systematic conceptual engineering design using graph representations

Systematic conceptual engineering design using graph

representations .

Research Objectives

Development of Systematic design methods to facilitate conceptual engineering design using discrete mathematical models called combinatorial representations that are based on graph theory as a medium for knowledge transfer.

• Design through Common Graph Representation.

• Design through Dual Graph Representation.

• Identification and usage of special properties obtained by graphs.

Satellite communications

Different problems from different domainsNot Really!

Chessboard problem

All can be represented by a common bipartite graph

Problem solving with Graph Representations

Tensegrity

Satellite communications

Chessboard problem

Common Graph Representation

Solving one of the problems in its domain solves the analogous problems using the graph to transfer the solution.

Special properties of the graph are reflected in the domains represented.

Tensegrity solved

Satellite problem solved

Chessboard problem solved

Special Properties

Tensegrity

Design using Common Graph Representations

It was found that the same type of graph representations, say G can

be associated with more than one engineering domain, say D1 and D2.

In this case, G can be used to transfer solution from D1 to D2 and vice-versa.

Original engineering

domain

Step 1: Defining engineering problem in original domain.•Function Definition – What it does. •Use of “Black Box” Function Definition (Pahl and Wallace, 1996)

Design Problem

Alternating angular velocity drive

V

t

Rectified angular velocity output

Design Problem

t

V



domain

CGRCommon Graph Representation

Step 2: Transforming problem to Graph Representation level .•Use of “common language” to describe system function.•Flow or Potential variables to describe system.

Design Problem

Alternating Potential = input

Rectified Potential =

output

t t

Design Problem

Alternating angular velocity drive

Rectified angular velocity output

V V

t t



domain


Secondary engineering

domain

Step 3: Locate a solution in another engineering domain .•Engineering domain must share common representation.•Flow or Potential variables translated to corresponding terminology of secondary engineering domain.

Design Problem

Alternating Potential = input

Rectified Potential =

output

t t

Secondary engineering domain – Electrical engineeringElectric circuit is found that rectifies an alternating

voltage source: The Full Wave rectifier

Design Problem

Alternating voltage source

Rectified voltage output

V V

t t



domain



domain

Step 4: Transfer solution from engineering domain to Graph Representation level .•Each structure element in the engineering level is translated into it’s equivalent element representation in the graph through deterministic steps.•Graph topology insures proper representation of properties and system behavior.

2

4

C

A

3

BB

1

0



domain



domain


domain



domain

Step 5: Building new design at the engineering level using the graph solution .•Each element in the graph representation is represented at the engineering level as an equivalent element through deterministic steps.

•Graph topology again insures that proper representation of properties and system behavior is transferred to engineering solution.

This structural procedure on the graph representation ensures:•Each edge corresponds to an element in the mechanical system.•Each vertex corresponds to a point in the mechanical system where velocity is measured.

C


Step 5: Building new design at the engineering level using the graph solution .

2

4

C

A

3

BB

1

0


domain



domain

C

A

1

0AA

AC

AC

2A

C

A

CAC AC AC



2

4

C

A

3

BB

1

0


domain



domain

3

BB

A

C

B

AB C 4

BB

C

BC

•C elements both possess the same potential.

C

C

C



2

4

C

A

3

BB

1

0


domain



domain

A

BC

A

BC

Linear to Angular Design

Mechanical Design process can be made simpler by first designing linear systems and then converting to angular systems.

2

4

C

A

3

BB

1

0


domain



domain

A

BC

•Potential ( ) can be represented as tangential velocity with edges possessing angular velocity.•Flow (F) can be represented as force acting around an axis (Moment).

C

A

1

0

C

B

C

AC


2

4

C

A

3

BB

1

0


domain



domain

3

B

0

B

A

1

A

1

3

C

A

1in

in 3

B

A

B

C


2

4

C

A

3

BB

1

0


domain



domain

0

1

3

C

2

0

C4

C

2A

C

•Edge 2 subject to •Linear element replaced by angular element

2

4BB

C

4

C0

•C elements both possess the same potential

A

B

2

C

AC

2C

4

C

C

0

C

A

B

C

B4

C

0

A

2

C

B4

C

0

A

2

C

Looking at the complete mechanical rectifier where the driving input gear is subject to direction change:

C Rotates Anti-clock wise.

C Rotates Clock wise.


The same systematic process resulted in design through knowledge transfer of another available solution from the electronic engineering domain.

C

A

B

0

Diode Bridge Graph


domain



domain

2

4

C

3

B

10

A

Full Wave Rectifier Graph


domain



domain

Design using special properties of Graph Representations

Self Duality

1

23

4 5

6

III

III

IV

4’3’ 2’

1’

5’

6’IV

II I

III4’

3’ 2’

1’

5’

6’

IV

III

III

IV

II I

III


Self Duality

Potential Law: Flow Law:

Every cutset has a dual circle and vice-versa

11

23

4 5

6

4’

3’ 2’

1’

5’

6’

5

2

0521

'1F

0521 ''' FFF

'2F'5F

0431

0431 ''' FFF

0632

0632 ''' FFF

0654 0654 ''' FFF

Potentials in Graph = Flows in Dual Graph

IV

II I

III


Self Duality

I

Flow Law Broken = Illegal duality operation

1

23

4 55

4

23

03254 Potential Law:

4’

3’ 2’

1’

5’

Cutset does not have a dual circle and vice-versaPotentials in Graph = Flows in Dual Graph

Flow Law:

'4F '5F

0'3'2'5'4 FFFF

'2F'3F

Two Engineering systems in the Engineering Domain are transformed to graphs in the Graph Domain.

The Graph Domain reveals properties that were not discovered at the Engineering level.

These special properties may be transferred back to the Engineering Domain where they reflect the special properties in the Graph Domain .

Gl

g1

Dj

Ts2

s1

g2 Special properties

Special properties


C

A

B

0

C

A

B

0

Special Properties of Dual Graphs2 types of “rectifier” graphs

Graph 1: Diode Bridge

Dual to itself

Potential Source can be automatically exchanged for Flow

Source

I

II

III

IV

II

I

III=

IV

Graph 2: Full Wave rectifier

Not Dual to itself

Potential Source cannot be automatically exchanged for Flow

Source

Resulting Graph is Illegal

II

III

I

C

A

B

0≠

C

A

B

0

C

A

B

0


Dual to itself


Not Dual to itself

C

0

A

B

C

0

A

Dual Statically Valid

B

Dual Statically

Non-Valid

Special Properties of Dual Graphs

C

A

B

0

C

A

B

0


Dual to itself


Not Dual to itself

C

0

A

B

C

0

A

Dual Statically Valid

B

Dual Statically

Non-Valid

Special Properties of Dual Graphs

B

Design domain of concepts

•Each element in the graph representation is represented at the engineering level as an equivalent element through deterministic steps.

•A graph element can be represented by different structures possessing the same behavior.

Graph element

Equivalent Engineering structure

Behavior

Y

XX

Y

X Y

CXY

XYXY .1

XY .2

C

1

1

A

3

B4

2

C5

0

B

A

C

3

2

4

5

5

0

1

3

4

B

2

C

5

A

6

D

0

0

1

3

4

2

5

1

AA

3

BB

2

4

C5

C0


1

0

AA

3

B

4BB

2

C5

01

3

4

B

2

C

5A

6

D

Mechanisms taken from :

Mechanisms and Mechanical Devices Sourcebook

By :Nicholas P. Chironis


systematic conceptual engineering design using graph representations

Documents

graph solution

graph topology

graph representation

engineering solution

graph theory

engineering level

engineering problem

dual graph representation