A relation between Structures and Network Flows Through Graph

Representation It was found that the same type of representation, a Graph,

can be associated with more than one domain, say Network Flow and One-Dimensional Structures

Then for each engineering system si’ Structure and siNetwork we can construct a system so that

T’(s’i) =si

Network

si

Ts’i

Structure

B

A

ts

1 4

23

5

P

1

4

2

3

5

P

B

Network Flow and its connection to structures

• The following slides will demonstrate the close relation between maximum flow in networks and applying a maximal force in a one dimensional structure.

One dimensional structures

• In 1965, William Prager established the relation between network flow and plastic theory for one dimensional structures.

• Prager used graph theory in order to establish the connection between the two seemingly different areas.

One dimensional structures

• A one-dimensional structure is a solid structure built of rods and discs, where all rods are parallel to each other.

Ground

Rods

Disc

P

Network Flow and its connection to structures

• The rules are pretty straight forward – each disc is replaced by a vertex and each rod by an edge.

structure network

A

B

ts

1 4

2

3

5

P

1

4

2

3

5

P

B

A

s

t

One dimensional structures

• Applying more than the allowable force will in turn transform the structure into a mechanism.

• Cable – A rod which preserves its length but cannot accept any compression (Recski)

• Strut – A rod which cannot accept any tension

P s

A B

t

Energy preservation law:

n

i

m

jFjpi ji

FP1 1

P

s

B

A

t

Network Flow - Example

3 5

1

10 5

A

B

st

Network Flow - Example

One can see that the maximum flow is 9.

3 51

10 5

A

B

st

One dimensional structures- and its link to networks

• Let’s assume now that we have the following one-dimensional structure:

p s

3 10 A

B 1 5 5

t

One dimensional structures- and its link to networks

• Where the numbers refer to the maximal allowable force in each rod.

p s

3 10 A

B 1 5 5

t

One dimensional structures- and its link to networks

• Now we have to find the maximal force that can be applied on the structure before any of its discs start moving.

p s

3 10 A

1

B t

5 5

One dimensional structures- and its link to networks

• The optimal solution here is 9, and one of the paths is marked in the graph:

A

B

s t

t

3 03 0

5 2 1

5 2 110 5 4

10 5 4

5 0

5 0

1 01 0

P

B

A

s

• Now, finding a cutset in the structure in which all rods are saturated means we have a mechanism:

Augmenting Path

Min Cut Max Flow

Isomorphic

Network Flow and its connection to structures

• What we get is the graph representation of the structure which is isomorphic to the network.

• Now, finding the maximal allowable force is the same objective as finding the maximum allowable flow.

structure Network

p

s

1 2 1 4

A 3

4 3 B 2 5

5 P

t

A

B

s t

A relation between Structures and Network Flows Through Graph

Representation It was found that the same type of representation, a Graph,

can be associated with more than one domain, say Network Flow and One-Dimensional Structures

Then for each engineering system si’ Structure and siNetwork we can construct a system so that

T’(s’i) =si

Network

si

Ts’i

Structure

B

A

ts

1 4

23

5

P

1

4

2

3

5

P

B

A relation between Structures and Linear Programming Through

Matroid theory It was found that the same type of representation, a

Matroid can be associated with more than one domain, say LP (Linear Programming) and Multi-Dimensional Structures

Then for each engineering system siLP we can construct a system si’ Structures so that

T(si)=mi =T’(s’i)Matroid

mi

LP

si

Ts’i

Structures

T’

n

iiiiXC

mibxa i

n

jjij ...1,

1

0ix

Max

st Q(M)*F=0B(M)*D=0

Frames

Trusses

Electronic circuits

Dynamical system

Static lever system

PGRPotential Graph Representation

LGRLine Graph

Representation

RGRResistance Graph

Representation

FGRFlow Graph

Representation

FLGRFlow Line Graph Representation

PLGRPotential Line Graph

Representation

Planetary gear systems

Determinate beams

Serial robots Stewart platformPillar

system

The map of domains

Dual

RMRResistance Matroid

Representation

Plane kinematical linkage

LPLinear Programming

Operational research

DualNetwork Flow

n

iiiiXC

mibxa i

n

jjij ...1,

1

0ix

Max

st

Q(G)*F=0

B(G)*D=0

F=k*D

Q(M)*F=0

B(M)*D=0

Matroid Representation

The scalar cutset matrix defines the matroid MQ=(S,F) where S is the set of columns of Q(G) and F is a family of all linearly independent subsets of S.

P

321S={1,2,3,P)

F={{1},{2},{3},{P},{1,2},{1,3},{1,P},{2,3},

{2,P},{3,P}}

Structures Network Flows Matroid

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Choosing determinate

structure

Choosing augmenting path

Choosing a base

Choosing a cutChoosing a cutRemoving these rods makes the structure not rigid

Choosing a self stress

Choosing a cycleChoosing a set of linearly dependant

members

Force law Flow law Q(M)*F=0

Allowable force in each rod

The weights in the edges

The maximum values of the

members

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The maximum values of the members

Choosing a base

B(M)*D=0

Q(M)*F=0

Choosing a set of linearly dependant members

A group of manufacturing

workers

A group of manufacturing workers and an administrative/non-manufacturing worker

The sum of hours that a

worker manufactures the

products equals to the sum

of his Working hours

A unit’s cost equals to the

sum of the hours multiplied

by the workers’ wages

The workers’ hour

constraint

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Matroid Operational Research

Multi-Dimensional Structures

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