机械CADChapter 4 Modelling of Solids

Chapter 4 Modeling of Solids

4.1 Introduction

So far we have represented objects with wireframe and surface modeling. Another

method of geometric modeling is to use vertices, edges and surfaces to define a solid.

Solid

modeling

systems

allow

users

to

create,

store

and

manipulate

unambiguous

models

of

physical

solid

objects,

In

recent

years,

solid

modeling

has

become

the

prevalent and favorable tool for applications in design and manufacturing.

Some of the major advantages of solid modeling are listed here.

(1)

Visualization

of

components

in

3D

space

or

in

realistic

surroundings

can

be

made easier.

(2)

Solid

models

can

be

used

as

data

input

to

other

systems,

like

FE- systems,

integration between solid model and FE-model provides great possibility to cut short

the product development process.

(3) Components can be machined straight from the files created by these systems.

A solid model contains

both

the

geometry

data

and topological

data. Topological

data

describe

the

connectivity

and

associability

of

the

object

entities.

Solid

model

generation is often not unique, that is, there are often several different ways to create a

solid model. CAD users need to generate solid models which can make the computer

storage small and suited for later utilize.

Boundary representation (B-rep) and Constructive Solid Geometry (CSG) are two

most popular schemes for solid modeling. B-reps are based on the topological notion

that an object is bounded by faces. CSG is based on that an object can be divided into

a set of primitives. Table 4.1 shows some widely used CAD systems.

4.2 Solid Representation

Most geometric objects we see every day are solids. Solids can be very simple like

a cube or very complex like a piston engine. To be processed by

computers, solids

must

have

some

representations

that

can

describe

the

geometry

and

characteristics

completely. In fact, a good representation should address the following issues.

①

Domain:

While

no

representation

can

describe

all

possible

solids,

a

representation should be able to represent a useful set of geometric objects. Domain

should give a useful set of physical objects that can be represented.

②

Unambiguity: A solid should be represented without any doubt. An unambiguous

representation is usually referred to as a complete one. Figure 4.1 shows an example

of ambiguous solids.

③

Uniqueness:

There

is

only

one

way

to

represent

a

particular

solid.

If

a

representation is unique, then it is easy to determine if two solids are identical since

one can just compare their representations.

④

Accuracy: A representation is said accurately if no approximation is required.

⑤

Validity:

A

representation

should

not

create

any

invalid

or

impossible

solids.

More precisely, a representation will not represent an object that does not correspond

to a solid.

⑥

Closure; Solids will be transformed and used with other operations such as union

and intersection,

solid.

⑦

Compactness and efficiency:

A

good representation

should be compact

enough

for

saving

space

and

allow

for

efficient

algorithms

to

determine

desired

physical

characteristics.

There

are

different

methods

to

create

models,

such

as

half-space,

boundary

representation

(B-rep),

constructive

solid

geometry

(CSG),

sweeping,

analytic

solid

modeling,

cell

decomposition.

Following

concepts

are

important

in

solid

modeling:

solid primitives, Boolean operations, geometry closure and regularized set operations.

(1) Primitives

Solid primitives are frequently used in solid modeling. CAD/CAM systems provide

various

primitives

such

as

block,

cylinder,

cone,

and

sphere.

Figure

4.2

shows

an

example of these solid entities.

(2) Boolean operations

Union,

difference,

complement

and

intersection

are

essential

Boolean

operations.

Figure 4.3 shows Boolean operations used in the solid modeling. Boolean operations

are intuitive to use and easy to understand, and they provide for rapid manipulation of

large

amounts

of

data.

Because

of

this,

Boolean

operations

are

also

used

in

many

non-CSG systems.

(3) Interior, exterior and closure

We

need

the

concepts

of

interior,

exterior

and

closure

to

understand

regularized

Boolean operators. Intuitively, the interior of a solid consists of all points lying inside

of the solid; the closure consists of all interior points and all points on the surface of

solids;

and

the

exterior

of

a

solid

is

the

set

of

all

points

that

do

not

belong

to

the

closure.

A open ball with centre (a, b, c) and radius r consists of all points that satisfy the

following relation,

(x-a)

2

+(y-b)

2

+ (z-c)

2

2

< br>

A point P is an interior point of a solid S if there exists a radius r such that the open

ball with centre P and radius r is contained in the solid S. The set of all interior points

of solid S is the interior of S, written as int (S). Based on this definition, the interior of

an open ball is the open ball itself.

On the other hand, a point Q is an exterior point of a solid S if there exists a radius r

such that the open ball with centre Q and radius r does not intersect S. The set of all

exterior point of solid S is the exterior of solid S, written as ext(S).

Those

points

that

are

neither

in

the

interior

nor

in

the

exterior

of

a

solid

S

constitutes the boundary of solid S, written as b(S). Therefore, the union of interior,

exterior and boundary of a solid is the whole space.

The closure of a solid S is defined to be the union of the interior and the boundary

of S, written as closure (S). Or, equivalently, the closure of solid S contains all points

that are not in the exterior of S.

(4) Regularized set operations

When set operations are used in geometry modeling, unwanted geometry may be

got. In Fig. 4.4, two cubes touch each other and their intersection is a rectangle shown

on the right. A rectangle is not a three-dimensional object and hence not a solid.

To

eliminate

these

lower

dimensional

branches,

the

three

set

operations

are

regularized as follows. The idea is simple.

Compute

the

result

as

usual

and

lower

dimensional

components

may

be

generated.

Compute

the

interior

of

the

result.

This

step

removes

all

the

lower

dimensional components.

Compute

the

closure

of

the

result

obtained

in

the

above

step.

This

adds

the

boundary back.

Let op* be the regularized operator and let A and B be two solids. Then,

A

∪

* B ,

A

∩

*B

and A-* B can be defined mathematically based on the above description.

A

∪

* B - c1osure(int(the set union of A and B))

A

∩

*B

- c1osure(int(the set intersection of A and B))

A-* B - c1osure (int (the set difference of A and B))

Based

on

this

definition,

the

intersection

of

the

two

cubes

shown

in

Fig.

4.4

is

empty. The intersection of these two cubes is a rectangle, which is a two-dimensional

object and has no interior. Hence, after taking interior (i.g. , int()), we get an empty set,

whose closure is also empty. Consequently, the intersection is empty.

4.3 Boundary Representation

Boundary models have a hierarchical format. As mentioned earlier, B-reps describe

objects

bounded by

a set

of faces.

Each face has

its

underlying closed (continuous)

and oriented surface as shown in Fig. 4.5 and the face is bounded by edges. Each edge

is bounded by vertices. A boundary model of an object is comprised of faces, edges

and vertices.

The surface of a solid consists of a set of well-organized faces, each of which is a

piece of some surface (e. g., a surface patch). Faces may share vertices and edges that

are

curve

segments.

Therefore,

a

B-rep

is

an

extension

to

the

wireframe

model

by

adding face information to the later.

The orientation of each face is important. Normally, a face is surrounded by a set of

vertices. Using the right- handed rule, the ordering of these vertices for describing a

particular face must

guarantee that the normal

vector of that

face is

pointing

to

the

exterior

of

the

sol-id.

Normally,

the

order

is

counter

clockwise.

Therefore,

by

inspecting normal vectors one can immediately tell the inside and outside of a solid

under B-rep. This orientation must be done for all faces.

The database of a boundary model contains

both

topological

and

geometric data.

Topological data provide the relationships among vertices, edges and faces similar to

that used in a wireframe model. In addition to connectivity, topological information

also

includes

orientation

of

edges

and

faces.

Geometric

information

is

usually

equations of the edges and faces. Topology is created by performing Euler operations

and geometry is created by per-form Euclidean calculations. Both the topological and

geometry validity should be checked to

avoid

nonsense object.

Euler operators also

provide designers with drafting functionality.

Boolean operators are often used to

create

and edit models. Since B-rep

requires

explicit

representation

of

the

boundary

of

the

solid,

boundaries

must

be

evaluated

after the operation. For CSG model, the Boolean operation is simply an addition to the

CSG tree.

The main

advantage of

B-rep is

its

ability to construct

solids that are difficult to

build

u-sing

primitives.

The

main

disadvantage

is

that

it

requires

large

amounts

of

storage because it stores the boundary explicitly.

4.3.1 Euler Formula

Objects that are often encountered in engineering are polyhedral or curved objects.

Figure 4.6 shows examples of polyhedral.

Eu1er

(or

Euler-Poincare)

law

is

that

a

polyhedron

is

topologically

valid

if

they

satisfy the following equation,

F-E+V-L=2(B-G)

(4.1)

Where F, E,

V

,

L,

B and G are the number of faces,

edges,

vertices, faces

’

inner

loop

bodies

and

genus

(handles

or

through

holes)

respectively.

For

a

simple

polyhedron, we have: F-E+V=2.

Euler law can be used to check the object

’

s validity, the system commands (Euler

operators) are based on Euler law and thus ensure the validity simultaneously. Euler

law given above applies to closed polyhedral objects. For open polyhedron, we have:

P-E+V-L=B-G

(4.2)

If objects have curved surface, such as cylinders and spheres, Euler law is still valid.

Topologically, one can always stretch curved edges and faces so that they become flat

without changing the relationships among them.

4.3.2 Euler Operators

Once

a

polyhedron

model

is

available,

one

might

want

to

edit

it

by

adding

or

deleting

vertices, edges

and faces to

create a new polyhedron. These operations are

called Euler operators.

Recall

from

the

discussion

of

the

Euler

formula

that

the

following

holds

for

all

polyhedra:

V-E+F-(L-F)-2(B-G) =0

(4.3)

Based

on

this

relation,

some

Euler

operators

have

been

selected

for

editing

a

polyhedron so that the Euler formula is always satisfied. There are two groups of such

operators: the Make group and the Kill group. Operators starting with M and K are

operators of the Make and Kill groups, respectively.

Euler operators are written as Mxyz and Kxyz for operations in the Make and Kill

groups, respectively, where x, y and z are elements of the model (e.g., a vertex, edge,

face, loop, shell and genus). For example, MEV means adding an edge and a vertex

while

KEV

means

deleting

an

edge

and

a

vertex.

The

user

is

not

free

to

construct

faces, edges or vertexes. For example, there are no such operators such as ME, MV or

MF as they violate Euler law.

Every topologically valid polyhedron can be constructed from an initial polyhedron

by

a

finite

sequence

of

Euler

operators.

Therefore,

Euler

operators

are

powerful

operations.

4.3.2.1 The Make Group of Euler Operators

The

Make

group

consists

of

four

operators

for

adding

some

elements

into

the

existing model creating a new one, and a Make-Kill operator for adding and deleting

some elements at the same time. The operators are as shown in Table 4. 2.

Table 4.2 shows the change of values of V

, E, F, L. B and C. Note that adding a face

produces a loop, the outer loop of that face. Therefore, when F is increased, L should

also be increased. This new loop and the new face will cancel each other in the sub

expression of L-F. Please verify that none of these operators would cause the Euler

formula to fail.

MBFV

is

often

used

to

begin

constructing

the

boundary

models,

so

it

could

be

through as the first vertex of the model. Table 4.3 illustrates the way of using Euler

operators

to

construct

a

tetrahedron,

in

seven

steps

or

seven

Euler

operators

a

tetrahedron is created.

MBG

simply

makes

a

body

with

a

hole.

After

this,

one

can

add

vertices,

edges,

faces, and loops. There must be loops, because the new hole penetrates at least one

face.

MEKL makes an edge and at the same time kills a loop. A commonly used MEKL is

adding an edge connecting the outer loop and the inner loop of a face. In this case, the

number of edges E is increased by 1and the number of loops L is decreased by 1 since

that loop is killed.

Higher level

Euler operators can be developed, such as MCUBE

(create

a cube),

MCYL(create a cylinder).

4.3.2.2 The Kill Group of Euler Operators

The Kill group just performs the opposite of what the Make group does. In fact,

replacing the M and K in all Make operators with K and M, respectively, would get

the operators of the Kill group. Therefore, the Kill group consists of the five operators

as shown in Table 4.4.

With

these

operators,

one

can

start

with

a

tetrahedron

and

reduce

it

to

nothing.

These

operators

are

the

opposites

of

the

Make

operators.

The

advantages

of

Euler

operators are that they ensure creating valid topology and reasonable is simple. Euler

operators offer a mechanism to check the topological validity of the models. However

the

models

are

not

unique

because

the

boundary

of

any

object

can

be

divided

into

faces, edges and vertices in many ways. Euler operators are difficult for designers to

use and usually not available to users at interface, only exist internal to software.

Earlier user interfaces used commands such as

although

such

interfaces

are

not

convenient.

Object-oriented

(feature- based)

user

interfaces are more acceptable by users. For example, command

than

4.3.3 The Winged-edge Data Structure

The winged data structure is useful data structure in B-rep. In this structure, all the

adjacency

relations

of

each

edge

are

described

explicitly.

The

winged-edge

data

structure uses edges to keep track almost everything.

For each edge, the following information is important: vertices of this edge, its left

and

right

faces,

the

predecessor

and

successor

of

this

edge

when

traversing

its

left

face, the predecessor and successor of this edge when traversing its right face.

Each

entry

in

the

edge

table

contains

above

information

mentioned

earlier:

edge