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MESTRADO EM ENGENHARIA MECÂNICA
November 2014
APPLICATION OF ALGORITHMS FOR AUTOMATIC
GENERATION OF HEXAHEDRAL FINITE ELEMENT MESHES
Luís Miguel Rodrigues Reis
Abstract. The accuracy of a finite element method analysis depends on the mesh quality of the domain
discretization. Additionally, quadrilateral and hexahedral meshes presents several numerical advantages
over triangular and tetrahedral ones. Thus, mesh generation of quadrilateral and hexahedral elements is a
very important, complex and time demanding issue in finite element problems. Furthermore, most finite
element commercial programs don’t have implemented the latest mesh generation algorithms. In this work
some recent quadrilateral and hexahedral mesh generation algorithms were implemented in order to import
meshes to commercial software or to use with in-house software. Receding front, level sets, medial axis and
transfinite mapping algorithms were used in order to generate meshes of several 2D and 3D geometries.
Element quality and distortion measure implemented uses the condition number of the Jacobian matrix.
Quadrilateral and hexahedral meshes were generated for several geometries and meshes with good quality
were obtained.
1 INTRODUCTION
Mesh generation is a very complex and time demanding issue in a finite element analysis. Additionally,
quadrilateral two-dimensional meshes present several numerical advantages when compared with
triangular meshes [1]. For three-dimensional analysis, hexahedral elements have also some advantages over
tetrahedral meshes [1]. On the other hand, in order to generate a quadrilateral or a hexahedral mesh, in
commercial software, user has a tedious manual work to sub-divide the domain in quad or hex meshable
sub-regions. But even so, sometimes quadrilateral or hexahedral meshes are not possible to generate or the
final result is different from the idealized one. Thus, the implementation of mesh generation algorithms can
be useful since import meshes in commercial software is usually possible and is also possible to use meshes
with in-house finite element programs.
Nowadays, none of the existent algorithms are robust and automatic in order to generate quadrilateral
or hexahedral meshes for any initial geometry [2]. Grid-based and advancing front methods are almost
automatic and robust. However both have some advantages and disadvantages. Grid-based methods
generate a good inner mesh, since algorithm starts in the inner part of domain. Near the outside boundary
elements have low-quality [2]. Thus, this method is not suitable for contact or blood flow problems, because
interface mesh quality is an important issue.
The advancing front family methods meshes are generated layer by layer following the shape of the
boundary surface. Consequently, boundary elements have good-quality. However, these methods are less
robust and automatic, because fronts can collide and create voids [2,3].
In the last decade, some new algorithms have been proposed. In fact, in order to avoid the main
disadvantages of both methods and also to combine advantages, Ruiz-Gironés [1,2] proposed the receding
front method. This method computes the layers of elements combining two solutions of the Eikonal
equation. This procedure needs an inner boundary, if the domain hasn’t any hole inside, then the medial
axis should be computed [4]. The receding front method wasn’t the first method using the Eikonal equation.
In 1994, Sethian [5] presents a 2D mesh generation method that uses level set fronts and curvature.
Other method, developed in the last decades, is the transfinite mapping (TFI) [6]. With this method is
possible to mesh geometries with curved sides, like, for instance, the S-shape. This method could be also
applied to domains with one dimension bigger than others and also with bifurcations.
Quality and distortion measure of meshes is also an important issue, in order to compare elements
quality. The condition number of the Jacobian matrix can measure the algebraic element quality in terms of
Luís Miguel Rodrigues Reis
the Jacobian of the mapping between an ideal and a physical element [7].
The objective of this work is to implement some quadrilateral and hexahedral mesh generation
algorithms. In this way, some recent mesh generators methods are available to create meshes outside
commercial finite element software. This could be important in order to have better meshes, since some
commercial software doesn’t have the latest methods available. These methods may also be relevant for
shape optimization analysis, in order to avoid stopping at all iterations to re-mesh the new geometry. It is
also essential to avoid commercial software and use with in-house finite element software.
2 METHODS
In this work the receding front method and transfinite mapping (TFI) method were implemented. Both
methods are meshing generation algorithms to define quadrilateral and hexahedral elements. A method to
define the medial axis was also implemented since it is necessary to define an inner boundary. Moreover,
level set method is used to define the layers of elements in both receding method and TFI. An algebraic
method to measure quality and distortion of finite elements was also applied.
2.1 Level set method
A level set is, generally, a level curve or a level surface, or, in other words, an isoline or an isosurface.
This curve, or surface, can move with a velocity f [8]. If velocity only depends on position the level set
method reduces to Eikonal equation:
 d  f in 


d U  0
(1)
where f=1,  is the euclidean norm, U is the 0-distance field and  is the domain. In this case (f=1) the
solution d is the distance from  [1]. The Eikonal equation is solved on a triangular or a tetrahedral mesh
by means of an edge-based solver [7].
To compute the node fronts, or the element layers, to be used in the receding front method, Eikonal
method is solved for inner boundary ( in ) and outer boundary ( out ). Finally the combined result is
obtained:
u
d out
d out  din
(2)
where din and dout are the solutions of (1) considering in and out , respectively. In figure 1 is possible to
observe an example of the combined final result (u).
Figure 1: a) triangular mesh to solve Eikonal problem, b) dout, c) din, d) u.
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Luís Miguel Rodrigues Reis
2.2 Receding front method
The receding front method combines the advancing front technique with the grid-based one. In the
advancing front methods mesh is generated from outside to the inside of the domain. In the grid-based
methods is the other way around, the mesh generation starts in the outer boundary to the inner limit. The
receding front method uses a set of fronts, computed using the Eikonal equation (1) and the combined
result presented in equation (2). Thus, with the receding front method is possible to generate good
quadrilateral and hexahedral meshes near both boundaries and also inside de domain.
If the domain hasn’t an inner boundary (an inner hole), the first step is to create an inner front or an
inner boundary. To do so the medial axis is computed. Then, the receding front method is used to create the
mesh between the outer boundary and the defined inner front (boundary). After this step, is necessary to
generate the mesh inside the defined inner front. This inner front geometry definition is a very important
issue for the final mesh quality. However, is still missing a perfect inner shape definition [1]. Could be tested
several shapes and keep the best final shape, or use an optimization procedure to move nodes and
maximize elements quality [9].
The receding front method generates a hexahedral mesh between an inner and an outer boundary. If
boundaries have vertices and edges is necessary to use some templates to expand the elements to the next
front [1]. This is one of the major disadvantages of the receding front method since is almost impossible to
define templates to all boundaries [1]. But, templates can be added to algorithm, and difficulties can be
avoided in future. Another problem is related with geometries with more than one interior hole. In this case
domain should be divided in sub-domains with one hole each.
Finally, the receding front algorithm has 5 steps [1]:
i. Compute the medial axis of the domain (if necessary);
ii. Generate the mesh inside de inner boundary (if necessary);
iii. Compute the seeds in the inner boundary;
iv. Compute level sets using equation (1) and (2);
v. Generate layers of elements from the inner boundary to the outer boundary using the fronts
computed in step iv.
2.3 Transfinite mapping method
Transfinite mapping method (TFI) allows meshing extruded volumes in which the opposite logical sides
should have the same number of nodes [6,10]. TFI has a starting surface and a target surface, and could also
have linking sides [6]. Starting from the source surface, the opposite node in the target surface minimizes
the distance between both nodes [6]. This method is well suited to the S-shape and all other similar curved
shapes [6,10]. More recently some other procedures were added to TFI in order to improve the mesh and to
increase the applicability to more domains, for instance with some vertices (not only in the starting and
ending points). Optimizing the mesh quality [9] or penalizing the large distances between two consecutive
nodes on the target surface, are two examples of other techniques that could be added to TFI.
2.4 Medial axis
Medial axis can be necessary to create an inner boundary in order to use the receding front method. But,
can also be essential to define an inner boundary to use TFI, especially if domain has one dimension bigger
than others and bifurcations, like 2D geometry in figure 2.
Figure 2: definition of medial axis.
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The definition of medial axis used in this work is the one of Tamal K. Dey [4]: “the medial axis of a curve
or a surface  is meant to capture the middle of the shape bounded by  ”.
Definition 1. The medial axis M of a curve (surface)   k is the closure of the set of points in k that
have at least two closest points in  [4].
2.5 Quality and distortion measure
For quality and distortion measure of elements the condition number of the Jacobian matrix [7,9,11]:
J 
1
J F  J 1
3
for 3D meshes, and J is the Jacobian matrix and 
(3)
F
F
represents the Frobenius norm.
For two-dimensional meshes the condition number is given by:
J 
1
J F  J 1
2
(4)
F
The element with the desired shape have   J  equal to one. All other elements have condition number
greater than one. Thus, the greater is the condition number less quality and more distortion the element is.
In order to have all results between zero and one the quality of an element is the inverse of the condition
number,
q
1
J
(5)
then 0  q  1, and the closer q is to 1, the better is the quality of the element.
3 RESULTS
The implemented mesh generation algorithms were applied to 4 different domains, in order to compare
results, validate methods and understand the aspects to improve. All meshes are parameterized in order to
make several meshes with the desired number of elements per edge or surface.
3.1 2D mesh – five-pointed star
The first domain is a benchmark domain [1], presented in figure 1, and receding front method was used.
Star angles are not constant. In the upper part the angles are near 55 degrees, and in the other 3 edges the
angles are greater than 90 degrees. In figure 1 is also possible to observe the level sets, or the layers of
quadrilateral elements. In this example, the number of level sets and the number of nodes in each star edge
is parameterized, so user can make their choice. In figures 3, 4 and 5 is possible to observe 3 different
meshes with 7, 10 and 12 level fronts, respectively. It is also possible to verify that meshes have less quality
near the upper part of star, due to the more acute angle.
Like we can verify in table 1, meshes are good, with an average q always greater than 0.8. However, the
minimum value is too small near the upper part of the star. To avoid this, a curvature flow, like the one
presented in Sethian [5], should be implemented.
Table 1: q for 2D – five-pointed star meshes.
Mesh
3.1-1
3.1-2
3.1-3
Figure
3
4
5
Nº Elements
540
720
840
Minimum q
0.220
0.128
0.110
4
Maximum q
0.998
0.999
0.999
Average q
0.859
0.852
0.817
Luís Miguel Rodrigues Reis
Figure 3: Mesh 3.1-1(5-pointed star) with 7 fronts and q.
Figure 4: Mesh 3.1-2 (5-pointed star) with 10 fronts and q.
Figure 5: Mesh 3.1-3 (5-pointed star) with 12 fronts and q.
3.2 2D mesh – medial axis
The second geometry was inspired in medial axis example from the book of Dey [4], see figure 2. The
medial axis was computed and then an inner boundary with straight lines was defined. Between the inner
and outer limits the TFI was used. Inside the inner boundary a simple box mesh was generated. In the
middle of the domain a small instability was applied, like in Sethian [5], in order to avoid triangular
elements. In the bottom part of domain large distances between two consecutive nodes on the target
surface were penalized. On the other regions no penalty was used, is a pure TFI algorithm. In this way is
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Luís Miguel Rodrigues Reis
possible to compare both techniques. In figures 6 and 7 is possible two observe to meshes and verify that
the number of nodes per edge is important in mesh quality. It is also possible to verify that the penalization
of large distance between two consecutive nodes in the target boundary improve the mesh quality. In table
2 is possible to verify the quality parameters, and q is less than 0.2 for the worst element, in the middle of
domain. This could be improved with other types of instability, maybe a circular shape in the middle region.
The average quality is always greater than 0.75.
Figure 6: Mesh 3.2-1 (medial axis) and q.
Figure 7: Mesh 3.2-2 (medial axis) and q.
Table 2: q for 2D – medial axis meshes.
Mesh
3.2-1
3.2-2
Figure
6
7
Nº Elements
821
705
Minimum q
0.193
0.188
Maximum q
1.000
1.000
Average q
0.805
0.768
3.3 3D mesh – sphere-cube
This example is a 3D case: a hexahedral mesh of a sphere with a cube hole inside. The receding front
method was applied to this example. In figure 8 is possible to observe the level set result (u computed with
equation 2).
Figure 8: level set (u) for sphere-cube mesh.
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Luís Miguel Rodrigues Reis
In figures 9 and 10 is possible to see two meshes. The first one has only 5400 elements, and the second
one has 65664, as presented in table 3. This is possible because mesh generation is parameterized. In this
example the average q is greater than 0.9 and the minimum value always larger than 0.6.
Figure 9: Mesh 3.3-1 (sphere-cube) and q.
Figure 10: Mesh 3.3-2 (sphere-cube) and q.
Table 3: q for 3D – sphere-cube meshes.
Mesh
3.3-1
3.3-2
Figure
9
10
Nº Elements
5400
65664
Minimum q
0.693
0.608
Maximum q
0.992
0.999
Average q
0.915
0.919
3.4 3D mesh – hip implant
In this last example is presented a more complex geometry, and with contact surfaces. Near the
interfaces, elements should have good quality to avoid errors in contact solution. In figure 11 is possible to
observe the three parts involved: prosthesis, cement and femur.
Figure 11: prosthesis (implant), cement and femur.
Medial axis has computed to define an inner boundary in prosthesis. The inner boundary chosen is an
ellipse, however maybe a rectangle is a better choice. The receding front method was used to generate all
the mesh. In figures 12 and 13 is possible to observe the mesh of prosthesis, cement and femur for two
distinct examples. In the first one (figure 12), prosthesis has a small flange in the proximal part. In the
second case (figure 13), the flange is greater and more elements are used. In tables 4 and 5 is possible to
verify the quality of both meshes. Tables have the quality of elements in each part, also. The minimum value
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Luís Miguel Rodrigues Reis
of q for prosthesis is a very low one. It should be tried other inner boundaries, different from ellipses, for
instance a rectangle. Nevertheless, the quality of the elements in the interface is almost always greater than
the average q, as is possible to observe in figures 12 and 13. In proximal part the interface elements are
almost red, that means that q is almost equal to 1. For cement and femur the average quality results are
greater than 0.5, and in femur the worst elements are far from the interfaces. Cement is a special case since
has two interfaces.
Figure 12: Mesh 3.4-1 (hip implant) and q.
Figure 13: Mesh 3.4-2 (hip implant) and q.
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Luís Miguel Rodrigues Reis
Table 4: q for 3D – hip implant mesh 3.4-1.
Part
prosthesis
Cement
Femur
Total
Nº Elements
22400
4864
14336
41600
Minimum q
0.011
0.179
0.112
0.011
Maximum q
0.792
0.974
0.974
0.974
Average q
0.269
0.556
0.577
0.409
Table 5: q for 3D – hip implant mesh 3.4-2.
Part
prosthesis
Cement
Femur
Total
Nº Elements
91800
24624
60480
176904
Minimum q
0.024
0.323
0.195
0.024
Maximum q
0.990
0.972
0.992
0.992
Average q
0.520
0.777
0.795
0.650
The hip implant example is a more difficult one, but the receding front method was able to generate
good hexahedral elements. However, different strategies for the definition of the inner boundary in the
prosthesis layers are necessary in order to improve the implant mesh.
With mesh generation algorithms implemented in this work is possible to define a mesh in several parts,
with a good interface definition. Observe in figures 12 and 13 that interfaces nodes are coincident for both
surfaces. This is also an essential issue to have better contact analysis. Additionally, the hip implant mesh
generation allows shape optimization procedures, since algorithm is parameterized to implant, cement and
femur geometric parameters.
4 FINAL REMARKS AND FUTURE WORK
In this work some algorithms to generate quadrilateral and hexahedral meshes were implemented.
These methods are recent and the major part of finite elements commercial software doesn’t have these
algorithms available. With these algorithms is also possible to use in-house finite element software or make
shape optimization analysis.
In order to study the methods some examples were performed. The two-dimensional meshes obtained
have good global quality. However, some improvements can be implemented. In fact, for the first example
(five-pointed star) if level set curvature has fast changes, mesh could be improved with a curvature term like
in Sethian work [5]. In the second example (medial axis) a penalization was introduced and good results
were obtained when compared with the unchanged TFI. The large distance between two consecutive nodes
penalization should be applied to all target line or surface. On the other hand, the mesh in the middle of the
inner boundary should be improved with a circle instead of instability.
Two examples of hexahedral meshes were also presented. The first one is an easy one in order to test
the algorithm. However, an Abaqus® expert, with more than 15 years of experience, made a similar spherecube mesh and the best average q obtained was 0.808 and a minimum q equal to 0.294. It is an easy mesh,
but even so the result obtained is much better compared with the mesh generated in Abaqus® commercial
software.
The last example is a more difficult one, due to geometric complexities and the two interfaces: femurcement and cement-prosthesis. Near the interface is important to have good elements and that was
achieved. It was also possible to obtain coincident nodes in the interface, for both surfaces. However,
prosthesis inner boundary, defined to make the level sets, should be changed to other solution, for instance
a rectangle instead an ellipse.
In the future, an optimization procedure should be added in order to change the nodes position to
maximize elements global quality. Additionally, curvature flow should be added to receding front algorithm
[5], and other shapes should be tried to describe inner boundary after medial axis definition. In the list of
future works is also the development of an algorithm to mesh volumes with more than one hole inside.
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Luís Miguel Rodrigues Reis
In conclusion, the implementation of these mesh generation methods is important to have better
quadrilateral and hexahedral meshes, and, consequently to have better finite element analysis.
REFERENCES
[1] Ruiz-Gironés E., Automatic Hexahedral Meshing Algorithms: From Structured to Unstructured Meshes,
PhD thesis, Universitat Politècnica de Catalunya, 2011.
[2] Ruiz-Gironés E., Roca X. and Sarrate J., “The receding front method applied to hexahedral mesh
generation of exterior domains”, Engineering with Computers, 28, 391-408, 2012.
[3] Ruiz-Gironés E., Roca X. and Sarrate J., “Unconstrained plastering—hexahedral mesh generation via
advancing-front geometry decomposition”, International Journal for Numerical Methods in Engineering,
81, 135-171, 2010.
[4] Dey T.K., Curve and Surface Reconstruction, Cambridge University Press, 2007.
[5] Sethian J.A., “Curvature flow and entropy conditions applied to grid generation”, Journal of
Computational Physics, 115(2), 440-454, 1994.
[6] Roca X., Paving the Path Towards Automatic Hexahedral Mesh Generation, PhD thesis, Universitat
Politècnica de Catalunya, 2009.
[7] Knupp P.M., “Algebraic mesh quality metrics”, SIAM Journal on Scientific Computing, 23(1), 193-218,
2001.
[8] Sethian J.A., Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999.
[9] Ruiz-Gironés E., Roca X. and Sarrate J., “Optimizing mesh distortion by hierarchical iteration relocation
of the nodes on the CAD entities”, Procedia Engineering, (available online), 2014.
[10] Gordon W.J. and Thiel L.C., “Transfinite mappings and their application to grid generation”, Numerical
Grid Generation, 171-192, 1982.
[11] Leng J., Xu G., Zhang Y. and Qian J., “Quality improvement of segmented hexahedral meshes using
geometric flows”, Image-Based Geometric Modeling and Mesh Generation, Zhang Y. (eds.), Springer,
195-222, 2013.
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