ∈ V , the Voronoi
cell of p is the set of points with distance to p not greater than to any other
3
1.1 Triangulations of Point Sets
z
p+
x/1.0e-01
p
Figure 2: The relation between Delaunay triangulation in Rd and convex
hull in Rd+1 (here d = 2). Left: Some 2d points and their corresponding 3d
lift points. Right: The Delaunay triangulation of a set of 2d points and the
lower convex hull of its 3d lifted points.
vertex of V , i.e. it is the set cell(p) = {x ∈ Rd ; kx−pk ≤ kx−qk, ∀q ∈ V },
where k · k stands for the Euclidean distance. The Voronoi diagram of V is a
subdivision of Rd into Voronoi cells (some of which may be unbounded) and
their faces [27]. It is a d-dimensional polyhedral complex. If the point set
V is in general position, there is a one-to-one correspondence between the
k-simplices of the Delaunay triangulation and the (d − k)-polyhedra of the
Voronoi diagram, where 0 ≤ k ≤ d. In R3 , the vertices of the Voronoi diagram
are the circumcenters of the tetrahedra of the Delaunay tetrahedralization.
There is a nice relation between a Delaunay triangulation in Rd and a
convex hull in Rd+1 . For any point p = (p0 , p1 , · · · , pd−1 ) ∈ Rd , define its
lifted point p+ = (p0 , p1 , · · · , pd−1 , pd ) ∈ Rd+1 , where pd = p20 + · · · + p2d−1 .
For any point set V ⊂ Rd , define V + = {p+ ; p ∈ V } ⊂ Rd+1 be the lifted
point set of V . All points in V + lie on a paraboloid in Rd+1 (see Figure 2
left). The convex hull of V + is a (d + 1)-dimensional convex polytope P .
A lower face of P is a face of P which is on the downside of P (visible by
points in V ). The Delaunay triangulation of V is the projection of the set of
lower faces of P onto d dimensions. Figure 2 right illustrates the relationship
when d = 2. A simplex σ is a Delaunay simplex if and only if there exists
a hyperplane in Rd+1 passing through the lifted vertices of σ such that no
other lifted vertices in V + lies below of it. Similarly, the Voronoi diagram of
V is the projection of the lower faces of a convex polytope Q ⊂ Rd+1 such
that P and Q are polar to each other [29].
1.1.2
Weighted Delaunay Triangulations, Power Diagrams
Weighted Delaunay triangulations are generalizations of Delaunay triangulations by replacing the Euclidean distance by “weighted distance”.
A weighted point, p0 = (p, p2 ) ∈ Rd × R, can be interpreted as a sphere
4
1 INTRODUCTION
Z
P
p
z
Figure 3: Left: The weighted
distance (left) of two weighted points (p, p2 )
Z
Figure
2.13:
Orthospheres
are
generalization
of circumspheres.
and (z, z 2p). Right:
The
orthosphere
of three
weighted
points. (Figures are
p
z
z
taken from Damrong Guoy’s PhD thesis.)
Figure 2.11: The weighted distance between two circles equals the length of the second
centered
at p with radius p. The weighted distance between p0 and z0 is
tangent line. The weighted distance from a point to a circle equals the length of the tangent
line. The weighted distance from one point to p
another is their Euclidean distance.
πp0 ,z0 = kp − zk2 − (p2 + z 2 ),
Euclidean
pointsfor
havean
zero
distance whenIn
they
are the same point.
Two in
weighted
seeTwo
Figure
3 left
example.
particular,
points
Rd can be considered
points p̂, ẑ have
zero weighted
when they are orthogonal, meaning that the two
weighted
points
with distance
zero weight.
2
spheres
intersect
at
right
angle
i.e.
�p
−
+ orthogonal
Z 2 ). We also define
a point to
be
Two weighted points p0z�
, 2z=0 (P
are
if their
weighted
distance is
zero,
i.e.,to a sphere when the sphere passes through the point, which means that the
orthogonal
kp
− zk2 =orthogonality
(p2 + z 2in).two dimensions.
weighted distance is also zero. Figure 2.12
demonstrates
The book by Pedoe [46] offers a comprehensive study on the system of orthogonal circles. For
We
say that two weighted points are farther than orthogonal when their
completeness,
we say two weighted
points are furthercloser
than orthogonal
when
their weighted
weighted distance
is positive,
thantriangulations
orthogonal
Figure
2.14: Twoand
weighted Delaunay
ofwhen
a cube.the distance
distance
is
positive,
and
closer
than
orthogonal
when
the
distance
becomes
an imaginary
becomes an imaginary number.
number.
In general, d+1 points in Rd define a unique circumsphere passing through
them.
Similarly, d + 1 weighted points in Rd define a unique common orIn general, four points in R3 define a unique circumsphere passing through them. Sim21
thosphere.
When all points have zero weights,
their orthosphere is just their
ilarly four weighted points p̂0 , p̂1 , p̂2 , p̂3 in three dimensions define a unique common orthocircumsphere. Figure 3 (right) gives an example of the orthosphere of three
sphere(orthogonal sphere) ẑ by the system of equations
weighted points in two dimensions.
Let V 0 ⊂ Rd �z
×−Rp �2be
a finite set of weighted points. We say a sphere
− (Z 2 + Pi2 ) = 0, 0 ≤ i ≤ 3.
i
is empty if all weighted points in V 0 are farther than orthogonal of it. The
19
weighted Delaunay triangulation
of V 0 is a simplicial complex D0 such that
every simplex has an orthosphere which is empty, and the underlying space
of D0 is the convex hull of V 0 . Obviously, if all the points have the same
weight, the weighted Delaunay triangulation is the same as the usual Delaunay triangulation. Note that, a weighted Delaunay triangulation does not
necessarily contain all points in V 0 .
The dual of a weighted Delaunay triangulation is a weighted Voronoi
diagram, also called the power diagram [1, 9] of the weighted point set V 0 .
Power diagrams can be similarly defined as the Voronoi diagram by using
the weighted distance instead of the Euclidean distance. If no d + 2 weighted
points of V 0 share a common orthosphere, i.e., it is in general position, then
1.1 Triangulations of Point Sets
5
the simplices of the weighted Delaunay triangulation and the cells of the
power diagram have a one-to-one correspondence. In R3 , the vertices of
the power diagram are the orthocenters of the tetrahedra of the weighted
Delaunay tetrahedralization.
A weighted Delaunay triangulation of V ⊂ Rd is also the projection
of the set of lower faces of a convex polytope P ⊂ Rd+1 . Any point in
p = {p0 , ..., pd−1 } ∈ V is lifted to a point p0 = {p0 , ..., pd−1 , pd } ∈ Rd+1 ,
where pd = p20 + · · · + p2d−1 − p2 (p is the weight of p). For p 6= 0, p0 does
not lie on a paraboloid in Rd+1 , but is moved vertically downward by p2 . A
simplex belongs to the weighted Delaunay triangulation of V (i.e., it has an
empty orthosphere) if and only if there exits a hyperplane passing through
the lifted weighted points of these simplex and no lifted weighted point of V
lie below the hyperplane.
Both weighted Delaunay triangulations and power diagrams are called
regular subdivisions of point sets [29]. Regular subdivisions have nice combinatorial structures. They are one of the important objects studied in higherdimensional convex polytopes [29, 5].
1.1.3
Algorithms
Algorithms for generating Delaunay (and weighted Delaunay) tetrahedralizations are well studied in computational geometry [7]. TetGen implemented
two algorithms, the Bowyer-Watson algorithm [3, 28] and the incremental flip
algorithm [9]. Both algorithms are incremental, i.e., insert points one after
another. Both have the worst case runtime O(n2 ). In most of the practical
applications, they are usually very fast. The expected running time of these
algorithms is O(n log n) if the points are uniformly distributed in [0, 1]3 .
The speed of incremental algorithms is very much affected by the cost of
point location. TetGen uses a spatial sorting scheme [2] to improve the point
location. The idea is to sort the points such that nearby points in space have
nearby indices. The points are first randomly sorted in different groups, then
points in each group are sorted along the Hilbert curve. Inserting points in
this order, each point location can be done in nearly constant time.
TetGen uses Shewchuk’s exact geometric predicates [15] for performing
the Orient3D, InSphere, and Orient4D tests. These suffice to guarantee
the numerical robustness of generating Delaunay and weighted Delaunay
tetrahedralizations. A simplified symbolic perturbation scheme [8] is used to
remove the degeneracies.
6
1 INTRODUCTION
A 3d PLC
non-PLCs
Figure 4: Left: A 3d piecewise linear complex. The left shaded area shows a
facet, which is non-convex and has a hole in it. It has also edges and vertices
floating in it. The right shaded area shows an interior facet separating two
sub-domains. Right: Configurations which are not PLCs.
1.2
Tetrahedral Meshes of 3d Spaces
A tetrahedral mesh is a 3d simplicial complex that is a discrete representation
of a 3d continuous space (domain), both in its topology and geometry. Note
that a Delaunay tetrahedralization is a tetrahedral mesh of the convex hull
of its vertex set. In general, a geometric domain may not be convex and may
have arbitrarily complex boundaries.
The input domain of TetGen is modeled by a piecewise linear complex
(Section 1.2.1). The focuses of TetGen are the representation of the geometry
(the boundary) and the quality of the mesh. TetGen generates several types
of tetrahedral meshes to achieve these goals. They are explained in the
following subsections.
1.2.1
Piecewise Linear Complexes (PLCs)
At first we need a model to represent a 3d domain such that it can be easily
described and handled. A 3d piecewise linear complex (PLC) X is a set of
cells, that satisfies the following properties:
(1) The boundary of each cell in X is a union of cells in X .
(2) If two distinct cells f, g ∈ X intersect, their intersection is a union of
cells in X .
It is first introduced by Miller, et al. [11], see Figure 4 left for an example.
The boundary of a 3d PLC is the set of cells whose dimensions are less
than or equal to 2. A 0-dimensional cell is a vertex. In particular, we call
a 1-dimensional cell (an edge) a segment, and a 2-dimensional cell a facet.
Each facet of a PLC is a 2d PLC. It may contain holes, segments and vertices
in its interior, see Figure 4 left for an example.
1.2 Tetrahedral Meshes of 3d Spaces
7
Figure 5: Examples of surface meshes of PLCs.
PLCs are flexible in describing 3d geometric features. For instance, they
permit facets, segments and vertices to float in a domain, or segments and
vertices to float in the facet. One purpose of these floating cells is to constrain
how the PLC can be meshed, so that boundary conditions may be applied
at those cells.
The definition of a PLC disallows illegal intersections of its cells, see Figure 4 right for examples. Two segments only can intersect at a common
vertex that is also in X . Two facets of X may intersect only at a union of
vertices and segments which are also in X .
S
The underlying space of a PLC X , denoted |X |, is f ∈X f , which is the
domain to be triangulated. A tetrahedral mesh of X , is a 3d simplicial complex T such that (1) X and T have the same vertices, (2) every cell in X is
a union of simplices in T , and (3) |T | = |X |.
Let T be a tetrahedral mesh of a 3d PLC X . The boundary of X is
respected by the elements of T , i.e., each segment of X is represented by a
union of edges in T , and each facet of X is represented a union of triangles
in T . To distinguish those edges and triangles of T which are on segments
and facets of X , we call them boundary edges and boundary faces.
TetGen uses a simple boundary representation (a surface mesh) to represent a 3d PLC. It is explained in Section 5.1.1 and in the file formats .poly
and .smesh of TetGen. Figure 5 shows two typical surface meshes of 3d
PLCs. The following points are useful to know.
• TetGen does not generate the surface mesh of the PLC. It must be
given by the users as the input of TetGen.
• TetGen is able to modify the surface mesh by further subdividing them.
This is necessary in order to conform to the constrained Delaunay prop-
8
1 INTRODUCTION
Figure 6: Polyhedra which can not be tetrahedralized without Steiner points.
Left: The Schönhardt polyhedron [14]. Right: The Chazelle’s polyhedron [4].
erty and improve the mesh quality. This is the default choice of the -p
switch.
• TetGen will preserve the surface mesh (does not subdivide it) when the
switch -Y is applied.
• If the input surface mesh contains self-intersections, TetGen will detect
them and stop the meshing process automatically.
• If the input surface mesh contains holes, i.e., it is not watertight, TetGen will finish the meshing process, but it returns an empty 3d tetrahedral mesh unless the -c switch (to keep the convex hull of the mesh)
is used.
Limitation of PLCs. A PLC only gives a piecewise linear approximation
of a 3d domain. It does not take the curvature of the surfaces into account.
When TetGen modifies the surface mesh, it only modifies the linear edges
and facets. This is unfortunately a limitation of using PLC.
1.2.2
Steiner Points
There are 3d polyhedra which may not be tetrahedralized with only its own
vertices, two typical examples are shown in Figure 6. Nevertheless, it is
always possible to tetrahedralize a polyhedron if Steiner points (which are
not vertices of the polyhedron) are allowed.
A Steiner tetrahedralization of a PLC X is a tetrahedralization of X ∪ S,
where S is a finite set of Steiner points (disjoint from the vertices of X ).
TetGen generates Steiner tetrahedralizations of PLCs. Two types of Steiner
points are used in TetGen:
1.2 Tetrahedral Meshes of 3d Spaces
9
• The first type of Steiner points are used in creating an initial tetrahedralization of PLC. These Steiner points are mandatory in order to
create a valid tetrahedralization.
• The second type of Steiner points are used in creating quality tetrahedral meshes of PLCs. These Steiner points are optional, while they
may be necessary in order to improve the mesh quality or to conform
the size of mesh elements.
In both cases, TetGen tries to generate the Steiner points efficiently and
limit the number of Steiner points as small as possible. The optimal locations
and the optimal number of Steiner points is still a topic of research.
1.2.3
Boundary Conformity
A fundamental problem in mesh generation is how to enforce a set of constraints, such as edges and triangles, to be preserved or represented by a
mesh. These constraints usually describe the special features in the domain
boundaries, such as the boundary complex of a PLC, and they are required
to be correctly represented in the generated meshes. It is generally referred
to the boundary conformity or boundary recovery problem.
Boundary conformity in 2d is very easy. One can enforce any edge (which
does not intersect any boundary) into a triangulation. Moreover, it does not
need any Steiner point. However, it is very difficult in 3d since it is not
always possible to enforce an edge or a triangle into a tetrahedralization
without using Steiner points.
TetGen always respects the boundary of the domain. TetGen can generate
different types of (Steiner) tetrahedralizations such that input segments and
facets of a PLC are respected.
• A conforming Delaunay tetrahedralization. It is a subcomplex of a Delaunay tetrahedralization, i.e., every tetrahedron is a Delaunay tetrahedron. It may contain Steiner points. Some Steiner points may lie on
the boundary of the PLC, i.e., the boundary triangulation of the PLC
may be a refinement of the original one.
• A constrained Delaunay tetrahedralization (CDT). Each of its tetrahedron satisfies a constrained Delaunay criteria, and it has many properties similar to those of a Delaunay tetrahedralization. It is further
explained in Section 1.2.4. A CDT may contain Steiner points. Moreover, most of the Steiner points lie on the segments of the PLC. Hence,
the boundary triangulation of the PLC may be a refinement of the
original one.
10
1 INTRODUCTION
e
p
b
c
f
c
b
d
a
a
q
d
Figure 7: Left: The tetrahedron abcd is constrained Delaunay. Right: The
triangle abc is locally Delaunay.
• A constrained tetrahedralization. It is a tetrahedralization which preserves the input surface mesh of the PLC. It may contain Steiner points,
but they must lie in the interior of the PLC. This type of tetrahedralization may be neither Delaunay nor constrained Delaunay.
These different types of tetrahedralizations produced by TetGen may find
use in different situations. For instances, conforming DTs are desired for applications which need the Delaunay property. While this type of tetrahedralization usually need a large number of Steiner points. CDTs require much
less Steiner points. They are an alternative choice of conforming DTs when
the applications can accept non-Delaunay elements. Constrained tetrahedralizations are useful in many engineering applications for which the input
domain boundaries need to be preserved.
1.2.4
Constrained Delaunay Tetrahedralizations
A constrained Delaunay tetrahedralization (CDT) is a variation of a Delaunay
tetrahedralization that is constrained to respect the edges and facets of X .
CDTs in the plane were introduced by Lee and Lin [10]. Shewchuk [16, 21]
generalized them into three or higher dimensions.
In the following, we give two equivalent definitions of constrained Delaunay tetrahedralizations.
The visibility between two vertices p, q ∈ |X | is called occluded if there is a
facet f ∈ X such that p and q lie on opposite sides of the plane that includes
f , and the line segment pq intersects this facet (see Figure 7). A tetrahedron
t whose vertices are in X is constrained Delaunay if its circumsphere encloses
no vertex of X , which is visible from any point in the relative interior of t
(see Figure 7 Left).
A tetrahedralization T is a constrained Delaunay tetrahedralization of X
if it is a tetrahedralization of X and every tetrahedron of T is constrained
Delaunay.
11
1.2 Tetrahedral Meshes of 3d Spaces
Regular
Needle
Spindle
Wedge
Cap
Sliver
Figure 8: Tetrahedra of different shapes.
Let s be a triangle in a tetrahedralization T of X . s is said to be locally
Delaunay if either it belongs to only one tetrahedron of T , or it is a face of
exactly two tetrahedra t1 and t2 and it has a circumsphere which does not
enclose any vertex of t1 and t2 . Equivalently, the circumsphere of t1 encloses
no vertex of t2 and vice versa (see Figure 7 Right). A tetrahedralization T
of P is a CDT of P if every triangle in T not included in any facet of P is
locally Delaunay.
The definitions of Delaunay tetrahedralization and constrained Delaunay
tetrahedralization are almost the same except that, for the CDT, we free
the requirement of being locally Delaunay for triangles in the facet. Hence
CDTs retain many nice properties of those of Delaunay tetrahedralizations,
see [21, 23]. Note that simplices (tetrahedra, triangles, and edges) in a CDT
are not always Delaunay.
A CDT of an arbitrary PLC X may not exist [21]. Steiner points are
necessary to ensure the existence of a CDT. A Steiner CDT of X is a CDT
of X ∪ S, where S ⊂ |X | is a set of Steiner points.
Compared to conforming Delaunay tetrahedralizations, (Steiner) CDTs
usually require much less Steiner points.
1.2.5
Mesh Quality, Tetrahedron Shape Measures
There is no unique definition of the term “mesh quality”. It depends on the
intended application and the numerical methods employed, see, e.g. [19].
As a general guideline, elements with very small and large angles (and
dihedral angels) should be avoided since they usually downgrade the accuracy
and performance of numerical methods.
Figure 8 shows six differently shaped tetrahedra. A tetrahedron shape
measure is a continuous function which evaluates the shape of a tetrahedron
by a real number. Various tetrahedron shape measures have been suggested,
some of them are equivalent.
The most general shape measure for a simplex is the aspect ratio. The
aspect ratio, η(τ ), of a tetrahedron τ is the ratio between the longest edge
length lmax and the shortest height hmin , i.e., η(τ ) = lmax /hmin . The aspect
12
1 INTRODUCTION
L
R
L
L
R
R
Figure 9: The radius-edge ratio ( R
) of tetrahedra. Most of the badly shaped
L
tetrahedra will have a large radius-edge-ratio except slivers (Right).
ratio
√ √measures the “roundness” of a tetrahedron in terms of a value between
2/ 3 and +∞. A low aspect ratio implies a better shape. Other possible
definitions of aspect ratio exist, such as the ratio between the circumradius
and inradius. These definitions are equivalent in the sense that if a tetrahedralization is bounded w.r.t. one of the ratios then it is bounded w.r.t. all
the others.
The tetrahedron shape measures used in TetGen are the face angles (angles between two edges) and dihedral angles (angles between two
faces) as the shape measures of tetrahedra. They work well with the Delaunay refinement algorithm used in TetGen. Also, the combination of them
achieve the same objective as the aspect ratio.
To bound the smallest face angle is equivalent to bound the radius-edge
ratio of the tetrahedron. The radius-edge ratio, ρ(τ ) of a tetrahedron τ is
the ratio between the radius r of its circumscribed ball and the length d of
its shortest edge, i.e.,
1
r
,
ρ(τ ) = ≥
d
2 sin θmin
where θmin is the smallest
face angle of τ , see Figure 9. The radius-edge
√
ratio ρ(τ ) is at least 6/4 ≈ 0.612, achieved by the regular tetrahedron.
Most of the badly shaped tetrahedra will have a big radius-edge ratio (e.g.,
> 2.0) except the sliver, which is a type of very flat tetrahedra having no
small
edges, but nearly zero volume. A sliver can have a minimal value
√
2/2 ≈ 0.707, hence the radius-edge ratio is not equivalent to aspect ratio
(due to the slivers). Nevertheless, the radius-edge ratio is a useful shape
measure. It can be shown that if a tetrahedral mesh has a radius-edge ratio
bounded for all of its tetrahedra, then the point set of the mesh is well spaced,
and each node of the mesh has bounded degree [11, 23].
Each of the six edges of a tetrahedron τ is surrounded by two faces.
At a given edge, the dihedral angle between two faces is the angle between
the intersection of these faces and a plane perpendicular to the edge. The
1.2 Tetrahedral Meshes of 3d Spaces
13
Figure 10: An adaptive tetrahedral mesh (left) and the calculated numerical
solution (right) of a heat conduction problem.
dihedral angle in τ is between 0◦ and 180◦ . The minimum dihedral angle
φmin (τ ) of τ is a tetrahedron shape measure used by TetGen.
1.2.6
Mesh Adaptation, Mesh Sizing Functions
The goal of adaptive mesh generation is to generate a mesh which achieves
the desired mesh quality with a small number of mesh elements. In numerical
methods, such a mesh gives a good balance between the solution time and
the accuracy of the solution, see Figure 10 for an example.
The total number of mesh element is determined by the mesh element
sizes, i.e., the diameters of the mesh elements. It is in general no possible to
determine a proper mesh element size in advance. It depends on the actual
applications and the numerical methods employed.
As a general guideline for adaptive mesh generation, TetGen uses a mesh
sizing function. Let X be a 3d PLC. A mesh sizing function H : |X | → R,
is a function that maps each point p ∈ |X | to a positive value H(p) which
specifies the desired mesh edge lengths at the point location p. Typically,
the mesh sizing function can be the geometrical features of the input PLC,
an error distribution function obtained from a previous numerical solution,
or a user-specified function.
A mesh sizing function H is isotropic if the edge length does not vary
with respect to the directions at p, otherwise, it is anisotropic. The current
version of TetGen only supports isotropic mesh sizing functions. An ideal
sizing function is C ∞ , ∀p ∈ |X |. However, in most cases, H is approximated
by a discrete function specified at some points in |X |. The size at other
points in |X | is obtained by means of interpolation.
TetGen supports several ways of defining a sizing function. It can be
defined automatically, explicitly on various sources of volumes, facets, line
segments, and points, or through user-defined sizing functions.
• By default, TetGen uses the local feature size [13] of the PLC. It is a
14
1 INTRODUCTION
distance function on |X | based on its boundary information.
• One can apply a bound of maximum volume (the -a switch) on every
tetrahedra of the mesh. It is the same as defining a global constant
sizing function. For a domain consisting of multiple sub-domains (materials), the volume bound can vary in each sub-domain, see the .poly
and .smesh file formats.
• One can apply a bound of maximum area on each facet, or apply a
bound of maximum edge length on each boundary segment of the input
PLC, see the .var file format.
• One can define a mesh sizing function directly on the input PLC. In
this way, to each vertex of the PLC a value for the mesh sizing function
is to be assigned, see the .mtr file format.
• One can use a background mesh to define a sizing function. The background mesh can be any tetrahedral mesh. Its underlying space must
cover the input PLC. To each mesh node of the background tetrahedral
mesh a value for the mesh sizing function is assigned, which contains
the desired edge length at that location in the PLC.
All the above ways of defining sizing function can be used at the same time.
TetGen will automatically choose the smallest mesh element size. In the last
two ways, i.e., defining mesh element sizes on nodes, it is possible to set the
size to zero at a node. In this case, the mesh element size at this location is
ignored.
1.2.7
Mesh Optimization
Mesh optimization is an essential way for further improving the mesh quality.
Typically, it improves one or several objective functions on mesh quality, such
as the aspect ratio, minimum or maximum dihedral angles, etc.
Mesh optimization is usually done by applying various local mesh operations which either change the node locations or change the mesh connections.
The most frequently used operations are: node smoothing, edge/face swapping, edge contraction, and vertex insertion. These operations are combined
according to a schedule to iteratively improve the mesh quality. One can decide to either optimize the whole mesh (global optimization) or only optimize
a part of the mesh (local optimization).
TetGen uses mesh optimization after the mesh generation. It locally optimizes the tetrahedral mesh to restore the Delaunay property and to improve
the mesh quality. The current version uses the maximum dihedral angle as
1.2 Tetrahedral Meshes of 3d Spaces
15
objective function. It provides options to choose local operations and to restrict the maximum number of iterations in the optimization procedure, see
the -O switch.
1.2.8
Algorithms
Algorithms for constructing 3d CDTs were first considered by Shewchuk.
In [16], a sufficient condition for the existence of a CDT of a PLC (or a polyhedron) is given. Based on this condition, several algorithms for constructing
Steiner CDTs are proposed [18, 20, 26, 25]. TetGen’s CDT algorithm is from
Si and Gärtner [26, 25].
The basic algorithm for generating quality tetrahedral meshes is the Delaunay refinement algorithm from Ruppert [13] and Shewchuk [17]. This
algorithm generates a quality mesh of Delaunay tetrahedra with no tetrahedra having a radius-edge ratio greater than 2.0 (equivalently, no face angle
less than 14.5o ). The sizes of tetrahedra are graded from small to large over a
short distance. TetGen implemented this algorithm for improving the mesh
quality of a CDT of a PLC. In practice, the algorithm generates meshes
generally surpassing the theoretical bounds and eliminates tetrahedra with
small or large dihedral angles efficiently.
There are two theoretical problems of the basic Delaunay refinement algorithm. First, it does not remove slivers due to the use of radius-edge ratio
as the sole tetrahedral shape measure. Second, it may not terminate if the
input PLC X contains sharp features, i.e., there are two edges of X meeting
at an acute angle, or two facets of X meeting at an acute dihedral angle.
TetGen uses the minimal dihedral angle of tetrahedron as a second shape
measure for the Delaunay refinement algorithm, hence slivers are found by
this measure and are removed by the above mentioned Delaunay refinement
iterations. Since TetGen works with CDTs, it can detect all the sharp features in the CDT in advance. Then it starts the Delaunay refinement process.
Tetrahedra at the sharp features are never removed. The modified algorithm
in TetGen always terminates. However some badly-shaped tetrahedra near
the sharp features may survive.
TetGen accepts a user-defined mesh sizing function to control the mesh
element size. If this is given, TetGen will generate an adaptive tetrahedral
mesh according to the input mesh sizing function. It uses a constrained
Delaunay refinement algorithm [22].
16
1 INTRODUCTION
Figure 11: The flowchart of the mesh generation process of TetGen.
1.3
Description of the Meshing Process
Figure 11 shows a graphic flowchart of the meshing process in TetGen.
Here are the general steps of TetGen to create a (quality) tetrahedral
mesh. Many of these steps can be skipped, depending on the command line
switches.
1. Initialize constants and parse the command line.
2. Read the vertices from a (.node) file and either
- create the corresponding Delaunay tetrahedralization (DT) (no
-r), or
- read an existing tetrahedral mesh from (.ele, .face, .edge) files
and reconstruct it (-r).
1.3 Description of the Meshing Process
17
3. Read the boundary informations (segments and facets) from (.poly or
.smesh, .edge) files and triangulate them (-p).
4. Read the background mesh from (.b.node, .b.ele, .b.mtr ...) files
(if it is provided) and interpolate the mesh element size from the background mesh to the current mesh (-m).
5. Insert the boundary segments and facets into the DT (-p) by either
– constructing a constrained Delaunay tetrahedralization (CDT) in
which the segments and facets may be split into smaller pieces (no
-Y), or
– recovering the segments and facets in a tetrahedralization which
contains them (-Y).
6. Read the holes (-p), regional attributes (-pA), and regional volume
constraints (-pa), and either
- remove the exterior tetrahedra (in the holes and concavities) (no
-c), or
- mark the exterior tetrahedra (-c),
and spread the regional attributes and volume constraints.
7. Coarsen the mesh (-R) by removing vertices which are either marked
or inconsistent according to a mesh sizing function (-m).
8. Read the list of additional vertices from a (.a.node) file (if it is provided) and insert them into the current mesh (-i).
9. Enforce the constraints on minimum quality bound (-q) and maximum,
volume (-a), and mesh sizing function (-m).
10. Optimize the mesh with respect to the optimization scheme (-O) based
on specified quality measures (-o).
11. Write the output files and print the statistics.
12. Check the consistency of the mesh (-C).
18
2
2 GENERAL INFORMATION
General Information
This section gives some general information about TetGen. These are not
required in order to use TetGen.
2.1
Language, Platforms
TetGen is written entirely in C++, and it only uses the standard C (ANSI)
libraries. Hence the code is easily portable and should be compiled by a
popular C++ compilers, such as GNU’s gcc/g++, Intel’s and Mircosoft’s C++
compilers. So far, TetGen has been successfully compiled on all major computer architectures and major operating systems (Unix/Linux, Windows,
Mac OS) with both 32-bit and 64-bit versions.
The current version of TetGen contains about 25, 000 source lines and
7, 000 comment lines. These numbers do not include those of Shewchuk’s
robust predicates.
2.2
Memory requirement
TetGen dynamically allocates memory when it is needed. There is no minimum memory requirement to run TetGen. The maximum memory is only
limited by the physical memory available from your system. The more memory you have, the larger the mesh you can generate.
For example, the 32-bit version of TetGen used about 694.5 Mega bytes
memory to generate the Delaunay tetrahedralization (DT) of a set of 2, 000, 000
(two million) points randomly distributed inside a unit cube. This DT has
13, 504, 899 tetrahedra. This is approximately 364 bytes per vertex or 54
bytes per tetrahedron. In other words, with 4 Giga byte memory, a maximum Delaunay tetrahedralization may be generated by the 32-bit version
of TetGen having approximately 11, 799, 360 (ca. 11 million) vertices or
79, 536, 431 (ca. 79 million) tetrahedra.
More memory is needed in generating quality tetrahedral meshes. The
extra memory is used to store boundary information and working arrays of
algorithms. For example, the 32-bit version of TetGen used about 770 Mega
bytes memory to generate a quality tetrahedral mesh with 2, 000, 000 (two
million) vertices and 12, 237, 300 tetrahedra.
So far, the largest quality tetrahedral mesh was generated by the 64-bit
version of TetGen. It contains 1, 007, 700, 944 (ca. 1 billion) vertices, and
6, 454, 556, 696 (ca. 6.45 billion) tetrahedra. TetGen used about 905.7 Giga
bytes memory on generating this mesh, see Table 2.
2.3 CPU time estimation
2.3
19
CPU time estimation
When generating Delaunay tetrahedraliations, the worst case running time
of TetGen is O(n2 ), where n is the input number of vertices. However, this
only happens for some very special point sets. For most point sets appearing
in applications, TetGen runs in almost O(n log n) time, see Table 1.
The CPU time required to obtain the quality tetrahedral mesh is obviously related to the complexity of the geometry and topology of the inputs,
as well as the command line switches and parameters specified. Nevertheless, the running time of TetGen increases almost linearly with respect to the
output number of mesh elements, see Table 2.
2.4
Performance
This section gives some practical information regarding the performance of
TetGen. In particular, Table 1 and Table 2 report the statistics of some test
runs of TetGen. The used version of TetGen was a 64-bit version compiled by
GNU gcc/g++ version 4.7.2 with the −O3 (optimization) switch. The used
computer was a computer of WIAS (erhard-03) with Intel(R) Xeon(R)
Ten-Core 2.40GHz CPU, 1, 024 Giga byte memory, and SuSE Linux. The
given CPU times exclude the file I/O time. Comparisons of TetGen with
other programs are available in paper [24].
Table 1 shows the statistics of TetGen on creating Delaunay tetrahedralizations. Three random point sets of different sizes are used in these tests.
They are generated by the tool rbox (command: rbox -D3 xxx) in the program qhull (http://www.qhull.org). It is well known that the Delaunay
tetrahedralizations of random point sets have linear complexity. Both memory usage and CPU time of TetGen are quasilinear.
# of points
# of tets used memory CPU time
(input)
(output) (Mega bytes) (seconds)
1, 000, 000
6, 748, 645
710
9.2
2, 000, 000 13, 504, 917
1, 420
19.0
20, 000, 000 135, 059, 867
14, 196
203.1
Table 1: Statistics of TetGen on generating Delaunay tetrahedralizations.
Table 2 shows the statistics of TetGen on creating quality tetrahedral
meshes. The input of these tests is a 3d unit cube with 8 vertices and 6
facets described in file cube.smesh. The resulting meshes were generated by
using the command: tetgen -pqa# -x1000000 cube.smesh, where “-a#”
20
2 GENERAL INFORMATION
specifies a very small tetrahedron volume constraint, and “ -x1000000” specifies a user-defined memory allocation size.
volume
# of points
# of tets used memory CPU time
constraint
(output)
(output) (Mega bytes) (seconds)
−7
1.0 × 10
3, 023, 518
18, 744, 549
2, 904
126
1.0 × 10−8
29, 412, 255
186, 090, 870
27, 599
1, 374
1.0 × 10−9
290, 323, 148 1, 854, 336, 524
268, 781
12, 124
−10
5.0 × 10
579, 301, 396 3, 706, 331, 655
534, 386
26, 872
2.87 × 10−10 1, 007, 700, 944 6, 454, 556, 696
927, 449
44, 261
Table 2: Statistics of TetGen on generating quality tetrahedral meshes.
2.5
Errors
When running TetGen, errors may happen. In this case, TetGen may fail to
generate a mesh. The typical reasons that cause failures of TetGen may be
one of the follows:
• The wrong use of command line switches and parameters.
• The input surface mesh contains self-intersections.
• Out of memory.
• A known bug of TetGen.
• An unknown bug of TetGen.
In most cases TetGen is able to detect the error. If TetGen is running as
a standalone program and if an error is detected, then TetGen will report a
message that describes the error possibly together with a suggestion to fix
it, and terminate. Below is an example of such a message.
Found two segments intersect each other.
1st: [13,7] 1.
2nd: [1114,1113] 1.
A self-intersection was detected. Program stopped.
Hint: use -d option to detect all self-intersections.
2.5 Errors
21
If TetGen is running as a library, i.e., it is called inside other program,
then the error message will not be seen, and TetGen will throw an error
index (integer), which can be catched by the standard C++ exception handler
try and catch. A list of error indices and messages are provided in the
Appendix.
When TetGen terminates on error, it automatically releases the used
memory before terminating itself. This avoids potential memory leak when
calling TetGen multiple times.
22
3
3 GETTING STARTED
Getting Started
TetGen is distributed in its source code (written in C++). The latest version
of TetGen is available at http://www.tetgen.org.
Section 3.1 briefly explains how to compile TetGen into an executable
program or a library.
Once TetGen is compiled, and assume you have the executable file, tetgen
(or tetgen.exe in Windows), you can start testing TetGen with the included
example file by following the tutorial in Section 3.2.
TetGen does not have a graphic user interface (GUI). The TetView program can be used to visualize the input and output of TetGen. Alternatively,
other popular mesh viewers are supported, see Section 3.3.
3.1
Compilation
The downloaded archive should include the following files:
README
LICENSE
tetgen.h
tetgen.cxx
predicates.cxx
makefile
CMakeLists.txt
example.poly
General information.
Copyright notices.
C++ header file of TetGen.
C++ source file of TetGen.
C++ source of Shewchuk’s predicates.
make file for compiling TetGen.
cmake file for compiling TetGen.
An example input file.
The file predicates.cxx is a modified C++ version of Shewchuk’s robust
geometric predicates http://www.cs.cmu.edu/~quake/robust.html.
To compile TetGen, use a C++ compiler for the system on which TetGen will be used, such as GNU’s g++, or Microsoft C++ on MS Windows
systems. TetGen may be compiled into an executable program or a library,
which can be embedded into another program.
3.1.1
Using make
The easiest way to compile TetGen is to edit and use the included makefile.
Before compiling, put all source files, tetgen.h, tetgen.cxx, and predicates.cxx
and the makefile into one directory (usually they are), read the makefile,
which describes your options, and edit it accordingly.
You should at least specify the C++ compiler and the level of optimization. By default, the GNU C++ compiler (g++) is used, and there is no
optimization is used.
3.1 Compilation
23
Once you have done this, type make to compile TetGen into an executable
program or type make tetlib to compile TetGen into a library. The executable file tetgen or the library libtet.a appears in the same directory as
the makefile.
Alternatively, the files are usually easy to compile directly on the command line. Assume you’re using g++, first compile the file predicates.cxx
to get an object file:
g++ -c predicates.cxx
To compile TetGen into an executable file, use the following command:
g++ -o tetgen tetgen.cxx predicates.o -lm
To compile TetGen into a library, the symbol TETLIBRARY is needed:
g++ -DTETLIBRARY -c tetgen.cxx
ar r libtet.a tetgen.o predicates.o
Some additional remarks to get an efficient executable version of TetGen.
• One should use the optimization options provided by the C++ compiler. Usually, an optimized version of TetGen may run double times
faster than an unoptimized one.
• There are some assertions inserted into the source code of TetGen.
They are used for catching program bugs. But they may slow down the
performance of TetGen. You can disable them by adding the symbol
-DNDEBUG in your commands of compilation.
Here is an example to get an optimized version of TetGen using GNU’s C++
compiler:
g++ -O3 -DNDEBUG -c predicates.cxx
g++ -O3 -DNDEBUG -o tetgen tetgen.cxx predicates.o -lm
3.1.2
Using cmake
As an alternative, one can compile TetGen using cmake (www.cmake.org).
It simplifies the compilation of TetGen with different compilers and architectures (Linux, MacOSX, and MS Windows) at the same time.
Using the provided file CMakeLists.txt, the simplest sequence of commands is:
24
3 GETTING STARTED
cd <tetgen-directory>
mkdir build
cd build
cmake ..
make
When the compilation is finished, you should get both of the executable
file tetgen and the library libtet.a under the directory build.
To get a debug version of TetGen, use the following commands
cd <tetgen-directory>
mkdir build
cd build
cmake -DCMAKE_BUILD_TYPE=Debug ..
make
The default is cmake -DCMAKE_BUILD_TYPE=Release ...
To specify a compiler, do
CXX=icpc cmake -DCMAKE_BUILD_TYPE=Debug ..
3.1.3
Remarks on Using Shewchuk’s Robust Predicates
TetGen default uses Shewchuk’s robust geometric predicates which perform
exact floating-point arithmetic (predicates.cxx). The arithmetic are based
on the IEEE 754 floating-point standard. However, some processors may not
default use this standard for floating-point representations and arithmetic.
If so, a configuration is needed to correctly execute the predicates. It is
described on its website http://www.cs.cmu.edu/~quake/robust.pc.html
for details.
Below are my own experience in using Shewchuk’s predicates.
• If TetGen fails with “a segmentation fault” during the construction of
the Delaunay tetrahedralization, it is most likely that the predicates
were not correctly configured.
• I used gcc/g++ version 4.4 on a MacOSX system with an Intel Core 2
Duo. There is no need to configure the predicates. Just use the default
settings in the predicates. TetGen runs correctly both with debug and
with optimization options. The same when I used gcc/g++ version 4.6
on a Linux (Ubuntu) system with an Intel Pentium 2.8GHz.
3.1 Compilation
25
• I did encounter failures of TetGen when I used the the gcc/g++ version
4.2 on a Linux system with an Intel Xeon CPU, and TetGen was compiled with an optimization option -O3. However, TetGen ran correctly
when it was compiled with the debug option -g. The same issue happened when TetGen was compiled using Intel’s C++ compiler (version
2012) with the optimization option -O3.
3.1.4
Using CGAL’s Robust Predicates
Alternatively, TetGen can use other robust predicates developed in computational geometry, such as the filtered robust predicates in CGAL (http:
//www.cgal.org). TetGen includes the interface to the CGAL’s predicates in
the file predicates.cxx. It uses the following kernel,
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
In this section, we show how to use CGAL’s predicates in TetGen. The
following steps are needed.
1. Download CGAL. The version I used was 4.1 (October 2012). Compile
and install CGAL by using the following commands:
cd CGAL-x.y # go to CGAL directory
cmake . # configure CGAL
make # build the CGAL libraries
If you have any problems in compiling CGAL, please see CGAL’s manual
about installation. Once CGAL is compiled, make sure that you get the
new file compiler config.h inside the directory CGAL-x.y/include/CGAL.
2. Set the following environment path variables:
BOOST=/opt/local/include
GMP_LIB=/opt/local/lib
CGAL_INC=$(HOME)/Programs/CGAL/CGAL-4.1/include
BOOST should point to the directory containing the boost library. Version 1.35 or later is required. GMP LIB should point to the directory
containing the library GMP, - the GNU Multiple Precision Arithmetic
Library. And CGAL INC should point to the C++ headers of CGAL’s
library.
26
3 GETTING STARTED
3. Compile predicates.cxx by using the variable -DUSE CGAL PREDICATES,
i.e.,
g++ -I$(BOOST) -I$(CGAL_INC) -DUSE_CGAL_PREDICATES \
-O3 -c predicates.cxx
4. Compile an executable version of TetGen by the following command:
g++ -O3 -o tetgen tetgen.cxx predicates.o -L$(GMP_LIB) -lgmp
3.2
A Short Tutorial
TetGen gives a short list of command line options if it is invoked without
arguments (that is, just type tetgen). A brief description of the usage is
printed by invoking TetGen with the -h switch:
tetgen -h
The enclosed example file, example.poly, is a simple 3d mesh domain (a
PLC), see Figure 24. Try out TetGen using:
tetgen -p example.poly
With the -p switch, TetGen will read the file, i.e., example.poly, and
generate its constrained Delaunay tetrahedralization. The resulting mesh is
saved in four files: example.1.node, example.1.ele, example.1.face, and
example.1.edge, which are a list of mesh nodes, tetrahedra, boundary faces,
and boundary edges, respectively. The file formats of TetGen are described
in Section 5. The screen output of the above command looks like this:
Opening example.poly.
Delaunizing vertices...
Delaunay seconds: 0.000864
Creating surface mesh ...
Surface mesh seconds: 0.000307
Constrained Delaunay...
Constrained Delaunay seconds: 0.000325
Removing exterior tetrahedra ...
Exterior tets removal seconds: 8.6e-05
Optimizing mesh...
Optimization seconds: 6.4e-05
Writing example.1.node.
Writing example.1.ele.
3.2 A Short Tutorial
27
Writing example.1.face.
Writing example.1.edge.
Output seconds: 0.000663
Total running seconds: 0.002451
Statistics:
Input
Input
Input
Input
Input
Mesh
Mesh
Mesh
Mesh
Mesh
points: 28
facets: 23
segments: 44
holes: 2
regions: 2
points: 28
tetrahedra: 68
faces: 160
faces on facets: 50
edges on segments: 44
The above mesh is pretty coarse, and contains many badly-shaped (e.g.,
long and skinny) tetrahedra. Now try:
tetgen -pq example.poly
The -q switch triggers the mesh refinement such that Steiner points are
added to remove badly-shaped tetrahedra. The resulting mesh is contained
in the same four files as before. However, now it is a quality tetrahedral mesh
whose tetrahedra have no small face angle less than about 14o (the default
quality value). The screen output of the above command looks like this:
Opening example.poly.
Delaunizing vertices...
Delaunay seconds: 0.000384
Creating surface mesh ...
Surface mesh seconds: 0.000181
Constrained Delaunay...
Constrained Delaunay seconds: 0.000163
Removing exterior tetrahedra ...
Exterior tets removal seconds: 0.000368
Refining mesh...
Refinement seconds: 0.009291
Optimizing mesh...
Optimization seconds: 0.000517
Writing example.1.node.
Writing example.1.ele.
Writing example.1.face.
28
3 GETTING STARTED
Writing example.1.edge.
Output seconds: 0.004716
Total running seconds: 0.015802
Statistics:
Input
Input
Input
Input
Input
points: 28
facets: 23
segments: 44
holes: 2
regions: 2
Mesh points: 216
Mesh tetrahedra: 689
Mesh faces: 1587
Mesh faces on facets: 430
Mesh edges on segments: 126
Steiner points inside domain: 1
Steiner points on facets: 105
Steiner points on segments: 82
Now try to run:
tetgen -pq1.2V example.poly
TetGen will again generate a quality mesh, which contains more points
than the previous one, and all tetrahedra have radius-edge ratio bounded by
1.2, i.e., the element shapes are better than those in the previous mesh. In
addition, TetGen prints a mesh quality report (due to the -V switch) which
looks as below:
Mesh quality statistics:
Smallest
Shortest
Smallest
Smallest
Smallest
volume:
edge:
asp.ratio:
facangle:
dihedral:
0.00059866
0.25
1.3255
24.898
8.6045
|
|
|
|
|
Largest
Longest
Largest
Largest
Largest
Aspect ratio histogram:
< 1.5
:
23
|
6
1.5 - 2
:
364
|
10
2 - 2.5
:
480
|
15
2.5 - 3
:
248
|
25
3 - 4
:
125
|
50
4 - 6
:
56
|
100
(A tetrahedron’s aspect ratio is its longest
volume:
edge:
asp.ratio:
facangle:
dihedral:
0.09363
1.4142
10.169
126.8698
163.4980
- 10
:
26
- 15
:
1
- 25
:
0
- 50
:
0
- 100
:
0
:
0
edge length divided by its
29
3.2 A Short Tutorial
smallest side height)
Face angle histogram:
0 - 10 degrees:
10 - 20 degrees:
20 - 30 degrees:
30 - 40 degrees:
40 - 50 degrees:
50 - 60 degrees:
60 - 70 degrees:
70 - 80 degrees:
80 - 90 degrees:
Dihedral
0 5 10 20 30 40 50 60 70 -
0
0
188
1059
1704
1865
1609
1105
736
|
|
|
|
|
|
|
|
|
90
100
110
120
130
140
150
160
170
-
100
110
120
130
140
150
160
170
180
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
465
132
54
5
0
0
0
0
0
angle histogram:
5 degrees:
0
10 degrees:
2
20 degrees:
150
30 degrees:
266
40 degrees:
643
50 degrees:
1056
60 degrees:
1213
70 degrees:
1121
80 degrees:
946
|
|
|
|
|
|
|
|
|
80
110
120
130
140
150
160
170
175
-
110
120
130
140
150
160
170
175
180
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
1831
274
193
105
62
52
24
0
0
Instead of using the -q switch to get a finer mesh, one can use the -a
switch to impose a maximum volume constraint on the resulting tetrahedra.
By doing so, no tetrahedron in the resulting mesh has volume bigger than
the specified value after -a. Try to run the following command.
tetgen -pq1.2Va1 example.poly
Now the resulting mesh should contain much more vertices than the previous one. Besides of -q and -a switches, TetGen provides more switches to
control the mesh element size and shape. They are described in Section 4.
To compute the Delaunay tetrahedralization and convex hull of the point
set of this PLC, try this:
cp example.poly example.node
tetgen example.node
The Delaunay tetrahedralization is saved in example.1.node and example.1.ele.
The convex hull is represented by a list of triangles in file example.1.face.
All these meshes and Delaunay tetrahedralizations can be visualized by
the programs introduced in the next section.
Detailed descriptions of the command line switches and file formats are
found in Section 4 and 5.
30
3 GETTING STARTED
3.3
3.3.1
Visualization
TetView
TetView is a graphic interface for viewing piecewise linear complexes and
simplicial meshes. It can read the input and output files of TetGen and
display the objects. It also shows other information as well, such as boundary
types and materials. The interactive GUI allows the user to manipulate (i.e.,
rotate, translate, zoom in/out, cut, shrink, etc.) the viewing objects easily
through either mouse or keyboard. TetView can save the current window
contents into high quality encapsulated postscript files. Most of the figures
of this document were produced by TetView.
TetView is freely available from http://www.tetgen.org/tetview.html.
You will find a list of precompiled executable versions on different platforms.
Download the one corresponding to your system.
To show the PLC in example.poly, first copy the executable file (tetview)
to the directory where you have this file. It is loaded by running:
tetview example.poly
And the following command will display the mesh (in files example.1.node,
example.1.ele, and example.1.face):
tetview example.1.ele
The instruction for using TetView can be found on the above website.
3.3.2
Medit and Paraview
TetGen can export its tetrahedral mesh into the .mesh format. It can be
then visualized by the software Medit, which is freely available from http:
//www.ann.jussieu.fr/~frey/logiciels.
For viewing mesh under Medit, add a -g switch in the command line.
TetGen will additionally output a file named example.1.mesh, which can be
read and rendered directly by TetGen. Try running:
tetgen -pg example.poly
medit example.1
Alternatively, TetGen can also output its tetrahedral mesh into the .vtk
format by adding the switch -k, i.e.,
tetgen -pk example.poly
It will output a file named example.1.vtk. It can then be visualized by
the software Paraview: http://www.paraview.org.
31
4
Using TetGen
This section describes the use of TetGen as a stand-alone program. It is invoked from the command line with a set of switches and an input file name.
Switches are used to control the behavior of TetGen and to specify the output files. In correspondence to the different switches, TetGen will generate
the Delaunay tetrahedralization, or the constrained (Delaunay) tetrahedralization, or the quality conforming (Delaunay) mesh, etc.
4.1
Command Line Syntax
tetgen [-pYrq_Aa_miO_S_T_XMwcdzfenvgkJBNEFICQVh] input_file
Underscores indicate that numbers may optionally follow certain switches.
Do not leave any space between a switch and its numeric parameter. These
switches are explained in Section 4.2.
input file can be different files depending on the switches you use. If
no command line switch is used, it must be a file with extension .node which
contains a list of 3d points and the Delaunay tetrahedralization of this point
set will be generated.
If the -p switch is used, input file must be a file with one of the following extensions: .poly, .smesh, .off, .stl, .ply, and .mesh, which describes the boundary (a surface mesh) of a 3d piecewise linear complex. The
boundary constrained (Delaunay) tetrahedralization of this object will be
generated. If the -q switch is used simultaneously, a boundary conforming
quality tetrahedral mesh will be generated.
If the -r switch is used, an existing tetrahedral mesh will be read. You
must supply .node and .ele files which describe the tetrahedral mesh. Optionally a .face and a .edge file can be supplied which contain the boundary
faces and edges of the mesh. input file can have no file extension.
If the switch -q is applied, the mesh will be refined with respect to the
new quality measure and variant constraints. Optionally, and a .vol, a .mtr,
and a .var file can be supplied which contain the mesh element size control
information.
File formats are described in Section 5.
4.2
Command Line Switches
An overview of all command line switches and a short description follow.
These switches are shown by invoking TetGen without any switch and in-
32
4 USING TETGEN
put file. Detailed descriptions of these switches are given in the following
subsections.
-p
-Y
-r
-q
-R
-A
-a
-m
-i
-O
-S
-T
-X
-M
-w
-c
-d
-z
-f
-e
-n
-v
-g
-k
-J
-B
-N
-E
-F
-I
-C
-Q
-V
-h
Tetrahedralizes a piecewise linear complex (PLC).
Preserves the input surface mesh (does not modify it).
Reconstructs a previously generated mesh.
Refines mesh (to improve mesh quality).
Mesh coarsening (to reduce the mesh elements).
Assigns attributes to tetrahedra in different regions.
Applies a maximum tetrahedron volume constraint.
Applies a mesh sizing function.
Inserts a list of additional points.
Specifies the level of mesh optimization.
Specifies maximum number of added points.
Sets a tolerance for coplanar test (default 10−8 ).
Suppresses use of exact arithmetic.
No merge of coplanar facets or very close vertices.
Generates weighted Delaunay (regular) triangulation.
Retains the convex hull of the PLC.
Detects self-intersections of facets of the PLC.
Numbers all output items starting from zero.
Outputs all faces to .face file.
Outputs all edges to .edge file.
Outputs tetrahedra neighbors to .neigh file.
Outputs Voronoi diagram to files.
Outputs mesh to .mesh file for viewing by Medit.
Outputs mesh to .vtk file for viewing by Paraview.
No jettison of unused vertices from output .node file.
Suppresses output of boundary information.
Suppresses output of .node file.
Suppresses output of .ele file.
Suppresses output of .face and .edge file.
Suppresses mesh iteration numbers.
Checks the consistency of the final mesh.
Quiet: No terminal output except errors.
Verbose: Detailed information, more terminal output.
Help: A brief instruction for using TetGen.
4.2 Command Line Switches
4.2.1
33
Delaunay and weighted Delaunay tetrahedralizations
Given a set of 3d points or weighted points, TetGen generates the Delaunay
tetrahedralization or the weighted Delaunay tetrahedralization of the point
set. It can also output the Voronoi diagram or the power diagram.
Save the set of points in a .node file, e.g., test.node. Run TetGen using
the command:
tetgen test.node
This command generates the Delaunay tetrahedralization (DT) of this
point set. Below is a screen output of TetGen:
Opening test.node.
Delaunizing vertices...
Delaunay seconds: 0.001695
Writing test.1.node.
Writing test.1.ele.
Writing test.1.face.
Output seconds: 0.001555
Total running seconds: 0.003615
Statistics:
Input points: 100
Mesh points: 100
Mesh tetrahedra: 514
Mesh faces: 1057
Mesh edges: 642
Convex hull faces: 58
Figure 12 shows an example of an input point set (100 vertices) and the
generated DT and its convex hull.
The default outputs of TetGen are three files listed in Table 3.
The set of all faces and edges of the DT can be obtained by adding the
output switches -f (output all faces) and -e (output all edges), respectively.
For example, by the following command
34
4 USING TETGEN
Figure 12: From left to right: a set of 100 randomly distributed points in a
unit cube, the Delaunay tetrahedralization, and the convex hull of the point
set, respectively.
test.1.node The list of vertices (same as input) of the DT.
test.1.ele
The list of tetrahedra of the DT.
test.1.face The list of convex hull faces of the point set.
Table 3: The default output files of TetGen.
tetgen -fe test.node
TetGen will output the four files listed in Table 4.
test.1.node The list of vertices (same as input) of the DT.
test.1.ele
The list of tetrahedra of the DT.
test.1.face The list of all faces of the DT.
Convex hull faces have a face marker ‘1’.
Interior faces have a face marker ‘0’.
test.1.edge The list of all edges of the DT.
Convex hull edges have an edge marker ‘1’.
Interior edges have an edge marker ‘0’.
Table 4: The output files by the command: tetgen -fe test.node.
The adjacency graph of the list of tetrahedra of the DT can be obtained by
adding the -n switch in the command line. An additional file, test.1.neigh,
will be output by TetGen, see file format .neigh for details.
The -w switch creates a weighted Delaunay tetrahedralization from a set
of weighted points. Remember that a weighted point is defined as p0 =
{px , py , pz , p2x + p2y + p2z − w} ∈ R4 , where w is the weight (a real value) of the
point p = {px , py , pz } ∈ R3 [7].
35
4.2 Command Line Switches
Figure 13: Left: a set of 164 randomly distributed points in a unit cube.
Right: the Delaunay tetrahedralization (shown in black edges) and Voronoi
diagram (shown in colored faces).
Save the set of weighted points in a .node file. The points in .node file
must have at least one attribute, and the first attribute of each point is its
weight. To generate a weighted DT of this point set, run TetGen with the
following command:
tetgen -w test.node
The weighted Delaunay tetrahedralization and its convex hull are saved
in the files with the same names as is listed in Table 3. Note that some of
the points in test.1.node may not belong to any tetrahedron.
The Voronoi diagram or the power diagram of the point set is obtained
by taking the dual of the generated Delaunay or weighted Delaunay tetrahedralization, see Figure 13 for an example.
By adding a -v switch in the command line, TetGen outputs the Voronoi
diagram or the power diagram in the four files shown in Table 5:
test.1.v.node
test.1.v.edge
test.1.v.face
test.1.v.cell
The
The
The
The
list
list
list
list
of
of
of
of
Voronoi
Voronoi
Voronoi
Voronoi
vertices (or orthocenters).
edges.
faces.
cells.
Table 5: The output files for Voronoi or power diagram.
The .v.node file has the file format as a .node file. The file formats of
.v.edge, .v.face, and .v.cell are described in the file format section.
Note that the switches -w and -v are only used for a point set.
36
4 USING TETGEN
4.2.2
Boundary conformity and recovery (-p, -Y)
The -p switch reads a boundary description (a surface mesh) of a 3d piecewise linear complex (PLC) stored in file .poly or .smesh and generates a
tetrahedral mesh of the PLC.
By default, TetGen generates a constrained Delaunay tetrahedralization
(CDT) of the PLC. Here is an example of creating a CDT of the PLC named
cami1a.poly (Figure 14 left). Run the following command:
tetgen -p cami1a.poly
This will produce the CDT of the PLC shown in Figure 14 middle. Below
is a screen output of TetGen:
Opening cami1a.poly.
Opening cami1a.node.
Delaunizing vertices...
Delaunay seconds: 0.019862
Creating surface mesh ...
Surface mesh seconds: 0.002374
Constrained Delaunay...
Constrained Delaunay seconds: 0.012435
Removing exterior tetrahedra ...
Exterior tets removal seconds: 0.000783
Optimizing mesh...
Optimization seconds: 0.000662
Writing
Writing
Writing
Writing
cami1a.1.node.
cami1a.1.ele.
cami1a.1.face.
cami1a.1.edge.
Output seconds: 0.003398
Total running seconds: 0.039744
Statistics:
Input
Input
Input
Input
Input
points: 460
facets: 884
segments: 690
holes: 0
regions: 0
Mesh points: 542
37
4.2 Command Line Switches
Mesh tetrahedra: 1678
Mesh faces: 3904
Mesh faces on facets: 1118
Mesh edges on segments: 772
Steiner points on segments:
82
From the mesh statistics of the output (the last line), we can see that
TetGen added 82 Steiner points on the segments of the PLC.
If the -Y switch is used simultaneously, the input boundary edges and
faces of the PLC are preserved in the generated tetrahedral mesh. Steiner
points (if there exists any) appear only in the interior space of the PLC. For
example, run the following command:
tetgen -pY cami1a.poly
This will produce a tetrahedral mesh of the PLC shown in Figure 14 right.
Below is a screen output of TetGen:
Opening cami1a.poly.
Opening cami1a.node.
Delaunizing vertices...
Delaunay seconds: 0.016072
Creating surface mesh ...
Surface mesh seconds: 0.001333
Recovering boundaries...
Boundary recovery seconds: 0.0432
Removing exterior tetrahedra ...
Exterior tets removal seconds: 0.001152
Suppressing Steiner points ...
Steiner suppression seconds: 0.001164
Recovering Delaunayness...
Delaunay recovery seconds: 0.016093
Optimizing mesh...
Optimization seconds: 0.004006
Jettisoning redundant points.
Writing
Writing
Writing
Writing
cami1a.1.node.
cami1a.1.ele.
cami1a.1.face.
cami1a.1.edge.
38
4 USING TETGEN
Figure 14: An input PLC (cami1a.poly, left), the generated Steiner CDT
(middle, -p switch) in which Steiner points are located on the boundary
edges of the PLC, and a constrained tetrahedralization (right, -pY switch)
in which Steiner points lie in the interior of the PLC.
Output seconds: 0.003188
Total running seconds: 0.086466
Statistics:
Input
Input
Input
Input
Input
points: 460
facets: 884
segments: 1349
holes: 0
regions: 0
Mesh points: 461
Mesh tetrahedra: 1516
Mesh faces: 3498
Mesh faces on facets: 954
Mesh edges on segments: 1349
Steiner points inside domain: 1
From the mesh statistics of the output (the last line), we can see that
TetGen only added 1 Steiner point in the interior of the PLC. The input
facets and segments are preserved.
The default outputs of TetGen are four files listed in Table 6:
cami1a.1.node
cami1a.1.ele
cami1a.1.face
cami1a.1.edge
The
The
The
The
list
list
list
list
of
of
of
of
vertices (including Steiner points) of the CDT.
tetrahedra of the CDT.
boundary faces of the CDT.
boundary edges of the CDT.
Table 6: The output files by command: tetgen -p cami1a.poly
4.2 Command Line Switches
39
Other output switches are available by adding the switches: -f (output all
faces including interior faces), -e (output all edges including interior edges),
and -n (output the adjacency graph of the tetrahedra).
4.2.3
Quality mesh generation (-q)
The -q switch adds new points to improve the mesh quality. It can be used
together with -p (to refine a CDT), or -r (to refine a previously generated
mesh), -a, or -m (to conform to a mesh sizing function).
TetGen enforces two quality constraints on tetrahedra: a maximum radiusedge ratio bound and a minimum dihedral angle bound. By default, these
two constraints are 2.0 and 0 degrees, respectively. These quality constraints
may be specified after the -q. The two constraints are separated by a slash
‘/’ (or ‘,’):
• the first constraint is the maximum allowable radius-edge ratio, default
is 2.0; and
• the second constraint is the minimum allowable dihedral angle, default
is 0 (degree);
of any tetrahedron. For example, -q1.2 specifies a maximum radius-edge
ratio of 1.2; -q1.2/10 specifies the same plus a minimum dihedral angle of
10 degrees. -q/7 specifies the default radius-edge ratio bound of 2 and a
dihedral angle bound of 7 degrees.
For example, the following command uses the default quality constraints.
It is equivalent to -pq2.0/0.
tetgen -pq thepart.smesh
The screen output of the command line is shown below. Figure 15 illustrates three quality tetrahedral meshes of a PLC generated by applying
different radius-edge ratio bounds.
Opening thepart.smesh.
Opening thepart.node.
Delaunizing vertices...
Delaunay seconds: 0.03408
Creating surface mesh ...
Surface mesh seconds: 0.004497
40
4 USING TETGEN
Constrained Delaunay...
Constrained Delaunay seconds: 0.025309
Removing exterior tetrahedra ...
Exterior tets removal seconds: 0.001419
Refining mesh...
Refinement seconds: 0.489247
Optimizing mesh...
Optimization seconds: 0.014569
Writing
Writing
Writing
Writing
thepart.1.node.
thepart.1.ele.
thepart.1.face.
thepart.1.edge.
Output seconds: 0.048593
Total running seconds: 0.618028
Statistics:
Input
Input
Input
Input
Input
points: 994
facets: 1995
segments: 1491
holes: 0
regions: 0
Mesh points: 8029
Mesh tetrahedra: 33773
Mesh faces: 73092
Mesh faces on facets: 11092
Mesh edges on segments: 5143
Steiner points inside domain: 2485
Steiner points on facets: 898
Steiner points on segments: 3652
Remarks
• The default output files (four files) have the same names as those in
Table 6.
• Adding a -V switch in the command line, TetGen will print a mesh
quality report (aspect ratios, radius-edge ratios, dihedral angles) of the
generated tetrahedral mesh on the screen, see Section 4.2.11.
• If there are no sharp features in the input PLC, the Delaunay refinement
algorithm used in TetGen is guaranteed to terminate successfully with
4.2 Command Line Switches
41
Figure 15: The quality tetrahedral meshes of a PLC (thepart.smesh) generated by the commands: -pq2/0, -pq1.4/0, and -pq1.1/0
.
a radius-edge ratio bound no smaller than 2.0, and with no bound on
the minimum dihedral angle. In practice, the algorithm behaves much
better, e.g., it usually succeeds for a radius-edge ratio of 1.2 and a
minimum dihedral angle of 18 degrees.
• If there are sharp features in the PLC, TetGen will ensure the desired
quality constraints on most of the tetrahedra, but leave some badquality tetrahedra in the final mesh. Usually, they are near the sharp
features.
4.2.4
Adaptive mesh generation (-a, -m)
TetGen supports several ways of generating adaptive tetrahedral meshes.
They have been described already in Section 1.2.6.
Impose volume constraints (-a) The -a switch is used in mesh refinement, i.e., together with -q. It imposes a maximum volume constraint on
all tetrahedra. If a number follows the -a, no tetrahedra is generated whose
volume is larger than that number. See Figure 18 for an example.
• One can impose both a fixed volume constraint and a varying volume
constraint for some sub-regions (defined in .poly or .smesh file) by
invoking the -a switch twice, once with and once without a number
following. Each volume specified may include a decimal point.
• If no number is specified and the -r switch is used, a .vol file is
expected, which contains a separate volume constraint for each tetrahedron. It is useful for refining a finite element or finite volume mesh
based on a posteriori error estimates.
42
4 USING TETGEN
Figure 16: Examples of applying facet and segment constraints (.var file).
Impose facet area and segment length constraints TetGen also supports other constraints such as the constraint of maximum face area and the
constraint of maximum edge length imposed on facets and segments of the
PLC, respectively.
Figure 16 shows two examples of the results of applying constraints on a
facet and a segment, respectively.
These constraints are imposed by using a .var file (Section 5.2.9).
Apply a mesh sizing function (-m) The -m switch is used in mesh
refinement, i.e., together with the -q switch. It applies a user-defined mesh
sizing function which specifies the desired edge lengths in the final mesh. It
aims to create an adaptive mesh whose edge lengths are conforming to this
function. At the moment, only isotropic mesh sizing functions are supported.
TetGen assumes that the mesh sizing function is specified on a set of
discrete points whose convex hull covers the mesh domain (i.e., the underlying
space of the PLC). The mesh element size at any point in the domain is
automatically computed by a linear interpolation from its adjacent points.
When the -m switch is used, TetGen will read a .mtr file, which stores
the nodal mesh element size, i.e., the desired edge length at the location of
the node in the mesh domain. There are two possible ways to specify the
sizing function.
• The mesh element size is directly defined on the nodes of the input
PLC (-p switch) or the nodes of the input mesh (-r switch). In this
case, its file name is xxx.mtr, where xxx is the base file name of the
input PLC or the input mesh, see Figure 17 for an example.
• The mesh element size is defined on the nodes of a background mesh.
In this case, there is a background mesh given by the files xxx.b.node,
xxx.b.ele, and the mesh element size file xxx.b.mtr.
4.2 Command Line Switches
43
Figure 17: The tetrahedral meshes of a PLC (L.smesh) generated by the
commands: -pqm. A sizing function (L.mtr) was applied on the nodes of the
PLC. Both input files are found in Section 5.2.8.
.
4.2.5
Reconstructing a tetrahedral mesh (-r)
The -r switch reconstructs an existing tetrahedral mesh. Usually, the purpose of using this switch is to refine the mesh to improve its quality, i.e., to
use it together with the -q switch. Other usages of the -r switch are possible,
such as inserting additional points (-i switch), mesh adaptation (-m switch),
and linear function interpolation (-m switch plus a background mesh).
• The tetrahedral mesh is read from a .node and an .ele file. These two
files must be supplied.
• If a .face file exists, TetGen will read it and use it to find boundary
faces in the tetrahedral mesh. Note: only those faces with a nonzero boundary marker are regarded as boundary faces. In either case,
TetGen will automatically identify the faces on the exterior of the mesh
domain and regard them as boundary faces. Interior boundary faces are
also identified by comparing the attributes of two adjacent tetrahedra.
• If an .edge file exists, TetGen will read it and use it to find boundary
edges in the mesh. Note: only those edges with a non-zero boundary
marker are regarded as boundary edges. TetGen will also automatically
identify boundary edges from the identified boundary faces.
• The reconstructed mesh is distinguished from its origin with a different iteration number. For example, tetgen -r xxx.1 reads the
mesh in files xxx.1.node, xxx.1.ele and possibly xxx.1.face and
xxx.1.edge if they exist; reconstructs the mesh; outputs it into three
44
4 USING TETGEN
files xxx.2.node, xxx.2.ele, xxx.2.face, and xxx.2.edge. Now,
xxx.2 can be used as input in the above command, the result is another
mesh saved in files xxx.3.node, and so on. Mesh iteration numbers allow you to create a sequence of successively finer meshes.
• -r should not be used together with the -I.
4.2.6
Mesh optimization (-O)
The -O switch specifies a mesh optimization level and chooses the operations.
Both are integers and are separated by a slash ‘/’.
The mesh optimization level is an integer ranged from 0 to 10, where 0
means no mesh optimization is executed. The larger the level is, the more
mesh optimization iterations will be performed, and TetGen may run very
slow. Default the mesh optimization level is 2.
There are three local operations available in TetGen for optimizing the
mesh, which are:
• Edge/face flips.
• Vertex smoothing.
• Vertex insertion/Deletion.
The integer for choosing operations is ranged from 0 to 7. Here 0 means
no operation is chosen (hence no mesh optimization will be done). Each
operation is enabled/disabled by setting the corresponding bit in this integer.
• The 1st bit (the least significant bit) enables/disables edge/face flips.
• The 2nd bit enables/disables vertex smoothing.
• The 3rd bit enables/disables vertex insertion/deletion.
The default is 7, i.e., all of these three operations are enabled.
For examples, the switch -O2/7 specifies the optimization level 2 and uses
all optimizing operations. These are the default switches in TetGen. The
switch -O/1 chooses only the edge/face flip operation and uses the default
optimization level.
The following switch is temporary (maybe changed in the future)
The current objective function to be optimized by TetGen is to reduce the
maximum dihedral angle of the tetrahedra. The default value is 165 degree.
One can set this value after the -o/. For example, -o/150 sets the maximum
dihedral angle to be 150 degree.
4.2 Command Line Switches
4.2.7
45
Mesh coarsening (-R)
The -R switch indicates that some vertices of an existing tetrahedral mesh
are to be removed. TetGen provides two ways to indicate those vertices to
be removed.
• Vertices whose boundary markers (see .node file format) are ‘-1’ are to
be removed.
• When a mesh sizing function is supplied (-m switch), vertices whose
mesh element sizes are too large are to be removed.
The -R switch only removes vertices which can be removed. In particular,
such vertices lie in the interior of the domain, or vertices lying in the interior
of a facet or a segment. Note that this switch does not guarantee that all
the marked vertices are successfully removed.
Once the mesh has been coarsened, the mesh quality may decrease. You
may use the -q switch together with the -R switch. It will trigger the mesh
improvement algorithm of TetGen to improve the mesh quality after the mesh
coarsening.
4.2.8
Inserting additional points (-i)
The -i switch indicates that a list of additional points is going to be inserted
into an existing tetrahedral mesh. The list of additional nodes is read from
files xxx.a.node, where xxx stands for the input file name (i.e., xxx.poly or
xxx.smesh, or xxx.ele, ...). This switch is useful for refining a finite element
or finite volume mesh using a list of user-defined points.
• Points which lie in the exterior of the mesh domain are simply discarded.
• TetGen uses a relative tolerance (set by -T switch) to check whether a
point lies on the domain boundary or not, default it is 10−8 .
4.2.9
Assigning region attributes (-A)
The -A switch assigns an additional attribute (an integer number) to each
tetrahedron that identifies to what facet-bounded region each tetrahedron
belongs. In the output mesh, all tetrahedra in the same region will get a
corresponding non-zero attribute.
Figure 18 shows an example of tetrahedral meshes of a PLC which contains several sub-domains.
46
4 USING TETGEN
Figure 18: The tetrahedral meshes of a PLC (ts80305 nd32 cell834.off)
generated by the commands: -pqAa1e-12
.
• A region attribute is an integer which can be either positive or negative.
It must not be a zero, which is used for the exterior of the PLC.
• User-defined attributes are assigned to regions by the .poly or .smesh
file (in the region section). If a region is not explicitly marked by the
.poly file or .smesh file, tetrahedra in that region are automatically
assigned a non-zero attribute.
• By default, the region attributes are numbered from 1, 2, 3, · · ·. If there
are user-assigned region attributes (by the .poly or .smesh file), the
region attributes are shifted by a number i, i.e., i + 1, i + 2, i + 3, · · ·,
where i is either 0 or the maximum integer of the user-assigned region
attributes.
• The -A switch has an effect only if the -p switch is used and the -r
switch is not.
4.2.10
Mesh output switches (-f, -e, -n, -z, -o2)
TetGen provides various switches to output its mesh. They are summarized
below.
-f The -f switch outputs all triangular faces (including interior faces) of the
mesh into a .face file. Without -f, only the boundary faces or the convex
hull faces are output.
In the .face file, interior faces and boundary (or convex hull) faces are
distinguished by their boundary markers. Each interior face has a boundary
marker ‘0’. A non-zero boundary marker means a boundary or convex hull
face.
47
4.2 Command Line Switches
-e The -e switch outputs all mesh edges (including interior edges) of the
mesh into a .edge file. Without -e, only the boundary edges are output.
In the .edge file, interior edges and boundary edges are distinguished by
their boundary markers. Each interior edge has a boundary marker ‘0’. A
non-zero boundary marker means a boundary edge.
-n The -n switch outputs the neighboring tetrahedra to a .neigh file. Each
tetrahedron has four neighbors. The first neighbor of this tetrahedron is
opposite to the first of its corner, and so on. The neighbors are given by
their indices in the corresponding .ele file. A ‘-1’ indicates that there is no
neighbor at this face of the tetrahedron.
If the -nn switch is used, TetGen also outputs the neighboring tetrahedra
to each face of the mesh in the corresponding .face file.
-z The -z switch numbers all output items starting from zero. This switch
is useful in case of calling TetGen from another program.
-o2 With the -o2 switch, TetGen will output the tetrahedral mesh with
quadratic elements which have 10 nodes per tetrahedron, 6 nodes per triangular face, and 3 nodes per edge. The positions of these extra nodes in each
element is shown in Figure 20.
4.2.11
Mesh statistics (-V)
The -V switch gives detailed information about what TetGen is doing. More
‘V’s are increasing the amount of detail.
Specifically, -V gives information on algorithmic progress and more detailed statistics including a rough mesh quality report. Below is a screen
output of the quality report.
Mesh quality statistics:
Smallest
Shortest
Smallest
Smallest
Smallest
volume:
edge:
asp.ratio:
facangle:
dihedral:
Aspect ratio histogram:
< 1.5
:
1.5 - 2
:
2 - 2.5
:
2.5 - 3
:
0.016741
0.30902
1.2927
15.352
5.587
|
|
|
|
|
Largest
Longest
Largest
Largest
Largest
5
105
228
215
|
|
|
|
6
10
15
25
-
volume:
edge:
asp.ratio:
facangle:
dihedral:
10
15
25
50
125.77
12.189
16.964
141.8279
163.9413
:
:
:
:
33
4
1
0
48
4 USING TETGEN
3 - 4
:
321
|
50 - 100
:
0
4 - 6
:
150
|
100 :
0
(A tetrahedron’s aspect ratio is its longest edge length divided by its
smallest side height)
Face angle histogram:
0 - 10 degrees:
10 - 20 degrees:
20 - 30 degrees:
30 - 40 degrees:
40 - 50 degrees:
50 - 60 degrees:
60 - 70 degrees:
70 - 80 degrees:
80 - 90 degrees:
Dihedral
0 5 10 20 30 40 50 60 70 -
angle histogram:
5 degrees:
10 degrees:
20 degrees:
30 degrees:
40 degrees:
50 degrees:
60 degrees:
70 degrees:
80 degrees:
0
122
556
700
1273
1085
1129
871
506
|
|
|
|
|
|
|
|
|
90
100
110
120
130
140
150
160
170
-
100
110
120
130
140
150
160
170
180
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
637
131
101
44
5
1
0
0
0
0
10
141
362
487
762
770
812
768
|
|
|
|
|
|
|
|
|
80
110
120
130
140
150
160
170
175
-
110
120
130
140
150
160
170
175
180
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
degrees:
1675
228
149
92
77
32
7
0
0
To get the statistics for an existing mesh, run TetGen on that mesh with
the -rNEF switches to read the mesh and print the statistics without writing
any file.
Moreover, -V also gives information on the memory usage of TetGen.
Below is a screen output of the memory usage report.
Memory usage statistics:
Maximum number of tetrahedra: 45309
Maximum number of tet blocks (blocksize = 8188): 6
Approximate memory for tetrahedral mesh (bytes): 8,920,640
Approximate memory for extra pointers (bytes): 1,775,824
Approximate memory for algorithms (bytes): 637,136
Approximate memory for working arrays (bytes): 2,092,580
Approximate total used memory (bytes): 13,426,180
-VV gives more details on the algorithms, and slows down the execution,
while -VVV is only useful for debugging.
4.2.12
Memory allocation (-x)
TetGen allocates memory in blocks. Each block is a chunk of memory al-
4.2 Command Line Switches
49
located once. It stores a number of mesh entities, i.e., vertices, tetrahedra,
boundary faces, and boundary edges. TetGen will dynamically allocate new
blocks when they are needed.
By default, each block consists of 8188 tetrahedra. This data size may be
too small for generating large meshes. This may slow down the performance
of TetGen. The -x switch allows users to set the desired number of elements
allocated in one block.
If the -V switch is used, TetGen will report its memory usage, see Section 4.2.11. A hint to enlarge the block size can be seen from the “Maximum
number of tet blocks” (the second line in this report). If this number is large
(for example 10000), it is reasonable to enlarge the block size.
4.2.13
Miscellaneous
-c The -c switch keeps the convex hull of the tetrahedral mesh. By default,
TetGen removes all tetrahedra which do not lie in the interior of the PLC
(the domain) which may have an arbitrary shape and topology, i.e., it may
be non-convex and may contain holes. If the -c switch is used, tetrahedra in
the exterior of the PLC are not removed. The union of the mesh elements is
a topological ball.
TetGen assigns to all exterior tetrahedra a region attribute ‘-1’, so that
they can be distinguished from the interior tetrahedra.
-S The -S switch specifies a maximum number of Steiner points (points
that are not in the input) which are added by mesh refinement to improve
the mesh quality. The default is to allow an unlimited number of Steiner
points.
-C The -C switch indicates TetGen to check the consistency of the mesh
on finish. If it is specified twice, i.e., -CC, TetGen also checks the constrained
Delaunay (for the -p switch) or conforming Delaunay (for -q, -a, or -i)
property of the mesh.
-I With the -I switch, TetGen does not use the iteration numbers. It
suppresses the output of .node file, so your input file will not be overwritten.
It cannot be used with the -r switch, because that would overwrite your input
.ele file. It shouldn’t be used with the -q switch if one is using a .node file
for input, because no .node file is written, so there is no record of any added
Steiner points.
50
4 USING TETGEN
-T The -T switch sets a user-defined tolerance used by many computations
of TetGen, default is 10−8 .
In principle, the vertices which are used to define a facet of a PLC should
be exactly coplanar. But this is very hard to achieve in practice due to the
inconsistent nature of the floating-point format used in computers.
TetGen accepts facets whose vertices are not exactly but ”approximately
coplanar”. Four points a, b, c and d are assumed to be coplanar as long
as the ratio v/l3 is smaller than the given tolerance, where v and l are the
volume and the average edge length of the tetrahedron abcd, respectively.
To choose a proper tolerance according to the input point set will usually
reduce the number of adding points and improve the mesh quality.
51
4
12
17
9
3
11
10
1
2
Figure 19: The facet (shown in pink) consists of four polygons and one hole.
The ordered vertex lists of the polygons are: (1, 2, 3, 4), (9, 10, 11, 12), (11, 3),
and (17). The last two polygons are degenerate.
5
File Formats
Files are used as input and output when using TetGen as a stand-alone
program. This section describes the input/output file formats of TetGen.
When using TetGen as a library, the data structure tetgenio (explained in
Section 6) is used to transfer the data stored in files.
5.1
5.1.1
Useful Things to Know
A Boundary Description of PLCs
In TetGen, a 3d PLC is described by a boundary discretization (e.g., a surface
mesh) of the PLC. This description can be viewed as a Boundary Representation (B-Rep) without topological information, i.e., there are no information
like incidences and orientations about edges and facets. This makes the
description simple and it can describe non-manifolds easily. TetGen will recover and validate the topological information from this description during
its meshing process.
The boundary description of a PLC contains the set of vertices and facets
of the PLC. Recall that each facet of a PLC is a 2d PLC. It may contain
holes, segments and vertices in its interior, see Figure 19 for an example.
TetGen describes a facet by a list of polygons and a list of holes. Each
polygon of a facet is described by an ordered list of vertices. The order of
the vertices can be in either clockwise or counterclockwise order. A polygon
may be degenerate, i.e., it may contain only one or two vertices. A degenerate
polygon is used to represent an isolated vertex or segment in this facet, see
Fig. 19.
A hole in a facet is described by specifying an arbitrary point p ∈ R3 ,
such that the projection of p onto this facet lies strictly in the interior of this
52
5 FILE FORMATS
hole. Note that p is not a vertex of the PLC.
Remark. By the definition of a PLC, all vertices of a facet must lie in
the same affine subspace of that facet. However, this requirement is generally
impossible to be satisfied in practice due to the floating-point numbers used in
computer. TetGen only requires that all vertices of a facet are approximately
coplanar.
In addition, a list of holes and sub-regions of the PLC can be defined. A
hole of the PLC is described by specifying an arbitrary point p ∈ R3 that
lies strictly in the interior of the hole. Sub-regions are described exactly the
same way. Note that the points used to define holes and sub-regions are not
vertices of the PLC.
This description of a PLC is further explained in the input file formats
of TetGen, i.e., the .node, .poly, and .smesh file formats.
5.1.2
Boundary Markers
In TetGen, the mesh entities like vertices, edges, and faces, are assigned with
a boundary marker. Boundary markers are tags (integers) used mainly to
identify which entities are associated with which boundary element of the
input PLC, such as, a segment or a facet. A common application is to determine where boundary conditions should be applied to a finite element mesh.
You can prevent boundary markers from being written into files produced by
TetGen by using the -B switch.
Mesh entities which are not on the boundary of the PLC must have the
boundary markers ‘0’.
Mesh entities which are on the boundary will be assigned to a boundary
marker that is the same as the boundary marker of that boundary of the
PLC. However, if a boundary of a PLC does not have a boundary marker or
have a marker ‘0’, TetGen will assign a ‘1’ to those entities belong to this
boundary in the output files. This way, TetGen is able to distinguish them
from other interior mesh entities.
5.2
TetGen’s File Formats
Table 7 lists all file formats that are used by TetGen. All files are of ASCII
form and may contain comments prefixed by the character ’#’. Points, tetrahedra, facets, edges, holes, and maximum volume constraints must be numbered consecutively, starting from either 1 or 0. However, all input files
must be consistent. TetGen automatically detects your choice while reading
the .node (or .poly or .smesh) file. When calling TetGen from another
program, use the -z switch if you wish to number objects from zero.
53
5.2 TetGen’s File Formats
.node
.poly
.smesh
.ele
.face
.edge
.vol
.mtr
.var
.neigh
input/output
input
input/output
input/output
input/output
input/output
input
input/output
input
output
a
a
a
a
a
a
a
a
a
a
list of nodes.
piecewise linear complex.
piecewise linear complex.
list of tetrahedra.
list of triangular faces.
list of edges.
list of maximum volumes.
mesh sizing function.
list of variant constraints.
list of neighbors.
Table 7: Overview of TetGen’s file formats.
Remark: in the following description ‘#’ stands for ‘number’ – it should
not cause confusion with the comment prefix.
5.2.1
.node files
A .node file contains a list of 3d points.
First line:
<# of points> <dimension (3)> <# of attributes>
<boundary markers (0 or 1)>
Remaining lines list # of points:
<point #> <x> <y> <z> [attributes] [boundary marker]
...
Each point has three coordinates (x, y and z), probably has one or several
attributes, and a boundary marker as well. The .node files are used as both
input and output files to represent the point set of a PLC, or the point set
of a mesh, or the set of additional points (for the -i switch) which need to
be inserted into a mesh. The example below demonstrates the layout of the
.node file.
# Node count, 3 dim, no attribute, no boundary marker
8 3 0 0
# Node index, node coordinates
54
1
2
3
4
5
6
7
8
5 FILE FORMATS
0.0
1.0
1.0
0.0
0.0
1.0
1.0
0.0
0.0
0.0
1.0
1.0
0.0
0.0
1.0
1.0
0.0
0.0
0.0
0.0
1.0
1.0
1.0
1.0
The attributes, which are typically floating-point values of physical quantities (such as mass or conductivity) associated with the points, are copied
unchanged to the output mesh.
If -p, -q, -a, or -i is selected, each Steiner point added to the mesh has
attributes zero.
If the -w (weighted Delaunay tetrahedralization) switch is specified, the
first attribute is regarded as the weight of that point.
If the <boundary marker> of the first line is 1, the last column of the
remainder of the file is assumed to contain boundary markers. Boundary
markers are used to identify boundary points (points resting on PLC facets).
The .node file produced by TetGen contains boundary markers in the last
column unless they are suppressed by the -B switch. The boundary marker
associated with each point in an output .node file is chosen as follows:
• If to a point a nonzero boundary marker is assigned in the input file,
then the same marker is assigned in the output .node file.
• Otherwise, if the node lies on a PLC facet with a nonzero boundary
marker, then to the node the same marker of that facet is assigned.
If the node lies on several such facets, one of the markers is chosen
arbitrarily.
• Otherwise, if the node occurs on a boundary of the mesh, then to the
node the marker 1 is assigned.
• Otherwise, the marker 0 is assigned to the point.
TetGen can determine which points are on the boundary. Input with the
boundary marker zero (or use no markers at all) will result in output with
boundary marker 1 for all points on the boundary.
If the -R (mesh coarsening) switch is used, points with boundary markers
equal to −1 will be removed.
5.2 TetGen’s File Formats
5.2.2
55
.poly files
A .poly file is a B-Rep description of a piecewise linear complex (PLC)
containing some additional information. It consists of four parts.
• The first part is a a list of points.
• The second part is a list of facets.
• The next part is a list of hole points.
• The fourth part is a a list of region attributes.
The first three parts are mandatory, but the fourth part is optional. They
are respectively described below.
Part 1 - node list Part 1 lists all the points, and is identical to the format
of .node files. <# of points> may be set to zero to indicate that the points
are listed in a separate .node file.
First line:
<# of points> <dimension (3)> <# of attributes>
<boundary markers (0 or 1)>
Remaining lines list # of points:
<point #> <x> <y> <z> [attributes] [boundary marker]
...
Part 2 - facet list The facet list is given by
One line: <# of facets> <boundary markers (0 or 1)>
Following lines list # of facets:
<facet #>
...
The format of a single facet is:
56
5 FILE FORMATS
One line: <# of polygons> [# of holes] [boundary marker]
Following lines list # of polygons:
<# of corners> <corner 1> <corner 2> ... <corner #>
...
Following lines list # of holes:
<hole #> <x> <y> <z>
...
Each facet is a polygonal region which may contain segments, single points
and holes. It consists of a list of polygons. Each polygon is specified by giving
the number of corners n, n ≥ 1, followed by the list of ordered indices of those
corners. It does not matter which order (counterclockwise or clockwise) you
choose to list the indices. It can be degenerate, i.e., n = 1 or n = 2 indicates
a single point or a segment, respectively.
A hole in a facet is specified by identifying a point inside the hole. The
list of hole points (consecutively) follows the list of polygons.
Boundary markers of facets are tags (integers) used mainly to identify
which faces of the tetrahedralization are associated with which PLC facet,
hence identify which faces occur on a boundary of the tetrahedralization. A
common application is to use them to determine where different boundary
condition types should be applied to a mesh.
If the [boundary marker] is 1, each facet is assumed to have a boundary
marker (an integer). TetGen will produce an additional boundary marker for
each face in the .face (output) file (in the last column of each record).
If the [boundary marker] is 0, TetGen will automatically assign a 1 to
all boundary faces (which belong to facets of the PLC) in the .face (output)
file.
You can prevent boundary markers from being written into the .face file
by using the -B switch.
Note that each line of indices should not be arbitrarily long because the
maximum characters per line read by TetGen is 1024. The list can be broken
into several lines.
Part 3 - hole list Holes in the volume are specified by identifying a point
inside each hole.
5.2 TetGen’s File Formats
57
One line: <# of holes>
Following lines list # of holes:
<hole #> <x> <y> <z>
...
After the constrained Delaunay tetrahedralization is formed, TetGen creates holes by removing tetrahedra. This exactly is the reason that TetGen requires a closed boundary of the PLCs. In case of non-closed PLC facets the
whole tetrahedralization will be “eaten” away. If two tetrahedra abutting a
boundary face are removed, the boundary face itself is also vanished.
Hole points have to be placed inside a region, else the rounding error
determines which side of the facet is being transformed into the hole.
Part 4 - region attributes list The optional fourth section lists regional
attributes (to be assigned to all tetrahedra in a region) and regional constraints on the maximum tetrahedron volume. TetGen will read this section
only if the -A switch is used or the -a switch without a number is invoked.
Regional attributes and volume constraints are propagated in the same manner as holes.
One line: <# of region>
Following lines list # of region attributes:
<region #> <x> <y> <z> <region number> <region attribute>
...
If two values are written on a line after the x, y and z coordinate, the
former is assumed to be a regional attribute (but will only be applied if
the -A switch is selected), and the latter is assumed to be a regional volume
constraint (but will only be applied if the -a switch is selected). It is possible
to specify just one value after the coordinates. It can serve as both an
attribute and a volume constraint, depending on the choice of switches. A
negative maximum volume constraint allows to use the -A and the -a switches
without imposing a volume constraint in this specific region.
An example In the following, a unit (1 × 1 × 1) cube is described by the
poly file format.
58
5 FILE FORMATS
# Part 1 - node list
# node count, 3 dim, no attribute, no boundary marker
8 3 0 0
# Node index, node coordinates
1 0.0 0.0 0.0
2 1.0 0.0 0.0
3 1.0 1.0 0.0
4 0.0 1.0 0.0
5 0.0 0.0 1.0
6 1.0 0.0 1.0
7 1.0 1.0 1.0
8 0.0 1.0 1.0
# Part 2 - facet list
# facet count, no boundary marker
6 0
# facets
1
# 1 polygon, no hole, no boundary marker
4 1 2 3 4
# front
1
4 5 6 7 8
# back
1
4 1 2 6 5
# bottom
1
4 2 3 7 6
# right
1
4 3 4 8 7
# top
1
4 4 1 5 8
# left
# Part 3 - hole list
0
# no hole
# Part 4 - region list
0
# no region
5.2.3
.smesh files
A .smesh file is also a B-Rep description of a PLC. It describes a simple
B-Rep model where each of its facet only has exactly one polygon, no holes,
no segment and no point inside. It is less flexible than the .poly file format,
while it is much simpler and useful when the boundary of PLC consists of
only simple polygons, i.e., triangles, quads, etc.
Analogously to the .poly file format, the .smesh file format consists of
5.2 TetGen’s File Formats
59
four parts, which are points, facets, holes and regions, respectively. Only the
second part, which describes facets, is different. It is described below.
Part 2 - facet list Each facet consists of exactly one polygon. The corner
list of each polygon can be distributed over a number of lines. The optional
boundary marker of each facet is given at the end of the corner list.
One line: <# of facets> <boundary markers (0 or 1)>
Following lines list # of facets:
<# of corners> <corner 1> ... <corner #> [boundary marker]
...
The following example demonstrates the layout of the facet part of the
unit cube.
# Part 2 - facet list
# facet count, no boundary marker
6 0
# facets
4 1 2 3 4
# front
4 5 6 7 8
# back
4 1 2 6 5
# bottom
4 2 3 7 6
# right
4 3 4 8 7
# top
4 4 1 5 8
# left
5.2.4
.ele files
An .ele file contains a list of tetrahedra.
First line: <# of tetrahedra> <nodes per tet. (4 or 10)>
<region attribute (0 or 1)>
Remaining lines list # of tetrahedra:
<tetrahedron #> <node> <node> ... <node> [attribute]
...
60
5 FILE FORMATS
4
5
6
9
3
10
1
8
7
2
Figure 20: The local numbering of the vertices (corners) and the second-order
nodes of a tetrahedron.
Each tetrahedron has four corners (or ten corners if the -o2 switch is
used). Nodes are indices into the corresponding .node file. The first four
nodes are the corner vertices. If -o2 switch is used, the remaining six nodes
are generated on the midpoints of the edges of the tetrahedron. Figure 20
shows how these corners and the second-order nodes are locally numbered.
Second order nodes are output only. They are omitted by the mesh reconstruction (the -r switch).
If the <region attribute> in the first line is 1, each tetrahedra has an
additional region attribute (an integer) in the last column. Region attributes
of tetrahedra are tags used mainly to identify which tetrahedra of the tetrahedralization are associated with which facet-bounded region (sub-domain) of
the PLC, set in the fourth part of a .poly or a .smesh file. Region attributes
do not diffuse across facets, all tetrahedra in the same region have exactly
the same region attribute. A common application of the region attribute is
to determine which material a tetrahedron has.
The .ele file is the default output file of TetGen. However, it can be
omitted by -E switch. If -r switch is used, TetGen reads a .ele file and
reconstructs a tetrahedral mesh from it.
The following example illustrates the layout of a .ele file.
154
4
1
0
4
107
3
50
61
5.2 TetGen’s File Formats
2
3
4
5
6
7
...
5.2.5
4
9
4
56
94
97
108
97
107
1
98
9
3
95
50
50
97
95
107
94
93
47
95
55
.face files
A .face file contains a list of triangular faces of the tetrahedralization.
First line: <# of faces> <boundary marker (0 or 1)>
Remaining lines list # of faces:
<face #> <node> <node> <node> ... [boundary marker] ...
...
In its basic form, each face has three corners and possibly has a boundary
marker. Nodes are indices of the corresponding .node file.
If the <boundary marker> in the first line is 1, each face has an additional
boundary marker (an integer) in the last column. Boundary markers of facets
are defined in the .poly or the .smesh files. The optional column of boundary
markers can be suppressed by the -B switch.
If the -o2 switch is used, each face has three corners and three secondorder nodes generated on the midpoints of the edges of this face. Figure 21
shows the local numbering of the corners and the second-order nodes of a
face. They are listed after the corners of this face, and before its boundary
marker. Second order nodes are output only. They are ignored during the
mesh reconstruction (the -r switch).
If the -nn switch is used, each face contains two additional indices (after
the boundary marker) in the corresponding .ele file. They are indices of
the tetrahedra containing this face. A −1 indicates that there is no adjacent
tetrahedron at this side, i.e., it is “outer space”.
TetGen default only outputs the boundary faces or the convex hull faces
into a .face file. If the -f switch is used, TetGen outputs all faces (including
interior faces) of the tetrahedralization. In this case, each interior face will
always have a ‘0’ as its boundary marker. This file can be omitted by using
the -F switch.
62
5 FILE FORMATS
5
3
1
4
6
2
Figure 21: The local numbering of the corners and the second-order nodes
of a face.
If the -r switch is used, TetGen can also read the .face file for identifying
boundary faces in a reconstructed mesh.
5.2.6
.edge files
An .edge file contains a list of edges of the tetrahedralization.
First line: <# of edges> <boundary marker (0 or 1)>
Remaining lines list # of edges:
<edge #> <endpoint> <endpoint> ... [boundary marker] ...
...
In its basic form, each edge has two endpoints and possibly a boundary
marker (if the <boundary marker> in the first line is 1). Endpoints are
indices in the corresponding .node file.
If the -o2 switch is used, each edge has a second order node (its midpoint).
It is listed after its endpoints, and before its boundary marker. Second order
nodes are output only. They are ignored during the mesh reconstruction (the
-r switch).
If the -nn switch is used, each edge contains one additional index (after
the [boundary marker]) in the corresponding .ele file. It is the index
of a tetrahedron which contain this edge. A −1 indicates that there is no
tetrahedron containing this edge.
If the -r switch is used, TetGen can also read the .edge file for identifying
boundary edges in a reconstructed mesh.
The .edge file is the default output of TetGen when -p or -r switch is
used. By default, it only contains a list of boundary edges (segments) of
the tetrahedralization. If the -e switch is used, it contains a list of all edges
5.2 TetGen’s File Formats
63
(including interior edges) of the tetrahedralization. The output of this file
can be suppressed by the -F switch.
5.2.7
.vol files
A .vol file associates with each tetrahedron a maximum volume that is used
for mesh refinement. It is read by TetGen in case the -r switch is used.
First line: <# of tetrahedra>
Remaining lines list # of maximum volumes:
<tetrahedron #> <maximum volume>
...
As with other file formats, every tetrahedron must be represented, and
they must be numbered consecutively. A tetrahedron may be left unconstrained by assigning it a zero or negative maximum volume to it.
5.2.8
.mtr files
The .mtr file assigns a mesh sizing function defined on the nodes of an input
PLC, or an input tetrahedral mesh, or a background tetrahedral mesh. It is
used when the -m switch is applied.
First line: <# of nodes> <size of metric (always 1)>
Remaining lines list # of point metrics:
<value>
...
Each node is assigned a value that is interpreted by TetGen as the desired
length for all edges connecting to the node. TetGen uses this size information
to generate an adaptive tetrahedral mesh.
Below is an example of an L-shaped PLC (L.smesh) (see Figure 17):
64
5 FILE FORMATS
# L.smesh
12 3 0 0
1 0 0 0
2 4 0 0
3 4 2 0
4 2 2 0
5 2 6 0
6 0 6 0
1 0 0 2
2 4 0 2
3 4 2 2
4 2 2 2
5 2 6 2
6 0 6 2
8 0
6 1 2 3 4 5 6
6 7 8 9 10 11 12
4 1 2 8 7
4 2 3 9 8
4 3 4 10 9
4 4 5 11 10
4 5 6 12 11
4 6 1 7 12
0
0
Below is an example file L.mtr to assign a sizing function on the nodes
of the PLC:
# L.mtr
12 1
0.25
0.25
0.25
0.025
0.25
0.25
0.25
0.25
0.25
0.025
0.25
0.25
5.2 TetGen’s File Formats
65
The following command generates the adaptive tetrahedral mesh shown
in Figure 17.
tetgen -pqm L.smesh
It is possible to set a size 0 to a node. In this case, TetGen ignores the
mesh element size at this node. For example, it is possible to use an L.mtr
file like the following;
# L.mtr
12 1
0
0
0
0.025
0
0
0
0
0
0.025
0
0
5.2.9
.var files
A .var file allows you to specify maximum area constraints on facets and
maximum length constraints on segments. They are used for mesh refinement.
One line: <# of facet constraints>
Remaining lines list # of facet constraints:
<constraint #> <boundary marker> <maximum area>
...
One line: <# of segment constraints>
Remaining lines list # of segment constraints:
<constraint #> <point1> <point2> <maximum length>
...
66
5 FILE FORMATS
A constraint of maximum area on facet is set by specifying the boundary
marker of that facet, which is the integer assigned to that facet in the corresponding .poly or .smesh file. A constraint of maximum length on segment
is set by specifying the indices of the two endpoints of that segment.
Figure 16 shows two examples of the results of applying constraints on a
facet and a segment, respectively.
5.2.10
.neigh files
A .neigh file associates with each tetrahedron its neighbors (adjacent tetrahedra), which are indices in the corresponding .ele file.
First line: <# of tetrahedra> 4
Following lines list # of neighbors:
<tetrahedra #> <neighbor> <neighbor> <neighbor> <neighbor>
...
An index of −1 indicates no neighbor (because the tetrahedron is on
boundary of a mesh domain). The first neighbor of the tetrahedron i is
opposite to the first corner of tetrahedron i, and so on. The .neigh file is
output by TetGen when the -n switch is used.
5.2.11
.v.node, .v.edge, .v.face, .v.cell
A .v.node file contains a list of vertices of the Voronoi diagram or the power
diagram ( -w switch). Each Voronoi vertex is the circumcenter (or the orthocenter) of a Delaunay (or weighted Delaunay) tetrahedron. The format
of .v.node is the same as that of a .node file.
A .v.edge file contains a list of edges of the Voronoi diagram or the power
diagram ( -w switch). Each edge corresponds to a face of the Delaunay (or
weighted Delaunay) tetrahedralization. Each Voronoi edge is either a line
segment connecting two Voronoi vertices or a ray starting from a Voronoi
vertex. The file format of a .v.edge file is
First line: <# of edges>
Following lines list # of edges:
5.2 TetGen’s File Formats
67
<edge #> <vertex 1> <vertex 2> <V_x> <V_y> <V_z>
...
<vertex 1> and <vertex 2> are two indices pointing to the list of Voronoi
vertices. <vertex 1> must be non-negative, while <vertex 2> may be −1
which means it is a ray. In this case, the unit vector of this ray is given by
<V x>, <V y>, and <V z>.
A .v.face file contains a list of faces of the Voronoi diagram or the power
diagram ( -w switch). Each face corresponds to an edge of the Delaunay
(or weighted Delaunay) tetrahedralization. It is formed by a list of Voronoi
edges, which may not be closed. The file format of a .v.face file is
First line: <# of faces>
Following lines list # of faces:
<face #> <cell 1> <cell 2> <# of edges> <edge 1> <edge 2> ...
...
<cell 1> and <cell 2> are two indices pointing into the list of Voronoi
cells, i.e., the two cells share this face. <edge 1>, <edge 2> ..., are indices
pointing into the edge list, i.e., they are the edges of this face, there are total
<# of edges>. If the face is not closed, the index of the last edge of this
face is −1.
A .v.cell file contains a list of cells of the Voronoi diagram or the power
diagram ( -w switch). Each cell corresponds to a vertex of the Delaunay (or
weighted Delaunay) tetrahedralization. A cell is formed by a list of Voronoi
faces, which may not be closed. The file format of a .v.cell file is
First line: <# of cells>
Following lines list # of cells:
<cell #> <# of faces> <face 1> <face 2> ...
...
68
5 FILE FORMATS
.off
.ply
.stl
.mesh
input
input
input
input/output
Geomview’s polyhedral file format.
Polyhedral file format.
Stereolithography format.
Medit’s mesh file format.
Table 8: Overview of supported file formats.
<face 1>, <face 2>, ... are indices pointing into the Voronoi face list.
There are total <# of faces> faces. If the cell is not closed, the index of
the last face in this cell is −1.
5.3
Supported File Formats
TetGen supports additional polyhedral file formats as well. Table 8 lists
the supported file formats. TetGen recognizes the file formats by the file
extensions.
5.3.1
.off files
The .off is the one of the file formats of Geomview http://www.geomview.
org - an interactive 3d viewing program for Unix/Linux. It represents collections of planar polygons with possibly shared vertices, a convenient way to
describe polyhedra. The polygons may be concave, but there’s no provision
for polygons containing holes.
The full description of the .off file format can be found elsewhere on the
internet. Below is a simple description of this file format.
OFF numVertices numFaces numEdges
x y z
x y z
... numVertices like above
NVertices v1 v2 v3 ... vN
MVertices v1 v2 v3 ... vM
... numFaces like above
Note that vertices are numbered starting at 0 (not starting at 1), and
that numEdges will always be zero.
5.3 Supported File Formats
5.3.2
69
.ply files
The .ply file format is a simple object description that was designed as a
convenient format for researchers who work with polygonal models. Early
versions of this file format were used at Stanford University and at UNC
Chapel Hill.
A description examples as well as codes of the PLY file format can be
found at http://paulbourke.net/dataformats/ply/.
5.3.3
.stl files
The .stl or stereolithography format is an ASCII or binary file used in
manufacturing. It is a list of the triangular surfaces that describe a computer
generated solid model. This is the standard input for most rapid prototyping
machines.
The description of the .stl file format can be found elsewhere on the
web. An elaborated description can be found at http://en.wikipedia.
org/wiki/STL_(file_format). Below is an example.
solid
...
facet normal 0.00 0.00 1.00
outer loop
vertex 2.00 2.00 0.00
vertex -1.00 1.00 0.00
vertex 0.00 -1.00 0.00
endloop
endfacet
...
endsolid
5.3.4
.mesh files
The .mesh is the file formats of Medit - an interactive 3d mesh viewing program http://www.ann.jussieu.fr/frey/software.html. This file format
is described in the documentation of Medit.
A repository of free 3d models in this file format are available at INRIA’s Free 3d Meshes Download http://www-roc.inria.fr/gamma/gamma/
70
5 FILE FORMATS
3
7
4
−1
−2
8
2
1
6
5
Figure 22: A bar having two boundary markers (−1 and −2) defined.
Accueil/.
5.4
File Format Examples
This section provides three examples. They are designed to support interactive learning. The topics are describing PLCs using TetGen’s .poly file
format and constructing different quality meshes through the command line
switches.
5.4.1
A PLC with Two Boundary Markers
Figure 22 shows the geometry of a rectangular bar. This bar consists of
eight nodes and six facets (which are all rectangles). In addition, there are
two boundary markers (−1 and −2) associated to the leftmost facet and the
rightmost facet, respectively. This simple model has its physical meaning. It
can be seen as the geometry of a typical heat transfer problem. The task is
to compute the temperature diffusion in the bar, in which the flow of heat
moves from the hot side to the cold side. The two boundary markers can
represent two different boundary conditions, one has high temperature and
the other has low temperature. Here is the input file bar.poly describing
the bar:
# Part 1 - the node list.
# A bar with 8 nodes in 3d, no attributes, no boundary marker.
8 3 0 0
# The 4 leftmost nodes:
1 0 0 0
2 2 0 0
3 2 2 0
4 0 2 0
# The 4 rightmost nodes:
5 0 0 12
6 2 0 12
7 2 2 12
8 0 2 12
5.4 File Format Examples
71
# Part 2 - the facet list.
# Six facets with boundary markers.
6 1
# The leftmost facet.
1 0 -1
# 1 polygon, no hole, boundary marker (-1)
4 1 2 3 4
# The rightmost facet.
1 0 -2
# 1 polygon, no hole, boundary marker (-2)
4 5 6 7 8
# Other facets.
1
4 1 5 6 2 # bottom side
1
4 2 6 7 3 # back side
1
4 3 7 8 4 # top side
1
4 4 8 5 1 # front side
# Part 3 - the hole list.
# There is no hole in bar.
0
# Part 4 - the region list.
# There is no region defined.
0
The command line is chosen as follows: first mesh the PLC (-p), then
impose the quality constraint (-q). This will result in a quality mesh of four
files: bar.1.node, bar.1.ele, bar.1.face, and bar.1.edge.
> tetgen -pq bar.poly
Here is the output file bar.1.node. It contains 47 points. The additional
points were added by TetGen automatically to meet the quality measure.
47
3 0 0
1
0 0 0
2
2 0 0
3
2 2 0
4
0 2 0
5
0 0 12
6
2 0 12
7
2 2 12
8
0 2 12
9
1.0000469999999999 0 0
10
0 0.999668 0
11
0 0.99944500000000003 12
12
1.000594 0 12
...
# Generated by tetgen -pq bar.poly
72
5 FILE FORMATS
Here is the output file bar.1.ele, which contains 83 tetrahedra.
83
4 0
1
18
33
20
34
2
9
2
3
25
3
17
18
20
34
4
43
32
18
37
5
19
20
30
33
6
14
41
13
42
7
12
26
7
6
8
10
28
1
9
9
28
33
18
34
10
35
41
38
45
11
10
9
25
28
12
3
25
19
30
...
# Generated by tetgen -pq bar.poly
Here is the output file bar.1.face with 90 boundary faces. Faces 1 and
2 are on the leftmost facet thus have markers −1; faces 3 and 4 have markers
−2 indicating they are on the rightmost facet. Other faces have the default
markers zero.
90
1
1
3
4
2
10
9
3
7
12
4
7
11
5
18
37
6
24
46
7
26
6
8
35
22
9
29
7
10
39
7
11
29
11
12
23
45
...
# Generated by tetgen
10
3
11
8
43
39
7
24
8
29
27
38
-1
-1
-2
-2
0
0
0
0
0
0
0
0
-pq bar.poly
However, the mesh above may be too coarse for numerical simulation
using finite element method or finite volume method. Using either the -q or
the -a switch or both of them will result in a more dense quality mesh:
> tetgen -pq1.414a0.1 bar.poly
TetGen generates a mesh with 330 points and 1092 tetrahedra. The
added points are due to both the -q and -a switches we have applied. To
see a quality report of the mesh, type:
> tetgen -rNEFV bar.1.ele
73
5.4 File Format Examples
11
3
7
12
−1
4
−2
8
2
10
6
1
5
9
−10
−20
Figure 23: A bar having two regions (with region attributes −10 and −20,
respectively) and two boundary markers (−1 and −2) defined.
5.4.2
A PLC with Two Sub-regions (Materials)
In this example, we add an internal facet into the bar (in Figure 22), so that
two sub-regions (separated by the newly added facet) in the bar are created.
Figure 23 shows the modified geometry. This bar consists of twelve nodes
(which I numbered) and seven facets (Note, some of them are not a single
polygon any more). In addition, there are two regions defined, which have
region attributes −10 and −20, respectively. Physically, you can associate
two different materials to each of these two regions, and the two boundary
markers (−1 and −2) in the last example still remain. Here is the input file
bar2.poly describing the modified bar:
# Part 1 - the node list.
# The model has 12 nodes in 3d, no attributes, no boundary marker.
12 3 0 0
# The 4 leftmost nodes:
1 0 0 0
2 2 0 0
3 2 2 0
4 0 2 0
# The 4 rightmost nodes:
5 0 0 12
6 2 0 12
7 2 2 12
8 0 2 12
# The 4 added nodes:
9 0 0 3
10 2 0 3
11 2 2 3
12 0 2 3
# Part 2 - the facet list.
# Seven facets with boundary markers.
7 1
# The leftmost facet.
1 0 -1
# 1 polygon, no hole, boundary marker (-1)
74
5 FILE FORMATS
4 1 2 3 4
# The rightmost facet.
1 0 -2
# 1 polygon, no hole, boundary marker (-2)
4 5 6 7 8
# Each of following facets has two polygons, which are
# one rectangle (6 corners) and one segment.
2
6 1 9 5 6 10 2 # bottom side
2 9 10
2
6 2 10 6 7 11 3 # back side
2 10 11
2
6 3 11 7 8 12 4 # top side
2 11 12
2
6 4 12 8 5 9 1 # front side
2 12 9
# The internal facet separates two regions.
1
4 9 10 11 12
# Part 3 - the hole list.
# There is no hole in bar.
0
# Part 4 - the region list.
# There are two regions (-10 and -20) defined.
2
1 1.0 1.0 1.5 -10 0.1
2 1.0 1.0 5.0 -20 -1
The command line tetgen -pqaA bar2.poly generates the file bar2.1.ele.
The first eight lines are listed next. It differs from bar.1.ele in that each
record has an additional region attribute.
431
4 1
1
32
57
50
2
51
23
50
3
88
138
116
4
76
96
95
5
29
55
56
6
132
138
88
7
65
138
132
8
16
54
53
...
# Generated by tetgen -pqaA
60
49
149
36
52
139
139
15
-20
-20
-10
-20
-20
-10
-10
-20
bar2.poly
Visualization of the resulting meshes (by TetView or other tools) shows
that the refinement in the region with attribute −10 is denser than in that
5.4 File Format Examples
75
with −20. This is due to the volume constraints, (0.1) defined in the file
bar2.poly, and due to the -aA switches.
5.4.3
A PLC with Two Sub-regions and Two Holes
This is the example file (example.poly) coming together with the TetGen’s
source distribution. The geometry as well as some generated mesh are shown
in Figure 24.
# Part 1 - the node list.
28 3 0 1
1 0 0 0 1
2 2 0 0 1
3 2 2 0 1
4 0 2 0 1
5 0 0 4 9
6 2 0 4 9
7 2 2 3 9
8 0 2 3 9
9 0 0 5 2
10 2 0 5 2
11 2 2 5 2
12 0 2 5 2
13 0.25 0.25 0.5 4
14 1.75 0.25 0.5 4
15 1.75 1.5 0.5 4
16 0.25 1.5 0.5 4
17 0.25 0.25 1 4
18 1.75 0.25 1 4
19 1.75 1.5 1 4
20 0.25 1.5 1 4
21 0.25 0 2 4
22 1.75 0 2 4
23 1.75 1.5 2 4
24 0.25 1.5 2 4
25 0.25 0 2.5 4
26 1.75 0 2.5 4
27 1.75 1.5 2.5 4
28 0.25 1.5 2.5 4
# Part 2 - the facet list
23 1
1 0 1 # 1
4
1 2 3 4
1 0 9 # 2
4
5 6 7 8
2 1 3 # 3
4
1 2 6 5
4
21 22 26 25
76
1 1 0 2.25
1 0 3 # 4
4
2 3 7 6
1 0 3 # 5
4
3 4 8 7
1 0 3 # 6
4
4 1 5 8
1 0 2 # 7
4
9 10 11 12
1 0 3 # 8
4
9 10 6 5
1 0 3 # 9
4
10 11 7 6
1 0 3 # 10
4
11 12 8 7
1 0 3 # 11
4
12 9 5 8
1 0 4 # 12
4
13 14 15 16
1 0 4 # 13
4
17 18 19 20
1 0 4 # 14
4
13 14 18 17
1 0 4 # 15
4
14 15 19 18
1 0 4 # 16
4
15 16 20 19
1 0 4 # 17
4
16 13 17 20
1 0 4 # 18
4
21 22 23 24
1 0 4 # 19
4
25 26 27 28
1 0 4 # 20
4
21 22 26 25
1 0 4 # 21
4
22 23 27 26
1 0 4 # 22
4
23 24 28 27
1 0 4 # 23
4
24 21 25 28
# Part 3 - the hole list
2
1 1 0.4 2.25
2 1 0.4 0.75
# Part 4 - the region list
2
1 1 0.25 0.1 10 0.001
2 1 0.5 4 20 0.01
5 FILE FORMATS
77
Figure 24: Left: A PLC with two sub-regions (materials) and two holes.
Middle: the CDT generated by the command line: -pA. Right: The quality
tetrahedral mesh generated by the command line: -pqAa.
6
Calling TetGen from Another Program
One can use TetGen as a library so that it can be called directly from another program. This section gives the necessary instructions for using the
TetGen library. Users are supposed to be able to use TetGen, i.e., know its
command line switches and the input and output file formats. We refer to
Section 3 for the instructions of how to compile TetGen into a library.
6.1
The Header File
Programs calling TetGen must include the header file tetgen.h.
#include "tetgen.h"
It includes all data types and function declarations of the TetGen library.
It defines the function tetrahedralize() and the data type tetgenio,
which are provided for users to call TetGen with all its functionality. They
are described in Section 6.2 and Section 6.3, respectively.
6.2
The Calling Convention
The function tetrahedralize() is declared as follows:
78
6 CALLING TETGEN FROM ANOTHER PROGRAM
void tetrahedralize(char *switches, tetgenio *in, tetgenio *out,
tetgenio *addin = NULL, tetgenio *bgmin = NULL);
The parameter switches is a string containing the command line switches
for this call. In this string, no initial dash ’-’ is required. The Q (quiet) switch
is recommended in the final code. Some file output switches, like I and g are
ignored.
The parameters in and out, which are two pointers pointing to objects
of tetgenio, describing the input and the output. in and out must not be
NULL.
Two additional parameters addin and bgmin may be supplied. When the
switch -i is used, addin contains a list of additional vertices to be inserted.
When the switch -m is used, bgmin contains a background mesh which is
used to provide a mesh sizing function.
6.3
The tetgenio Data Type
The tetgenio structure is used to pass data into and out of the tetrahedralize() procedure. It replaces the input and output files of TetGen by
a collection of arrays, which are used to store points, tetrahedra, boundary
markers, and so forth. It is a C++ class including data fields and functions.
The data fields of tetgenio:
int firstnumber; // 0 or 1, default 0.
int mesh_dim;
// must be 3.
REAL *pointlist;
REAL *pointattributelist;
REAL *pointmtrlist;
int *pointmarkerlist;
int numberofpoints;
int numberofpointattributes;
int numberofpointmtrs;
int *tetrahedronlist;
REAL *tetrahedronattributelist;
REAL *tetrahedronvolumelist;
int *neighborlist;
int numberoftetrahedra;
int numberofcorners;
int numberoftetrahedronattributes;
facet *facetlist;
int *facetmarkerlist;
int numberoffacets;
6.4 Description of Arrays
79
REAL *holelist;
int numberofholes;
REAL *regionlist;
int numberofregions;
REAL *facetconstraintlist;
int numberoffacetconstraints;
REAL *segmentconstraintlist;
int numberofsegmentconstraints;
int *trifacelist;
int *trifacemarkerlist;
int numberoftrifaces;
int *edgelist;
int *edgemarkerlist;
int numberofedges;
6.4
Description of Arrays
In all cases, the first item in any array is stored starting at index [0]. However,
that item is item number firstnumber (0 or 1) unless the z switch is used,
in which case it is item number ‘0’. Now the description of arrays follows.
pointlist An array of point coordinates. The first point’s x coordinate is
at index [0], its y coordinate at index [1], and its z coordinate at index [2],
followed by the coordinates of the remaining points. Each point occupies
three REALs.
pointattributelist An array of point attributes. Each point’s attributes
occupy numberofpointattributes REALs.
pointmarkerlist An array of point markers; one int per point.
pointmtrlist An array of metric tensors at points. Each point’s tensor
occupies numberofpointmtrs REALs.
tetrahedronlist An array of tetrahedron corners. The first tetrahedron’s
first corner is at index [0], followed by its other three corners, followed
80
6 CALLING TETGEN FROM ANOTHER PROGRAM
by any other nodes if the ’-o2’ switch is used. Each tetrahedron occupies
numberofcorners (4 or 10) ints.
tetrahedronattributelist An array of tetrahedron attributes. Each tetrahedron’s attributes occupy numberoftetrahedronattributes REALs.
tetrahedronvolumelist An array of tetrahedron volume constraints; one
REAL per tetrahedron. Input only.
neighborlist An array of tetrahedron neighbors; four ints per tetrahedron.
Output only.
facetlist An array of PLC facets. Each facet is an object of type facet
(see Section 6.4.2).
facetmarkerlist An array of facet markers; one int per facet.
holelist An array of holes. The first hole’s x, y and z coordinates are at
indices [0], [1] and [2], followed by the remaining holes. Three REALs per
hole.
regionlist An array of regional attributes and volume constraints. The
first constraints’ x, y and z coordinates are at indices [0], [1] and [2], followed
by the regional attribute at index [3], followed by the maximum volume at
index [4], followed by the remaining volume constraints. Five REALs per
volume constraint. Each regional attribute is used only if the A switch is
used, and each volume constraint is used only if the a switch (with no number
following) is used, but omitting one of these switches does not change the
memory layout.
facetconstraintlist An array of facet maximum area constraints. Two
REALs per constraint. The first one is the facet marker (cast the type to
integer), the second is its maximum area bound. Note the ’facetconstraintlist’
is used only for the ’q’ switch.
segmentconstraintlist An array of segment length constraints. Two REALs per constraint. The first one is the index (pointing into pointlist) of
the node, the second is its maximum length bound. Note the ’segmentconstraintlist’ is used only for the ’q’ switch.
6.4 Description of Arrays
81
trifacelist An array of triangular faces. The first face’s corners are at
indices [0], [1] and [2], followed by the remaining faces. Three ints per face.
trifacemarkerlist An array of face markers; one int per face.
edgelist An array of segment endpoints. The first segment’s endpoints are
at indices [0] and [1], followed by the remaining segments. Two ints per
segment.
edgemarkerlist An array of segment markers; one int per segment.
6.4.1
Memory Management
Two routines defined in tetgenio are used for memory initialization and cleaning. They are:
void initialize();
void deinitialize();
initialize() initializes all fields, that is, all pointers to arrays are initialized to NULL, and other variables are initialized to zero except the variable
’numberofcorners’, which is 4 (a tetrahedron has 4 nodes). Initialization is
implicitly called by the constructor of tetgenio. For an example, the following
line creates an object of tetgenio named io, all fields of io are initialized:
tetgenio io;
The next step is to allocate memory for each array which will be used.
In C++ the memory allocation and deletion can be done by the new and
delete operators. Another pair of functions (preferred by C programmers)
are malloc() and free(). Whatever you use, you must stick with one of
these two pairs, e.g., ’new’/’delete’ and ’malloc’/’free’ cannot be mixed. For
example, the following line allocates memory for io.pointlist:
io.pointlist = new REAL[io.numberofpoints * 3];
deinitialize() frees the memory allocated in objects of tetgenio by using ’delete’. It is automatically called on deletion of the tetgenio objects. If
the memory was allocated by using the function malloc(), the user is responsible to free it. After having freed all memory, one call of initialize()
disables the automatic memory deletion.
To reuse an object is possible: first call deinitialize(), then call initialize()
before the next use.
82
6 CALLING TETGEN FROM ANOTHER PROGRAM
6.4.2
The facet Data Structure
The facet data structure defined in tetgenio can be used to represent any
facet of a PLC. The structure of facet shown below consists of a list of
polygons and a list of hole points.
typedef struct {
polygon *polygonlist;
int numberofpolygons;
REAL *holelist;
int numberofholes;
} facet;
A polygon is again an object of type polygon. It consists of a list of
corner points (vertexlist). The structure is shown below.
typedef struct {
int *vertexlist;
int numberofvertices;
} polygon;
The structure of a facet corresponds to the facet description in a .poly
file format, described in Section 5.2.2. The front facet of Figure 23 serves an
example for setting a PLC facet into an object of facet. It has two polygons,
one has six vertices, and the other is a segment, no holes, the ASCII data is:
2
6
2
4 12 8 5 9 1 # front side
12 9
The following C++ code does the translation. Assume the object of
tetgenio is io and has already be created.
tetgenio::facet *f;
// Define a pointer of facet.
tetgenio::polygon *p; // Define a pointer of polygon.
// All indices start from 1.
io.firstnumber = 1;
...
// Use ’f’ to point to a facet of ’facetlist’.
f = &io.facetlist[i];
// Initialize the fields of this facet.
//
There are two polygons, no holes.
f->numberofpolygons = 2;
// Allocate memory for polygons.
6.5 A Complete Example
83
f->polygonlist = new tetgenio::polygon[2];
f->numberofholes = 0;
f->holelist = NULL;
// Set the data of the first polygon into facet.
p = &f->polygonlist[0];
p->numberofvertices = 6;
// Allocate memory for vertices.
p->vertexlist = new int[6];
p->vertexlist[0] = 4;
p->vertexlist[1] = 12;
p->vertexlist[2] = 8;
p->vertexlist[3] = 5;
p->vertexlist[4] = 9;
p->vertexlist[5] = 1;
// Set the data of the second polygon into facet.
p = &f->polygonlist[1];
p->numberofvertices = 2;
p->vertexlist = new int[2]; // Alloc. memory for vertices.
p->vertexlist[0] = 12;
p->vertexlist[1] = 9;
6.5
A Complete Example
This section gives an example of how to call TetGen from another program
by using the tetgenio data structure and the function tetrahedralize().
The input PLC in Section 5.4.1 (Figure 22) is used again.
The complete C++ source code is given below. It is also available on
TetGen’s website:http://www.tetgen.org/files/tetcall.cxx. The code
illustrates the following basic steps:
• at first it creates an input object in of tetgenio containing the data of
the bar;
• then it calls function tetrahedralize() to create a quality mesh of
the bar with output in out.
In addition, it outputs the PLC in the object in into two files (barin.node
and barin.poly), and outputs the mesh in the object out into three files
(barout.node, barout.ele, and barout.face).
This example can be compiled into an executable program.
• Compile TetGen into a library named libtet.a (see Section 3.1 for
compiling);
84
6 CALLING TETGEN FROM ANOTHER PROGRAM
• Save the file tetcall.cxx into the same directory in which you have
the files tetgen.h and libtet.a;
• Compile it using the following command:
g++ -o test tetcall.cxx -L./ -ltet
which will result an executable file test.
The complete source codes are given below:
#include "tetgen.h" // Defined tetgenio, tetrahedralize().
int main(int argc, char *argv[])
{
tetgenio in, out;
tetgenio::facet *f;
tetgenio::polygon *p;
int i;
// All indices start from 1.
in.firstnumber = 1;
in.numberofpoints = 8;
in.pointlist = new REAL[in.numberofpoints * 3];
in.pointlist[0] = 0; // node 1.
in.pointlist[1] = 0;
in.pointlist[2] = 0;
in.pointlist[3] = 2; // node 2.
in.pointlist[4] = 0;
in.pointlist[5] = 0;
in.pointlist[6] = 2; // node 3.
in.pointlist[7] = 2;
in.pointlist[8] = 0;
in.pointlist[9] = 0; // node 4.
in.pointlist[10] = 2;
in.pointlist[11] = 0;
// Set node 5, 6, 7, 8.
for (i = 4; i < 8; i++) {
in.pointlist[i * 3]
= in.pointlist[(i - 4) * 3];
in.pointlist[i * 3 + 1] = in.pointlist[(i - 4) * 3 + 1];
in.pointlist[i * 3 + 2] = 12;
}
in.numberoffacets = 6;
in.facetlist = new tetgenio::facet[in.numberoffacets];
in.facetmarkerlist = new int[in.numberoffacets];
6.5 A Complete Example
// Facet 1. The leftmost facet.
f = &in.facetlist[0];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 1;
p->vertexlist[1] = 2;
p->vertexlist[2] = 3;
p->vertexlist[3] = 4;
// Facet 2. The rightmost facet.
f = &in.facetlist[1];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 5;
p->vertexlist[1] = 6;
p->vertexlist[2] = 7;
p->vertexlist[3] = 8;
// Facet 3. The bottom facet.
f = &in.facetlist[2];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 1;
p->vertexlist[1] = 5;
p->vertexlist[2] = 6;
p->vertexlist[3] = 2;
// Facet 4. The back facet.
f = &in.facetlist[3];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
85
86
6 CALLING TETGEN FROM ANOTHER PROGRAM
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 2;
p->vertexlist[1] = 6;
p->vertexlist[2] = 7;
p->vertexlist[3] = 3;
// Facet 5. The top facet.
f = &in.facetlist[4];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 3;
p->vertexlist[1] = 7;
p->vertexlist[2] = 8;
p->vertexlist[3] = 4;
// Facet 6. The front facet.
f = &in.facetlist[5];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 4;
p->vertexlist[1] = 8;
p->vertexlist[2] = 5;
p->vertexlist[3] = 1;
// Set ’in.facetmarkerlist’
in.facetmarkerlist[0]
in.facetmarkerlist[1]
in.facetmarkerlist[2]
in.facetmarkerlist[3]
in.facetmarkerlist[4]
in.facetmarkerlist[5]
=
=
=
=
=
=
-1;
-2;
0;
0;
0;
0;
// Output the PLC to files ’barin.node’ and ’barin.poly’.
in.save_nodes("barin");
in.save_poly("barin");
6.5 A Complete Example
// Tetrahedralize the PLC. Switches are chosen to read a PLC (p),
//
do quality mesh generation (q) with a specified quality bound
//
(1.414), and apply a maximum volume constraint (a0.1).
tetrahedralize("pq1.414a0.1", &in, &out);
// Output mesh to files ’barout.node’, ’barout.ele’ and ’barout.face’.
out.save_nodes("barout");
out.save_elements("barout");
out.save_faces("barout");
return 0;
}
87
88
A BASIC DEFINITIONS
(a)
(b)
(c)
Figure 25: (a) A two-dimensional simplicial complex K consists of 2 triangles(which are shaded), 7 edges, and 5 vertices. (b) The underlying space.
(c) A subcomplex, which is a 1-dimensional simplicial complex, consists of
4 edges, and 4 vertices.
A
Basic Definitions
This section gives simplified explanations of some basic notions of combinatorial topology as a quick reference.
A.1
Simplices, Simplicial Complexes
Convex Hull A point set V ⊂ Rd is convex if it contains every line segment
whose end points are in this set. There are infinitely many convex sets
containing V . The smallest convex set containing V is called the convex
hull of V , denoted as convV . The dimension of the convex hull of V is the
dimension of the affine space of V .
Simplex A simplex σ is the convex hull of an affinely independent set S
of points. The dimension of σ is one less than the number of points of S.
Specifically, in R3 the maximum number of affinely independent points is 4,
so we have non-empty simplices of dimensions 0, 1, 2 and 3 referred to as
vertices, edges, triangles, and tetrahedra, respectively. For any subset T ⊆ S,
the simplex τ = convT is a face of σ and we write τ ≤ σ. τ is a proper face
of σ if T is a proper subset of S.
simplicial Complex A simplicial complex K is a finite set of simplices
such that (i) any face of a simplex in K is also in K, and (ii) the intersection
of any two simplices in K is a face of both. Condition (ii) allows for the case
in which two simplices are disjoint because the empty set is the unique (-1)dimensional simplex, which is a face of any simplex. Figure 25 (a) illustrates
a two-dimensional simplicial complex.
A.2 Polyhedra and Faces
89
Underlying Space The underlying
space of a set of simplices L, denoted
S
|L|, is the union of simplices, σ∈L σ (see an illustration in Figure 25 (b)).
|L| is a topologically closed set if and only if L is a simplicial complex.
Subcomplex A subcomplex of K is a subset of simplices of K that is also
a simplicial complex. For an example see Figure 25 (c).
A.2
Polyhedra and Faces
Convex polyhedra and their faces are well defined objects. It is a central
topic in discrete geometry to study their structures and properties, see [29].
However, general polyhedra which are not necessarily convex are much complex objects. There exist various definitions in the literature. We adopt the
definitions given by Edelsbrunner [7].
A polyhedron P is the union of convex polyhedra and the space of P is
connected. It is not necessarily convex, see Figure 26 left for an example. The
dimension of P is the dimension of the smallest affine space that contains P .
The interior of P , denoted as int(P ) is the point set such that every point
p ∈ int(P ) has a neighborhood (e.g., an open ball centered at this point)
which is a subset of P . The boundary of P is the point set bd(P ) = P −int(P ).
A face F of P satisfies the following three conditions:
(1) F is the closure of a connected set of points;
(2) F is contained in the boundary of P ; and
(3) F is equal to the intersection of P with the minimal affine space containing F .
F is also a polyhedron whose dimension is the dimension of the affine space
that determines F . 0-, 1-, and 2-dimensional faces of P are called vertices,
edges, and facets of P . Each facet is a polygonal region which may not be
convex and may contain holes in it, as illustrated in the right of Figure 26.
A.3
CSG and B-Rep Models of 3d Domains
In geometric and solid modeling, constructive solid geometry (CSG) and
boundary representation (B-Rep) are two popular representations for threedimensional objects.
A CSG model implicitly describes the domain as a combination of simple
primitives or other solids in a series of Boolean operations. It can describe
rather complicated shapes simply. However, the domain boundary must be
90
B LIST OF ERROR CODES AND MESSAGES
Figure 26: A three-dimensional non-convex polyhedron. In left, the vertices
(0-faces) and edges (1-faces) of the polyhedron are shown. All its facets
(2-faces) are shown in right.
calculated numerically in order to find the intersecting points, curves, and
patches. This involves solving many non-linear equations in three variables.
Obtaining a PLC from a CSG model is generally not a simple task.
A B-Rep model explicitly describes the domain boundary by a set of
non-overlapping facets (may be curved surfaces) together with topological
information (such as incidence and adjacency) between the facets. The volume of the domain is implicitly bounded by them. However, it is not trivial
to correctly define such a model for a complex object. Nevertheless, The
B-Rep model is popularly used in describing 3d geometries.
B
List of Error Codes and Messages
The list of error codes and messages can also be found in the function
terminatetetgen() defined in the file tetgen.h.
code 1
message Error: Out of memory
code 2
message Please report this bug to [email protected] Include
the message above, your input data set, and the exact
command line you used to run this program, thank you
description This failure was caused by a known bug of TetGen.
91
REFERENCES
code 3
message A self-intersection was detected. Program stopped.
Hint: use -d option to detect all self-intersections.
description This failure was caused by an input error.
code 4
message A very small input feature size was detected. Program stopped.
Hint: use -T option to set a smaller tolerance.
description This failure was caused by a possible input error.
For example, there are two segments nearly intersecting each.
other. If you want to ignore this possible error, set a smaller
tolerance by the -T switch, default is 10−8 .
References
[1] F. Aurenhammer. Voronoi diagrams – a study of fundamental geometric
data structure. ACM Comput. Surveys, 23:345–405, 1991.
[2] J.-D. Boissonnat, O. Devillers, and S. Hornus. Incremental construction
of the Delaunay triangulation and the Delaunay graph in medium dimension. In Proc. 25th Annual Symposium on Computational Geometry,
2009.
[3] A. Bowyer.
Computing Dirichlet tessellations.
24(2):162–166, 1987.
Comp. Journal,
[4] B. Chazelle. Convex partition of polyhedra: a lower bound and worstcase optimal algorithm. SIAM Journal on Computing, 13(3):488–507,
1984.
[5] J. A. de Loera, J. Rambau, and F. Santos. Triangulations, Structures for
Algorithms and Applications, volume 25 of Algorithms and Computation
in Mathematics. Springer Verlag Berlin Heidelburg, 1 edition, 2010.
[6] B. N. Delaunay. Sur la sphère vide. Izvestia Akademii Nauk SSSR,
Otdelenie Matematicheskikh i Estestvennykh Nauk, 7:793–800, 1934.
[7] H. Edelsbrunner. Geometry and topology for mesh generation. Cambridge University Press, England, 2001.
[8] H. Edelsbrunner and M.P. Mücke. Simulation of simplicity: A technique
to cope with degenerate cases in geometric algorithm. ACM Transactions on Graphics, 9(1):66–104, 1990.
92
REFERENCES
code 5
message Two very close input facets were detected. Program stopped.
Hint: use -Y option to avoid adding Steiner points on boundary.
description This failure was caused by a possible input error.
For example, there are two facets nearly overlapping each
other. Once Steiner points are added to one of the facets,
it will cause a self-intersection.
code 10
message An input error was detected. Program stopped.
description This failure was caused by an input error.
[9] H. Edelsbrunner and N. R. Shah. Incremental topological flipping works
for regular triangulations. Algorithmica, 15:223–241, 1996.
[10] D. T. Lee and A. K. Lin. Generalized Delaunay triangulations for planar
graphs. Discrete and Computational Geometry, 1:201–217, 1986.
[11] G. L. Miller, D. Talmor, S.-H. Teng, N. J. Walkington, and H. Wang.
Control volume meshes using sphere packing: Generation, refinement
and coarsening. In Proc. 5th Intl. Meshing Roundtable, 1996.
[12] V. T. Rajan. Optimality of the Delaunay triangulation in Rd . Discrete
and Computational Geometry, 12:189–202, 1994.
[13] J. Ruppert. A Delaunay refinement algorithm for quality 2-dimensional
mesh generation. Journal of Algorithms, 18(3):548–585, 1995.
[14] E. Schönhardt. Über die zerlegung von dreieckspolyedern in tetraeder.
Mathematische Annalen, 98:309–312, 1928.
[15] J. R. Shewchuk. Adaptive precision floating-point arithmetic and fast
robust geometric predicates. Discrete and Computational Geometry,
18:305–363, 1997.
[16] J. R. Shewchuk. A condition guaranteeing the existence of higherdimensional constrained Delaunay triangulations. In Proc. 14th Ann.
Symp. on Comput. Geom., pages 76–85, 1998.
[17] J. R. Shewchuk. Tetrahedral mesh generation by Delaunay refinement.
In Proc. 14th Ann. Symp. on Comput. Geom., pages 86–95, 1998.
REFERENCES
93
[18] J. R. Shewchuk. Constrained Delaunay tetrahedralizations and provably good boundary recovery. In Proc. 11th International Meshing
Roundtable, pages 193–204. Sandia National Laboratories, 2002.
[19] J. R. Shewchuk. What is a good linear element? interpolation, conditioning, and quality measures. In Proc. 11th International Meshing
Roundtable, pages 115–126, Ithaca, New York, September 2002. Sandia
National Laboratories.
[20] J. R. Shewchuk. Updating and constructing constrained Delaunay and
constrained regular triangulations by flips. In Proc. 19th Ann. Symp.
on Comput. Geom., pages 86–95, 2003.
[21] J. R. Shewchuk. General-dimensional constrained Delaunay and constrained regular triangulations, i: combinatorial properties. Discrete
and Computational Geometry, 39:580–637, 2008.
[22] H. Si. Adaptive tetrahedral mesh generation by constrained delaunay refinement. International Journal for Numerical Methods in Engineering,
75(7):856–880, 2008.
[23] H. Si. Three dimensional boundary conforming Delaunay mesh generation. PhD thesis, Institut für Mathematik, Technische Universität
Berlin, Strasse des 17. Juni 136, D-10623, Berlin, Germany, August
2008. Available online: http://opus.kobv.de/tuberlin/volltexte/
2008/1966/.
[24] H. Si. TetGen, towards a quality tetrahedral mesh generator. WIAS
Preprint No. 1762, 2013. submitted to ACM TOMS.
[25] H. Si and K Gaertner. 3d boundary recovery by constrained Delaunay tetrahedralization. International Journal for Numerical Methods in
Engineering, 85:1341–1364, 2011.
[26] H. Si and K. Gärtner. Meshing piecewise linear complexes by constrained
Delaunay tetrahedralizations. In Proc. 14th International Meshing
Rountable, pages 147–163, 2005.
[27] G. Voronoi. Nouvelles applications des parametrès continus à la théorie
de formas quadratiques. Reine Angew. Math., 133:97–178, 1907.
[28] D. F. Watson. Computing the n-dimensional Delaunay tessellations
with application to Voronoi polytopes. Comput. Journal, 24(2):167–
172, 1987.
94
REFERENCES
[29] G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in
Mathematics. Springer-Verlag, New York, second edition edition, 1997.
Index
anisotropic, 13
aspect ratio, 11
attribute
of point, 35, 54
of region, 45, 57, 60
-k, 30
-m, 39, 42, 43, 63
-n, 34, 39, 47
-nn, 61, 62
-o2, 47, 60–62
-p, 31, 36, 39
background mesh, 14
-q, 31, 39, 41, 45
boundary
-r, 31, 39, 41, 43
of PLC, 6
-v, 35
boundary conformity, 9
-w, 34
boundary edges, 7
-x, 49
boundary faces, 7
-z, 47
boundary marker, 52, 56
command line syntax, 31
of face, 61
compile
of point, 54
cmake, 23
boundary recovery, see boundary conCGAL’s predicates, 25
formity
CMakeLists.txt, 23
make, 22
CDT, see constrained Delaunay tetramakefile, 22
hedralization
Shewchuk’s predicates, 5, 24
circumsphere, 2
constrained Delaunay tetrahedralizacommand line switches, 31
tion, 10
-A, 45
constrained Delaunay tetrahedron, 10
-B, 52, 54, 56, 61
constrained tetrahedralization, 10
-C, 49
constraint
-F, 61, 63
area, 42
-I, 44, 49
length, 42
-N, -E, -F, 48
volume, 41
-O, 44
convex, 88
-R, 45
-S, 49
convex hull, 88
-T, 45, 50
convex polyhedra, 89
-V, 40, 47
convex polytope, 89
-Y, 37
degeneracy, 2
-a, 39, 41
degenerate polygon, 51
-c, 49
Delaunay refinement, 14
-e, 33, 39, 47, 63
Delaunay simplex, 2
-f, 33, 39, 46, 61
Delaunay tetrahedralization, 2
-g, 30
conforming, 9
-i, 43, 45
95
96
constrained, 9, 10
generation, 33
Delaunay triangulation, 2
dihedral angle, 12
dimension, 88
facet, 6, 51, 56
data structure, 80, 82
planarity, 52
polygon, 51
file formats, 52
.edge, 62
.ele, 59
.face, 61
.mesh, 69
.mtr, 42, 63
.neigh, 66
.node, 52, 53
.off, 68
.ply, 69
.poly, 52, 55
.smesh, 52, 58
.stl, 69
.v.cell, 67
.v.edge, 66
.v.face, 67
.v.node, 66
.var, 42, 65
.vol, 41, 63
general position, 2
hole
of facet, 52, 56
of PLC, 52, 56
isotropic, 13
lifted point, see lifting map
lifting map, 3
locally Delaunay, 11
lower face, see lifting map
INDEX
memory allocation, 48
memory usage report, 48
mesh adaption, 13, 41
mesh coarsening, 45
mesh element size, 13
mesh iteration number, 44
mesh optimization, 14, 44
level, 44
local operations, 44
mesh process, 16
mesh quality, 11
report, 28, 47
mesh reconstruction, 43
mesh refinement, 39
mesh size, 13
mesh sizing function, 13, 42
mesh statistics, 47
mesh validation, 49
orthogonal, 4
orthosphere, 4
piecewise linear complex, 6
PLC, seepiecewise linear complex6
polar, see lifting map
polygon, see polygon of facet, 56
data structure, 82
degenerate, 56
polyhedron, 89
boundary, 89
face, 89
interior, 89
power diagram, 4
radius edge ratio, 12
region attribute, see attribute of region
regular subdivision, 5
regular tetrahedron, 12
segment, 6
sharp feature, 15
INDEX
sharp features, 40
simplex, 88
face, proper face, 88
simplical complex, 88
subcomplex, 89
sliver, 12
solid modeling
B-Rep, 90
CSG, 90
Steiner point, 8, 49
Steiner tetrahedralization, 8
sub-region, 52
terminatetetgen() function, 90
tetgen.h C++ source, 77
tetgenio structure, 51, 78
tetrahedral mesh, 6
of a 3d PLC, 7
of a 3d point set, 6
tetrahedralize() function, 77
tetrahedron shape measure, 11
triangulation, 2
underlying space, 7, 89
vertex, 6
visible, 10
visualization
Medit, 30, 69
Paraview, 30
TetView, 30
Voronoi cell, 2
Voronoi diagram, 3
weight, see attribute of point
weighted Delaunay triangulation, 4
weighted distance, 4
weighted point, 3
weighted Voronoi diagram, see power
diagram
97
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