finite element mesh generation for subsurface simulation

ORIGINAL ARTICLE

Finite element mesh generation for subsurface simulation models

Antonio Carlos de Oliveira Miranda • William Wagner Matos Lira •

Ricardo Cavalcanti Marques • Andre Maues Brabo Pereira •

Joaquim B. Cavalcante-Neto • Luiz Fernando Martha

Received: 14 March 2013 / Accepted: 7 January 2014 / Published online: 19 January 2014

� Springer-Verlag London 2014

Abstract This paper introduces a methodology for cre-

ating geometrically consistent subsurface simulation mod-

els, and subsequently tetrahedral finite element (FE)

meshes, from geometric entities generated in gOcad

software. Subsurface simulation models have an intrinsic

heterogeneous characteristic due to the different geome-

chanics properties of each geological layer. This type of

modeling should represent geometry of natural objects,

such as geological horizons and faults, which have faceted

representations. In addition, in subsurface simulation

modeling, lower-dimension degenerated parts, such as

dangling surfaces, should be represented. These require-

ments pose complex modeling problems, which, in general,

are not treated by a generic geometric modeler. Therefore,

this paper describes four important modeling capabilities

that are implemented in a subsurface simulation modeler:

surface re-triangulation, surface intersection, automatic

volume recognition, and tetrahedral mesh generation.

Surface re-triangulation is used for regenerating the

underlying geometric support of surfaces imported from

gOcad and of surface patches resulting from intersection.

The same re-triangulation algorithm is used for generating

FE surface meshes. The proposed modeling methodology

combines, with some adaptation, meshing algorithms pre-

viously published by the authors. Two novel techniques are

presented, the first for surface intersection and the second

for automatic volume recognition. The main contribution

of the present work is the integration of such techniques

through a methodology for the solution of mesh generation

problems in subsurface simulation modeling. An example

illustrates the capabilities of the proposed methodology.

Shape quality of generated triangular surface and tetrahe-

dral meshes, as well as the efficiency of the 3D mesh

generator, is demonstrated by means of this example.

Keywords Geometric modeling � Subsurface modeling �Mesh generation � Surface intersection � Finite elements

A. C. de Oliveira Miranda (&)

Department of Civil and Environmental Engineering, University

of Brasılia, SG-12 Building, Darcy Ribeiro Campus, Brasilia,

DF 70910-900, Brazil

e-mail: [email protected]

W. W. M. Lira

Laboratory of Scientific Computing and Visualization-LCCV,

Technology Center, Federal University of Alagoas, Maceio,

AL 57072-970, Brazil


R. C. Marques � L. F. Martha

Department of Civil Engineering and Tecgraf - Technical-

Scientific Software Development Institute, Pontifical Catholic

University of Rio de Janeiro, Rua Marques de Sao Vicente 225,

Rio de Janeiro 22451-900, Brazil


L. F. Martha


A. M. B. Pereira

Engineering School, Fluminense Federal University, Rua Passo

da Patria 156, Niteroi 24210-240, Brazil


J. B. Cavalcante-Neto

Department of Computing, Federal University of Ceara, Campus

do Pici, Bloco 910, Fortaleza 60455-760, Brazil


123

Engineering with Computers (2015) 31:305–324

DOI 10.1007/s00366-014-0352-3

1 Introduction

Ultra-deepwater rock drilling for oil exploration has been

demanding the research and development of new technol-

ogies, including new approaches to computational three-

dimensional (3D) subsurface simulation modeling. This

work describes a methodology for creating finite element

(FE) models of heterogeneous objects, including degener-

ated parts. Examples of these types of objects occur in

subsurface models of hydrocarbonate reservoir simulation,

in which the horizon (limiting) surfaces of the several

geological layers are considered, as well as fault (discon-

tinuity) surfaces and drilled well curves. These models

have an intrinsic heterogeneous characteristic due to the

different geomechanics properties of each geological layer.

In addition, this type of model considers lower-dimension

degenerated parts, such as a fault surface dangling inside a

geological layer or a well bore represented by a curve.

One of the main challenges in this type of modeling is

that it should represent natural geometric entities, such as

geological horizons and faults. Usually, the geometry of

man-made entities is described in a model through math-

ematical parametric representation of curves and surfaces.

On the other hand, the geometry of nature entities has a

faceted representation, such as polygonal curves or poly-

hedral surfaces.

A 3D subsurface simulation model needs a consistent

topological representation composed by vertices, curves,

surfaces, and regions, as well as associated geometric

information (coordinates of the vertices and mathematical

representation of the curves). Differently from classical

CAD/CAE, a surface in this modeling environment has a

discrete faceted representation: it is composed by a set of

triangles. In addition, the geometric model may contain

several regions. In this environment, the attributes of the

simulation, such as material properties, loads, and dis-

placement restrictions, are associated to geometric entities.

In this framework, mesh entities of a FE model (nodes and

elements) automatically receive the attributes of their

corresponding geometric entities. With this approach, it is

possible to generate new FE meshes without losing

attributes.

It is important to make a distinction between an

underlying faceted geometric representation of a surface

and a surface represented by a finite element mesh. In the

present context, geometric surfaces have discrete faceted

representations. On top of a geometric surface, a FE mesh

may be generated with an arbitrary refinement strategy

usually geared for numerical analysis purposes. Since FE

meshes of adjacent surfaces have to match, the underlying

geometric faceted representations of these surfaces must be

consistent, at least geometrically. In other words, to gen-

erate consistent surface and volume finite element models,

the underlying surface faceted representations have to

match geometrically, but not necessarily topologically. On

the other hand, FE meshes of adjacent surfaces have to

match perfectly, node by node, at the intersection curves.

These FE modeling constraints impose significant topo-

logical and geometrical requirements on a subsurface

simulation modeler for FE numerical analysis.

Subsurface geological modelers, such as gOcad [1],

usually do not achieve these minimum requirements for FE

simulation modeling. In the gOcad modeling environment

proposed by Mallet [2], natural objects (or geological

entities) also have a discrete faceted representation, in

which any object is represented by a graph similar to sur-

face and volumetric FE meshes. However, this discrete

representation is used solely for geometric representation

and is not appropriate for numerical analyses such as in FE

simulations. Considered a geological modeler, gOcad

focuses on the following areas:

1. Structural geology, such as in balanced unfolding,

discrete fracture networks, and fault characterization

2. 3D subsurface modeling, such as for constructing

geological surfaces, modeling geological structures,

and modeling geological volumes

3. Reservoir engineering, such as in petrophysical

modeling, reservoir uncertainty, discrete fracture net-

works, and streamline simulation

4. Geophysics, such as in seismic interpretation, velocity

model building, ray tracing through various media

representations, and visualization of 3D seismic attri-

butes and

5. Sedimentary geology, such as in applications for

sequence stratigraphy and channel modeling.

Because gOcad was not originally designed for FE

simulations, the proposed modeling and mesh generation

methodology intends to create FE models using gOcad data

files as input. The types of FE analysis envisioned are stress

analysis due to gravity loads of geological layers, pressure

in oil reservoirs, and stability analysis during oil well

drilling, among others.

Some works in the literature address the problem of

generating subsurface simulation models. However, they

do not describe in detail their modeling strategy. Gemmer

et al. [3] present a 3D finite element model that includes

thermomechanical processes in the lithosphere as well as

sedimentation, erosion and eustasy, but without providing

much detail about the modeling process. Wu and Xu [4]

use a framework [5] to model and visualize 3D geological

faults, where a mixed-mesh model provides the spatial

integration between the regular mesh and the irregular

Delaunay triangulation. Martelet et al. [6] model 3D

geometries with surfaces of a segment of the Hercynian

suture zone in the Champtoceaux area of Europe, but they

306 Engineering with Computers (2015) 31:305–324

123

do not describe the 3D model. Lixin [7] classifies 3D

(including pseudo-3D) models for geosciences as three

classes: facial models, volumetric models, and mixed

models. The author also presents a framework for 3D

geoscience modeling and spatial analysis. Wu et al. [8]

propose an architecture to hold geological information

attached to geometric objects (points, lines, polygons,

faces, surfaces, blocks), but without topological entities.

Qu et al. [9] present a method for three-dimensional

computerized modeling of geological objects from sets of

intersected cross sections. Butscher and Huggenberger [10]

present a 3D geological model of a hydrological model

based on a set of subsurface surfaces. Kaufmann and

Martin [11] propose a methodology developed to process

geological information that includes gOcad software,

ArcGIS (Geographic Information System) software, and a

database. Zanchi et al. [12] present an approach that is

similar to the previous work. Ming et al. [13] summarize

related works and some software used for geological

modeling; in addition, they propose a geological modeling

system called GSIS (Geological Spatial Information Sys-

tem). Xu et al. [14] propose a methodology to build com-

plex geological models for rock engineering projects that

require numerical stress analysis using tetrahedral elements

and considering mesh refinement and optimization. In

summary, most of the cited works resort to mixed model-

ing, i.e., they combine different types of representations.

However, there are few details regarding topological

description of the published models. The present work also

uses mixed modeling, as well as a topological representa-

tion of each geometric entity. Explicit topological repre-

sentation is essential to allow efficient regeneration of

curve and surface geometric descriptions, simplifying the

modeling process.

The aim of this paper is to present a methodology that

involves the integration of various technologies for sub-

surface simulation modeling. The methodology consists of

several steps of geometric modeling and mesh generation,

as shown in Fig. 1. Initially, surfaces from gOcad, com-

posed of arbitrary triangulations, are imported. Each set of

connected triangles is interpreted as the underlying geo-

metric support of a topological surface. This way, curves

are automatically created from the input surface bound-

aries. In addition, to improve shape quality of the imported

triangulations, an optional re-triangulation can be per-

formed using a surface mesh generation algorithm [15].

Although gOcad provides surface intersection tools, the

proposed methodology uses a more robust intersection

algorithm, which is achieved with exact arithmetic tech-

nology. In addition, new surfaces may be generated using

existing curves and curves created by surface intersections.

A subsequent step is the generation of triangular FE

meshes on the model surfaces. The creation of consistent

FE volume mesh for analysis requires closed volumes.

Therefore, an algorithm to automatically detect all volu-

metric regions in the subsurface model is also required.

Finally, tetrahedral finite elements are generated in each

region using a procedure that considers the constraints

imposed by internal geologic faults [16]. The proposed

methodology is implemented in a geometric modeler called

MTool3D (Mesh Tool 3D).

The proposed modeling methodology combines, with

some adaptation, several algorithms previously published

by the authors. The main contribution of the present work

is the integration of such techniques through a methodol-

ogy that tries to solve the mesh generation problem, that is,

to construct valid and good quality FE triangular and tet-

rahedral meshes for subsurface simulations. One of the

most important aspects of this methodology is that it may

handle arbitrary geometries of subsurface models. This is

achieved because arbitrary triangulations are used as geo-

metric support of surfaces (underlying faceted geometric

representation) and because these triangulations are

regenerated in an efficient and robust way. In the present

methodology, surface re-triangulation is used to fix input

surface data and to create the geometric support of new

surface patches resulting from surface intersections. It

should be emphasized that this methodology is appropriate

for the specific objective of modeling subsurface objects, in

which a faceted and non-precise geometric representation

is adequate. Of course, re-triangulating an existing trian-

gulated surface with no knowledge on the original surface

analytical description will introduce some geometric

approximations. However, in the specific case of subsur-

face modeling this works well because the existing trian-

gulation is the only available geometric information for a

Fig. 1 Proposed methodology for subsurface simulation modeling

Engineering with Computers (2015) 31:305–324 307

123

surface, which is in general not precise, and the algorithm

approximately preserves areas with high curvatures. In

addition, as reported in previous work, special care is taken

in the current implementation to obtain good performance

in re-triangulation time. Another important aspect of the

proposed methodology is that a triangular FE surface mesh

is independent of the triangulation that constitutes the

geometric support of a surface. Naturally, the FE surface

mesh is generated on top of the surface geometric support,

but not necessarily matching FE nodes with vertices of the

underlying triangulation. As mentioned, the underlying

surface faceted representation is used to represent geome-

try and the FE surface mesh is used to generate consistent

tetrahedral FE models. The same surface discretization

algorithm is used to generate surface geometric support

triangulations and triangular FE surface meshes. Shape

quality of generated triangles is demonstrated in this paper

for subsurface simulation models (Sect. 5). This double

triangulation of surfaces is one of the key features of the

proposed subsurface modeling, which turned out to be very

versatile.

The paper is organized as follows. Section 2 presents the

techniques used to import geological surfaces represented

by triangular meshes and the tools used to model surfaces

in the proposed methodology. These tools are used to treat

surface input data, to generate both surface geometric

support and FE surface mesh, and to create support trian-

gulation of new surface patches resulting from surface

intersections. In addition, this section presents the

description of the algorithm used to intersect surfaces.

Section 3 describes a technique used to automatically

detect volumetric regions in the model, whereas Sect. 4

presents the algorithm used to generate tetrahedral ele-

ments in each region. Finally, Sect. 5 provides examples to

validate the proposed methodology, and Sect. 6 draws

concluding remarks.

2 Surface modeling tools

In the proposed modeling methodology, each surface has a

double triangulation, one for the underlying surface faceted

representation (geometric support) and other for FE surface

mesh. Therefore, a triangulation algorithm may be used to

generate both the geometric support and the FE surface

mesh of a surface. Essentially, this is a generic algorithm

for generating a triangulation on a surface, which has been

published previously by the authors [15]. Here, it will be

generically referred to as the ‘‘surface mesh generation’’

algorithm. Although shape quality of the triangles gener-

ated by this algorithm has been proved in the previous

work, some shape quality measurements are shown here to

demonstrate that this characteristic is maintained in the

current application.

The surface mesh generation algorithm is used in three

different operations: (1) importing arbitrary triangulations

from gOcad; (2) remeshing an existing surface; and (3)

intersecting surfaces. These three modeling operations are

described in the following subsections. In the first and third

operations, the algorithm is used to regenerate the under-

lying surface geometric support triangulation. In the sec-

ond, it is used to generate a FE mesh on a surface that has a

generic underlying geometric representation (in the present

case, this is a faceted triangular representation).

2.1 Importing arbitrary triangulations

In the present context, an arbitrary triangulation is com-

posed by a list of vertices, defined by their coordinates, and

a list of triangles that are defined by vertex incidences.

gOcad adopts a file format called Tsurf for surfaces defined

by a set of triangles, as shown in Fig. 2. In this example,

the key VRTX is used to define four vertices and the key

TRGL to define two triangles that represent the surface.

As can be observed, the Tsurf format is very similar to a

generic FE mesh representation of a surface. This generic

representation does not contain complete information about

the topology of the mesh. This poor information makes it

very difficult to identify topological features of the surface,

such as the list of vertices that are on the boundary of the

surface patches or the list of triangles of disconnected tri-

angular patches. In addition, some vertices may be dis-

connected in space, i.e., they may not be referenced by any

triangle. Moreover, triangulated surfaces generated by

gOcad may contain some errors, such as inverted triangles,

overlapping of triangle, etc. Unfortunately, these errors

exist and usually are related to lack of user knowledge on

the modeling tools of gOcad. In spite of these problems,

these surface representations are sufficient to the types of

applications addressed by gOcad. The quality of the tri-

angulations in gOcad may be improved, but they are still

Fig. 2 Example of a ‘‘Tsurf’’ surface adopted by gOcad


123

not appropriate to FE numeral analysis. Therefore, to

handle adequately arbitrary triangulations imported from

gOcad, first, several verifications to filter any triangulation

inconsistencies are performed. Then, curves and vertices

on the boundaries of the surface patches are automatically

created to set the complete topological information of the

model. Finally, the geometric representations of surfaces

are redefined, using the algorithm described in the next

subsection.

A topological data structure is used here to identify

input triangulation inconsistencies (hanging and isolated

vertices, inverted triangles, overlapping, etc.). Basically,

this data structure should perform the following procedures

in relation to an imported triangulation: (a) identify the

triangular faces adjacent to a given edge; (b) identify the

edges adjacent to a triangular face following a given ori-

entation; (c) identify the next/previous edge from an edge

based on a triangular face; (d) identify edges adjacent to a

vertex; and (e) identify external and internal loops on a

triangulation. Such procedures may be performed by a

complete topological data structure based on edges. In the

literature, the most consolidated edge-based data structures

are the winged-edge [17], the half-edge [18], the radial-

edge [19], and the DCEL [20]. In MTool3D, a simple

topological data structure, similar to DCEL, was developed

with the capability of also representing and identifying

internal loops, disconnected and dangling parts, and other

non-manifold situations, such as more than two triangles

framing into an edge. This way, it is possible to identify

several triangular surface patches that result from a trian-

gulation imported from gOcad.

2.2 Surface mesh generation

In general, triangles on arbitrary triangular meshes

imported from gOcad are not adequate to be used as input

for tetrahedral mesh generation, because in the FE method

the quality of the elements is an important factor as it

directly affects the results. Since the size of elements must

conform to the ones required by a FE analysis, a triangular

FE surface mesh is generated, and often regenerated, on top

of the underlying surface geometric support, as described

below.

This work adopts an algorithm based on an advancing

front strategy [21–26] for generating unstructured trian-

gulations for arbitrarily shaped three-dimensional surfaces

[15]. These generic surfaces can be a triangulated mesh, a

set of points, or an analytical surface such as NURBS. To

be generic, as shown in Fig. 3, the algorithm requires three

functions as input: (1) given the position of a point, the

function returns the desired size of the element; (2) given

the current edge in the boundary contraction algorithm, the

function returns the ideal opposite point that forms a

triangle; and (3) given a point outside the generic surface,

the function returns the position of the closest point on the

surface.

Boundary information is given by a list of nodes (ver-

tices defined by their 3D coordinates and their normal

vectors on the surface), and a list of boundary segments (or

edges) defined by their node connectivity (Fig. 3, top left).

The triangulation algorithm requires this input boundary

information. Therefore, the definition of the boundary

nodes is not a role of the algorithm and, for better results,

the boundary segment sizes should be consistent with local

surface curvatures (i.e., the boundary should be relatively

more refined in areas of relatively high curvature). When

the algorithm is used to generate the underlying support

triangulation of a surface, the discretization of a boundary

curve is generated automatically, even for the curves

resulting from surface intersections. This is a procedure

analogous to the surface re-triangulation that takes into

account local curvature information of the boundary curve.

This procedure is simpler than the surface re-triangulation

and is described in details elsewhere [27]. On the other

hand, for FE surface mesh generation, a surface boundary

curve discretization is specified by the user, as explained in

Sect. 5. In the latter case, the user may specify a curve

subdivision that also takes into account curvatures.

The given boundary edges form the initial front that

advances as the algorithm progresses. At each step of this

meshing procedure, a new triangle is generated for each

base edge of the front. The front advances replacing the

base edge with new triangle edges. Consequently, the

domain area is contracted, possibly into several subareas.

The process stops when all contracted areas result in single

triangles. As the front advances, the generic methods are

called to obtain the size of a new element and the position

of an ideal node (Fig. 3, right). After this process, the mesh

is improved using Laplacian smoothing methods [28]. In

general, the surface mesh smoothing moves nodes to a

position off of the surface. The 3rd input function is

employed after each smoothing procedure as a ‘‘pull back’’

operation, moving the target node back to the geometric

supporting surface.

In this algorithm, the advancing front process is divided

into two phases to ensure the generation of valid triangu-

lations. In the first phase, a geometry-based element gen-

eration is pursued to obtain elements of optimal shapes.

After this phase is exhausted and no more optimal elements

can be generated, a topology-based element generation

takes place, creating valid, but not necessarily well shaped,

elements in the remaining area.

To provide the three input functions of the algorithm, a

background octree [29] and an R-tree structure are used.

The background octree has two main objectives. The first is

to develop local policies used to define the discretization of


123

the surface mesh. The second is to define the sizes of the

triangular elements to be generated during the advancing

front procedure, which is essentially a mechanism to

implement the first generic function required as input to the

surface re-triangulation algorithm. R-trees [30] are tree

data structures similar to B-trees [31], but they are used for

spatial access methods, i.e., for indexing multi-dimensional

information such as 3D coordinates The R-tree structure is

used here to ensure that the algorithm responds fast and to

identify high curvatures in the original surface. These

locations require a smooth element transition.

The creation of the background octree from an input

surface follows five steps: (1) octree initialization based on

given boundary edges; (2) refinement to force maximum

cell size; (3) refinement to provide minimum size disparity

for adjacent cells; (4) refinement to account for surface

curvatures; and (5) refinement to provide minimum size

disparity for adjacent cells (Step 3 is repeated for the

updated octree). The authors have demonstrated [15] that

the background octree tends to provide better control over

the quality of the generated mesh and to decrease the

number of heuristic, clean-up procedures.

The second generic method returns the ideal apex point

that forms a candidate equilateral triangle for a given

current base edge in the boundary contraction algorithm

(Fig. 4a). In addition to the base edge, the height of the

candidate triangle and a ‘‘surface intersection’’ direction

are given, as shown in Fig. 4b.

Two different approaches to this task are described in

the literature. Lohner [32] tries to find a host triangle,

where the ideal node lies, using a neighbor-to-neighbor

search. If this fails, octrees are employed. Finally, if this

approach fails again, a brute force search over all the

surface elements is performed. Schreiner et al. [33] use a

procedure that guarantees that the new triangle sides have

exactly the desired length. This procedure considers the

equation of a sphere, and finds the intersections between

the sphere and the mesh. The approach used in this work is

Fig. 3 Process overview of the

surface mesh algorithm


123

similar to this one, employing the circle equation in a 2D

coordinate system. Fig. 5.

Figure 4 illustrates how to obtain the target apex point in

the present implementation. A local search based on the

auxiliary R-tree structure creates a list of all neighborhood

elements around the middle base edge point. A plane of

intersection, containing the given surface intersection

direction, is created at this point (Fig. 4b). The intersection

of the neighbor triangles with the plane of intersection

forms a polyline. A 2D local coordinate system is created

on the plane of intersection and points of the intercepting

polyline are transformed to this coordinate system. The

ideal node position is computed by intersecting a circle,

whose center is the base edge middle point and radius is the

height of the candidate triangle, with this polyline. Finally,

the point is transformed to the 3D coordinate system.

The third generic method returns the point on the geo-

metric supporting surface that is closest to a given point

near to, but not necessarily on, the surface. During the nodal

smoothing phase, points may be moved off the surface. This

method is used to move the points back to the surface. As

input, the method receives the current node location and its

current normal vector. The implementation of this function

is quite simple. First, a local search based on the auxiliary

R-tree structure creates a list of all neighborhood triangles

of the geometric supporting surface around the given node.

Second, the node is projected on each neighbor triangle

plane, using the given projection direction. Finally, the

selected new node location is the one that lies inside one of

the neighbor triangles. Nonetheless, the node is only moved

to the new position if all neighborhood triangles are valid

after the smoothing operation. This naturally avoids the

generation of inconsistencies such as inverted triangles,

overlapping, mesh folder, etc. Our experience is that, when

surface and boundary curve curvatures are considered in the

triangulation, the node pull-back operation, at least in all the

created models, always generates valid triangles. Usually

the nodal smoothing phase is repeated five times.

The shape quality and time performance of the surface

meshing algorithm have been presented previously [15].

However, Sect. 5 demonstrates the quality of the surface

mesh algorithm in the current application by showing

shape metrics of generated triangles.

Fig. 4 Obtaining ideal node position in second generic method (top view on the left and lateral view on the right)

Fig. 5 Example of remesh


123

2.3 Surface intersection

Surface–Surface intersections were pioneered by Lo [34].

His method seems to be quite accurate and efficient for

manufactured objects, which are initially built from ana-

lytical equations and then transformed into a discrete rep-

resentation. The meshes generated in this process are

extremely regular and smooth which is not the case of

natural objects. Therefore, in geological models, a robust

method for intersecting triangular surfaces is still a

challenge.

In geological models, the spatial coordinates of mesh ver-

tices usually present high orders of magnitude, in contrast with

the (relatively) tiny size of triangles, especially those near

reservoirs. Typically, horizons surfaces have hundreds of

thousands of kilometers of extension, while reservoirs thick-

nesses are around a few dozens of meters. Since a drill could

be stuck in an oil well by a few centimeters, the higher

acceptable precision is in millimeters. The IEEE double

floating-point numbers have precisions of about 16 decimal

digits only, which may lead to some numerical errors in our

model. This numerical imprecision may not only calculate

intersection points erroneously, but, the worst, incorrectly

discard intersection points. As a result, some intersection

points may be misplaced in the final intersection curve(s) and

others may be missing, which causes modeling inconsisten-

cies. To avoid numerical problems, Lo [34] suggests a mesh

refinement in the most critical areas of the surface.

In the latter years, the search for robustness on surfaces

intersection has focused on the problem of intersecting

segments with triangles. Segura and Feito [35] developed a

fast algorithm for testing segment-triangles intersections

which uses signed volumes. However, the intersection

point is calculated using a classical segment-plane inter-

section algorithm. Jimenez et al. [36], using barycentric

coordinates and signed volumes, built an algorithm that

calculates segment-triangle intersection points even faster

than Segura and Feito’s [35]. A curious fact is that Jimenez

et al. [36], in search of robustness, minimized the number

of arithmetic operations to be performed when calculating

the determinants. Besides, they avoided divisions, which

are a relatively slow and imprecise operation, and used a

tolerance range in their comparisons.

To gain robustness, exact arithmetic [37, 38] could be

introduced in those algorithms. However, to fully benefit

from exact arithmetic, a new surface intersection algo-

rithm, based on a robust segment-triangle intersection

method, was developed in this work. Our segment-triangle

algorithm consists primarily of intersecting a segment with

a triangle plane. Although all the algorithms mentioned

before use signed volumes to identify if there is intersec-

tion or not, none of them reuse this information to define

intersection points, what, certainly, would improve the

performance of these algorithms. A signed volume may be

accurately calculated using Shewchuk’s predicate orient3d

[37, 38]. If the points A, B, and C define a plane on 3D

space and D is a point above or below this plane, orient3d

will return the signed volume of the parallelepiped defined

by the points A, B, C, and D. If point D is perfectly

coplanar to A, B, and C, then orient3d will return exactly

zero. The predicate orient3d is defined as:

orient3dðA;B;C;DÞ ¼

Ax Ay Az 1

Bx By Bz 1

Cx Cy Cz 1

Dx Dy Dz 1

��

��

¼Ax � Dx Ay � Dy Az � Dz

Bx � Dx By � Dy Bz � Dz

Cx � Dx Cy � Dy Cz � Dz

��

��

¼ ðA� DÞ � ðB� DÞ � ðC � DÞ

ð1Þ

In our algorithm, there will be no intersection only if the

signed volumes concerning each segment endpoints are non-

zero and they have the same sign. After being sure that the

segment intersects the triangle plane, we classify the inter-

section case by checking again the signed volumes. If the

endpoint of a segment is the intersection point, then, its

signed volume must be zero. This must happen to none, one

or to both endpoints of the segment. And, once we are using

exact arithmetic, no tolerance is needed. This modification

increases a lot the reliability of our algorithm, since prob-

lems such as parametric values very close to 0 or to 1 will

not occur in the parametric equation of the segment:

P ¼ Rþ t � S� Rð Þ; 0 � t� 1; ð2Þ

where R and S are the segment endpoints, P is a point in

this segment and t is the parametric value associate to point

P. Typical problems related to tolerance tests when using

parametric values are:

• Very small negative values: should it be zero? Or this

intersection point is to be discarded?

• Very small positive values: should it be zero? Or this

intersection point is really not a segment endpoint?

• Slightly greater than 1 value: should it be one? Or this

intersection point is to be discarded?

• Slightly smaller than 1 value: should it be one? Or this

intersection point is really not a segment endpoint?

If there is a unique intersection point between both

segment endpoints, the signed volumes are both non-zero

and have different sign. There is a simple way to calculate

the parametric value of Eq. (2) reusing the signed volumes:

t ¼ orient3dðA;B;C;RÞorient3dðA;B;C; SÞ � orient3dðA;B;C;RÞ ð3Þ


123

Next, all we have to do is make sure that the intersection

point lies inside the triangle borders. There are three

delimiting planes for each triangle intersected, as shown in

Fig. 6. If p is the plane defined by the three vertices of a

triangle, we can define the points.

np ¼ C � Að Þ � B� Að ÞA0 ¼ Aþ np

B0 ¼ Bþ np

ð4Þ

Then, the delimiting planes q, r and s will be defined by

the points.

q :R0 ¼ B

R1 ¼ B0

R2 ¼ C

8

<

:; r :

S0 ¼ C

S1 ¼ A0

S2 ¼ A

8

<

:; s :

T0 ¼ A

T1 ¼ A0

T2 ¼ B

8

<

:ð5Þ

These points are given counterclockwise, so any point D

inside these delimiting planes will satisfy.

orient3dðR0;R1;R2;DÞ� 0

orient3dðS0; S1; S2;DÞ� 0

orient3dðT0; T1; T2;DÞ� 0

8

><

>:

ð6Þ

where orient3d is defined by Eq. (1). Our segment-triangle

intersection algorithm is presented in Fig. 7.

It is assumed that there are no coplanar triangles in

intersection. This situation is easily verified using the

geometric predicates with the exact arithmetic technology.

However, since this situation is undesired for the tetrahe-

dral mesh generation, coplanar triangles in intersection are

Fig. 6 A triangle’s delimiting planes q, r, and s

Fig. 7 Segment-triangle

intersection algorithm


123

avoided. In our modeler, when this is detected, surface

intersection is aborted, prompting the user to treat the

original surfaces in gOcad.

An important issue of surface intersection methods is

how to establish the correct sequence of segment-triangle

intersection points in the final intersection curve(s). Using a

topological data structure (for instance, a half-edge [18]), it

is possible to walk on the mesh by adjacencies. So, if we

walk on the mesh in a smart way, always intersecting the

current triangle with the current edge, we will generate

intersection curve(s) with its points already sorted. This

‘‘smart walk on the mesh’’ will be our search for inter-

secting triangle-edge pairs (or, in short, our search for

intersections), which is very similar to a depth-first search

in a graph.

Our search for intersections is based on two facts:

• Whenever a triangle edge intersects another triangle,

the edges adjacent to this edge may also intersect this

triangle.

• The triangles adjacent to the intersected triangles are

candidates for intersection.

Therefore, from a pair of intersecting triangles, we can

start searching recursively for triangle-edge intersections,

as shown in Fig. 8. Since our Segment_Triangle_Intersec-

tion algorithm automatically stores intersection points, in

case there is any, our Search_Intersections algorithm

builds the intersection curve at the same time as it walks

through the intersecting pair of triangles. This search, when

triggered from a pair of triangle in intersection, will stop

only when all intersection points were included in their

respective intersection curve. If the search for intersection

is triggered by a pair of triangles without intersection, these

triangles will be marked as visited and no curve will be

generated.

Since it is not desirable to call our search for intersec-

tions for every pair of triangles, we, first, create a list of

triangle pairs probably in intersection, using an R-tree. An

R-tree [30] structure is also used here to improve the

efficiency of searches for intersecting triangles. In this

algorithm, a 3D bounding box enclosing a triangle is used

for indexing the auxiliary R-tree.

For surface–surface intersections, a good heuristic is to

start searching for intersections from all pairs of intersecting

boundary triangles and, then, from all pairs of intersecting

internal triangles. Doing this, we generate, firstly, all

intersection curves that start and end at the boundary of a

surface. If a surface contains holes, the borders of the holes

are considered as boundaries of the surface and holes are

naturally handled properly. Then, we generate all intersec-

tion curves starting from pairs of intersecting internal tri-

angles, using the same Search_Intersections method

previously described. In this case, the intersection

curve(s) must circumvent two colliding peaks in different

surfaces, as shown in Fig. 9, and the intersection curve

becomes a closed polyline. To close this polyline, the first

intersection point must be revisited once more and included

as the last intersection point. Every time a search for

intersections is started from a different pair of triangles, new

curves (polylines) of intersected surfaces are obtained.

When all pairs of triangles in our list were visited, the

process of creating intersection curves ends up, generating

Fig. 8 Search intersections

algorithm


123

all possible intersection curves between the two considered

surfaces. These intersection curves are still polylines con-

taining all calculated intersection points. After that, the

intersection curves are automatically redefined, so that

sections with higher curvatures will be more refined, as

described by the authors in another work [27]. Then, the

borders of the original surfaces are divided according to the

new intersection curves, creating new boundary curves for

the new surfaces. Note that, so far, the original support

triangulations were not modified; only the borders of the

surfaces were redefined. Finally, using the surface reme-

shing algorithm described in the previous section, new

geometric support triangulations are generated on these

surfaces patches based on the original underlying surface

triangulation.

The algorithm for defining the intersection curves is

O(n log n), with n being the total number of triangles on

both meshes. The cost of building the R-tree is O(n log n).

After that, an edge (in any mesh) will be visited once, twice

or it will never be visited. Most of the edges will not be

visited at all and a very few (if any) will be visited twice.

So, in the worst case, there will be O(m) checks of edge-

triangle intersections, where m is the total number of edges.

In practice, number of checks performed is much smaller

than m. If it is assumed that, in a triangular mesh, m = 3n/

2, then the total cost of checking all edge-triangle inter-

sections is O(n). Therefore, the computational complexity

of this algorithm is O(n log n).

3 Volume recognition

The tetrahedral mesh generation algorithm, described in

the next section, requires as input a set of connected tri-

angles, given by a list of nodes and a list of faces (FE

triangular faces), and which must form a closed volume.

This section describes an algorithm to recognize closed

volumes in a subsurface model. Since each volume is

formed by a group of surfaces composed by FE connected

triangles, this task requires three steps: (1) transforming all

selected surfaces (which can be all surfaces of the model)

into a single triangulation; (2) building a list of face-uses

for the two faces of all triangles (one face-use lies on the

positive normal side of the face, upper face-use, and the

other lies on the negative normal side of the face, lower

face-use [19]), in which each face-use points to its adjacent

face-uses; and (3) identifying closed volumes using a

linked list of face-uses.

The first step is to group all FE triangles of a set of

selected surfaces into a single triangular mesh, in which the

triangles share a single node list. A set of nodes that occupy

a single geometric location from different selected surfaces

are recognized as a single node, and all adjacent triangles

refer to this node. This operation can be performed using a

special data partitioning structure such as an R-tree or a

B-tree.

In the current implementation, an R-tree [30] structure is

also used here to improve the efficiency of node queries. In

this algorithm, a 3D bounding box enclosing a node is used

for indexing the auxiliary R-tree (the bounding box can be

determined by a percentage of the smallest edge size

among all triangles in the model, for example).

The creation of a single triangular mesh follows some

sub-steps. Initially, the R-tree structure is created with an

empty numbering of nodes and their coordinates. Then, all

triangles of all selected surface meshes are visited. Each

node of a triangle is geometrically verified to see whether it

is already in the R-tree structure, using a bounding box of

the node. If the node is not in the R-tree, it is inserted in the

structure and a new identifier is assigned to this new node.

Otherwise, the existing identifier of that node is recovered.

In both cases, the current node identifier is updated in the

Fig. 9 Intersection of two

colliding peaks in

(a) perspective view and

(b) orthogonal view


123

connectivity of the corresponding triangle. This operation

creates a single list of nodes and triangles.

In the second step of the volume recognition algorithm,

another auxiliary structure is created with the face-uses of

each triangle face, in which each face-use points to its

adjacent face-uses (Fig. 10). For each face (Fig. 10a), two

face-uses are created: one at the side of the normal vector’s

direction and the other at the opposite normal vector’s side.

The adjacencies of each face-use are obtained by ordering

the faces on a local coordinate system (x, y), where plane

x–y is perpendicular to the adjacent edge in Fig. 10b.

Going through each triangle face in order and using the

normal vectors of the faces, it is possible to define appro-

priate pointers to each triangle’s three adjacent face-uses

(Fig. 10c, d).

In the third step, volumes may be identified by walking

through all face-uses and their adjacent face-uses, as shown

in the algorithm of Fig. 11. A face-use can have two states:

(a) unmarked, when the volume recognition algorithm has

not visited the face-use, and (b) marked, when the face-use

has been visited. Initially, all face-uses are unmarked. The

first step is to look for each face-use and check whether it is

marked. For each face-use, it verifies whether it is marked.

If not, the current face-use is inserted in a list, which is

initially empty, and at the end will contain all face-uses of a

detected volume. This list is updated according to the

following steps: (1) the algorithm marks the face-use that is

inserted in the list; (2) then, it visits all face-uses adjacent

to the current one that are not marked and inserts them in

the list, recursively repeating step 1 for each of these

adjacent face-uses; (3) if there are no more face-uses to

visit in the recursive traversal by adjacency, the algorithm

returns to the original list of face-uses to detect another

volume; and (4) if all the face-uses are marked, the process

finishes. At the end, each face-use list identifies a detected

Fig. 10 Simplified process to

build face-uses for volume

recognition: (a) faces of each

triangle and their normal

vectors; (b) faces in an auxiliary

coordinate system; (c) pointers

to adjacent face-uses; and

(d) walking through face-uses

Fig. 11 Volume recognition algorithm


123

volume. There are two important remarks. Firstly, this

algorithm also identifies the boundaries of the exterior

domain of a model, which should not be considered. Sec-

ondly, internal surfaces that are completely disconnected

from other surfaces will also be discarded for volume

detection. These surfaces are considered in the tetrahedral

mesh generation as internal constraints inside a volume.

As this volume recognition strategy is just based on

adjacency information, it will not be able to identify cases in

which a domain is completely enclosed by another. How-

ever, this problem can be solved using a point location (point

in region) algorithm. Some algorithms for point location are

described by De Berg et al. [20] and Preparata and Shamos

[39]. An alternative to this could be to connect, with an

auxiliary surface, the enclosed domain to the external one,

using interactive user tools, as described in Sect. 5.

4 Tetrahedral mesh generation

The 3D mesh generation in each closed region of the model

is based on a 2D technique presented by the authors [40]

and is used to obtain tetrahedral elements in arbitrary 3D

domains [16]. Similarly to the procedure applied to gen-

erate surface meshes, this one is based on an advancing

front technique coupled with a recursive spatial decom-

position technique (octree). The algorithm was designed to

meet a default requirement (to generate meshes conforming

to existing FE triangular meshes at the boundary of a

domain) and three specific requirements: (1) to avoid

producing elements with poor aspect ratios; (2) to generate

meshes exhibiting good transitions between regions of

different element sizes; (3) to work properly for cases in

which distinct boundary nodes are geometrically coinci-

dent (e.g. nodes on opposite faces of a crack). The

advancing front technique is based on standard procedures

in the literature [21–26], with two additional steps used to

ensure the generation of a valid tetrahedral mesh for almost

any domain.

The input data for the tetrahedral mesh generation are

described by a list of nodes defining their coordinates and a

list of triangular faces defined by their nodal connectivity.

Among other advantages of this data structure, it can rep-

resent any geometry, including holes, cavities and cracks,

in a simple manner, so it can easily be incorporated to any

finite element system.

Similar to surface mesh generation, a background octree

is used to control the distribution of the node points gen-

erated in the interior. Its generation follows these steps:

1. Octree initialization based on given boundary faces:

each face of the input boundary data is used to

determine the local subdivision depth of the octree;

2. Refinement to force maximum cell size: the octree is

refined to guarantee that no cell in its interior is larger

than the largest cell at the boundary. This will avoid

excessively large elements in the domain interior,

resulting in better quality of elements [16];

3. Refinement to obtain minimum size disparity for

adjacent cells: this additional refinement enforces only

one level of tree depth between neighboring cells and

provides a natural transition between regions with

different degrees of mesh refinement.

The input to the algorithm is a triangular surface mesh,

which describes the domain to be meshed. The algorithm

steps are as follows:

• A two-pass advancing front procedure is applied to

generate elements. On the first pass, elements are

generated based on geometrical criteria, producing

well-shaped elements. The background octree is used to

control the sizes of the elements and the position of the

interior nodes. The octree determines an ideal position

for an optimal node to form a new element. This ideal

position defines a search region where the best accept-

able node location for the new element may be located.

This region is a sector of a sphere whose center is the

ideal position and whose radius is proportional to the

octree cell size. If one or more existing nodes are inside

this region, they are ranked based on a solid angle

criterion, to get the best node for the new element.

However, if no existing node is found, a new node is

inserted at the ideal position and an element is

generated using this node. In the second pass, elements

are generated based only on the criterion that they have

valid topology. Here, any existing node that forms a

valid new element can be used, regardless of whether it

is close to the ideal position or not. However, the same

quality criterion is used and the node that forms the best

solid angle is chosen for the generation of the new

element.

• If the advancing front procedure cannot progress, a

back-tracking strategy is employed to delete some

elements, and the procedure is restarted only in the

created local cavities. The back-tracking strategy is

explained in details in another work [41]. It consists

basically of back tracking a few steps in the mesh

generation and deleting faces that hinder the front from

converging. This creates better regions where valid

elements can be then generated. It is possible that the

process of finding better regions may fail, for instance,

if faces to be removed are part of the original boundary.

When this occurs, other elements are deleted instead

and the procedure is restarted. If a mesh still cannot be

generated for this region, the algorithm fails and

terminates. In principle, it is possible to create a


123

boundary input mesh that enforces the failure of the

tetrahedral mesh generation. Such failure, however, has

not yet been observed in ‘‘non-contrived’’ input, i.e., in

any realistic input boundary meshes. In our modeler,

when the algorithm fails, the tetrahedral mesh gener-

ation is aborted, prompting the user to redefine the

boundary FE mesh of that volume.

• Once a valid mesh is created, the quality of the

elements’ shapes is improved using the standard

Laplacian smoothing technique and locally deleting

poorly shaped elements and those adjacent to them.

Finally, the algorithm generates a mesh of tetrahedral

elements that considers cracks and the restrictions imposed

by the subsurface model. Figure 12 shows an example of

tetrahedral mesh generation. The various tetrahedral ele-

ments generated in many domains, with attributes resulting

from the analysis, complete the desired finite element

mesh. Then, this mesh may be examined by a FE solver.

Shape quality of generated tetrahedral elements and

time performance of presented solid mesh generation

algorithm has been published previously [16]. Shape met-

rics of generated tetrahedral elements are shown in the next

section to demonstrate the quality of this algorithm.

5 A didactic example

To validate the presented ideas and to verify the perfor-

mance of the proposed methodology, the procedures

described in this work were implemented in a geometric

modeler that was originally called MG [42]. Later, it was

extended to generate volumetric FE meshes and now it is

called MTool3D, improving the capability of the modeler

to deal with complex geometric models. Besides some

conventional CAD/CAE software tools, new ones were

implemented to simplify our specific modeling needs.

Some of these tools are:

• Creation of embedded or non-embedded curves on a

surface.

• Intersection curves and surfaces with a defined plane.

• Extension of curve to another curve, of curve to a

surface, and of surface border to a surface.

• Creation of a ‘‘black-list’’: a list to which defective

entities are sent when problems are detected. These

problems can be: entities occupying identical geometric

position, near tangency, topological incoherencies,

isolated entities, triangles with poor shape on surfaces,

volumetric regions without mesh, etc.

• Smoothing of surfaces by Laplacian or Taubin [43]

technique, which reduces high curvature variations and

does not produce shrinkage.

• Refining or coarsening the triangle geometric support

of a surface.

In relation to the ‘‘black-list’’ tool, it is well known that

computational geometry algorithms are susceptible to

failure, even considering that these algorithms have been

designed to attend robustness requirements. Based on

experience of the authors, most of these errors can be

solved by user intervention. Hence, the authors have

designed a dedicated environment to automatically sepa-

rate modeling entities involved in algorithm failure, topo-

logical inconsistencies, or geometric problems. Then,

through the modeler graphics interface, the user may

modify or delete these entities.

To illustrate the new features described in this work, this

section presents a didactic example, illustrated in Figs. 13–

15. The first step is to generate a geometric subsurface

model, which is done in gOcad. The imported Tsurf sur-

faces from gOcad as shown in Fig. 13, where two surfaces

represent faults, and the other two represent the top and

Fig. 12 Example of tetrahedral mesh generation


123

bottom limiting horizon surfaces of an oil reservoir. The

overall sizes of these surfaces are on the order of

12,500 9 10,000 m and the reservoir thickness is about

35 m. Note that, when the surfaces (sets of triangles) are

imported into MTool3D, curves and vertices on the

boundaries of the surfaces are also incorporated to the

topological model. Topological vertices are created in two

ways: (a) from a Tsurf file, when vertices are created in

gOcad and identified by the label ‘‘BORDER STONES’’,

or (b) from the corners on the boundaries of surfaces that

are automatically identified when two adjacent boundary

edges form a kink angle. Topological curves have polyline

geometric representations separated by vertices, as shown

in Fig. 13 (left). Figure 13 (right) shows details of the

triangles that compose the support surfaces (triangulations

that define the underlying geometry representation of sur-

faces). Note in Fig. 13 (right) that the triangulations of

faults and horizons do not match. The reason is that the

intersections between these support surfaces have not been

treated yet. These operations are performed in a further

step.

After importing the fault and horizon surfaces, it is

necessary to limit the domain of the subsurface model.

Figure 14 shows some details of surface intersections per-

formed to define the limiting domain of the initial model,

and of intersections between fault and horizon surfaces.

Initially, a bounding box (5,000 9 5,000 9 2,500, in

meters) is inserted in the model to limit its size (Fig. 14a).

Then, intersections between surfaces are defined using the

triangles of each surface. For this specific example, the

intersections were performed between pairs of surfaces:

first, the top reservoir surface was intersected with each

surface of bounding box; second, the same operations were

performed with the bottom reservoir surface. New triangles

on the surfaces are obtained after each intersection step, as

shown in Fig. 14b. These new triangles respect the original

geometry of the support surfaces. Note that there is a

concentration of nodes on curves created by intersection

due to the adaptive refinement of curves to consider its

curvature [27]. Figure 14c shows the model after perform-

ing intersections with the limiting bounding box and

excluding spurious parts of reservoir surfaces. Note that

new topological entities (vertices and curves) are generated

at the intersection of surfaces. In addition, new support

surfaces are also generated from surface intersections, as

shown in Fig. 14d: six new surfaces were obtained from the

original ones, and four new intersection curves were gen-

erated between surfaces. These intersection curves, which

divide the fault surfaces in Fig. 14d, are the ones that appear

on reservoir surfaces in Fig. 14e. These curves represent

internal restrictions in fault and horizon surfaces that should

be considered in surface mesh generation. As mentioned,

the geometry of an intersection curve is automatically

redefined to improve quality of generated support triangu-

lations of adjacent surface patches, i.e., edge subdivision of

an intersection curve is regenerated automatically, and this

takes into account local curvature information. After

defining the intersections between support surfaces, a con-

sistent topological model is built in terms of vertices, curves

and surfaces. The triangles generated in the previous steps

serve only to form the support surface, because the triangles

that are input data to tetrahedral mesh generation are

described in the next paragraph. The triangulations shown

in Fig. 14 correspond to the underlying geometric support

triangulation of surfaces, which present local concentration

of triangles. The refinement of geometric support of curves

(polylines) and surfaces (triangulations) is affected by local

curvatures. This is the reason for local concentration of

vertices. As explained, the support triangulation of a surface

is regenerated when it is imported from gOcad and when

there are surface intersections.

Triangular surface and tetrahedral finite elements can be

generated from the model shown in Fig. 14c. This process

is similar to many CAD/CAE systems: edges are created on

Fig. 13 Initial imported

surfaces in Example 1


123

curves by the user of Mtool3D, and then triangular ele-

ments are generated on each surface, using the algorithm

described in Sect. 2.2. The boundary edge sizes for FE

surface mesh generation are defined by giving a number of

curve subdivisions or by specifying an average size,

including a size ratio, using many tools implemented in

MTool3D. In both cases, boundary edge of a FE surface

mesh respect boundary curve geometries. The inputs to the

surface mesh generation are these boundary edges on

curves and a surface as support. The edges on curves and

the FE surface meshes are shown in Fig. 15a. The next step

identifies the set of triangles of each closed volume. This

set of triangles serves as input to the tetrahedral mesh

generation algorithm described in Sect. 4. Figure 15b

shows three volumes detected in the model. Note that

dangling fault surfaces are also part of the closed volumes.

The sequence of geometric modeling in Figs. 13–15

shows some important aspects that should be highlighted:

• The consistency between the geometric and the topo-

logical representation of vertices, curves, and surfaces

is automatically updated in all modeling steps, such as

when surfaces are imported and intersected. However,

topological volume consistency is not enforced auto-

matically. This is performed by the user in the domain

recognition process. The authors have already pub-

lished a work for creating and maintaining a spatial

subdivision [44] that could be used to check the

consistency of topological volumes. However, the

authors realized that this approach is very expensive

for subsurface models. For this reason, the volume

recognition is used here as it is proposed.

Fig. 14 Model after inserting

limiting surfaces and surface

intersections in Example 1

Fig. 15 Meshes on surfaces

after all necessary intersection

procedures in Example 1


123

• The surface mesh generator, described in Sect. 2, is

used for two purposes: to reconstruct the faceted

geometric support of a surface and to generate the

finite elements on surfaces. Geometric support recon-

struction may be forced on demand, usually to improve

the quality of the input surface’s triangulation, or

automatically, for example after surface intersection. In

any situation, the geometric re-triangulation of a

surface uses the current triangulation as geometric

support [15]. Furthermore, the algorithm identifies

locations of high curvatures to generate triangles

compatible with the surface’s geometry. Another

feature is the possibility of treating internal constraints,

such as internal boundaries associated to other curves

or surfaces. This characteristic is essential for meshing

over reservoir surfaces, as shown in Figs. 14e and 15a.

• The volume recognition process also takes into account

dangling surfaces inside volumetric regions, since these

dangling objects belong to the region’s boundary, as

shown in Fig. 15b. The concept of face-use is important

in the volume recognition process (see Sect. 3): an

internal face on a dangling surface is traversed twice,

one for each of its face-uses.

• Finally, the tetrahedral mesh generation algorithm,

described in Sect. 4, is also capable of dealing with

dangling triangles in volumetric regions.

There is one possible weak point related to the use of

surface mesh algorithm to regenerate the surface FE mesh.

The algorithm works well when the size of elements of the

new mesh is similar or larger than the size of elements of

the supporting mesh. In case a finer mesh is required, the

new generated triangles will conform geometrically to the

coarser triangles of the supporting mesh. This means that

the generated refined mesh will present a ‘‘piecewise-pla-

nar’’ geometry. However, this problem is solved by

applying the following steps: refinement of the support

triangulation, which uses Taubin smoothing; and genera-

tion of a FE mesh on top of the surface geometric support.

For our purposes, this procedure works very well.

To estimate the performance of the proposed method-

ology, Fig. 16 shows the elapsed processing times of 3D

mesh generator for the examples. Here only the volumetric

refinement step is included, because this is the step where it

takes longer to generate the final finite element model.

These processing times were measured on a Pentium Core

i7-1.73 GHz PC with 6 GB of RAM, under Windows 7

operating system. Note that the graph is in log–log scale.

Using this graph, it is possible to estimate the time required

for 1 million elements, for example, which would take

nearly 17 min. The extrapolation of the mesh generation

run time shows that, for large models, a serial version of

the algorithm might not be sufficient. In fact, a

computational complexity of O(n log n) has been reported

in the literature for advancing front techniques, n being the

number of generated elements [21, 45, 46], and it has also

been reported that a performance of O(c np log n), with

p slightly larger than one and c being a constant, is even a

more realistic measure [23]. In the present work, the

complexity of the serial algorithm is reported as O(c np log

n), where c * 0:000269–0:0000629 and p * 1.2 [16].

Therefore, to generate 1 million elements, it is better to

generate 10 times 100,000 elements in approximately

7 min, using a parallel 3D mesh generator, for example.

The authors are currently working in a parallel version of

this algorithm that will be published in the near future.

The well-known refinement and mesh improvement

algorithms from the literature achieved running times on

the same range of the proposed algorithm or worse (for

some of them). Although some of them are based on dif-

ferent approaches, such as Delaunay, and they were tested

in different models (many times more simple then the ones

presented in this work), they were included here to give an

idea of running times for models of similar sizes. Gosselin

and Ollivier-Gooch [47] developed a fast algorithm that

consisted basically of creating a mesh from a point cloud,

then refining and improving the mesh. The refinement

phase for a mesh of the ridged torus using R = 2 and

G = 10 yielded around 330,000 tetrahedra and required

16.3 s, therefore inserting slightly \4,000 vertices every

second. The same insertion rates were observed for the

other meshes presented in their work. When considering

the mesh improvement phase, however, the insertion rates

dropped considerably but even then they were able to mesh

a model made of roughly 25,000 tetrahedra connecting

random vertices placed inside a cube in 9.1 s, using their

mesh improvement algorithm. In the proposed methodol-

ogy, the generation of the same number of tetrahedra, for

more complex models, took around the same time

Fig. 16 Time performance of 3D mesh generator


123

(Fig. 16). The work of Klingner and Shewchuk [48], on the

other hand, attains its best possible results at a very high

computational cost. For the same model, Klingner and

Shewchuk’s implementation took between 1,430 and

4,658 s. Another important work by Alliez et al. [49]

produced meshes with remarkably high average quality and

reasonable number of elements (275,000 elements for one

of their models) in 4 min. Extrapolation from these timings

indicates that they can generate a mesh containing about

one million tetrahedra in around 15 min, which is the same

range of the proposed algorithm (Fig. 16).

The shape quality of generated triangle surface and

tetrahedral meshes is shown in the original and complete

descriptions of mesh generators [15, 16]. However, for the

sake of completeness, four configurations of mesh refine-

ment are generated in this work to demonstrate the quality

of the presented meshing algorithms: (a) edges on all

curves of didactic model were created with average size of

200 m and only edges on curves on reservoir were created

with average size of (b) 100, (c) 50, e (d) 25 m. Note that

using theses sizes, curves along thickness of reservoir

(about 35 m) have only one edge. The quality of the shapes

of the generated triangles on surface is measured by a

normalized ratio quality measure, c/c* [15, 50]. In this

measure c is the ratio between the root mean square of the

lengths (Si) of a triangle’s edges and the triangle’s area, and

c* is the corresponding value for an equilateral triangle:

c ¼ 1

3

X3

i¼0

S2i

,

Area ð7Þ

where the c/c* quality measure has a valid interval between

1.0 and infinity, with high quality elements, those close to

an equilateral triangle, having values close to 1.0. For

tetrahedral elements, the mean-ratio metric is adopted [51].

The mean-ratio l of a matrix S n 9 n with det (S) [0 is.

l Sð Þ ¼ ndet Sð Þ2=n.

Sk k2F ð8Þ

where Sk k2F¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tr ST Sð Þp

is the Frobenius norm. This metric

has a valid interval 1.0 C l(S) [ 0.0. Let xi be the coor-

dinates of vertex i, and let xk be the coordinates of another

vertex in the element connected to vi by an edge. Construct

the matrix A(i) whose columns are the vectors xk–xi for each

adjacent vertex vk in the element. For tetrahedral elements,

the matrix is 3 9 3. For a regular tetrahedron, the mean-

ratio is about 0.79.

Table 1 Quality of triangles on surfaces for different mesh refinements

Mesh ref. No. of nodes No. of triangle c/c* [15, 50]

Average SD Max. [1.15 (%) [1.50 (%)

(a) 4,961 7,956 1.15 0.48 6.62 17.07 5.67

(b) 10,303 17,505 1.08 0.18 3.51 13.21 5.44

(c) 30,123 54,902 1.05 0.11 4.38 10.28 1.60

(d) 102,942 195,946 1.03 0.08 9.04 5.25 0.77

Table 2 Quality of tetrahedral meshes for different mesh refinements

Mesh ref. Region No. of nodes No. of elements l [51]

Average SD Min. \0.3 (%) \0.2 (%) \0.1 (%)

(a) 1 6,350 27,958 0.65 0.16 0.07 3.34 1.10 0.15

2 1,711 5,033 0.30 0.11 0.05 54.94 18.18 1.05

3 5,785 23,169 0.62 0.18 0.05 5.92 1.94 0.22

(b) 1 13,836 63,536 0.66 0.16 0.05 2.97 1.17 0.15

2 5,925 17,722 0.46 0.14 0.07 12.94 2.90 0.27

3 11,904 51,709 0.65 0.16 0.05 3.74 1.34 0.11

(c) 1 44,238 210,604 0.67 0.15 0.04 2.33 0.81 0.10

2 22,232 67,203 0.65 0.15 0.06 1.62 0.53 0.07

3 37,022 171,491 0.66 0.16 0.04 3.04 1.08 0.13

(d) 1 170,180 836,355 0.68 0.15 0.01 2.18 0.79 0.09

2 107,816 402,661 0.70 0.13 0.05 1.52 0.68 0.11

3 151,413 739,831 0.68 0.15 0.01 2.47 0.90 0.11


123

Table 1 shows the quality of triangles on surfaces for

different mesh refinements. This table gives the number

of nodes and triangles, along with the average, standard

deviation, maximum, percentage of element [1.15 and

1.50, for the normalized ratio quality measure. A triangle

with metric 1.15 is an isosceles triangle with base 1.0

and height 0.5, for example. A triangle with metric 1.50

is an isosceles triangle with base 1.0 and height 0.33, for

example. One can see that the number of nodes and

triangles growth exponentially and the quality of trian-

gles increases with the refinement of reservoir curves.

The quality of triangle in mesh refinement (a), 17.07 %

of triangles [1.15, is due to the small thickness of

reservoir.

Table 2 shows the quality tetrahedral meshes for dif-

ferent mesh refinements. This table gives the number of

nodes and tetrahedrons, along with the average, standard

deviation, minimum, percentage of element smaller than

0.3, 0.2 and 0.1, for the mean-ratio quality in the regions.

For example, considering a regular tetrahedron with an

isosceles triangular base and height H, mean-ratio values of

0.3, 0.2 and 0.1 means this tetrahedron with height 0.16H,

0.088H, and 0.025H, respectively. Note that the quality of

elements on reservoir (Region 2) is lower when compared

to region 1 and 3, due again to the small thickness of

reservoir. However, most of cases the average mean-metric

is close to a regular tetrahedron meaning a good mesh for

FE geomechanics analysis.

6 Conclusion

This paper presented a methodology based on the inte-

gration of several advanced technologies for subsurface

simulation modeling using finite elements. The methodol-

ogy involves all stages of geometric modeling and mesh

generation. The process starts with arbitrary surfaces rep-

resented by a set of triangles, which are imported. The

triangular meshes can be improved by an advancing front

algorithm applied to arbitrary surfaces, including surfaces

with accentuated curvature. The same algorithm is used to

redefine meshes after the surface intersection process. After

this step, an automatic recognition of multi-volumes is then

performed. This step is an important procedure in the

simulation of complex subsurface models, so it must be

robust, forcing geometric and topological consistency.

Finally, volumetric meshes can be generated inside the

regions bounded by FE surface meshes. An algorithm to

generate tetrahedral elements in arbitrary domains is used,

consistently treating internal restrictions (faults, for

example).

The proposed methodology was implemented in a geo-

metric modeler, being put to use in the development and

simulation of complex subsurface problems, including

reservoir simulation and flow in fractured rocks.

A subsurface simulation model was built using several

computational geometry technologies, with some important

requirements:

• Geometric support when importing and intersecting

surfaces;

• Surface mesh generation algorithm that considers high

curvatures and internal constraints;

• Volume recognition algorithm that also identify dan-

gling surfaces;

• Tetrahedral mesh generation algorithm that also con-

siders internal constraints of dangling surfaces.

Obviously, the current modeling strategy needs

improvements. For example, there are new features under

development: parallel 3D mesh generator; refinement of

curves, surfaces and volumes using automatic adaptive

approach [27]; volume recognition strategy to identify

domain completely included into other one; hexahedral

mesh generation algorithm; optimization-based smoothing

[52], that gives better results than Laplacian smoothing, for

triangles on surface and tetrahedral elements.

Acknowledgments The authors would like to thank the National

Council for Scientific and Technological Development (CNPq),

University of Brasılia, the Technical-Scientific Software Develop-

ment Institute (Tecgraf/PUC-Rio), and Pontifical Catholic University

of Rio de Janeiro (PUC-Rio) for the financial support and for pro-

viding the necessary space and resources used during the development

of this work.

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http://dx.doi.org/10.1007/s00366-008-0119-9

http://dx.doi.org/10.1115/1.4024106

http://dx.doi.org/10.1115/1.4024106

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http://www.tecgraf.puc-rio.br/~lula/manual/mg.pdf

finite element mesh generation for subsurface simulation

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