7. Solid Modeling

Assoc.Prof.Dr. Ahmet Zafer Şenalpe-mail: azsenalp@gmail.com

Mechanical Engineering DepartmentGebze Technical University

ME 521Computer Aided Design

• 1967 Surface Modeling(S.A. Coons) • 1973 Solid Modeling–

Constructive Solid Geometry(Laning et al., Draper Lab.)

• 1973 Solid Modeling– Boundary Representation(Ian Braid, Cambridge U.)

• 1985 Feature Based Modeling(Pratt & Wilson)

• 1990 Parametric Modeling(PTC)

History of Geometric Modeling

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7. Solid Modeling

• Using volume information– weight or volume calculation, mass center calculation, moment of inertia calculation,– Finite element analysis model preperation

• Using volume and boundary information– CNC code generation, robotics and assemblies

Solid Modeling

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7. Solid Modeling

• Unbounded geometric entities• Each one of them divides the representation space into infinite portions, one filled

with material and the other empty• Surfaces can be considered half-space boundaries and half spaces can be

considered directed surfaces• An object is defined by the volume space contained within the defined boundary

of the objectIntroduces the direction into the modeling, thus enables the topological information

be stored in a geometric model

Half Space

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7. Solid Modeling

• By specifying different boundary surface, we can have any half-spaces;• The most commonly used half spaces:• Planar• Cylindrical• Spherical• Conical• Toroidal• By combining half-spaces (using Boolean operations) in a building block fashion,

various solids can be constructed.

Half Space

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7. Solid Modeling

The properties that a solid model or an abstract solid should capture mathematically can be stated as follows:

1. Rigidity. This implies that the shape of a solid model is invariant and does not depend on the model location or orientation in space.

2. Homogeneous three-dimensionality. Solid boundaries must be in contact with the interior. No isolated or dangling boundaries should be permitted.

3. Finiteness and finite describability. The former property means that the size of the solid is not infinite while the latter ensures that a limited amount of information can describe the solid. The latter property is needed in order to be able to store solid models into computers whose storage space is always limited. It should be noted that the former property does not include the latter and vice versa. For example, a cylinder that may have a finite radius and length may be described by an infinite number of planar faces.

Properties of Solid Model

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7. Solid Modeling

4. Closure under rigid motion and regularized Boolean operations. This property ensures that manipulation of solids by moving them in space or changing them via Boolean operations must produce other valid solids.

5. Boundary determinism. The boundary of a solid must contain the solid and hence must determine distinctively the interior of the solid.

Properties of Solid Model

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7. Solid Modeling

• Constructive Solid Geometry: CSG• Boundary Representation: B-rep• Hybrid (Feature Based Modeling)

Solid Modeling Approaches

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7. Solid Modeling

• Based on simple geometric primitives– box– cone– sphere– etc.

Constructive Solid Geometry

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7. Solid Modeling

• Primitives are placed and combined by using boolean operations boolean operations:– Union

denoted by

– Difference or subtraction denoted by

– Intersectiondenoted by

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7. Solid ModelingConstructive Solid Geometry

B

A

C

A-B(A-B) C

Constructive Solid GeometryExample:

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7. Solid Modeling

Constructive Solid GeometryExample:

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7. Solid Modeling

• All the operations are stored as a boolean tree

Constructive Solid Geometry

sphere cylinder

cylinder

rotation

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7. Solid Modeling

Boundary Representation (B-Rep)

• The boundary representation method represents a solid as a collection of boundary surfaces. The database records both of the surface geometry and the topological relations among these surfaces.

• This representation is used mainly for graphical displays.• Solids are represented by faces, edges and vertices.• To enable the validity of the object topological rules should be guaranteed;

– Faces are bounded with edges– Each edge should be shared with exactly 2 faces– Each edge should have a vertex at each end – At least 3 edges should connect at each vertex

• The surface must be closed• Euler operations are used to construct B-rep models by combining faces, edges and vertices.

Euler operation are lower level operations than boolean operations.

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7. Solid Modeling

VerticesEdges

Faces

Boundary Representation (B-Rep)

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7. Solid Modeling

Boundary Representation (B-Rep)

Geometry should obey Euler rules: V: Number of verticesE: Number of edgesF: Number of faces

V-E+F=2

• B-rep does not guarantee that a group of boundary surfaces should form a closed solid.• Most CAD software uses both CSG and B-rep methods (hybrid structure).

16 – 24 + 10 – 2 = 2

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7. Solid Modeling

Solid modelers store more information (geometry and topology) than wireframe or surface modelers (geometry only).

Geometry is the actual dimensions that define the entities of the object. The geometry that defines the object shown in Figure1 is the lengths of lines L1, L2 and L3,

the angles between the lines, and the radius R and the center P1 of the half-circle. Topology (sometimes called combinatorial structure), on the other hand, is the connectivity

and associativity of the object entities. The topology of the object shown below can be stated as follows;

L1 shares a vertex (point) with L2 and C1 L2 shares a vertex with L1 and L3 ; L3 shares a vertex with L2 and C1. L1 and L3 do not overlap

Boundary Representation (B-Rep)

Same geometry but different topology(P1 is inside/outside) Same topology but different geometry

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7. Solid Modeling

• Based on these definitions, neither geometry nor topology alone can completely model objects.

• Wireframe and surface models deal only with geometrical information of objects, and are therefore considered incomplete and ambiguous.

• From a user point of view, geometry is visible, and topology is considered to be nongraphical relational information that is stored in solid model databases and is not visible to users.

Boundary Representation (B-Rep)

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7. Solid Modeling

Geometry construction commands:

Geomerty Construction With B-Rep

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7. Solid Modeling

CSG• Simple representation• Limited to simple objects• Stored as binary tree• Difficult to calculate• Rarely used anymore

B-Rep• Flexible and powerful representation• Stored explicitly• Can be generated from CSG

representation• Used in current CAD systems

CSG-B-Rep

Most of the CAD system uses B-rep or hybrid method.

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7. Solid Modeling

• Parts modelled by adding features to a base part• Features represent manufacturing “operations”

– hole– fillet– round– rib– chamfer– slot– pocket– etc.

• Material can be added or subtracted, similar to CSG• Features are not limited to simple primitives, and can be created by Extrusion Sweeping Revolving etc.• A history tree is created, similar to a CSG boolean tree• Today most of the CAD systems use feature based modelling

Feature Based Modelling

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7. Solid Modeling

Feature Based Modelling

Features represent manufacturing “operations”

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7. Solid Modeling

Fillet

Slot

Hole

Shell

Feature Based Modelling

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7. Solid Modeling

Final Part

Part 3

Hole

Shell

Part 2

FilletsPart 1

SlotBase

Featuresadded

Increasingpartcomplexity

Feature Based ModellingHistory Tree

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7. Solid Modeling

• The part is created from the history tree• Features can be added, deleted and re-ordered• Feature dimensions can be changed• Feature parameters can be changed

– eg. From protrusion to cutout

Feature Based ModellingModifying Parts

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7. Solid Modeling

Feature Based Modelling

Feature-based, Parametric Models – Pro/E

Feature-based, Parametric Solid Modeling system represents the recent advance of computer geometric modeling. It is used as the foundation of Pro/ENGINEER, etc.

Feature-based, parametric solid modeling eliminated the direct use of common geometric primitives such as cone, cylinder, sphere, etc, since these primitives only represent low-level geometric entities. In designing and manufacturing mechanical parts, one would always refer to mechanical features.

The modeling approach uses sweeping to form the main shape of the part, and build-in mechanical features to specify the detailed geometry of the model. These features include holes (through, blind, sink), rounds, chamfers, slots, etc. Operations to solid model, such as cut and shell (change a solid model into a hollow shell) are also supported.

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7. Solid Modeling

Feature Based Modelling

Feature-based, Parametric Models – Pro/E

To create the 2D cross-section for sweeping, a 2D sketch needs to be generated in the 2D Sketcher. A user can sketch the rough shape of the closed shape. The system will automatically assign a dimension value of the sketched feature. The dimensions of the sketched feature can be changed at any time by simply entering the desired value, or kept as a variable, allowing even more convenient change of its value. The user has to provide all necessary dimensions to pass the section of cross-section generation. Problems of under- or over- dimensioning can be identified.

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7. Solid Modeling

Sweeping Operations 3D solids are formed by using 2D cross sections Sweeping consists of :

- extrude- revolve- sweep

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7. Solid Modeling

Feature based modeling – Base Features :

can be a datum plane/coordinate axis or a sketched feature. All future geometry will refer to this feature directly/indirectly.

– Sketched Features: These are user created features using a sweep/blend technique from 2D sections. Sweep => extrude, revolve ...

– Referenced Features: These features reference existing features and do not have to be drawn. e.g.

holes, chamfers ... – Datum Features:

Features used only to provide a reference for other features.

Modeling with Pro/E

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7. Solid Modeling

Constraint Based Modelling

• User constraints geometry based on Design Intent

• Design variations can be generated by changing a few key dimensions

• Geometry is automatically regenerated based on constraints

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7. Solid Modeling

D1

D2

D4

D3

D5

Constraint Based Modelling

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7. Solid Modeling

Constraint Based Modelling

Design Intent:

• The part should be twice as long as it is wide• The hole should be centred in both directions• The hole diameter should be 50mm

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7. Solid Modeling

Parametric and Variational Modelling

• Parametric modelling– constraints defined sequentially– each constraint calculated based on previously defined constraints– order of constraint specification is important

• Variational modelling– constraints solved simultaneously– order of constraint specification doesn’t matter

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7. Solid Modeling

Parametric Definition

User specifies dimension D1, other dimensions calculated sequentially

50

2/

2

2/

5

34

13

12

1

D

DD

DD

DD

xD

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7. Solid Modeling

Variational Definition

Solve system of simultaneous equations:

0

02

02

02

050

1

43

31

21

5

xD

DD

DD

DD

D

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7. Solid Modeling

Constraint Types

• Ground constraints• Dimensional constraints• Geometric constraints

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7. Solid Modeling

Ground Constraints

• Horizontal• Vertical• Both ends fixed• Point location• X of point• Y of point• Angle of line

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7. Solid Modeling

Dimensional Constraints

• Horizontal dimension• Vertical dimension• Linear dimension• Angular dimension• Radial dimension

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7. Solid Modeling

Geometric Constraints

• Parallel• Perpendicular• Tangent• Collinear, coincident, coplanar

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7. Solid Modeling

Modelling Approach

• Sketch approximate geometry• Generate solids and features• Add constraints and dimensions afterwards

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7. Solid Modeling