Hexahedral Mesh Generation for Tubular Shapes using Skeletons and

Connection Surfaces

P. Viville, P. Kraemer and D. Bechmann

ICube, Université de Strasbourg, CNRS, France

Keywords:

Mesh Generation, Skeleton, Hexahedral Meshing.

Abstract:

We propose a pipeline for the generation of hexahedral meshes for domains whose shape can be properly

represented by their 1-dimensional skeleton. By leveraging this representation, the resulting mesh is aligned

with the geometry of the domain and its connectivity is as regular as possible. The main challenge lies in

the management of the connectivity around branching points that can have an arbitrary number of incident

branches. We propose a new solution with the construction of connection surfaces that encode the connectivity

of the ﬁnal volume mesh around each vertex of the skeleton. Each vertex is processed independently in a

mutually compatible way, thus sparing the resolution of a global constraint problem. This leads to a pipeline

that does not need particular handling in the presence of cycles and in which most steps are able to process the

cells in a parallel way.

1 INTRODUCTION

The construction of a volume mesh from the surface

of a geometric domain has been a widespread sub-

ject of research for several decades. Most methods

attempt to solve this problem for domains of arbitrary

geometries. However, depending on the purposes we

deﬁne for our meshes, these generic methods might

provide unsatisfactory results. Concerning the shape

of cells, some methods generate cells of lower qual-

ity, in particular near the boundaries of the domain,

which is generally an area of greater importance. As

for the nature of cells, a few methods generate only

tetrahedrons whereas others cannot guarantee the pu-

rity of the results and produce mixed meshes which,

depending on context, might not fulﬁll the require-

ments. Purely hexahedral meshes are still notoriously

difﬁcult to build even though they are necessary or at

least highly appreciated for many applications.

Specialized methods can be proposed for the par-

ticular case of tubular objects, i.e. shapes whose cross

section with respect to their 1-dimensional skeleton is

roughly circular. The goal then is to create meshes

whose quality and regularity are higher than those

produced by generic methods, by taking advantage of

the intermediate skeletal representation whilst offer-

ing a simpler and more efﬁcient implementation.

The whole process chain includes the extraction of

a skeleton, the re-sampling of the skeleton, the gener-

ation of the volume mesh initial connectivity, reﬁne-

ment and adaptation to the geometric domain, as well

as the optimization of the shape of the cells. One of

the main challenges in this process is the handling

of the connectivity of the mesh around the branch-

ing points of the skeleton. This is where our main

contribution lies. We propose a technique that can be

applied on a branching point of any degree. Depend-

ing on the local conﬁgurations, a few variations can

also be applied. Because these different techniques

are all mutually compatible and each local choice has

no effect on the remainder of the object, many types

of skeletons can be directly managed even in the pres-

ence of cycles, and cells can be processed in a parallel

way through our devised algorithm.

2 EXISTING METHODS

Several general purpose hex-meshing methods have

been proposed. They are based on different ap-

proaches such as grid based (Schneiders, 1996),

volumetric parametrizations (Nieser et al., 2011),

(Livesu et al., 2013), advancing fronts (Kremer et al.,

2014), centroidal voronoi tessellations (Lévy and Liu,

2010), frame ﬁelds generation (Kowalski et al., 2016),

(Sokolov et al., 2016) and block decomposition, ei-

ther with manual input (Lu et al., 2017) or automat-

ically guided by a surface cross ﬁeld (Livesu et al.,

2020). These methods are well proven, foresight con-

Viville, P., Kraemer, P. and Bechmann, D.

Hexahedral Mesh Generation for Tubular Shapes using Skeletons and Connection Surfaces.

DOI: 10.5220/0010222000450054

In Proceedings of the 16th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2021) - Volume 1: GRAPP, pages

45-54

ISBN: 978-989-758-488-6

cerning the nature, shape, or alignment, of the cells is

often limited. The issues are compounded in the case

of purely hexahedral meshing which can be a require-

ment of computation tools we might want to apply to

the mesh.

A few methods, building either surface or volume

meshes, have been developed to handle the special

case of skeletonizable tubular shapes.

(Hijazi et al., 2010) constructed an hybrid trian-

gle/quad surface mesh around a given skeleton. The

branches are thickened as triangular prisms. On each

branch point, the convex hull of the projection on a

sphere of the incident branches triangular section is

constructed. Branches are ﬁnally connected to these

intersection meshes to form the ﬁnal mesh.

(Livesu et al., 2016) offered an extension of the

quadrilateral surface mesh generation described in

(Usai et al., 2015) to produce hexahedral meshes. A

hexahedron is placed on each vertex of the skeleton

with a degree higher than 2. It is then oriented and

subdivided according to incoming branches before

being extruded along the skeleton branches. In spite

of good results, this method suffers from signiﬁcant

issues. The cuts created on the branching point hex-

ahedra have to be mutually compatible, and a global

optimization problem has to be solved. Such a solu-

tion does not always exist, thus leading to some ar-

bitrary choices of connectivity, particularly when the

skeleton contains cycles. The propagation of the cuts

along the skeleton branches during extrusion also cre-

ates extraneous hexahedra in the initial crude mesh.

Finally, depending on the degree and angles of the

incident branches, the cube ﬁtting approach do not al-

ways lead to the most symmetric or natural results.

(Panotopoulou et al., 2018) and (Fuentes Suárez

and Hubert, 2018) presented a method to generate a

minimal quadrilateral surface mesh around a skele-

ton. On each branching point, a sphere is partitioned

into as many faces as incident branches. To sleeve the

branches with exclusively quadrilateral faces without

inserting irregular vertices, the sphere faces on ei-

ther side of a branch have to expose identical degrees.

Several steps in the method we propose are inspired

from these methods that target surface meshes.

(Verhetsel et al., 2019) offered a method to build a

hexahedral mesh contained inside a sphere partitioned

into quads. Even though this work could be of partic-

ular interest for our process, its scope is fairly limited

and we have developed a solution that is considerably

simpler and faster.

3 PROPOSED METHOD

This section describes our hex-meshing method. We

ﬁrst give a general description of the algorithm, with

generic handling of the branching points in mind.

Branching points processing and their compatible

specializations will then be explained in more detail.

Terminology In the skeleton graph, we call extrem-

ity a vertex of degree 1, joint a vertex of degree 2, and

branching point a vertex of higher degree. Extrem-

ities and branching points are the ends of branches

along which only joints can be found. We call a chunk

the set of hexahedra associated to each edge of the

skeleton.

3.1 Process Overview

Connection Surfaces. First, a connection surface is

created for each vertex of the skeleton. The connec-

tivity of those surfaces will contain the information

required to connect the hexahedral cells that are gen-

erated on each edge along the branches of the skele-

ton, serving as a scaffold for the geometry and con-

nectivity of the ﬁnal mesh.

For each vertex, a partition of a sphere in a quadri-

lateral surface mesh is created with as many faces as

branches incident to the skeleton vertex. Regardless

of the number of branches incident to the vertex – and

therefore quads in the partition – such a connectivity

can always be found.

Indeed, according to Euler’s formula for surfaces

of genus 0 we have v − e + f = 2, with v the number

of vertices, e the number of edges, and f the number

of faces. f being set and the number of edges per face

being 4, we get e = 2 f and v = 2 f − f + 2 = f + 2

which has an integer solution for any value of f .

Requiring the sphere partition to present only

quadrilateral faces will allow us at a later stage to gen-

erate a ﬁrst hexahedral mesh that exhibits no irregular

vertices along the branches.

Figure 1: Quad partitions of a sphere around vertices of

degree 1, 2, and 4.

For joints, this step is trivial, a dihedron composed

of 2 quads can be inserted onto the vertex (Figure 1 b).

For extremities, the same dihedron can be used while

GRAPP 2021 - 16th International Conference on Computer Graphics Theory and Applications

simply treating one of the quads as boundary (Figure

1 a). The algorithm used to partition the sphere for

higher degree branching points (Figure 1 c: example

of degree 4) is detailed in the following section.

Geometry. For joints and extremities, the vertices

are set on the plane orthogonal to the tangent of the

skeleton at that vertex. Rotation around that tangent

axis remains as a degree of freedom. For branching

points, the position of the vertices on the sphere are

constrained by the partitioning process and can thus

be used as a basis to set the rest of the geometry.

An initial frame is computed for each quad of the

branching point connection surfaces. These frames

are then propagated along branches using a rotation

minimizing frame scheme (Wang et al., 2008). If a

branch has branching points at both ends, the angle

formed between the frame propagated from one end

and the frame computed on the other end is computed.

This angle, bounded by [−

], is then back propa-

gated along the branch using an arc length parameter-

ization (Figure 2). For every joint along the branch

and on extremities, the propagated frames enable us

to set the geometry.

Figure 2: Twist redistribution between two branching

points.

Hexahedra Generation. For each edge of the

skeleton graph, a set of four hexahedra – a chunk –

is generated and linked to the edge. Both sides of the

chunk exhibit an interface composed of 4 quads (Fig-

ure 3).

Figure 3: Hexahedra chunk around an edge of the skeleton.

Each edge of the skeleton is incident to two ver-

tices. During the ﬁrst step a connection surface is built

for each vertex of the skeleton. Each incident branch

of the vertex is associated to a quad of this surface

(Figure 4b). The four front-facing quads of the cor-

responding chunk are paired with the four edges of

the connection surface quad (Figure 4c and d). By the

end, whatever the degree of the branching point, each

edge of the connection surface is paired with exactly

two hexahedra (Figure 4e). To complete the connec-

tivity of the hexahedral mesh, those pairs of hexahe-

dra are connected together through the face they re-

spectively exhibit to the connection surface edge (Fig-

ure 4f). This process demonstrated on a vertex of de-

gree 3 in Figure 4 can be generalized to vertices of

any degree as long as a proper quad connection sur-

face has been built.

Figure 4: The connection surface built on each skeleton ver-

tex is used as a scaffold for the geometry and connectivity

of the hexahedra. (a) a vertex incident to three branches

(b) the connection surface composed of three quads (c,d,e)

incident chunks are paired (indicated by colour) with their

associated quad edges (f) pairs of hexahedra are connected.

3.2 Branching Points Processing

As stated in the previous section, to handle complex

branching points of degree higher than 2, we strive to

construct the partition of a sphere into a given number

of quads each containing an entry point correspond-

ing to the direction of one of the incident branches

on the sphere. The positions of these entry points are

ﬁxed and cannot be modiﬁed.

The fact that a sphere can be partitioned into any

number of quads implies that there exists a dual mesh

of this partition in which all vertices are of degree 4.

Our ﬁrst goal is then to create a mesh whose vertices

are the entry points and are all of degree 4. The dual

of that mesh will be the quadrilateral partition of a

sphere that we are looking for, with each face con-

Hexahedral Mesh Generation for Tubular Shapes using Skeletons and Connection Surfaces

taining its entry point.

Initially, we compute the convex hull of the entry

points. Since the points are on a sphere this convex

hull corresponds to the Delaunay triangulation of the

set of points (Na et al., 2002). In a second step, we

update the connectivity of this triangulation to set all

degrees to 4.

This remeshing is performed in two stages. First,

a degree reduction stage for all vertices of degree

higher than 4, followed by a degree increasing stage

for all pairs of vertices of degree 3. If we do not cre-

ate any vertex of degree lower than 3 during the ﬁrst

stage we are certain that there will be 0 or an even

number of vertices of degree 3. This can be shown

again with Euler’s formula, here applied to the primal

mesh. If we have f = q + t with q number of quads

and t the number of triangles, then e =

t + 2q and

v =

t + q + 2 which only has integer solutions for

an even number of triangles. Therefore the dual mesh

has an even number of vertices of degree 3 which can

be eliminated by pairs by using a method derived from

(Peng et al., 2011).

We aim to minimize the number of remeshing op-

erations in order for the result to be as close to the

Voronoï diagram of the entry points as possible. This

will also help to preserve the convexity of the spheri-

cal quads and the inclusion of the entry points in their

respective face of the partition. Since the number of

vertices and their positions cannot be modiﬁed, the

only operations at our disposal are adding an edge (or

cutting a face) and deleting an edge (or merging two

faces).

Figure 5: Angle criterion: γ < β < δ < ε < α.

Our main criterion for prioritizing the remesh-

ing operations is the angle between edges incident

to a vertex. The heuristic will carry out the opera-

tions in order to minimize angles created when delet-

ing edges (Figure 5) and to maximize angles created

when adding edges. Focusing on angles seemed to be

the most relevant criterion to minimize the degree of

the polygons created in the ﬁnal mesh and maximize

their convexity.

We propose the following algorithm:

Degree Reduction Stage:

• Deletion of edges incident to two vertices of de-

gree higher than 4, ordered by the smallest created

angles.

• Deletion of edges incident to a vertex of degree

higher than 4 and a vertex of degree 4, ordered by

the smallest created angles.

Removal of 3 - 3 Pairs

• Add an edge between two non-adjacent vertices

of degree 3 of a face.

• Add an edge between two adjacent vertices of de-

gree 3 of a face.

• Search all the shortest paths between pairs of ver-

tices of degree 3, sorted by descending length. A

set of operations is then applied depending on the

parity of the path length. For paths of odd lengths

(Figure 6): by adding an edge and deleting the

next one, the vertex of degree 3 is moved along

the path by 2 steps. Once both vertices of degree

3 are adjacent, their degrees are increased to 4 by

adding an edge between them. For paths of even

lengths (Figure 7): by using a vertex adjacent to

the path the vertex of degree 3 is shifted by one

step along the path and the scheme used for odd

length paths can be applied.

Figure 6: Connectivity modiﬁcation between two vertices

of degree 3 (in red) separated by an odd length path (a).

Doubling an edge and deleting the next one moves a degree

3 vertex of two steps along the path (b). Once the degree 3

vertices are adjacent, a last edge increases both their degree

to 4 (c).

Figure 7: Connectivity modiﬁcation between two vertices

of degree 3 (in red) separated by an even length path (a).

An adjacent vertex is used to shift a degree 3 vertex by one

step along the path (b). Then we are left with the odd length

case (c-d).

Once this mesh is only composed of vertices of

degree 4, we compute its dual to obtain the desired

quad mesh. Vertices of the quad mesh are placed at

the barycenters of the faces. As a ﬁnal result we ob-

tain a sphere partitioned in quadrilaterals with an en-

try point associated to each face. Figure 8 shows the

GRAPP 2021 - 16th International Conference on Computer Graphics Theory and Applications

Figure 8: Quad partitioning algorithm on a sphere. Given

an initial set of points, we obtain a quad mesh where each

quad contains one of the given points. (a) Intersection of the

branches with a sphere (b) Construction of the Delaunay tri-

angulation (c) Remeshing of the triangulation (d) Construc-

tion of the dual mesh (e) Resulting quad mesh connection

surface.

different steps of the algorithm on an example of de-

gree 7.

This algorithm yields good results – i.e. a parti-

tion of the sphere with convex quads each contain-

ing one entry point – in most of the cases encoun-

tered when working on skeletons extracted from usual

shapes. Some challenging, mostly artiﬁcial, branch-

ing points geometries can prove to be problematic and

do not allow us to obtain such a partition. Figure 9

shows a conﬁguration that results in a partition where

some points are out of their quad. These irregularities

can be mostly corrected in the following geometry op-

timization step but the resulting mesh is likely to be

of lower quality. However, improvements can still be

made to obtain better results in the most challenging

conﬁgurations.

Figure 9: A failure example of the quad partitioning of the

sphere: not all quads contain their respective point.

3.3 Special Cases

In addition to the generic algorithm presented in the

previous section, we chose to distinguish two cases

in which a specialized, yet globally compatible, al-

ternative can be used to generate the connectivity of

the branching points. When their required conditions

are fulﬁlled, these specializations are applied in prior-

ity and allow the generation of an initial coarse mesh

with higher quality elements.

Flat Cases. For ﬂat branching points where all en-

try points are on or near a common plane, an "orange

slice" type connectivity is generated. This is inspired

by the initial step of the sphere partition method of

(Panotopoulou et al., 2018). For such a point with n

branches, we create a connection surface quad mesh

composed of two vertices of degree n, the poles, and n

vertices of degree 2 along the meridians (Figure 10).

Figure 10: "Orange slice" handling of a ﬂat branching point

of degree 3.

Cube Cases. For branching points whose degree is

between 3 and 6, and for incident branches direc-

tions that are either mutually orthogonal or pairwise

aligned, a subdivided cube is inserted (Figure 11). As

in (Usai et al., 2015), a cube is ﬁtted at the branching

point location, keeping the incident branches direc-

tion as orthogonal as possible to the cube faces. If a

face of the ﬁtted cube is pierced more than once, or an

angle between a branch and the corresponding cube

face normal exceeds a chosen threshold, this special-

ization is discarded and we revert to one of the pre-

vious methods. The subdivided cube exposes a com-

patible interface for the connection with the incoming

branches chunks, allowing this special case to be in-

tegrated seamlessly in the overall process.

Figure 11: Examples of orthogonal intersections.

Figure 12 illustrates two different conﬁgurations

of a degree 3 vertex. In the ﬁrst one (a) the three

branches form 120

◦

angles. Using a sphere partition

(left) generates better shaped elements than ﬁtting a

cube (right). In the second one (b) the branches form

a T shape. In this case, ﬁtting a cube (right) yields

better shaped elements. The versatility of our method

allows it to use the best method on each skeleton ver-

tex depending on the local conﬁguration.

Hexahedral Mesh Generation for Tubular Shapes using Skeletons and Connection Surfaces

Figure 12: Comparison of sphere partitioning and cube

insertion technique on two degree 3 vertices. Minimum

scaled Jacobian: a) 0.802 (left) 0.409 (right); b) 0.533 (left)

1 (right).

4 MESH GENERATION PIPELINE

We present here the whole process we have devel-

oped, illustrated with a simple example.

4.1 Skeleton Extraction

The entry point of the hexahedral mesh generation

algorithm is a 1-dimensional skeleton with a posi-

tion and a radius on each vertex. This skeleton can

be given, or extracted from an input surface using a

method such as Mean Curvature Flow (Tagliasacchi

et al., 2012). In this case, the skeleton is subsequently

resampled (Figure 13) in order for edge lengths to be

related to the local shape radius. This allows elements

of the generated coarse mesh to be as regular as pos-

sible. However, if desired, anisotropic elements can

easily be obtained by changing the parameter that ties

the subdivision of the skeleton to the local shape ra-

dius. When the enclosing surface geometry is com-

plex, the quality of the skeleton and its ability to cap-

Figure 13: Extraction and resampling of the skeleton.

ture the features of the shape will strongly impact the

quality of the resulting hexahedral mesh. Of course,

the skeleton can also be manually processed if the ap-

propriate tools and required time are available.

4.2 Initial Mesh Generation

The connection surfaces are built independently on

each vertex of the skeleton (Figure 14 left). The ge-

ometry is set, then propagated from the branching

points to the joints and extremities. A rough hexahe-

dral mesh is generated by creating the chunks for each

skeleton edge and connecting them following the con-

nection surfaces (Figure 14 middle). The vertices are

then projected along their normals onto the enclosing

surface, if any (Figure 14 right). This projection step

can cause potential issues with self intersections that

can be solved using mapping methods such as the one

described by (Komaritzan and Botsch, 2018).

Figure 14: Generation of the connection surfaces and rough

volume mesh.

The locality of each step allows the following

steps to process the cells in parallel: generation of

connection surfaces for each skeleton vertex; gener-

ation of hexahedral chunks for each skeleton edge;

gluing of the hexahedral elements for each connec-

tion surfaces edge; projection of the vertices on the

surface.

4.3 Reﬁnement and Geometry

Optimization

The resulting mesh depends on the sampling of the

skeleton and is as rough as our algorithm can produce.

In order to get a better and usable resulting mesh, the

mesh will have to go through several steps of process-

ing and optimization.

Subdivision and Padding. A global primal subdi-

vision step, or local subdivisions like cutting chunks

GRAPP 2021 - 16th International Conference on Computer Graphics Theory and Applications

Figure 15: Reﬁnement and geometry optimization.

along a given branch, can be executed to reﬁne the

mesh and allow better ﬁtting to a detailed surface.

Hexahedra that expose more than one face to the

boundary of the mesh can lead to badly shaped ele-

ments when constrained to an enclosing surface. A

classical way of dealing with this issue is to add a

padding layer to the whole surface of the mesh. Each

element of the mesh then has at most four vertices on

the boundary leaving more freedom to the subsequent

geometry optimization process.

Geometry Optimization. Considerable studies on

geometric optimization of hexahedral meshes has al-

ready been carried out (Livesu et al., 2015), (Gao and

Chen, 2016). The goal is usually to modify the posi-

tion of the vertices to increase the quality of the hex-

ahedra by ﬁtting their shape closer to that of a rectan-

gular cuboid, while preserving the outer surface ge-

ometry. Several quality measures exist (Gao et al.,

2017), the most prevalent one being the scaled Jaco-

bian, computed for each element of the mesh. A neg-

ative value indicates an inverted element – which can

make a whole mesh unusable for physical simulation

– and rectangular cuboids have a value of 1. The worst

and the average value over the whole mesh are usually

highlighted as good indicators of the global quality

of a model. In our experiments, we used the edge-

cone rectiﬁcation approach described in (Livesu et al.,

2015) whose iterative process improves the quality of

the elements and converges to an inversion-free mesh.

5 RESULTS

Figure 18 shows some hexahedral meshes obtained

with our method. Number of cells, minimum and av-

erage scaled Jacobian of the hexahedra of these mod-

els are presented in Table 1.

Compared to previous methods such as (Livesu

et al., 2016) and (Livesu et al., 2020) the quality of

Table 1: Minimum and average scaled Jacobian of the gen-

erated meshes presented in Figure 18. Values reported in

the previous works come from (Livesu et al., 2016) for

the Dinopet, Cactus, Fertility and Santa models and from

(Livesu et al., 2020) for the Metatron model.

Previous works Ours

Model Min/Avg SJ #Hex Min/Avg SJ #Hex

Dinopet .177/.920 18080 .540/.919 13120

Horse .449/.912 5568

Cactus .526/.919 4128 .635/.930 2400

Fertility .500/.892 7456 .543/.933 6848

Cycles .316/.937 8192

Vessel .615/.930 24592

Santa .367/.941 26240 .514/.965 21312

Metatron .682/.941 4544 .820/.981 6624

Figure 16: On the left, results obtained by (Livesu et al.,

2016) (Santa) and (Livesu et al., 2020) (Metatron), on the

right, our results. Combining cube ﬁtting and sphere parti-

tioning techniques depending on the angles in the branch-

ing points improves the symmetry and the overall elements

quality of the obtained hexahedral meshes.

the constructed meshes is generally higher while be-

ing composed of fewer elements. Average quality

could be improved further by applying some of the

described subdivision techniques. However, our main

point here is not on purely numerical quality results.

What we emphasize is the ability of our method to

process each skeleton branching point individually.

Branching points in which a cube does not naturally

ﬁt are processed with the more general sphere quad

partition approach. This allows complex branching

points to be handled without propagating additional

cuts and constraints along the incident branches (such

as in the Cycles model for which the central connec-

tion surface is depicted on its right).

More symmetric results are also obtained, such as

in the Santa model in which the torso branching point

is handled with a cube while the hips branching point

is handled with a sphere partition (Figure 16). A com-

bination of cube and sphere partition techniques is

Hexahedral Mesh Generation for Tubular Shapes using Skeletons and Connection Surfaces

also used in the Metatron model, here compared side-

by-side with the result obtained with the more gen-

eral method presented in (Livesu et al., 2020). The

regularity of the result is improved by using the most

suitable connection surface depending on the angles

in the branching points.

The pipeline is yet to be made fully automatic and

the different steps are triggered manually, therefore

giving exact ﬁgures of execution times is not possible.

Resampling of the skeleton, generation of the con-

nection surfaces and generation of the rough hexahe-

dral mesh are all near immediate, even with the larger

models we have used. This efﬁciency is mainly due to

the fact that the algorithm processes each branching

point independently and, unlike most existing meth-

ods, does not have to set up and resolve any global

constraints problem. Time spent in the ﬁtting to the

surface and iterative geometry optimization steps is

dependent on the requirements of the ﬁnal user. In the

examples used in 18, the optimization time ranged in

the order of a few seconds in total. In comparison, the

estimated times range from 1.5 second to 1 minute in

(Livesu et al., 2016) for the examples we have used

and a total time of 121 seconds is given for the Meta-

tron model in (Livesu et al., 2020).

As we already stated, due to the radial nature of

the generated structure, our method is particularly

suited to shapes whose cross section with respect to

the skeleton is roughly circular. Figure 17 shows

some shapes that do not meet these requirements,

such as the elephant ears or the rocker arm model

that has large ﬂat features that expand in different

directions, or the mechanical piece that has a more

complex cross section. The meshes generated by our

method cannot properly ﬁt the surface and the ﬁnal

result contains poor quality or highly anisotropic ele-

ments.

6 CONCLUSIONS AND FUTURE

WORK

We have presented a new skeleton based purely hex-

ahedral mesh generation algorithm. It leverages the

skeletal representation to construct a high quality

mesh whose elements are aligned with the geome-

try of the domain and has a mostly regular connec-

tivity. The scope is limited to objects whose cross

section with respect to their 1-dimensional skeleton

is roughly circular, which includes a large variety of

natural shapes.

Our main contribution is the management of the

connectivity of the mesh around the skeleton branch-

ing points using connection surfaces. Created for each

Figure 17: Shapes that will typically be poorly recon-

structed with our method, as the radial nature of the gen-

erated structure is not able to capture the ﬂat areas or com-

plex cross section. Top: Elephant and Mechanical piece.

Bottom: Rocker arm.

vertex of the skeleton, these surfaces encode the con-

nection plan for the hexahedral elements created on

the incident branches. The general sphere partitioning

algorithm can be used on vertices of arbitrary degree

and builds a quad mesh that exposes one face per in-

cident branch. Depending on the local conﬁguration,

other specializations can be used. The presence of

cycles within the skeleton has no impact on the pro-

cessing of the rest of the mesh.

These connection surfaces being all compatible,

the vertices can be processed independently. In a sec-

ond step, the orientation frames can be propagated

independently for each branch. The construction of

the chunks for each skeleton edge and their connec-

tion around each skeleton vertex using the connection

surfaces can in the same way be performed indepen-

dently on a per cell basis. Each of these steps can thus

process the cells in parallel which leads to a highly

efﬁcient generation of the topological structure of the

coarse hexahedral mesh.

This coarse volume mesh generation algorithm is

part of a processing pipeline that includes: extraction

of the skeleton from the domain surface, resampling

of the skeleton, generation of a rough mesh and its

GRAPP 2021 - 16th International Conference on Computer Graphics Theory and Applications

Figure 18: Some meshes obtained with our method. The skeleton (black) extracted from the surface mesh (blue) and connec-

tion surfaces (red) are displayed along with the hexahedral cells (yellow). Quality measures for these meshes are indicated in

Table 1. From top-left to bottom-right: Dinopet, Horse, Cactus, Fertility, Cycles, Vessel, Santa and Metatron.

subsequent reﬁnement, adaptation to the domain ge-

ometry, and cells shape optimization.

Improvements on this work are of course still pos-

sible to get better results. (Livesu et al., 2016) pro-

poses a method to adapt the mesh locally based on

radius variations along branches through the insertion

of patterns of hexahedra that lead to an increase or

decrease of the number of hexahedra within a chunk.

This resolution adaptation has a strong impact on the

ﬁnal result quality and usability. Our perspectives in-

clude its adaptation to our method.

At the core of our proposition, the sphere parti-

tioning method can also be improved. Our current

algorithm is led by local choices based on angles. It

produces acceptable results, but a canonical solution

of the sphere partitioning problem could be highly

valuable.

Hexahedral Mesh Generation for Tubular Shapes using Skeletons and Connection Surfaces

ACKNOWLEDGEMENTS

Thanks to the Hexalab platform (Bracci et al., 2019)

for providing tools and datasets from previous studies.

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