ISO 10303-52:2011

INTERNATIONAL ISO
STANDARD 10303-52
First edition
2011-03-01
Industrial automation systems and
integration — Product data
representation and exchange —
Part 52:
Integrated generic resource: Mesh-based
topology
Systèmes d'automatisation industrielle et intégration — Représentation
et échange de données de produits —
Partie 52: Ressources génériques intégrées: Topologie fondée sur la
maille
Reference number
©
ISO 2011
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ii © ISO 2011 – All rights reserved

Contents Page
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Normative references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3 Terms, definitions and abbreviated terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.1 Terms defined in ISO 10303-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.2 Terms defined in ISO 10303-110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.3 Other terms and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.4 Abbreviated terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4 Mesh topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.1 Fundamental concepts and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.1.1 Structured mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.1.2 Unstructured mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2 mesh_topology_schema type definitions . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2.1 cell_shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2.2 cell_shape_0D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2.3 cell_shape_1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2.4 cell_shape_2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2.5 cell_shape_3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2.6 indices_group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2.7 mesh_location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2.8 mesh_maths_space_type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2.9 structured_mesh_type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.3 mesh_topology_schema entity definitions . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3.1 array_based_unstructured_mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3.2 array_based_unstructured_mesh_and_vertices . . . . . . . . . . . . . . . . . . 14
4.3.3 cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.4 cell_with_explicit_boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.5 cell_of_structured_mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.6 explicit_unstructured_mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.7 extraction_of_structured_submesh . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3.8 extraction_of_submesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3.9 extraction_of_submesh_by_cells . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3.10 extraction_of_submesh_by_vertices . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3.11 indices_list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3.12 indices_range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3.13 mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3.14 mesh_derived_maths_space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3.15 product_of_mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
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4.3.16 rind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3.17 structured_mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3.18 structured_mesh_with_rind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.19 submesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.20 unstructured_mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.21 vertex_defined_cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 mesh_topology_schema function definitions . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.1 all_mesh_vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.2 cell_counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4.3 shorten_array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.4.4 this_schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Mesh connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Fundamental concepts and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3 mesh_connectivity_schema type definitions . . . . . . . . . . . . . . . . . . . . . . . 42
5.3.1 mismatched_region_type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 mesh_connectivity_schema entity definitions . . . . . . . . . . . . . . . . . . . . . . 42
5.4.1 matched_mesh_connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4.2 mesh_connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4.3 mesh_overset_hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4.4 mismatched_donor_mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4.5 mismatched_mesh_connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4.6 mismatched_mesh_region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4.7 multiple_mesh_block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4.8 structured_donor_mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4.9 unstructured_donor_mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 Mesh function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.2 Fundamental concepts and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3 mesh_function_schema entity definitions . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3.1 mesh_function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3.2 mesh_function_basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.4 mesh_function_schema subtype constraint definitions . . . . . . . . . . . . . . . . . . 55
6.4.1 sc1_application_defined_function . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.4.2 sc1_unary_generic_expression . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Annex A (normative) Short names of entities . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Annex B (normative) Information object registration . . . . . . . . . . . . . . . . . . . . . 58
B.1 Document identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
B.2 Schema identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Annex C (informative) Computer-interpretable listings . . . . . . . . . . . . . . . . . . . . . 59
Annex D (informative) EXPRESS-G diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 60
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Annex E (informative) Additional information . . . . . . . . . . . . . . . . . . . . . . . . . 73
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Figures
Figure 1 Schema relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Figure 2 Example convention for a 2-D cell centre . . . . . . . . . . . . . . . . . . . . . . . 5
Figure 3 Example mesh with rind vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Figure 4 A 1-D rectangular_mesh or pentahedral_mesh or pyramidal_mesh or tetrahedral_-
mesh (withi=5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Figure 5 A 2-D rectangular_mesh (withi=5,j =4) . . . . . . . . . . . . . . . . . . . . . 11
Figure 6 A 3-D rectangular_mesh (withi=5,j =4,k =3) . . . . . . . . . . . . . . . . . 11
Figure 7 A 2-D pentahedral_mesh or pyramidal_mesh or tetrahedral_mesh (withi=5,j =4) 12
Figure 8 A 3-D pentahedral_mesh (withi=5,j =4,k =3) . . . . . . . . . . . . . . . . . 12
Figure 9 A 3-D pyramidal_mesh (withi=5,j =4,k =3) . . . . . . . . . . . . . . . . . . 13
Figure 10 A 3-D tetrahedral_mesh (withi=5,j =4,k =3) . . . . . . . . . . . . . . . . . 13
Figure 11 Parametric coordinate system for a 1-D structured mesh . . . . . . . . . . . . . . . 24
Figure 12 Parametric coordinate system for a 2-D structured mesh . . . . . . . . . . . . . . . 24
Figure 13 Parametric coordinate system for a 3-D structured mesh . . . . . . . . . . . . . . . 24
Figure 14 Linear, quadratic and cubic line cells . . . . . . . . . . . . . . . . . . . . . . . . . 27
Figure 15 Linear, quadratic and cubic triangle cells . . . . . . . . . . . . . . . . . . . . . . . 28
Figure 16 Linear, quadratic and cubic quadrilateral cells . . . . . . . . . . . . . . . . . . . . 29
Figure 17 Linear, quadratic and cubic hexahedron cells . . . . . . . . . . . . . . . . . . . . . 30
Figure 18 Linear, quadratic and cubic wedge cells . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 19 Linear, quadratic and cubic tetrahedron cells . . . . . . . . . . . . . . . . . . . . . 32
Figure 20 Linear, quadratic and cubic pyramid cells . . . . . . . . . . . . . . . . . . . . . . 33
Figure 21 A 1-to-1 abutting interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Figure 22 A mismatched abutting interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Figure 23 An overset interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Figure D.1 Entity level diagram of mesh_topology_schema schema (page 1 of 10) . . . . . . 60
Figure D.2 Entity level diagram of mesh_topology_schema schema (page 2 of 10) . . . . . . 61
Figure D.3 Entity level diagram of mesh_topology_schema schema (page 3 of 10) . . . . . . 62
Figure D.4 Entity level diagram of mesh_topology_schema schema (page 4 of 10) . . . . . . 63
Figure D.5 Entity level diagram of mesh_topology_schema schema (page 5 of 10) . . . . . . 64
Figure D.6 Entity level diagram of mesh_topology_schema schema (page 6 of 10) . . . . . . 65
Figure D.7 Entity level diagram of mesh_topology_schema schema (page 7 of 10) . . . . . . 65
Figure D.8 Entity level diagram of mesh_topology_schema schema (page 8 of 10) . . . . . . 66
Figure D.9 Entity level diagram of mesh_topology_schema schema (page 9 of 10) . . . . . . 67
Figure D.10 Entity level diagram of mesh_topology_schema schema (page 10 of 10) . . . . . . 68
Figure D.11 Entity level diagram of mesh_connectivity_schema schema (page 1 of 3) . . . . . 69
Figure D.12 Entity level diagram of mesh_connectivity_schema schema (page 2 of 3) . . . . . 70
Figure D.13 Entity level diagram of mesh_connectivity_schema schema (page 3 of 3) . . . . . 71
Figure D.14 Entity level diagram of mesh_function_schema schema (page 1 of 1) . . . . . . . 72
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Tables
Table 1 Number of vertices in a structured_mesh . . . . . . . . . . . . . . . . . . . . . . . 25
Table 2 Edges of triangle, quadrilateral and polygon cells . . . . . . . . . . . . . . . . . . 27
Table 3 Edges of hexahedron, wedge, tetrahedron and pyramid cells . . . . . . . . . . . . . 27
Table 4 Faces of hexahedron, wedge, tetrahedron and pyramid cells . . . . . . . . . . . . . 32
Table 5 Domain of the control values table for a mesh_function . . . . . . . . . . . . . . . 52
Table A.1 Short names of entities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Table E.1 Elements of mesh_topology_schema used by other schemas . . . . . . . . . . . . 73
vi °c ISO 2011 — All rights reserved

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical com-
mittee has been established has the right to be represented on that committee. International organizations,
governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates
closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical stan-
dardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Stan-
dards adopted by the technical committees are circulated to the member bodies for voting. Publication
as an International Standard requires approval by at least 75% of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this part of ISO 10303 may be the
subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
International Standard ISO 10303-52 was prepared by Technical Committee ISO TC184/SC4. Automa-
tion systems and integration, Subcommittee SC4 Industrial data.
ISO 10303 is organised as a series of parts, each published separately. The structure of ISO 10303 is
described in ISO 10303-1.
Each part of ISO 10303 is a member of one of the following series: description methods, implementa-
tion methods, conformance testing methodology and framework, integrated generic resources, integrated
application resources, application protocols, abstract test suites, application interpreted constructs, and
application modules. This part is a member of the integrated generic resource series.
The integrated generic resources and the integrated application resources specify a single conceptual
product data model.
A complete list of parts of ISO 10303 is available from Internet:

Should further parts of ISO 10303 be published, they will follow the same numbering pattern.
°c ISO 2011 — All rights reserved vii

Introduction
ISO 10303 is an International Standard for the computer-interpretable representation and exchange of
product data. The objective is to provide a neutral mechanism capable of describing products throughout
their life cycle. This mechanism is suitable not only for neutral file exchange, but also as a basis for
implementing and sharing product databases and as a basis for archiving.
This part of ISO 10303 is a member of the integrated resources series. Major subdivisions of this part of
ISO 10303 are:
— mesh_topology_schema;
— mesh_connectivity_schema.
— mesh_function_schema.
The relationships of the schemas in this part of ISO 10303 to other schemas that define the integrated
resources of this International Standard are illustrated in Figure 1 using the EXPRESS-G notation.
EXPRESS-G is defined in ISO 10303-11. The schemas identified in the bold boxes are specified in
this part of ISO 10303. The support_resource_schema is specified in ISO 10303-41. The topology_-
schema is specified in ISO 10303-42. The mathematical_constructs_schema and the mathematical_-
functions_schema are specified in ISO 10303-50. The mathematical_description_of_distribution_-
schema is specified in ISO 10303-51. The structural_response_representation_schema is specified
in ISO 10303-104. The ISO13584_generic_expressions_schema is specified in ISO 13584-20. Except
for ISO13584_generic_expressions_schema, the schemas illustrated in Figure 1 are components of the
integrated resources.
There are many applications that have to deal with massive amounts of data, which is normally numerical
in nature. The quantity of data may be measured in gigabytes and in some cases terabytes. Examples
include computational fluid dynamics, dynamic simulation of vehicle behaviour, and experimental data
of many kinds ranging from high energy physics to global weather measurements.
A major concern in dealing with such data is to optimise the data representation and structure with respect
to data transmission and storage. As part of the optimisation, the data tends to be maintained in large
arrays where any particular data element can be referenced by a simple index into the array. When the
data is part of a computer simulation the data is usually associated with a mesh of some kind — either
structured or unstructured. The data can be bound to the vertices of the mesh or to the cells of the mesh.
In any case, it is also possible to represent the simpler kinds of meshes by an indexing scheme. Within
this part illustrative examples have been principally taken from the field of computational fluid dynamics.
This part of ISO 10303 provides general, application independent, means of representing indexible data
and meshes.
viii °c ISO 2011 — All rights reserved

structural response representation schema
topology schema
mathematical constructs schema mathematical functions schema
mesh topology schema mesh connectivity schema
mesh function schema
mathematical description of distribution schema
ISO13584 generic expressions schema support resource schema
Figure 1 – Schema relationships
°c ISO 2011 — All rights reserved ix

INTERNATIONAL STANDARD ISO 10303-52:2011(E)

Industrial automation systems and integration — Product data
representation and exchange —
Part 52:
Integrated generic resource: Mesh-based topology
1 Scope
This part of ISO 10303 provides general and application-independent means of representing structured
and unstructured meshes, and mathematical functions and numeric data defined over such meshes. The
schemas in this document are specified in the EXPRESS language; EXPRESS is defined in ISO 10303-
11.
The following are within the scope of this part of ISO 10303:
— mesh-based topologies;
— cell connectivity and multiblock mesh interfaces;
— mathematical functions defined over meshes;
— the association of numeric data with the cells, faces, edges, and vertices of a mesh.
The following are outside the scope of this part of ISO 10303:
— applications of mesh topologies;
— applications of mesh interfaces;
— the semantics of data associated with a mesh.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For updated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 10303-1, Industrial automation systems and integration — Product data representation and ex-
change — Part 1: Overview and fundamental principles.
ISO 10303-11, Industrial automation systems and integration — Product data representation and ex-
change — Part 11: Description method: The EXPRESS language reference manual.
°c ISO 2011 — All rights reserved 1

ISO 10303-41, Industrial automation systems and integration — Product data representation and ex-
change — Part 41: Integrated generic resource: Fundamentals of product description and support.
ISO 10303-42, Industrial automation systems and integration — Product data representation and ex-
change — Part 42: Integrated generic resource: Geometric and topological representation.
ISO 10303-50, Industrial automation systems and integration — Product data representation and ex-
change — Part 50: Integrated generic resource: Mathematical constructs.
ISO 10303-51, Industrial automation systems and integration — Product data representation and ex-
change — Part 51: Integrated generic resource: Mathematical description.
ISO 10303-104, Industrial automation systems and integration — Product data representation and ex-
change — Part 104: Integrated application resource: Finite element analysis.
ISO 10303-110, Industrial automation systems and integration — Product data representation and ex-
change — Part 110: Integrated application resource: Mesh-based computational fluid dynamics.
ISO 13584-20, Industrial automation systems and integration — Parts library — Part 20: Logical re-
source: Logical model of expressions.
3 Terms, definitions and abbreviated terms
3.1 Terms defined in ISO 10303-1
For the purposes of this document, the following terms defined in ISO 10303-1 apply.
— application protocol (AP)
— integrated resource
— product
3.2 Terms defined in ISO 10303-110
For the purposes of this document, the following term defined in ISO 10303-110 applies.
— rind
2 °c ISO 2011 — All rights reserved

3.3 Other terms and definitions
For the purposes of this document, the following terms and definitions apply.
3.3.1
cell
manifold of dimensionality one or higher that is a part of, or the whole of, a mesh
3.3.2
cell edge
one-dimensional manifold that is on the boundary of a cell and that joins two cell vertices
3.3.3
cell face
two-dimensional manifold that is on the boundary of a cell and that is enclosed by one or more cell edges
3.3.4
cell vertex
vertex that is at the end of one or more cell edges
3.3.5
mesh
arrangement of cells with connectivity between the cells defined by the possession of common cell faces
or cell edges
3.3.6
topological region
point set with a single topological dimension
3.3.7
vertex
point within, or on the boundary of, a cell
NOTE 1 A vertex can, but need not, be a cell vertex.
NOTE 2 In some applications, particularly in finite element analysis, the word node is used as an equivalent term
to vertex.
3.4 Abbreviated terms
CFD computational fluid dynamics
URL Universal Resource Locator
°c ISO 2011 — All rights reserved 3

4 Mesh topology
The following EXPRESS declaration begins the mesh_topology_schema and identifies the necessary
external references.
EXPRESS specification:
)
*
SCHEMA mesh_topology_schema;
REFERENCE FROM mathematical_description_of_distribution_schema -- ISO 10303-51
(property_distribution_description);
REFERENCE FROM mathematical_functions_schema -- ISO 10303-50
(maths_space);
REFERENCE FROM structural_response_representation_schema -- ISO 10303-104
(element_order,
element_representation,
fea_model);
REFERENCE FROM support_resource_schema -- ISO 10303-41
(identifier,
label,
text);
REFERENCE FROM topology_schema -- ISO 10303-42
(topological_representation_item,
vertex, vertex_point);
(
*
NOTE The schemas referenced above can be found in the following parts of ISO 10303:
mathematical_description_of_distribution_schema ISO 10303-51
mathematical_functions_schema ISO 10303-50
structural_response_representation_schema ISO 10303-104
support_resource_schema ISO 10303-41
topology_schema ISO 10303-42
4.1 Fundamental concepts and assumptions
A mesh is defined by its vertices and the connections between the vertices. A mesh is a connected graph.
4.1.1 Structured mesh
In a structured mesh the cells are arranged in a regular pattern and their shapes are implied by the
particular kind of mesh.
A 3–D rectangular mesh is a mesh of hexahedral cells. Each cell is a dimensionality 3 hexahedral region
defined by eight vertices forming the corners of the hexahedron. Each cell is bounded by six faces, where
each face is the quadrilateral defined by four vertices. A face is limited by the four edges that connect
the four vertices.
4 °c ISO 2011 — All rights reserved

(i;j+1) (i+1;j+1) (i+2;j+1)
² ² ²
(i;j) (i+1;j)
² ² ²
(i;j) (i+1;j) (i+2;j)
Figure 2 – Example convention for a 2-D cell centre
(5;4)
£ ² ² ² ² ² £
£ ² ² ² ² ² £
£ ² ² ² ² ² £
£ ² ² ² ² ² £
(0;1) (1;1) (5;1) (6;1)
Figure 3 – Example mesh with rind vertices
A 2–D rectangular mesh is a mesh of quadrilaterals. Each cell is a dimensionality two quadrilateral
region defined by four vertices forming the corners of the quadrilateral. Each cell is limited by the four
edges that connect the four vertices.
A 1–D mesh is of linear form. Each cell is a dimensionality one linear region bounded by two vertices.
Indices describing a structured mesh are ordered: for 3–D (i;j;k); (i;j) is used for 2–D; and (i) for
1–D.
Cell centres, face centres, and edge centres are indexed by the minimum of the connecting vertices.
EXAMPLE 1 For example a 2-D cell center (or face centre on a 3-D mesh) would have the conventions shown
in Figure 2.
In addition, the default beginning vertex for a regular mesh is(1;1;1); this means the default beginning
cell centre of a regular mesh is also(1;1;1).
There may be locations outside the mesh itself. These are referred to as ‘rind’ or ghost points and may
be associated with fictitious vertices or cell centres. They are distinguished from the vertices and cells
making up the mesh (including its boundary vertices), which are referred to as ‘core’ points.
EXAMPLE 2 Figure 3 shows a 2–D mesh with a single row of ‘rind’ vertices at the minimum and maximum
i-faces. The mesh size (i.e., the number of ‘core’ vertices in each direction) is5£4. ‘Core’ vertices are designated
by ‘²’, and ‘rind’ vertices by ‘£’. Default indexing is also shown for the vertices.
°c ISO 2011 — All rights reserved 5

For a mesh, the minimum faces in each coordinate direction are denoted i-min, j-min and k-min; the
maximum faces are denoted i-max, j-max and k-max. These are the minimum and maximum ‘core’
faces.
EXAMPLE 3 i¡min is the face or mesh plane whose core vertices have minimumi index (which if using default
indexing is 1).
4.1.2 Unstructured mesh
An unstructured mesh is composed of cells, where the cells need not form a regular pattern and the shape
of the cells is not restricted to be uniform throughout the mesh. Cells have vertices at their corners and
may also have nodes on cell edges, cell faces, and in the interior of the cell.
Each cell in an irregular mesh has at least one vertex in common with at least one other cell in the mesh.
The connectivity and adjacency of the cells may be determined from the common vertices.
Each cell in an unstructured mesh is explicitly represented in terms of its shape and an ordered list of its
vertices. The vertices are implied rather than being explicitly represented. Essentially all the vertices in
a mesh can be mapped to a sequential list, and reference to a vertex is then equivalent to specifying the
particular position in the list.
4.2 mesh_topology_schema type definitions
4.2.1 cell_shape
A cell_shape is an identifier of an unstructured mesh cell shape.
EXPRESS specification:
)
*
TYPE cell_shape = EXTENSIBLE SELECT
(cell_shape_0D,
cell_shape_1D,
cell_shape_2D,
cell_shape_3D);
END_TYPE;
(
*
4.2.2 cell_shape_0D
A cell_shape_0D is an identifier of a topologically 0–D unstructured mesh cell shape.
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EXPRESS specification:
)
*
TYPE cell_shape_0D = EXTENSIBLE ENUMERATION OF
(single);
END_TYPE;
(
*
Enumerated item definitions:
single: singleton vertex.
4.2.3 cell_shape_1D
A cell_shape_1D is an identifier of a topologically 1–D unstructured mesh cell shape.
EXPRESS specification:
)
*
TYPE cell_shape_1D = EXTENSIBLE ENUMERATION OF
(line);
END_TYPE;
(
*
Enumerated item definitions:
line: a topological line requiring 2 vertices.
4.2.4 cell_shape_2D
A cell_shape_2D is an identifier of a topologically 2–D unstructured mesh cell shape.
EXPRESS specification:
)
*
TYPE cell_shape_2D = EXTENSIBLE ENUMERATION OF
(quadrilateral,
triangle);
END_TYPE;
(
*
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Enumerated item definitions:
quadrilateral: four sided cell requiring 4 vertices;
triangle: three sided cell requiring 3 vertices;
NOTE 1 This type is defined as EXTENSIBLE to enable other 2D cell shapes to be added to the list as required
by particular applications.
4.2.5 cell_shape_3D
A cell_shape_3D is an identifier of a topologically 3–D unstructured mesh cell shape.
EXPRESS specification:
)
*
TYPE cell_shape_3D = EXTENSIBLE ENUMERATION OF
(hexahedron,
wedge,
tetrahedron,
pyramid);
END_TYPE;
(
*
Enumerated item definitions:
hexahedron: hexahedral (six quadrilateral faces) requiring 8 vertices;
wedge: pentahedral (three quadrilateral faces and two triangular faces) requiring 6 vertices;
tetrahedron: tetrahedral form (four triangular faces) requiring 4 vertices;
pyramid: pyramidal form (one quadrilateral face and four triangular faces) requiring 5 vertices.
NOTE 1 This type is defined as EXTENSIBLE to enable other 3D cell shapes to be added to the list as required
by particular applications.
4.2.6 indices_group
An indices_group is a selection of a group of indices into a multi-dimensional array.
EXPRESS specification:
)
*
TYPE indices_group = SELECT
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(indices_list,
indices_range);
END_TYPE;
(
*
4.2.7 mesh_location
A mesh_location is an enumeration of locations with respect to a mesh.
EXPRESS specification:
)
*
TYPE mesh_location = EXTENSIBLE ENUMERATION OF
(unspecified,
application_defined,
vertices,
cell_centre,
face_centre,
iface_centre,
jface_centre,
kface_centre,
edge_centre);
END_TYPE;
(
*
Enumerated item definitions:
unspecified: not specified;
application_defined: specified via an external agreement between the data creator and the data user;
vertices: at cell vertices for cells within the mesh;
cell_centre: the centre of a cell; this is also appropriate for entities associated with cells but not neces-
sarily with a given location in a cell;
face_centre: the centre of a generic face which can point in any coordinate direction;
iface_centre: the centre of a face in 3-D whose computational normal points in thei direction;
jface_centre: the centre of a face in 3-D whose computational normal points in thej direction;
kface_centre: the centre of a face in 3-D whose computational normal points in thek direction;
edge_centre: the centre of an edge.
°c ISO 2011 — All rights reserved 9

4.2.8 mesh_maths_space_type
A mesh_maths_space_type is an enumeration of the kinds of associations of a mesh_derived_maths_-
space and a mesh.
EXPRESS specification:
)
*
TYPE mesh_maths_space_type = EXTENSIBLE ENUMERATION OF
(cells,
vertices);
END_TYPE;
(
*
Enumerated item definitions:
cells: data is associated with mesh cells;
vertices: data is associated with mesh vertices.
4.2.9 structured_mesh_type
A structured_mesh_type is an enumeration of the kinds of structured meshes.
EXPRESS specification:
)
*
TYPE structured_mesh_type = EXTENSIBLE ENUMERATION OF
(rectangular,
pentahedral,
pyramidal,
tetrahedral);
END_TYPE;
(
*
Enumerated item definitions:
rectangular: a structured mesh that is topologically linear in 1–D, quadrilateral in 2–D, hexahedral in
3–D, etc.
In 2–D the cells are all quadrilateral. In 3–D the cells are all hexahedral.
NOTE 1 Illustrations of rectangular mesh topologies are shown in Figure 4, Figure 5 and Figure 6.
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[1] [i]
Figure 4 – A 1-D rectangular_mesh or pentahedral_mesh or
pyramidal_mesh or tetrahedral_mesh (withi=5)
[1;j] [i;j]
[1;1] [i;1]
Figure 5 – A 2-D rectangular_mesh (withi=5,j =4)
pentahedral: a structured mesh that is topologically linear in 1–D, triangular in 2–D, and pentahedral,
with 2 triangular and 3 quadrilateral faces forming a wedge-like shape, in 3–D. (where one of the edges
between a pair of rectangular faces is analogous to the axis of a sector of a cylinder).
It is convenient to think of this kind of mesh as like a sector of a circle in 2–D where one apex point is
anolagous to the centre point of the circle, and like a sector of a cylinder in 3–D where one of the edges
between a pair of rectangular faces is analogous to the axis of the cylinder.
In 2–D the cells adjacent to the apex are triangular; the rest are quadrilateral.
In 3–D the cells adjacent to the axis edge are pentahedral; the rest are hexahedral.
NOTE 2 Illustrations of pentahedral mesh topologies are shown in Figure 4, Figure 7 and Figure 8.
[i;j;k]
[i;1;k]
[1;j;k]
[i;1;1]
[1;j;1]
[1;1;1]
Figure 6 – A 3-D rectangular_mesh (withi=5,j =4,k =3)
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[i;j]
[1;1] [i;1]
Figure 7 – A 2-D pentahedral_mesh or pyramidal_mesh or
tetrahedral_mesh (withi=5,j =4)
[i;j;k]
[1;1;k] [i;1;k]
[1;1;1] [i;1;1]
Figure 8 – A 3-D pentahedral_mesh (withi=5,j =4,k =3)
pryamidal: a structured mesh that is topologically linear in 1–D, triangular in 2–D, and pyramidal in
3–D.
It is convenient to think of this kind of mesh as like a sector of a circle in 2–D where one apex point is
analogous to the centre point of the circle, and like a sector of a sphere in 3–D where the apex point is
analogous to the centre point of a sector of the sphere.
In 2–D the cells adjacent to the apex are triangular; the rest are quadrilateral.
In 3–D the cells adjacent to the apex are pyrimidal; the rest are hexahedral.
NOTE 3 Illustrations of pyramidal mesh topologies are shown in Figure 4, Figure 7 and Figure 9.
tetrahedral: a structured mesh that is topologically linear in 1–D, triangular in 2–D, and tetrahedral in
3–D.
It is convenient to think of this kind of mesh as like a sector of a circle in 2–D where one apex point is
analogous to the centre point of the circle, and like
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[i;j;k]
[i;1;k]
[1;1;1] [i;1;1]
Figure 9 – A 3-D pyramidal_mesh (withi=5,j =4,k =3)
[i;j;k]
[i;j;1]
[1;1;1] [i;1;1]
Figure 10 – A 3-D tetrahedral_mesh (withi=5,j =4,k =3)
In 2–D the cells adjacent to the apex are triangular; the remainder are quadrilateral.
In 3–D the cells adjacent to the apex are tetrahedral and the cells adjacent to one edge from the apex are
pentahedral; the remainder are hexahedral.
NOTE 4 Illustrations of tetrahedral mesh topologies are shown in Figure 4, Figure 7 and Figure 10.
4.3 mesh_topology_schema entity definitions
4.3.1 array_based_unstructured_mesh
An array_based_unstructured_mesh is a representation of an unstructured_mesh designed to min-
imise the amount of data by not requiring explicit identification of the vertices of the cells in the mesh.
°c ISO 2011 — All rights reserved 13

EXPRESS specification:
)
*
ENTITY array_based_unstructured_mesh
SUBTYPE OF (unstructured_mesh);
cells : ARRAY [1:cell_count] OF vertex_defined_cell;
WHERE
wr1 : SELF\mesh.index_count = 1;
END_ENTITY;
(
*
Attribute definitions:
cells: the vertex_defined_cells forming the mesh;
cell_count: (inherited) the number of cells in the mesh;
index_count: (inherited) the number of indices required to uniquely identify a vertex or cell in the mesh.
Formal propositions:
wr1: the value of index_count shall be 1.
4.3.2 array_based_unstructured_mesh_and_vertices
An array_based_unstructured_mesh_and_vertices is a kind of array_based_unstructured_mesh
where the vertices of the mesh are explicity identified and ordered.
EXPRESS specification:
)
*
ENTITY array_based_unstructured_mesh_and_vertices
SUBTYPE OF (array_based_unstructured_mesh);
vertex_count : INTEGER;
vertices : ARRAY [1:vertex_count] OF UNIQUE vertex;
WHERE
wr1 : all_mesh_vertices(SELF);
END_ENTITY;
(
*
Attribute definitions:
vertex_count: the number of unique vertex in the mesh;
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vertices: an array of unique vertexs for the cells of the mesh.
Formal propositions:
wr1: the elements of vertices shall be all and only the unique vertices in the mesh.
4.3.3 cell
A cell is a topological_representation_item that is a manifold with a boundary.
EXPRESS specification:
)
*
ENTITY cell
SUPERTYPE OF (ONEOF(cell_of_structured_mesh, vertex_defined_cell))
SUBTYPE OF (topological_representation_item);
description : text;
dimension : INTEGER;
END_ENTITY;
(
*
Attribute definitions:
description: annotation;
dimension: the topological dimension of the region.
4.3.4 cell_with_explicit_boundary
A cell_with_explicit_boundary is a cell that has a specified boundary.
EXPRESS specification:
)
*
ENTITY cell_with_explicit_boundary
SUBTYPE OF (cell);
boundary : SET [1:?] OF topological_representation_item;
END_ENTITY;
(
*
°c ISO 2011 — All rights reserved 15

Attribute definitions:
boundary: the elements forming the boundary of the region.
4.3.5 cell_of_structured_mesh
A cell_of_structured_mesh is an identified cell of a structured_mesh.
EXPRESS specification:
)
*
ENTITY cell_of_structured_mesh
SUBTYPE OF (cell);
the_mesh : structured_mesh;
cell_identifier : ARRAY [1:index_count] OF INTEGER;
DERIVE
index_count : INTEGER := the_mesh\mesh.index_count;
END_ENTITY;
(
*
Attribute definitions:
the_mesh: the structured_mesh;
cell_identifier: the indices of the cell;
index_count: the number of indices required to uniquely identify a vertex or cell in the mesh.
4.3.6 explicit_unstructured_mesh
An explicit_unstructured_mesh is a representation of an unstructured_mesh that is similar, but not
entirely identical to, that specified in ISO 10303-104.
EXPRESS specification:
)
*
ENTITY explicit_unstructured_mesh
SUBTYPE OF (unstructured_mesh);
explicit_model : fea_model;
cells : ARRAY [1:cell_count] OF UNIQUE element_representation;
END_ENTITY;
(
*
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Attribute definitions:
explicit_model: the finite element model;
cell_count: (inherited) the number of element_representations;
cells: the set of element_representations comprising the mesh.
Informal propositions:
ip1: every element_representation in the cells shall belong to the explicit_model.
4.3.7 extraction_of_structured_submesh
An extraction_of_structured_submesh is a type of extraction_of_mesh and is a relationship between
two structured_meshes that indicates one is part of the other.
EXPRESS specification:
)
*
ENTITY extraction_of_structured_submesh
SUBTYPE OF(extraction_of_submesh);
lower_vertex : ARRAY [1:whole_indices] OF INTEGER;
used_indices : ARRAY [1:part_indices] OF INTEGER;
used_senses : ARRAY [1:part_indices] OF BOOLEAN;
DERIVE
whole_indices : INTEGER := SELF\extraction_of_submesh.whole\mesh.index_count;
part_indices : INTEGER := SELF\extraction_of_submesh.part\mesh.index_count;
WHERE
WR1: (’MESH_TOPOLOGY_SCHEMA.STRUCTURED_MESH’ IN TYPEOF(
SELF\extraction_of_submesh.whole));
WR2: (’MESH_TOPOLOGY_SCHEMA.STRUCTURED_MESH’ IN TYPEOF(
SELF\extraction_of_submesh.part));
END_ENTITY;
(
*
Attribute definitions:
part: the structured_mesh that is part of the whole;
whole: the structured_mesh that contains the part;
lower_vertex: the position of the vertex in the whole that is the origin of the part. This is specified
with respect to each index of the whole.
°c ISO 2011 — All rights reserved 17

used_indices: the indices of the whole that are also indices of the part in the order that they are used in
the part;
used_senses: the sense for each index of part as:
— TRUE if the part uses the index of the whole in the same direction;
— FALSE if the part uses the index of the whole in the reverse direction;
whole_indices: the number of indices required to uniquely identify a vertex or cell in the whole;
part_indices: the number of indices required to uniquely identify a vertex or cell in the part;
Formal propositions:
WR1: The mesh referenced as whole shall be of type structured_mesh.
WR1: The submesh referenced as part shall be of type structured_mesh.
4.3.8 extraction_of_submesh
An extracrtion_of_submesh is a relationship between a mesh and a submesh that defines the submesh
as being part of the mesh.
EXPRESS specification:
)
*
ENTITY extraction_of_submesh;
whole: mesh;
part: submesh;
END_ENTITY;
(
*
Attribute definitions:
whole: the mesh from which the submesh is extracted.
part: the resulting submesh.
4.3.9 extraction_of_submesh_by_cells
An extraction_of_submesh_by_cells is a type of extraction_of_submesh that extracts a submesh by
listing the cells.
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EXPRESS specification:
)
*
ENTITY extraction_of_submesh_by_cells
SUBTYPE OF (extraction_of_submesh);
cell_count: INTEGER;
cells : ARRAY [1:cell_count] OF cell;
END_ENTITY;
(
*
Attribute definitions:
cell_count: the number of cells extracted to form the submesh.
cells: the collection of cells defining the submesh.
4.3.10 extraction_of_submesh_by_vertices
An extraction_of_submesh_by_vertices is a type of extraction_of_submesh that extracts a submesh
by listing the vertices.
NOTE 1 A submesh of lower topological dimension than the parent (whole) mesh can be specified by extrac-
tion_of_mesh_by_vertices.
EXPRESS specification:
)
*
ENTITY extraction_of_submesh_by_vertices
SUBTYPE OF (extraction_of_submesh);
vertex_count: INTEGER;
vertices : ARRAY [1:vertex_count] OF vertex;
END_ENTITY;
(
*
Attribute definitions:
vertex_count: the number of vertices extracted to form the submesh.
vertices: the collection of vertexs defining the submesh.
4.3.11 indices_list
An indices_list specifies a list of indices into a multi-dimensional array.
°c ISO 2011 — All rights reserved 19

EXPRESS specification:
...