APPENDIX A—VIRTUAL GEOMETRY

A.1 Introduction

GAMBIT geometry operations comprise a comprehensive assortment of tools that allow you to create and modify solid models. They involve three general types of entities:

• Real
• Virtual
• Faceted

Real entities possess their own geometrical descriptions—that is, they are defined by mathematical formulae that describe their locations and shapes. Virtual entities do not possess their own geometrical descriptions-instead, they derive their geometry by reference to one or more real entities. Faceted entities are defined in reference to an underlying mesh.

NOTE: The GAMBIT GUI references only real and virtual geometry. To apply GAMBIT geometry operations to faceted geometry, you must treat the faceted geometry as if it were virtual.

The purpose of this appendix is to describe the fundamental differences between real and virtual geometry operations (Section A.2) and to outline the following characteristics of virtual geometry:

• Fundamentals (Section A.3)
• Operations (Section A.4)
• Applications (Section A.5)

A.2 Differences Between Real and Virtual Operations

There are two basic types of GAMBIT geometry operations:

• Real
• Virtual

Table A-1 and Table A-2 list some of the basic tasks that are included in GAMBIT real and virtual geometry operations, respectively.

Chapter 2 of this guide describes the procedures and specifications required to create and/or modify real and virtual entities.

NOTE: Throughout this appendix, the labels of topological entities reflect the GAMBIT default labeling conventions. That is, vertices, edges, faces, and volumes are labeled vertex.a, edge.b, face.c, and volume.d, respectively, where a, b, c, and d represent integer numbers—for example, vertex.5 or face.12. Virtual-entity labels are similar to real entity labels but include the prefix "v_"—for example, v_edge.3 or v_volume.9.

A.3 Virtual Geometry Fundamentals

A.3.1 Model Foreground and Background

To understand the basic purpose of virtual geometry operations, it is useful to think of a GAMBIT model as possessing two different logical domains:

• Foreground
• Background

The model foreground consists of topological entities that are observable upon direct examination of the model. Such entities reflect the outward appearance of the model both in shape and in structure. The model background consists of topological entities that are not directly observable but the mathematical definitions of which define the overall shape and structure of the model.

The following sections describe and illustrate the differences between the foreground and background of GAMBIT models and the fundamental role of the model foreground in GAMBIT display and meshing operations.

Foreground vs. Background—Example

As an example of the difference between the foreground and background of a model, consider the simple, 2-D model shown in Figure A-1. The model consists of six real edges arranged in the form of an irregular, planar hexagon. Each edge shares its endpoint vertices with its neighboring edges and is, therefore, "connected" to those edges. All six edges and vertices exist in the foreground of the model.

Figure A-1: Irregular hexagon—before merge operation

Each edge and vertex shown in Figure A-1 possesses its own geometrical description; the edges are defined as curves, and the vertices are defined as specific points in the modeling space. The combined definitions of all six edges and vertices constitutes the overall geometrical description of the model.

If you perform a virtual "merge" operation (see "Merge Operations," below) that involves edge.1 and edge.2 in Figure A-1, GAMBIT replaces them in the model foreground with a single virtual edge, labeled v_edge.7 (see Figure A-2).

Figure A-2: Irregular hexagon-after merge operation

The virtual edge, v_edge.7, does not possess its own geometrical description. Instead, its shape is defined only by reference to the geometrical descriptions of edge.1 and edge.2. In that sense, v_edge.7 constitutes an "overlay" entity that represents the specific set of edges to which it refers.

When GAMBIT performs the merge operation illustrated in Figure A-1 and Figure A-2, it shifts edge.1, edge.2, and their common vertex, vertex.2, to the background of the model and replaces them in the model foreground with the single virtual edge, v_edge.7. Consequently, the model display retains its original hexagonal shape but includes only five topological edges-one of which constitutes a virtual edge.

The following table summarizes the foreground and background components of the model before and after the merge operation illustrated in Figure A-1 and Figure A-2.

Model Foreground in Display and Meshing Operations

The distinction between the model foreground and background is important to the GAMBIT user for the following reason:

GAMBIT display and meshing operations involve only those topological components that exist in the foreground of the model.

For example, if you mesh the curve represented by edge.1 and edge.2 in Figure A-1, you must apply mesh node grading schemes independently to each edge, because both edges exist in the foreground of the model. Because GAMBIT is constrained by its meshing rules to create mesh nodes at the endpoint vertices of meshed edges, it necessarily creates a mesh node at vertex.2—which constitutes the bending point in the curve (see Figure A-3(a)).

Figure A-3: Irregular hexagon-mesh node spacing

By contrast, if you mesh the curve as represented by the virtual edge, v_edge.7, GAMBIT allows you to apply a single grading scheme to the curve. Furthermore, because v_edge.7 constitutes an individual topological entity, GAMBIT is not constrained to create a mesh node at the bending point in the curve (see Figure A-3(b)). Therefore, the existence of v_edge.7 in the foreground of the model GAMBIT imposes fewer overall constraints on mesh node placement than are imposed by the existence of edge.1 and edge.2.

The fundamental purpose of GAMBIT is to create arrays of mesh nodes the locations of which represent specific points in a physical model. In that sense, GAMBIT geometry operations represent only intermediate steps in the overall process of creating a usable model. Because GAMBIT meshing procedures involve only those components that exist in the foreground of the model, virtual geometry operations provide the user with a convenient and powerful means of controlling the shape and density of the mesh in localized regions of the model and, therefore, in the model as a whole.

A.3.2 Virtual Entity Categories

There are two general categories into which entities that participate in GAMBIT virtual geometry operations can be grouped:

• Relationship
• Class

The relationship category defines which specific real and/or virtual entities are associated with each other by means of a given virtual geometry operation. The class category describes the nature of the association—that is, the manner in which a given virtual entity is defined by one or more real entities to which it refers.

The following sections outline the general rules of nomenclature that are employed with respect to the categories listed above.

Relationship Category

The relationship category includes two general classifications:

• Host
• Guest

Host entities are real or virtual entities that are in some way referenced by one or more virtual entities. In most cases, they exist in the background of the model. Guest entities are virtual entities that reference one or more real or virtual entities. They exist in the foreground of the model (see NOTE below).

NOTE: If a virtual entity serves as the host for another virtual entity, the host virtual entity exists in the model background.

Class Category

There are five basic classes of virtual entities, each of which is defined by the nature of the relationship between the virtual (guest) entity and its real (host) entity (or entities). The five classes of virtual entities are as follows:

• Superset
• Subset
• Interpolant
• Parasite
• Orphan

The following sections describe each of the entity classes listed above.

Superset Entities

A superset entity is a virtual entity that references two or more real entities. For example, the virtual edge, v_edge.7, shown in Figure A-2, above, constitutes a superset entity, because its shape is defined in reference to two real entities (edge.1 and edge.2 in Figure A-1). Superset entities occupy the foreground of the model, and the real entities that comprise the elements of their sets occupy the background of the model (see NOTE below).

NOTE: If a superset entity constitutes one component of a higher superset entity, the lower superset entity occupies the model background. For example, if you merge edge.6 and v_edge.7 in Figure A-2 to create the superset entity v_edge.8, GAMBIT places v_edge.7 in the model background.

Subset Entities

A subset entity is a virtual entity that constitutes one element in a set of entities that reference a single host entity. As an example of a subset entity, consider the topological configuration shown in Figure A-4(a). The configuration consists of a straight real edge (edge.1) and its endpoint vertices (vertex.1 and vertex.2).

Figure A-4: Subset virtual entities-example

If you perform a virtual split operation (see "Split Operations," below) on edge.1 using the split point indicated in Figure A-4(a), GAMBIT creates a virtual vertex (v_vertex.3) at the split point and replaces edge.1 with two virtual edges, v_edge.2 and v_edge.3 (see Figure A-4(b)). The virtual edges constitute subset entities, because they are each part of a set of entities that reference a single real entity—that is, edge.1.

Interpolant Entities

An interpolant entity is a virtual entity the geometrical description of which represents an average of two or more real entities to which it refers. As an example of an interpolant entity, consider the topological configuration shown in Figure A-5(a). The configuration consists of two real, NURBS edges and their respective endpoint vertices. The edges exist in a single plane and are located near to each other but are not connected to each other.

Figure A-5: Interpolant virtual entity—example

If you perform a virtual connect operation (see "Connect Operations," below) that involves edge.1 and edge.2, GAMBIT replaces the edges with a single virtual edge, v_edge.3 (see Figure A-5(b)). The geometry of the virtual edge represents a composite average of the geometries of edge.1 and edge.2. Similarly, its endpoint vertices, v_vertex.5 and v_vertex.6, are located at points that represent the average positions of the vertex pairs [vertex.1, vertex.3] and [vertex.2, vertex.4], respectively.

The virtual edge, v_edge.3, in Figure A-5(b), constitutes an interpolant entity that references edge.1 and edge.2. Similarly, the virtual vertices, v_vertex.5 and v_vertex.6, constitute interpolant entities each of which references a distinct pair of real vertices.

Parasite Entities

A parasite entity is a virtual entity that references a single, higher-order host entity such that its geometry is defined by that of the host entity. The location of a parasite virtual vertex is defined by reference to a host edge or face; the shape and orientation of a parasite virtual edge is defined by reference to a host face.

The following examples illustrate the operations and properties that are associated with the construction and modification of parasite vertices and edges.

Parasite Vertices

Constructing a Parasite Vertex

As an example of the construction of a parasite vertex, consider the topological configuration shown in Figure A-6(a). The configuration consists of a straight real edge that is identical in geometry to that shown in Figure A-4(a).

Figure A-6: Parasite virtual vertex-example

If you perform a virtual construct operation (see "Construct Operations," below) using the construction point indicated in Figure A-6(a), GAMBIT creates a parasite virtual vertex, v_vertex.3, at the construction point (see Figure A-6(b)). The virtual vertex, v_vertex.3, is constrained to lie on edge.1 but is not topologically connected to edge.1.

The configuration shown in Figure A-6(b) differs from that of Figure A-4(b) in that v_vertex.3 does not constitute an endpoint vertex for any virtual edge that derives its geometry from edge.1. Instead, v_vertex.3 exists as an independent entity that directly references a higher-topology host entity—that is, edge.1.

Modifying a Parasite Vertex

You can modify the position of any parasite vertex by means of the GAMBIT Slide Virtual Vertex operation. The Slide Virtual Vertex operation allows you to relocate a parasite vertex anywhere along the curve or surface to which it refers. For example, GAMBIT allows you to reposition v_vertex.3 in Figure A-6(b) anywhere along the curve represented by edge.1. (For a description of the procedures and specifications required to modify the position of a virtual vertex, see "Slide Virtual Vertex" in Chapter 2 of this guide.)

Parasite Edges

Constructing a Parasite Edge

As an example of the construction of a parasite edge, consider the topological configuration shown in Figure A-7. The configuration consists of a real four-sided face (face.1) that possesses a curved, 3-D surface and is bounded by two straight real edges (edge.2 and edge.4), two curved real edges (edge.1 and edge.3).

Figure A-7: Curved, four-sided face

If you construct virtual vertices at the two construction points indicated in Figure A-7, GAMBIT creates the two parasite vertices shown in Figure A-8. The parasite vertices are constrained to lie on the surface of face.1 but are not topologically connected to face.1.

Figure A-8: Curved, four-sided face with parasite virtual vertices

If you perform a virtual construct operation to create a virtual edge that employs v_vertex.5 and v_vertex.6 as its endpoints and face.1 as its host entity, GAMBIT creates the parasite virtual edge shown in Figure A-9.

Figure A-9: Curved, four-sided face with parasite virtual edge

The parasite edge, v_edge.5, is constrained to lie along the surface of face.1 but is not topologically connected to face.1. The geometry of v_edge.5 represents the projection onto the surface of face.1 of a straight line drawn between v_vertex.5 and v_vertex.6.

Modifying the Geometry of a Parasite Edge

As noted above, the GAMBIT Slide Virtual Vertex operation allows you to reposition an existing parasite vertex under the constraint that it remains on its host entity. If you reposition a parasite vertex that constitutes one endpoint of a parasite virtual edge, GAMBIT redefines the geometry of the parasite edge when it repositions the parasite vertex (for example, see Figure A-10).

Figure A-10: Repositioned parasite vertex and edge

For a description of the procedures and specifications required to modify the position of a virtual vertex, see "Slide Virtual Vertex" in Chapter 2 of this guide.

Orphan Entities

An orphan entity is a virtual entity that does not reference any host entity. Orphan entities derive their geometries only from the lower-topology components that comprise their boundaries.

Constructing an Orphan Edge

As an example of an orphan entity, consider again the topological configuration shown in Figure A-8, above. The configuration consists of a curved, non-planar face upon which two parasite virtual vertices have been constructed.

If you perform a virtual construct operation to create a virtual edge that includes v_vertex.5 and v_vertexmgimage6 as its endpoints of a virtual edge but do not specify a host entity for the edge, GAMBIT creates the orphan virtual edge, v_edge.5, shown in Figure A-11. Unlike the parasite edge shown in Figure A-9, the geometry of the orphan edge consists of a straight line drawn between its two endpoint vertices.

Figure A-11: Curved, four-sided face with orphan virtual edge

Modifying the Geometry of an Orphan Edge

As noted above, the GAMBIT Slide Virtual Vertex operation allows you to reposition an existing parasite vertex under the constraint that it remains on its host entity. If you reposition a parasite vertex that constitutes one endpoint of an orphan virtual edge, GAMBIT redefines the geometry of the orphan edge when it repositions the parasite vertex.

For a complete description of the procedures and specifications required to reposition a virtual vertex, see "Slide Virtual Vertex" in Chapter 2 of this guide.

Constructing an Orphan Face

It is possible to construct an orphan virtual face from edges that constitute the boundaries of a real face. Although the orphan virtual face shares common edges with the real face, its surface may differ in shape from that of the real face. The difference is due to the fact that, while the surface of the real face possesses its own geometrical description, the surface of the orphan face represents an interpolation based on the geometrical descriptions of its bounding edges.

A.4 Virtual Geometry Operations

There are two general types of GAMBIT virtual geometry operations:

• Low-level
• High-level

Low-level virtual operations are specialized operations that operate on individual topological entities or on pairs of entities. High-level virtual operations consist of two or more low-level operations that are grouped according to specific purposes.

The following sections describe the various low- and high-level virtual geometry operations available in GAMBIT.

A.4.1 Low-Level Operations

GAMBIT provides the following types of low-level virtual operations.

The following sections describe the operations listed above.

Merge Operations

When you merge two entities, GAMBIT replaces the entities with a single virtual entity the geometry of which represents the combination of the geometries of the merged entities. The virtual (guest) entity constitutes a superset of the merged (host) entities.

The following examples illustrate the principles of GAMBIT merge operations as they apply to face and edge entities.

Merging Faces

As an example of a merge operation, consider the two four-sided faces (face.1 and face.2) shown in Figure A-12(a). The faces share a common edge (edge.1).

Figure A-12: Merging of two real faces

If you merge face.1 with face.2, GAMBIT replaces them in the model foreground with a single virtual face, v_face.3 (see Figure A-12(b)). The virtual face does not possess its own geometrical description but derives its geometry from the mathematical definitions of the surfaces that describe face.1 and face.2.

When you merge two entities, GAMBIT shifts both entities to the model background and also shifts the entity by which they are connected. For example, if you perform the merge operation illustrated in Figure A-12, GAMBIT shifts face.1, face.2, and edge.1 to the model background.

The following table summarizes the entity populations of the model foreground and background before and after the merge operation illustrated in Figure A-12.

Merging Edges

As a second example of the merge operation, consider the topological configuration shown in Figure A-13(a). The configuration is identical to that shown in Figure A-12(b) but includes labels for each of the six real edges that circumscribe the virtual face and for the two real vertices that exist at the outer bending points of the virtual face.

Figure A-13: Merge operation-merging of real edges

If you merge edge2. with edge.7 and edge.4 with edge.5, GAMBIT creates the topological configuration shown in Figure A-13(b). The merged configuration consists of a single virtual face (v_face.3) that is circumscribed by four edges. Two of the edges constitute straight, real edges (edge.3 and edge.6), and two constitute bent, virtual edges (v_edge.8 and v_edge.9). The virtual edges do not possess their own geometrical descriptions but derive their geometries from the mathematical definitions of the curves that describe the real edges to which they refer.

The following table summarizes the entity populations of the model foreground and background before and after the merge operation illustrated in Figure A-13.

Merge Operation Rules and Restrictions

GAMBIT virtual merge operations are subject to the following rules and restrictions.

1. Each merge operation must involve exactly two entities to be merged.
2. The topological orders of the entities to be merged must be identical to each other. For example, it is not possible to merge a vertex with an edge or to merge an edge with a volume.
3. The entities to be merged must be connected by means of a common lower-order entity. For example, if two edges do not share a common endpoint vertex, they cannot be merged. Similarly, two faces can be merged only if they are connected by means of a common edge, and two volumes can be merged only if they are connected by means of a common face.
4. If you merge two entities that constitute subcomponents of a higher-order entity, GAMBIT replaces the higher-order entity with a virtual entity of the same order. For example, consider the five-sided, planar face shown in Figure A-14(a). If you merge edge.4 with edge.5, GAMBIT replaces them in the model foreground with v_edge.6 (see Figure A-14(b)). At the same time, however, GAMBIT replaces face.1 with v_face.2.



Figure A-14: Merge operations—replacement of higher-order entities

5. You cannot merge two edges the common endpoint vertex of which is also shared by one or more other edges in the model. For example, consider the topological configuration shown in Figure A-15. GAMBIT does not allow you to merge edge.1 with edge.2, because their common endpoint vertex (vertex.1) is also shared by edge.3.

Figure A-15: Merge operations-common-endpoint restriction

Similar rules apply to faces and volumes. For example, you cannot merge two faces a common edge of which is shared by another face.

Split Operations

When you split an entity, GAMBIT replaces the entity with two virtual entities (for exceptions, see NOTE, below). The virtual (guest) entities that result from the split constitute subsets of the split (host) entity, and they are connected to each other by means of a common lower-order virtual entity.

To perform a virtual split operation, you must specify two parameters:

• The entity to be split
• The split tool

The entity to be split can be any real or virtual edge, face, or volume that currently exists in the foreground of the model. The split tool constitutes the entity that defines the location of the split.

NOTE: If the entity to be split is a real entity, GAMBIT allows you to perform either a real or virtual split operation involving the entity. Real and virtual split operations differ from each other as follows: If you perform a real split operation, GAMBIT replaces the real entity with two real entities and deletes the original real entity from the model. If you perform a virtual split operation, GAMBIT replaces the real entity in the model foreground with two virtual entities and shifts the original real entity to the model background. For a complete description of real split operations, see Chapter 2 of this guide.

Splitting an Edge

As an example of a split operation, consider the topological configuration shown in Figure A-16(a). The configuration consists of a real, elliptical arc edge and two real endpoint vertices.

Figure A-16: Virtual split operation—real, elliptical arc edge

If you split the edge (edge.1) at the split point indicated in Figure A-16(a), GAMBIT replaces the edge with two virtual edges and connects the virtual edges by means of a common virtual vertex (v_vertex.3) which is located at the split point (see Figure A-16(b)). The virtual edges do not possess geometrical descriptions of their own but are defined in reference to the mathematical definition of the original edge.

The following table summarizes the foreground and background entity populations before and after the virtual split operation illustrated in Figure A-16.

Splitting a Face

As a second example of a virtual split operation, consider the topological configuration shown in Figure A-17(a). The configuration consists of a real, planar, quadrilateral face that is bounded by four real edges and four real vertices.

Figure A-17: Virtual split operation—real, four-sided face

To split the face, you must first create and/or specify the split tool. Figure A-17 illustrates the procedure required to split the face vertically using an edge as the split tool. The procedure involves the following steps.

When you split face.1 according to the procedure outlined above, GAMBIT replaces face.1 with v_face.2 and v_face.3 (see Figure A-17(d)). Note that, in the process of splitting the face.1, GAMBIT also splits edge.1 and edge.3 and replaces them with the virtual-edge pairs [v_edge.6, v_edge.7] and [v_edge.8, v_edge.9], respectively.

Split Operation Rules and Restrictions

GAMBIT virtual split operations are subject to the following rules and restrictions.

1. Each virtual split operation must involve only one entity to be split and one split tool.
2. When you perform a virtual split operation, you must employ a split tool that is of lower order than the entity to be split. Specifically, the rules regarding the available split tools are as follows.

NOTE: If you perform a real split operation on a face, GAMBIT allows you to use a separate face as the split tool.

Connect Operations

When you perform a virtual connect operation involving two entities, GAMBIT replaces the entities with a single virtual entity. The geometrical description of the virtual entity represents an average of the geometrical descriptions of the entities that are connected by means of the connect operation. The virtual (guest) entity is an interpolant entity that references the connected (host) entities.

NOTE: If the entities to be connected are both real entities, GAMBIT allows you to perform either a real or virtual connect operation. Real and virtual connect operations differ from each other as follows: If you perform a real connect operation, GAMBIT replaces the real entities with a single real entity and deletes the original entities from the model. If you perform a virtual connect operation, GAMBIT replaces the real entities with a single virtual entity and shifts the original entities to the model background.

Connecting Vertices

As an example of a virtual connect operation, consider the two real edges shown in Figure A-18(a). The endpoint vertices vertex.2 and vertex.3 are located in proximity to each other but are not connected to each other.

Figure A-18: Virtual connect operation—two real vertices

If you perform a virtual connect operation that involves vertex.2 and vertex.3, GAMBIT replaces them in the model foreground with a single virtual vertex, v_vertex.5 (see Figure A-18(b)). In the process, GAMBIT replaces edge.1 and edge.2 with v_edge.3 and v_edge.4, respectively.

The following table summarizes the foreground and background entity populations before and after the vertex connect operation illustrated in Figure A-18.

Connecting Edges

As a second example of a virtual connect operation, consider the two real, planar faces shown in Figure A-19. Each face is bounded by four real edges, three of which are straight and one of which is curved. The curved edges are located near each other but are not connected to each other.

Figure A-19: Virtual connect operation—two real, four-sided faces

If you perform a virtual connect operation that involves edge.2 and edge.8, GAMBIT replaces them in the model foreground with a single virtual edge, v_edge.9 (see Figure A-20). The virtual edge is bounded by the virtual endpoint vertices v_vertex.9 and v_vertex.10. In the process of the connect operation, GAMBIT also replaces face.1 and face.2 with two virtual faces (v_face.3 and v_face.4) that are connected to each other by means of their common virtual edge, v_edge.9.

Figure A-20: Virtual connect operation-connected virtual faces

The following table summarizes the foreground and background entity populations before and after the connect operation illustrated in Figure A-19 and Figure A-20.

Construct Operations

When you perform a virtual construct operation, GAMBIT creates either of two types of entities:

• Parasite
• Orphan

If you specify a host entity for the construct operation, GAMBIT creates a parasite virtual entity. Parasite entities derive their geometries from the host entity but are not directly connected to such entities. If you do not specify a host entity, GAMBIT creates an orphan entity. Orphan entities derive their geometries only from the entities that comprise their boundaries.

For examples of the types of entities that result from GAMBIT virtual construct operations, see "Parasite Entities" and "Orphan Entities" in Section A.2, above.

A.4.2 High-Level Operations

GAMBIT provides the following types of high-level virtual operations.

The following sections describe the operations listed above.

Collapse Operations

When you perform a virtual collapse operation, GAMBIT overlays the collapsed face with two or more virtual faces that represent the merging of the collapsed face with its neighboring faces. Collapse operations require that the face to be collapsed is connected by means of a common edge to each of the other faces that are involved in the operation.

As an example of a collapse operation, consider the topological configuration shown in Figure A-21. The configuration consists of three square, real faces arranged in the shape of an "L". The central face (face.2) is connected to each of its neighboring faces (face.1 and face.3) by means of two common edges—that is, edge.3 and edge.5, respectively.

If you perform a virtual collapse operation in which face.2 is collapsed between face.1 and face.3, GAMBIT performs the two-step process illustrated in Figure A-22. The process consists of the following steps.

1. Split face.2 along curve a to create two intermediate virtual faces, A and B (see Figure A-22(a)).
2. Merge face A with face.1 and face B with face.3 to create two virtual faces—that is, v_face.4 and v_face.5 (see Figure A-22(b)).

Figure A-21: Three square faces arranged in an "L" shape

Figure A-22: Virtual collapse operation—three square faces

T-Connect Operations

When you perform a virtual T-connect operation on an edge, GAMBIT performs the following operations:

• Split the edge using the projections of any vertices that exist near the edge to within a specified tolerance
• Connect the split entities

The following paragraphs illustrate the effect of T-connect operations on lone edges—that is, edges that do not constitute boundaries of a face-and boundary edges.

T-Connecting Lone Edges

As an example of a T-connect operation, consider the topological configuration shown in Figure A-23. The configuration consists of two edges, edge.1 and edge.2, neither of which constitutes a boundary edge for any higher-order entity. One of the endpoint vertices of edge.2 (vertex.3) is located near the curve that describes edge.1 to within a specified tolerance.

Figure A-23: Two perpendicular, unconnected edges

If you perform a virtual T-connect operation involving edge.1 and edge.2 in Figure A-23, GAMBIT executes the following steps (see Figure A-24).

1. Split edge.1 at Point a to create two intermediate virtual edges, A and B (see Figure A-24(a)). (NOTE: Point a represents the projection of vertex.3 onto edge.1.)
2. Connect the virtual vertex at Point a with vertex.3 to create a single virtual vertex (v_vertex.5) that constitutes the common endpoint for three virtual edges-v_edge.3, v_edge.4, and v_edge.5 (see Figure A-24(b)). (NOTE: The location of v_vertex.5 constitutes the average locations of the intermediate virtual vertex at Point a and vertex.3.)

Figure A-24: Virtual T-connect operation—two perpendicular edges

The following table summarizes the foreground and background entity populations before and after the T-connect operation illustrated in Figure A-24.

NOTE: If either edge constitutes part of a face, GAMBIT replaces the face in the model foreground with a virtual face. If the face constitutes part of a volume, GAMBIT creates corresponding a virtual volume.

T-Connecting Boundary Edges

If you perform a virtual T-connect operation that involves two edges each of which constitutes a boundary edge for an individual face, GAMBIT replaces the edges with virtual edges and replaces the faces with virtual faces. For example, consider the topological configuration shown in Figure A-25. The configuration includes two coplanar, rectangular faces (face.1 and face.2) the adjoining edges of which (edge.3 and edge.5) are offset slightly from each other.

Figure A-25: Two offset rectangular faces edges

If you perform a virtual connect operation that involves edge.3 and edge.5 and allow the formation of T-connections, GAMBIT splits both edges and connects them to each other such that they form three virtual edges (see Figure A-26). In the process of the connect operation, GAMBIT also replaces face.1 and face.2 with v_face.3 and v_face.4, respectively.

Figure A-26: T-connected boundary edges

The following table summarizes the foreground and background entity populations before and after the T-connect operation illustrated in Figure A-24.

NOTE: After you perform the connect operation illustrated in Figure A-26, above, you can merge v_face.2 and v_face.3 to create a single virtual face the geometry of which reflects the overall geometry of the original, unconnected faces—that is, face.1 and face.2. It is then possible to mesh the overall geometry of the original faces by means of a single meshing operation.

A.5 Virtual Geometry Applications

The virtual geometry operations described in Section A.4 allow you to perform several important tasks that are related to GAMBIT modeling and meshing operations. The four primary tasks are as follows:

• Clean up imported geometry
• Simplify geometry
• Decompose geometry
• Adjust the mesh

The following table describes each of the tasks listed above.

The following sections describe and illustrate the basic principles associated with each of the tasks listed above.

A.5.1 Cleaning Up Imported Geometry

Geometry that is created outside GAMBIT sometimes violates GAMBIT validity criteria. GAMBIT meshing operations do not apply to invalid geometry; therefore, you must clean up such geometry before attempting to mesh an imported model.

GAMBIT Validity Criteria

To constitute valid GAMBIT geometry, the topological components of a model must satisfy two criteria:

• Consistency
• Completeness

Consistent geometry is that for which topological components of a given entity are coincident with each other and are correctly connected to each other. Complete geometry is that which includes surface shape definitions and for which connectivity information is available.

The following sections describe the conditions of consistency and completeness and outline the use of virtual geometry operations to correct imported geometry that is inconsistent or incomplete.

Consistency Criteria

As noted above, consistent geometry is that for which topological components of a given entity satisfy the following criteria:

• They are coincident with each other—that is, they are located within proximity of each other to within a specified tolerance.
• They are correctly connected to each other.

As an example of the criteria described above, consider the topological configurations shown in Figure A-27(a) and (b). Each configuration consists of a curved face and four edges that constitute the boundaries of the face.

Figure A-27: Consistent vs. inconsistent geometry

Although the numbers and general geometries of the components in both configurations are similar, the configurations differ with respect to whether or not they satisfy GAMBIT consistency criteria. Specifically, the configuration shown in Figure A-27(a) is not consistent; whereas the configuration shown in Figure A-27(b) is consistent.

The configuration shown in Figure A-27(a) is inconsistent for the following reasons.

• The curves that define the edges are not coincident with the surface of the face at its boundaries.
• The edges are not connected to the face.
• The edges are not connected to each other at their endpoint vertices.

Conversely, the configuration shown in Figure A-27(b) satisfies the GAMBIT consistency criteria for the following reasons.

• The curves that define the edges are coincident with the surface of the face at its boundaries.
• The edges are connected to the face.
• The edges are connected to each other at their endpoint vertices.

Sources of Inconsistent Geometry

Computer automated design (CAD) programs often generate geometry data that is inconsistent with respect to GAMBIT operations. The inconsistencies arise due to the differences in tolerance values employed by the two types of programs. CAD programs are designed to create a set of mathematical descriptions that can be used to generate visual representations of geometry or to serve as a software blueprint for computer-aided manufacturing (CAM) operations. Consequently, their tolerance and consistency criteria are less strict than those employed by GAMBIT. GAMBIT operations are designed to employ geometry that is exact enough to be used for mathematical modeling operations. Therefore, they require that topological components of the model meet strict tolerance and consistency criteria.

Completeness Criteria

Model geometry is complete if it includes the following information:

• Connectivity
• Shape definitions

Connectivity information describes the relationship between, for example, the boundary edges of a given face or the endpoint vertices of a set of edges. Shape definitions describe curves or surfaces that are associated with the model edges or faces, respectively.

Example Application—Clean-up of Unconnected Edges

As an example of the use of GAMBIT virtual geometry operations to clean up geometry, consider the topological configuration shown in Figure A-28. The configuration includes 25 edges-only some of which are connected to others by means of common vertices. Although the configuration does not include any face entities, the edges are arranged such that they represent the general outline of a curved surface.

Figure A-28: Geometry clean-up example-unconnected edges

GAMBIT virtual geometry operations allow you to create a single, meshable face from the configuration shown in Figure A-28 without altering the underlying topology. To do so, you must perform the following steps:

1. Connect the edges to each other such that they form a continuous, wireframe outline of a curved surface
2. Create virtual faces from the connected edges
3. Merge the virtual faces to create a single, meshable face

Step (1)—Connecting Edges

If you perform a virtual connect operation that involves all edges shown in Figure A-28 and allow the T-connect operation, GAMBIT creates a web of connected virtual edges that forms a general outline of a curved surface (see Figure A-30).

Figure A-29: Geometry clean-up example-connected edges

Most of the virtual edges that exist in the interior of the wireframe shown in Figure A-30 constitute interpolant entities.

Step (2)—Creating Virtual Faces

If you construct virtual faces from the edge loops that result from the connect operation of Step (1), GAMBIT creates a patchwork of virtual faces the combined geometries of which generally describe a single curved surface (see Figure A-30). Each of the faces constitutes an orphan entity and is connected to its neighboring faces by means of two or more common virtual edges.

Figure A-30: Geometry clean-up example-patchwork of connected faces

Step (3)—Merging Virtual Faces

If you perform a virtual merge operation involving the virtual faces shown in Figure A-30, GAMBIT constructs a single, virtual face, v_face.8, the surface of which represents the shape of the original arrangement of unconnected edges (see Figure A-31). Because v_face.8 exists as a single entity, you can apply a single set of mesh specifications to the face, thereby creating a smooth, unified mesh, such as that shown in Figure A-31.

Figure A-31: Geometry clean-up example-single meshed surface

NOTE: It is possible to further simplify the geometry shown in Figure A-31 by merging the edges that comprise its boundaries. If you do so, GAMBIT eliminates the virtual vertices at the non-corner positions and thereby frees the meshing procedures from the constraint of having to locate mesh nodes at those vertices.

A.5.2 Simplifying the Geometry

Some models that represent detailed descriptions of real-world objects include details that complicate the process of creating a mesh. Such details may be important to processes that require an exact description of the object-such as computer-aided manufacturing (CAM) systems-but are often irrelevant to mathematical analyses for which a mesh is created. They include, but are not limited to, fillets and chamfered edges, sliver-shaped faces, holes, and small depressions or bumps on the surface of model faces.

There are two basic methods of simplifying complex geometry descriptions so that they can be used to create a mesh that is suitable for mathematical analyses. The two methods are as follows:

• Changing the existing real geometry itself
• Using virtual operations to overlay the existing real geometry with virtual counterparts

The process of changing the existing real geometry involves procedures that are often complicated and costly, such as the redefinition of curves and surfaces. The redefined geometry that results from the process often represents an approximation of the original geometry, thereby reducing the accuracy of the model. Furthermore, in some cases, the process itself is incapable of producing meshable geometry.

The process of using virtual operations, on the other hand, involves simple procedures that simplify the model without compromising or altering its underlying real geometry. Furthermore, virtual operations retain the exact geometrical descriptions of the original geometry, thereby circumventing the loss in accuracy that results from the process of changing the existing real geometry.

The following examples illustrate the use of virtual operations to simplify geometry and improve or enable GAMBIT meshing operations. Specifically, the examples illustrate the following operations.

Example 1—Simplifying a Square Surface

As an example of the use of virtual geometry operations to simplify and/or improve meshing operations, consider the topological configuration shown in Figure A-32. The configuration consists of two triangular faces and one quadrilateral face all of which are arranged such that they form a square, planar surface. The faces face.1 and face.2 are connected to each other by means of their common edge, edge.7, and face.2 and face.3 are connected to each other by means of edge.8.

Figure A-32: Three real faces arranged in the shape of a square

If you mesh each face separately by means of GAMBIT default meshing schemes and parameters, GAMBIT creates Tri Primitive meshes on the two triangular faces and a Quad-Map mesh on the quadrilateral face. (For a complete description of GAMBIT face-meshing schemes and parameters, see Chapter 3 of this guide.) The resulting mesh for the entire square surface is shown in Figure A-33.

Figure A-33: Example mesh-three meshed faces

The mesh shown in Figure A-33 is highly irregular with respect to the square surface. Such irregularity is a consequence of the fact that each face is meshed separately and that GAMBIT meshing operations require the creation of mesh nodes on all edges, including those that are common to multiple faces that describe a single surface.

In order to create a mesh that is smooth and continuous across the square surface shown in Figure A-32, you must form a single face the surface of which represents a combination of the three faces that make up the square. GAMBIT merge operations applied to the faces and edges of the configuration allow you to form such a face. Specifically, the procedure required to form a single, square face is as follows:

1. Merge face.2 with face.3 to create v_face.4.
2. Merge v_face.4 with face.1 to create v_face.5.
3. Merge the edges of v_face.5 to remove intermediate vertices.

The resulting virtual face includes only four edges and four vertices in the model foreground, therefore, GAMBIT meshing operations produce a smooth Map mesh for the square surface (see Figure A-34).

Figure A-34: Simplification of a square surface—Step 3

Example 2—Removing a Bump

As an example of the use of virtual geometry operations to remove irrelevant details and improve mesh quality, consider the topological configuration shown in Figure A-35. The configuration consists of two real faces one of which (face.2) constitutes a spherical bump in an otherwise planar, square surface.

Figure A-35: Planar square face with circular bump

To mesh the entire surface represented by the two faces shown in Figure A-35, you must mesh each face separately. The configurations of the faces dictate that GAMBIT typically applies a Pave meshing scheme to each face. Figure A-36 shows the overall mesh grid that results from the GAMBIT meshing operations.

Figure A-36: Pave-mesh grids on square face with circular bump

If you merge face.1 with face.2 by means of GAMBIT virtual geometry operations, GAMBIT replaces them with a single, virtual face, v_face.3 (see Figure A-37). The geometry of the virtual face represents that of the combined real faces, but its lower-topology components include only the outer edges and vertices of face.2. As a result, v_face.3 constitutes a simple quadrilateral face and can be meshed according to the map meshing scheme shown in the figure.

Figure A-37: Map-mesh grids on square face with circular bump

Note that the mesh node spacing is much more regular for the map meshing scheme shown in Figure A-37 than for the pave meshing schemes shown in Figure A-36. In addition, it is possible to employ a mapped mesh that is courser than a paved mesh of similar mesh quality.

Example 3—Simplifying a Cube with a Cutout Corner

As an example of the use of virtual geometry operations to enable meshing of a volume, consider the topological configuration shown in Figure A-38. The configuration consists of a regular cubic volume from which one corner has been removed. Each of the inside edges of the cutout section has been rounded by means of a volume blend operation, as has the vertex at which they intersect.

Figure A-38: Cube with cutout corner

Although each face of the volume is meshable on its own by means of a standard GAMBIT face-meshing scheme, the volume itself cannot be meshed as a single entity by any standard volume-meshing scheme. It is possible, however, to simplify the volume by means of GAMBIT virtual geometry operations and, thereby, to render it meshable.

The steps involved in simplifying the volume are as follows:

1. Collapse the three curved faces that constitute the inside corners of the cutout. The following table lists each face in relation to the neighboring faces between which it is collapsed.

Face to be Collapsed	Neighboring Faces
face.10	face.7 and face.8
face.11	face.8 and face.9
face.12	face.9 and face.7

When you perform the virtual collapse operations described in the table above, the volume appears as shown in Figure A-39.



Figure A-39: Simplifying a cube with cutout corner—Step 1

2. Collapse face.13 between v_face.14, v_face.15, and v_face.16.

The final form of the simplified volume consists of six rectangular faces and three L-shaped faces and can be meshed by means of the Submap volume meshing scheme (see Figure A-40).

Figure A-40: Simplifying a cube with cutout corner—Submap mesh

A.5.3 Decomposing the Geometry

It is sometimes necessary to decompose complex models into meshable subvolumes. GAMBIT provides two general types of operations that allow you to perform such decompositions:

• Real Boolean operations
• Virtual split operations

Real Boolean operations and virtual split operations exhibit the following characteristics.

In addition to the characteristics noted above, virtual faces often constitute more usable split entities than do real faces, because their surfaces do not exhibit the high curvature that is sometimes characteristic of real faces formed from existing edges.

A.5.4 Adjusting the Mesh

In addition to the applications noted above, you can use virtual geometry operations to create "flexible" topology that can, in turn, be used to control the shape and density of an existing mesh. As an example of the use of such geometry, consider the topological configuration shown in Figure A-41. The configuration consists of a curved, 3-D surface identical to that shown in Figure A-7. Its surface is represented in the model foreground by two virtual faces (v_face.2 and v_face.3) that are connected by means of a common virtual parasite edge (v_edge.5). Two virtual parasite vertices (v_vertex.5 and v_vertex.6) comprise the endpoints of the parasite edge and are constrained to lie on edge.1 and edge.3, respectively.

Figure A-41: Curved 3-D face with virtual parasite entities

The virtual faces shown in Figure A-41 are each meshed according to identical meshing parameters, and the virtual vertices that comprise the endpoints of their common edge are located at the midpoints of their respective edges.

As noted in Section A.2, above, GAMBIT allows you to reposition parasite vertices along or across their host edges or faces. Consequently, you can change the shape of the mesh shown in Figure A-41 simply by repositioning either or both of the parasite virtual vertices. When you do so, GAMBIT repositions the parasite edge, thereby changing the shape of the two virtual faces that represent the surface. In the process, GAMBIT repositions any existing mesh nodes on each of the faces.

Figure A-42 shows an example mesh that results from repositioning both parasite vertices from their positions as shown in Figure A-41.

Figure A-42: Curved 3-D face—repositioned parasite vertices

Note that, although the sizes and shapes of individual mesh elements change when you reposition the virtual vertices, the connectivity of the mesh remains unchanged.

© Fluent, Inc. 10/22/01