There are many functions in Patran that rely on the mathematical representation of the geometry. These functions are:

For every curve, surface or solid in a user database, information is stored on its Parameterization, Topology and Connectivity which is used in various Patran functions.

The concepts of parameterization, connectivity and topology are easy to understand and they are important to know when building a geometry and an analysis model.

The following sections will describe each of these concepts and how you can build an optimal geometry model for analysis.

All Patran geometry are labeled one of the following:

Depending on the order of the entity - whether it is a one-dimensional curve, a two-dimensional surface, or a three-dimensional solid - there is one, two or three parameters labeled , , that are associated with the entity. This concept is called “parameterization”.

Parameterization means the X,Y,Z coordinates of a curve, surface or solid are represented as functions of variables or parameters. Depending on the dimension of the entity, the X,Y,Z locations are functions of the parameters , , and .

An analogy to the parameterization of geometry is describing an , location as a function of time, t. If and , as changes, and will define a path. Parameterization of geometry does the same thing - as the parameters , , and change, it defines various points on the curve, surface and solid.

The following describes how a point, curve, surface and solid are parameterized in Patran.

A Point in Patran is a point coordinate location in three-dimensional global XYZ space.

Since a point has zero-dimensions, it has no associated parameters, therefore, it is not parameterized.

A Curve in Patran is a one-dimensional point set in three-dimensional global XYZ space. A curve can also be described as a particle moving along a defined path in space.

Another way of defining a curve is, a curve is a mapping function, , from one-dimensional parametric space into three-dimensional global XYZ space, as shown in Figure 1‑3.

A curve has one parametric variable, , which is used to describe the location of any given point, , along a curve, as shown in Figure 1‑2.

The parameter, , has a range of , where at , is at endpoint and at , is at endpoint .

A straight curve can be defined as:

(1‑1) of our straight curve can be represented as:

The derivative of in (1‑2), would give us (1‑3) which is the tangent of the straight curve.

Because the curve is straight, is a constant value. The tangent, , also defines a vector for the curve, which is the positive direction of .

For any given curve, the tangent and positive direction of at any point along the curve can be found. (The vector, , usually will not have a length of one.)

A surface in Patran is a two-dimensional point set in three-dimensional global XYZ space.

A surface has two parameters, and , where at any given point, , on the surface, can be located by and , as shown in Figure 1‑4.

A surface generally has three or four edges. Trimmed surfaces can have more than four edges. For more information, see Trimmed Surfaces, 20.

Similar to a curve, and for a surface have ranges of and . Thus, at , , is at and at , , is at .

A surface is represented by a mapping function, , which maps the parametric space into the global XYZ space, as shown in Figure 1‑5.

The first order derivatives of results in two partial derivatives, and :

where is the tangent vector in the direction and is the tangent vector in the direction.

At any point for a given surface, and which define the tangents and the positive and directions can be determined.

Usually and are not orthonormal, which means they do not have a length of one and they are not perpendicular to each other.

A solid in Patran is a three-dimensional point set in three-dimensional global XYZ space.

A solid has three parameters, , , and , where at any given point, , within the solid, can be located by , , and , as shown in Figure 1‑6.

A solid generally has five or six sides or faces. (A B-rep solid can have more than six faces.)

The parameters , and have ranges of , , and . At (0,0,0) is at and at (1,1,1), is at .

A solid can be represented by a mapping function, , which maps the parametric space into the global XYZ space, as shown in Figure 1‑7.

If we take the first order derivatives of , we get three partial derivatives, , and , shown in (1‑5):

Where is the tangent vector in the direction, is the tangent vector in the direction, and is the tangent vector in the direction.

At any point within a given solid, , and , which define the tangents and positive , and directions can be determined.

Topology identifies the kinds of items used to define adjacency relationships between geometric entities.

Every curve, surface and solid in Patran has a defined set of topologic entities. You can reference these entities when you build the geometry or analysis model. Examples of this include:

Topology is invariant through a one-to-one bicontinuous mapping transformation. This means you can have two curves, surfaces or solids that have different parameterizations, but topologically, they can be identical.

To illustrate this concept, Figure 1‑8 shows three groups of surfaces A-D. Geometrically, they are different, but topologically they are the same.

The types of topologic entities found in Patran are the following:

The connectivity for a curve, surface and solid determines the order in which the internal vertex, edge and face IDs will be assigned. The location of a geometric entity’s parametric axes defines the point where assignment of the IDs for the entity’s vertices, edges and faces will begin.

Figure 1‑9 and Figure 1‑10 show a four sided surface and a six sided solid with the internal vertex, edge and face IDs displayed. If the connectivity changes, then the IDs of the vertices, edges and faces will also change.

For example, in Figure 1‑9, the edge, ED3, of Surface 11 would be displayed as:

Surface 11.3

The vertex, V4, in Figure 1‑9 would be displayed as:

Surface 11.3.1

V4 has a vertex ID of 1 that belongs to edge 3 on surface 11.

The face, F1, of Solid 100 in Figure 1‑9 would be displayed as:

Solid 100.1

The edge, ED10, in Figure 1‑10 would be displayed as:

Solid 100.1.3

ED10 has an edge ID of 3 that belongs to face 1 on solid 100.

The vertex, V6, in Figure 1‑10 would be displayed as:

Solid 100.1.2.2

V6 has a vertex ID of 2 that belongs to edge 2 on face 1 on solid 100.

When meshing adjacent surfaces or solids, Patran requires the geometry be topologically congruent so that coincident nodes will be created along the common boundaries.

Figure 1‑11 shows an example where surfaces 1 through 3 are topologically incongruent and surfaces 2 through 5 are topologically congruent. The outer vertices are shared for surfaces 1 through 3, but the inside edges are not. Surfaces 2 through 5 all have common edges, as well as common vertices.

There are several ways to correct surfaces 1 through 3 to make them congruent. See Building a Congruent Model for more information.

For a group of surfaces or solids to be congruent, the adjacent surfaces or solids must share common edges, as well as common vertices.

(MSC.Software Corporation’s Patran software product required adjacent surfaces or solids to share only the common vertices to be considered topologically congruent for meshing.)

Another type of topological incongruence is shown in Figure 1‑12. It shows a gap between two pairs of surfaces that is greater than the Global Model Tolerance. This means when you mesh the surface pairs, coincident nodes will not be created along both sides of the gap.

MSC recommends two methods for closing surface gaps:

For more information on meshing, see Introduction to Functional Assignment Tasks (Ch. 1) in the Patran Reference Manual.

Non-manifold topology can be simply defined as a geometry that is non-manufacturable. However, in analysis, non-manifold topology is sometimes either necessary or desirable. Figure 1‑13 shows a surface model with a non-manifold edge.

This case may be perfectly fine. A non-manifold edge has more than two surfaces or solid faces connected to it. Therefore, two solids which share a common face also give non-manifold geometry (both the common face and its edges are non-manifold).

In general, non-manifold topology is acceptable in Patran. The exception is in the creation of a B-rep solid where a non-manifold edge is not allowed. The Verifying Surface Boundaries option detects non-manifold edges as well as free edges.

In Figure 1‑2, Figure 1‑4, and Figure 1‑6 in Parameterization, the axes for the parameters, , , and , have a unique orientation and location on the curve, surface and solid.

Depending on the orientation and location of the , , and axes, this defines a unique connectivity for the curve, surface or solid.

For example, although the following two curves are identical, the connectivity is different for each curve (note that the vertex IDs are reversed):

For a four sided surface, there are a total of eight possible connectivity definitions. Two possible connectivities are shown in Figure 1‑15. (Again, notice that the vertex and edge IDs are different for each surface.)

For a tri-parametric solid with six faces, there are a total of 24 possible connectivity definitions in Patran - three orientations at each of the eight vertices. Two possible connectivities are shown in Figure 1‑16.

Patran can plot the location and orientation of the parametric axes for the geometric entities by turning on the Parametric Direction toggle on the Geometric Properties form, under the Display/Display Properties/Geometric menu. See Preferences>Geometry (p. 467) in the Patran Reference Manual for more information.

For most geometric entities, you can modify the connectivity by altering the orientation and/or location of the parametric axes by using the Geometry application’s Edit action’s Reverse method. See Overview of the Edit Action Methods.

For solids, you can also control the location of the parametric origin under the Preferences/Geometry menu and choose either the Patran Convention button or the PATRAN 2.5 Convention button for the Solid Origin Location.

The geometry’s parameterization and connectivity affect the geometry and finite element analysis model in the following ways:

The parameterization and connectivity for a curve, surface or solid define the order of the internal IDs of their topologic entities. Patran stores these IDs internally and displays them when you cursor select a vertex, edge or face. See Vertex, Edge and Face ID Assignments in Patran for more information.

Using right hand rule by crossing a surface’s direction with its direction, it defines the surface’s positive normal direction ( direction). This affects many areas of geometry and finite element creation, including creating B-rep solids. See Building An Optimal Geometry Model for more information.

The parameterization and connectivity of a curve, surface or solid define the positive direction for a pressure load, and it defines the surface’s top and bottom locations for an element variable pressure load. See Create Structural LBCs Sets (p. 21) in the Patran Reference Manual for more information.

If you reference a field function that was defined in parametric space, when creating a varying loads/BC or a varying element or material property, the loads/BC values or the property values will depend on the geometry’s parameterization and the orientation of the parametric axes. See Fields Forms (p. 204) in the Patran Reference Manual for more information.

The Patran mapped mesher, IsoMesh, will use the geometric entity’s parameterization and connectivity to define the order of the node and element IDs and the element connectivity. (The parameterization and connectivity will not be used if the mesh will have a transition or change in the number of elements within the surface or solid.) See IsoMesh (p. 13) in the Reference Manual - Part III for more information.

Patran uses the Global Model Tolerance when it imports or accesses geometry, when it creates geometry, or when it modifies existing geometry.

The Global Model Tolerance is found under the Preferences/Global menu. The default value is 0.005.

When creating geometry, if two points are within a distance of the Global Model Tolerance, then Patran will only create the first point and not the second.

This rule also applies to curves, surfaces and solids. If the points that describe two curves, surfaces or solids are within a distance of the Global Model Tolerance, then only the first curve, surface or solid will be created, and not the second.

For more information on the Global Model Tolerance, see (p. 73) in the Patran Reference Manual.