An Unstructured Semantic Mesh Definition Suitable for Finite Element Method
Marek Gayer, Hannu Niemistö and Tommi Karhela
VTT Technical Research Centre of Finland
Vuorimiehentie 3, Espoo, P.O. Box 1000, 02044, Finland
marek.gayer@vtt.fi , hannu.niemisto@vtt.fi , tommi.karhela@vtt.fi
Abstract—Numerical methods, such as finite element method
(FEM), are widely used for computations on meshes in variety
of simulation related software applications. As these methods
become more popular in engineering, also information
management requirements for these models are growing.
Neutral data formats and standards are needed in order to
achieve vendor independence and transformation between
different tools. Semantic data modelling is one modern and
potential technique that can be used for this purpose.
In this paper, we propose our Semantic mesh data
structure that can be used in development of semantic software
applications, which use meshes. The proposed structure is
universal and can be used for representing meshes used in
wide number of popular FEM applications such as e.g. Abaqus
FEA, I-DEAS or MSC Adams. Our Semantic mesh proposal
can be also used for mesh based simulations with other
methods than FEM, such as e.g. finite volume method used in
computational fluid dynamics (CFD). Also it can be used
outside focus of numerical simulations, e.g. for projects related
to scientific data visualization and computer graphics.
We provide a description of our Semantic mesh, together
with an example of a very simple 2D mesh, illustrating our
proposal. The implementation is using the semantic database
in Simantics software platform.
Keywords-mesh format; semantic modelling; finite element
method; numerical simulation on meshes
I.
I
NTRODUCTION
A.
Finite element method and meshing
The finite element method (FEM) is currently one of
most often used techniques for numerical solutions of partial
differential equations on complex modelled domains. FEM
has become a de-facto industry standard for solving wide
variety of multi-disciplinary engineering problems. There are
many applications using this method, such as solid
mechanics, fluid mechanics, heat transfer, acoustics,
electromagnetics and computational fluid dynamics (CFD).
For numerical computation using the finite element
method on general 3D modelled objects, we need to provide
a suitable mesh representation of the continuum. This means
to find a suitable discretization of continuous domain to
simple volume cell elements (e.g. triangles or tetrahedrons).
Such discretization forms a mesh. In the cells, partial
differential equations (PDE’s) can be replaced by systems of
non-linear algebraic equations. Using the finite element
basis, discrete algebraic systems are assemblied into sparse
matrices and then solved. Computed characteristics are
determined in the nodes of elements.
Computation of general partial differential equations
using FEM is rather complex to design and implement, and
solid understanding of mathematics, numerical methods and
computer engineering is required for designing and
implementing new performance optimal solutions of real
problems based on this method. Therefore very often, instead
of making such implementations, existing software solver
libraries and commercial FEM solvers are used.
One of important advantages of FEM is availability of
software tools, pre- and post-processors as well as software
components based on this method. This allows for instance
to automatically generate unstructured meshes on arbitrary
structures, and there are some good open source mesh
generators for FEM. One of advantages of FEM is in general
lower requirements for the quality of the mesh than finite
volume method. However, for various application tasks, such
as flow problems, finite volume model is more natural and
suitable.
B.
Semantic data model
Our work is based on use of semantic graph as a
unifying data model. By defining semantics, we are adding
meaning of data objects by specifying their relations and by
annotating those using statements. Data consists of
resources, statements and values attached to some resources.
Each statement connects two resources and belongs to some
relation, so the graph can be seen as a collection of binary
relations.
Another view is to see the set of all statements as a
ternary relation of subject, predicate and object, where
predicate is the relation where the statement belongs to.
Seen this way, each statement is an atomic fact about the
model that the graph describes. Naming of the relations tries
to follow this viewpoint, so that for example the statement
(Pressure, HasUnit, Pascal) can be easily interpreted. A
special relation InstanceOf associates a resource with its
type.
Fig. 1 depicts a very simple example of semantic data
model, describing a pipe with diameter 0.142m.
Fig. 2 demonstrates another example, a semantic data
model of a simple automation model in Apros
1
process
simulation software, based on Simantic software platform
2
.
1
Apros software website:
http://www.apros.fi
2
Simantics software platform website:
https://www.simantics.org
Figure 1.
A very simple example of semantic data model
This automation model example contains an Analog
Switch, which switches between two Analog Signal inputs.
The Analog Switch is controlled by Binary Signal and
passes its output as Analog Signal. To one of input Analog
Signals is connected a Set Point, which can generate a
concrete output value.
In Simantics, each resource must have at least one type.
A resource with type Literal has a value as a data content.
Value may be number, text, array, record or tagged union of
values. In this respect, the data model is tailored to represent
typical simulation configuration and result data efficiently
compared to representing data with XML that is more
common choice with semantic graphs.
What makes the data model special is that types and
relations are also resources. This enables semantic data
modeling, where meaning of the concepts can be described
as a data and not just hardcoded to the application logic
manipulating the data. It also increases flexibility. When
integrating data from multiple sources, data schemas are
imported to the graph as easily as the data itself. Simple
uniform representation makes it also possible to build data
transformation and validation frameworks on top of the data
model.
The semantic graph is organized into models and
ontologies. Ontologies contain concepts shared by all
models. Typically the ontologies describe types and
relations and models use these concepts, but there is not so
clear separation between schema and instance as in many
other data modeling approach: ontologies may contain for
example templates, viewpoints to data, and constant value
and in some cases it is useful to define new types and
relations in models.
II.
R
ELATED WORK
We are not aware of any published work, which would
be proposing to use semantic data modelling for description
of whole mesh data representation. Nevertheless, there were
numerous efforts proposing to use semantic descriptions for
detections and segmentation of either volumetric or surface
parts of the mesh, such as e.g. for parts of human body. A
survey of such efforts can be found in [17]. In [1],
segmentations of 3D surface meshes and semantic annotation
of detected parts is presented. Another related work is a
proposal of simplification of meshes, based on semantic
mesh parts gained by face detector [3].
6
08/06/2011
Example
–
Ontology based simulation model
configuration in APROS 6
Figure 2.
A semantic data model of a simple automation model in Apros software
There are many different ontology definition languages.
For semantic graphs, the best known is OWL [2]
3
. It is
however based on open world assumption, while in the
context of simulation modeling closed world assumption is
more useful. Therefore the ontology definition language of
Simantics resembles more MOF
4
, which also makes UML
class diagrams suitable for visualizing the ontologies.
The development of the Semantic Web
5
and its
technologies has increased activity and interest on semantic
data management and its applications. Technologies such as
the Resource Description Framework (RDF) [12], OWL,
SPARQL Protocol and RDF Query Language (SPARQL)
[15], and Semantic Web Rule Language (SWRL) [5],
introduce a set of basic technologies to describe the data, to
model ontologies, to describe semantic database queries,
and to model rules and complex restrictions into ontologies
respectively. The overall concept that these technologies
introduce has inspired domains outside the Web technology
to consider applying these methods for data management.
The foundations of core of Simantics software platform,
which we use as the implementation environment for our
Semantic mesh proposal, are in detail described in [8, 20, 10,
6]. Some Simantics related methods and concepts, using
semantic and ontological representation have been
demonstrated and documented separately. Using ontology-
based 2D vector graphics, based on scalable vector graphics
(SVG) was proposed in [9], a proposal of 3D models based
on ontologies in large-scale process simulation can be found
in [11], a marker based learning environment for detection of
equipment in plants was described in [14]. Process
simulation software applications, developed at VTT
Technical Research Centre of Finland and widely used in
industry, Apros [18, 13] and the next version of BALAS
6
is
integrated to Simantics. A computational fluid dynamics
software environment, based on semantic modelling
approach, designed to be suitable for integration with large-
scale system level simulation tools for industry process
simulation of plants was introduced in [4].
III.
O
UR PROPOSAL OF THE
S
EMANTIC MESH
A.
Need for a general Semantic mesh datastructure
Features for different mesh formats differ, however the
most basic concepts remain the same for all of them. A mesh
(suitable not only for FEM) is a collection of nodes and
elements (cells). Each of the elements is composed by at
least one node. The cells can have dimension 0D (e.g.
vertices), 1D (e.g. beams), 2D (e.g. triangles) or 3D (e.g.
tetrahedrons). In practise, currently most commonly used are
3D and 2D meshes.
3
OWL Web Ontology Language:
http://www.w3.org/TR/owl-features/
4
MOF Version 2.0:
http://www.omg.org/spec/MOF/2.0/
5
Semantic Web:
http://www.w3.org/standards/semanticweb/
6
BALAS software website:
http://balas.vtt.fi
Meshes can have certain regularities. These regularities
can be used for forming and describing structured meshes. In
cases of arbitrary topology, unstructured meshes can be
instead used. These meshes do not contain any regularity and
therefore can be used to describe general structures and their
corresponding meshes. Our current efforts as well as
examples of our Semantic mesh in this paper use
unstructured meshes.
The concept of nodes and elements is the most
elementary and can be found also in applications, such as
ParaView
7
[19] based on VTK scientific visualization
library
8
[7, 16], which both we are using for external and
internal visualization of meshes and data fields defined on
them.
Another important aspect in mesh descriptions is
organizing parts of meshes in groups. The group is basically
a subset of nodes or elements. Groups are defined on nodes
or elements (cells). For example, in a mesh representing a
car, there can be a group on nodes or a group on elements
present at the engine of the car.
The aim of this article is to propose a general, universal
Semantic mesh data structure. Because of the flexibility of
the semantic data representation, such structure could be
used as a general base for eventual extension of Semantic
mesh with other, for example less common features, which
do not have to be present in all applications working with
meshes (e.g. loads, boundary conditions, materials, etc.). For
other, more specific needs and projects, these features might
be added by extending the proposed Semantic mesh. For
example, one of possible extensions includes also material
databases, since e.g. simple FEM applications do not have to
necessarily use these databases. Also, for solely visualization
reasons, these databases also may not be necessary.
Although the proposal and examples were directed for
FEM applications, the proposed Semantic mesh can be used
also for simulation software applications using other
numerical methods, such as e.g. finite volume method.
Our Semantic mesh definition unifies the organization
and storage format of the meshes, but not the semantics of
different features and properties, that may be format and
application specific. We see two benefits in this approach: it
allows importing data from many existing mesh formats,
because new features and properties can be added and
described when needed. On the other hand, it enables
describing transformations between different mesh formats
on a higher level: transformations do not need to consider the
way the mesh data is serialized and transformations can be
described by composing descriptions of how different
properties are transformed.
B.
Requirements for a proposal of Semantic mesh
One of most important problems to address is possible
large amount of components in the mesh. Because there can
be millions of nodes and elements, this can cause practical
7
ParaView website:
http://www.paraview.org
8
VTK website:
http://www.vtk.org
difficulties. Due to performance reasons and memory
consumption requirements, it may not be possible to use
semantic model, which would use too many (e.g. tens of
millions) semantic statements. It would be necessary to
propose a model with such features, which would use
reasonable amount of semantic resources and statements, yet
it would keep advantages that semantic modeling and
abstraction offers.
Simplicity is another requirement – even in cases of very
large meshes, we would like to use limited number of
semantic resources and statements, yet at the same time
utilize advantages of semantic modeling for rich description
of the mesh data model.
One of means necessary to achieve such simplicity would
be grouping of common features and operations by using
means of data abstraction. Examples of such similarities in
the mesh are nodes and elements. Both nodes and elements
are specified using arrays of numerical values, both of them
can be formed in groups; there are values assigned to them
(such as physical properties gained by simulation analysis).
Therefore, awareness and recognition of such similarities and
their utilization is important for us, so we could achieve the
requirement of simplicity.
Our proposal of the Semantic mesh also efforts to be
extendible for various mesh data related features.
C.
Our Semantic mesh datastructure definition
Our Semantic mesh model ontology is formed as a
graph, see Fig. 3. The root element of the graph is
FEMModel, which is a starting point to all other succeeding
elements, which are children of this node, and which contain
all data of the mesh model. We will now describe the most
important features of our model.
1)
Abstract data features notation
The proposed model is designed to be general. The same
ontology can be used for various data, such as lists of nodes
(NodeList), elements (ElementList) and coordinate systems
(CoordinateSystemList - for cases when coordinates in
elements and nodes can use different coordinate systems).
There may also be other kind of features, which can be in
case of necessity added by a developer. This is reflected by
the abstract class MeshItemList, which is an abstraction of
data features of the model. Each feature item has an integer
identifier. The identifiers are stored in array Indices, which
dimension reflects total count of features.
The further described concepts can be easily used for all
classes inherited from MeshItemList. It is important to note
that none of such classes actually contains any numerical
data or fields. All the data, including e.g. nodes coordinates
and element connectivity are contained in assignments
relations, which we will describe below.
2)
Groups
Groups allow us to properly focus on some subset of
mesh features. These can be used for defining e.g. groups on
mesh nodes or elements. The groups can be used in
subsequent assignments: e.g. we can use the same group to
assign two different physical properties. There is an abstract
class Group, which is used to specify the subset to which a
property will be assigned. MeshItemList is itself a group
representing the whole mesh. Smaller groups can be defined
as GroupSelection by enumerating the items of the group.
Groups as well as all other data items can be named. The
model could be extended to allow definition of groups in
some other way than GroupSelection, by inheriting from
Group.
Figure 3.
Semantic mesh abstraction described by an UML class diagram
3)
Property assignments
The actual data of MeshItemList, is described using
assignments. An assignment points to a Group. Subset of a
mesh feature selected using the group will be assigned.
Data are attached as literals and specify concrete values
that will be assigned to the items in the group, to which the
assignment is attached. Assignments link to a Property
(each of these is pointed with its corresponding relation
AssignsProperty), which specify type and name of the
corresponding data to be assigned.
Assignments are inherited from abstract class
PropertyAssignment. In case of ConstantAssignment, all
features will be assigned the same value. In case of
PerItemAssignment, the data is an array of values that assign
different values to all features in the selection in the same
order the features are listed in the selection. All the various
attached data can be in this way provided as an array (list)
of values with type (e.g. double or integer).
Assignments are ordered. If two assignments assign
different values of same property to the same item, the value
that was assigned later is chosen.
More structured types can be used as well. For example,
for a node it could be a structure with x, y, z coordinates and
an identifier specifying chosen coordinate system for the
node.
New property assignments can be added to our Semantic
mesh proposal, for example to describe a property by a
formula that depends on other properties.
IV.
A
N EXAMPLE MODEL OF OUR
S
EMANTIC MESH
We present an example of a very simple Semantic mesh,
on which we will illustrate described mesh Semantic mesh
definition. Fig. 4 illustrates a simple 2D mesh with five
nodes, four elements and two groups on nodes. Due to
limited space and effort to keep the example simple and
brief, assignments of colour property to GroupSelection
groups is demonstrated only on nodes, although it’s possible
to use these assignments with the same approach also on
elements. The example contains three assignments to nodes
and one assignment to elements. Two assignments of colour
property overlap on nodes 10 and 40. Because the
assignment that sets the green colour is defined later, the
final colour of both nodes is green.
V.
T
RANSFORMATION OF
S
EMANTIC MESHES
An advantage of ontology description and semantic
approach is in possibility of obtaining software and data
abstraction, suitable for using with various types of mesh
generators and other applications used in FEM modelling
Because various mesh based software applications do not
rely on semantic software related approach, it is necessary to
provide code that transforms our Semantic mesh to a format
used by other software. For example, a code would make
export of the Semantic mesh to an XML file, which can be
imported to a FEM application.
By means of special semantic transformation features, we
can utilize advanced means to help us with that. For example
in Simantics software platform, there is Simantics
Constraints Language (SCL), which can be used for this
purpose. It is a programming language that can transform a
data model using an ontology description, to another data
model with a different ontology, or transform a data model
into a different, non-semantically based format, such as a e.g.
plain text or XML file. It can be e.g. used for exporting to
binary or text file formats and databases, used by FEM
applications for storage of meshes. At the same time, it is
possible to use other means, such as e.g. using Java code to
generate these files.
Figure 4.
An example of Semantic mesh for a simple 2D Mesh
VI.
I
MPLEMENTATION
The implementation of our Semantic mesh definition
proposal is realised in our CFD/FEM integration application,
designed and implemented for working with various CFD
and FEM solvers and mesh generators.
The application is using Simantics software platform and
is implemented within the Rich Client Platform (RCP) of
Eclipse Java development environment. The RCP
environment provides suitable framework for creating
appropriate user interface with similar look and feel for
applications based on this technology. The application has
simple pre- and post-processing features which utilize VTK
visualization library.
The application reads meshes from I-DEAS UNV format
(Universal File Format) to Semantic mesh. The application
allows making changes to the mesh model. It can visualize
whole mesh, groups of nodes or elements, and assigned
properties. It also allows exporting Semantic mesh model to
the I-DEAS UNV format. User interface of this application
with Semantic mesh ontologies displayed in the object tree
of graph explorer is depicted on Fig. 5.
Figure 5.
Inspecting a Semantic mesh model in our application based on Simantics software platform
VII.
F
UTURE WORK
Our on-going and also future work includes creating
importers and exporters for the proposed Semantic mesh
representation also from other FEM software applications
than I-DEAS UNV format. We are currently working on
supporting file formats used in Abaqus FEA. Another work
is aimed to provide external visualization features, by using
ParaView XML formats, so that we would be able to
visualize the Semantic mesh and its data in ParaView.
Also we consider proposing of special mesh ontologies,
which would be suitable for concrete FEM software
applications. These would be different for each software
applications (e.g. Abaqus FEA, I-DEAS and MSC Adams).
An ontology oriented programming language, would be able
to perform transformation from such a concrete ontology
format into a more general ontology description, which
would be same or similar as the ontology format described in
this paper. Our Semantic mesh development is also a part of
our projects dedicated for creation of a new CFD/FEM
environment, with special impact of integration of 1D
industry process simulators (especially Apros) with 3D
simulations. The project is expected to provide pre-
processing features (such as an interactive geometry editor
based on OpenCASCADE
9
geometry modeling kernel),
mesh integration (free and commercial meshers), simulation
model configuration using semantic approach, solver
integration (such as OpenFOAM and FLUENT), controller
for synchronization and data transfer between the 1D and 3D
solvers, storage and management of data sets with results and
post-processing features (namely visualization of data-sets
with simulation results).
VIII.
C
ONCLUSION
We have presented our proposal for Semantic mesh
representation, suitable not only for meshes for finite
element method applications. The proposal is described by
an UML schema and we also provided an example model of
the schema. The proposal was tested with meshes imported
and exported from/to I-DEAS UNV format (Universal File
Format) files, but can be used also with other mesh file
formats. The Semantic mesh is proposed in the way that
number of semantic components necessary to allocate in a
semantic database is limited, and therefore can be used for
even very large meshes with millions of nodes and elements
with reasonable memory requirements. Our proposal has
interesting features resulting in generality and extendibility.
Our proposal was tested in the semantic database in
Simantics software platform and in our FEM integration
software application.
A
CKNOWLEDGMENT
Our research work had received financial support from
Tekes (the Finnish Funding Agency for Technology and
Innovation), from VTT Technical Research Centre of
Finland and from Finnish industry.
9
OpenCASCADE website:
http://www.opencascade.org
We thank our colleagues from VTT, especially Paul
Klinge and Juha Kortelainen for their advices related to the
structural analyses methods and tools.
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