Discrete optimization of Discrete optimization of trusses using ant colony trusses using ant colony

metaphormetaphor

Saurabh SamdaniSaurabh Samdani, , Vinay Belambe,Vinay Belambe,B.Tech Students, Indian Institute Of Technology B.Tech Students, Indian Institute Of Technology

Guwahati, Guwahati –781 039 India.Guwahati, Guwahati –781 039 India.

IntroductionIntroduction

Design of trusses- active area of Design of trusses- active area of research in search and optimizationresearch in search and optimization

Various classical techniques have Various classical techniques have been developed been developed

Ant colony metaphor relatively new Ant colony metaphor relatively new metaheuristic for solving metaheuristic for solving combinatorial optimization problemscombinatorial optimization problems

Truss optimization problemTruss optimization problem

Objectives Objectives

1.1. Minimize Material cost Minimize Material cost

2.2. Ease of fabricationEase of fabrication

3.3. Service lifeService life

4.4. Construction timeConstruction time

Classification on basis of Classification on basis of variablesvariables

Sizing – cross sectional areas.Sizing – cross sectional areas.

Configuration –nodal coordinates.Configuration –nodal coordinates.

Topology – connectivity between nodes.Topology – connectivity between nodes.

This work – only sizing is considered.This work – only sizing is considered.

Problem formulationProblem formulation

MinimizeMinimize

Subject to j=1….m Subject to j=1….m

and and k=1….n k=1….n`̀

SSjj --stress in member j ,s --stress in member j ,saa allowable stress and allowable stress and

uukk--displacement at node k and u--displacement at node k and uaa--allowable --allowable displacementdisplacement

01j

a

s

s

1

( )m

j ji

f x A l

01j

a

s

s

01k

a

u

u

Modified objective functionModified objective function

Where K is the penalty factor and C is the Where K is the penalty factor and C is the

cumulative constraint violation calculated cumulative constraint violation calculated as as

]1)[()( KCxfxP

otherwise )( and

0)( if 0 where1

xgc

xgc

cC

ii

ii

l

i

i

Why ant colony metaphor?Why ant colony metaphor?

Uses discrete variables Uses discrete variables Can avoid local optima easilyCan avoid local optima easily Easy to implementEasy to implement Finds good solutions quicklyFinds good solutions quickly Gives a number of solutions from Gives a number of solutions from

which the best solution can be which the best solution can be chosenchosen

What is ant colony What is ant colony optimization?optimization?

Introduced by Dorigo et al.Introduced by Dorigo et al.

First application to Travelling SalesmanFirst application to Travelling Salesman

Problem (TSP).Problem (TSP).

TSP -If a traveling salesman must visit TSP -If a traveling salesman must visit a given number of cities, being sure to a given number of cities, being sure to visit each city onlyvisit each city only once, what is the shortest possible path between all cities?

Ant colony optimization Ant colony optimization (ACO)(ACO)

ACO for TSPACO for TSP

Simulation of the autocatalytic positive feedback process exhibited by ants.

Virtual substance called trail which is analogous to pheromone in real ants

Ants can communicate with one another wholly through indirect means by making modifications to the pheromone level in their immediate environment.

Step 1:[initialization]Step 1:[initialization]

set t=0,nc=0;set t=0,nc=0;

initiainitia

• Pheromone increment calculated asPheromone increment calculated as

Ant colony approach to truss Ant colony approach to truss designdesign

Ants walking along the members!Ants walking along the members! Imagine multiple paths between two Imagine multiple paths between two

nodes in a truss .nodes in a truss . Length of each path corresponds to the Length of each path corresponds to the

volume of the materialvolume of the material Simulated ants would travel via one of Simulated ants would travel via one of

the virtual paths.the virtual paths. Complete traverse over the truss gives Complete traverse over the truss gives

a design to be evaluated!a design to be evaluated!

Possible virtual paths for a Possible virtual paths for a trusstruss

Probability of selecting jth cross section at member i is given by

Hence The number of ants passing through cross section i at member j in iteration t is

Which ant passes through which cross section is decided randomly to get distinct designs.

1

[ ( )] [ ]( )

[ ( )] [ ]

ij ijij m

il ill

tp t

t

( ) ( ).ij ijnextstate t p t ANTS

All the members are thus traversed and every ant passes through a cross section at a member

Having obtained the cross-section areas along with the member length fixed apriori, structural analysis of the different truss models is carried out making use of the Finite Element Method. Stress as well as deflection considerations are handled using constraints in the form of penalty functions as previously explained.

Trail is updated using the modified objective function

where if tour of ant k constitutes cross section j at member i.

= 0 otherwise And Wk is the objective function for ant k as explained

previously.

The modified values of pheromone create bias in the next iteration for the number of ants passing through a particular cross-section at a member. The cross section that corresponded to the best design of previous generation has a greater probability of getting selected. This way after a number of iterations the ants find out good solutions.

1

( 1) . ( )m

kij ij ij

k

t t

kij

W

Qt )(

The ACO TRUSS algorithmThe ACO TRUSS algorithmProcedure_ACO_for_truss_optimizaion()rocedure_ACO_for_truss_optimizat

ion()StartInput parameters;Initialize design variables;initialize trail; docycle=1;find number of ants in nextstate(i,j);

randomly allot cross sections to ants;structural analysis of designs(); compute penalty and evaluate objective function;store the best design;update trail;cycle =cycle +1; while(termination criteria not satisfied) print best design;end

ExamplesExamples Example 1 Six Node-Ten Bar TrussExample 1 Six Node-Ten Bar Truss

Data assumedData assumed

E=703700 kg/cm2. E=703700 kg/cm2. uuaa=5.08cm ,s=5.08cm ,sa a =1759 kg/cm=1759 kg/cm22.. The control parameters were The control parameters were The number of ants were set as 1000 and The number of ants were set as 1000 and

the number of cycles were set to 750. The the number of cycles were set to 750. The minimum weight found was 1911.89 kg minimum weight found was 1911.89 kg

Details are in table Details are in table

5.0,0,1

Table no 1Displacements And Stresses

Node X Y Node X Y

1 0.9335516 -4.990372 2 -1.522189 -5.057585

3 0.812347 -1.785223 4 -0.88787 -2.763437

5 0.0 0.0 6 0.0 0.0

Member Area sq.cm Stress kg/cm2 Member Area cm2 Stress kg/cm2

0 89.68 -819.85 5 46.58 866.02

1 74.19 -585.82 6 193.55 -449.18

2 24.77 62.070 7 19.94 903.37

3 13.74 111.89 8 141.94 433.03

4 141.94 750.14 9 11.61 -187.28

Example 2:41 bar 18 node trussExample 2:41 bar 18 node truss

DataData E=2100000 kg/cm2. E=2100000 kg/cm2. uuaa=8 mm ,s=8 mm ,sa a =1250 kg/cm=1250 kg/cm22.. The member section areas are allowed to take The member section areas are allowed to take

values between 2 and 64 cmvalues between 2 and 64 cm2 2 in step of 2 cmin step of 2 cm22. . The control parameters were The control parameters were The number of ants were set as 1200 and the The number of ants were set as 1200 and the

number of cycles were set to 1250. number of cycles were set to 1250. The minimum volume found was 90977.21cm3The minimum volume found was 90977.21cm3 Details are in table Details are in table

5.0,0,1

#A # A # A

1 28 606.57 15 30 -1187 29 4 1111.04

2 34 818.91 16 6 -956.03 30 2 -647.86

3 44 868.02 17 8 892.10 31 2 102.97

4 60 616.07 18 40 -842.82 32 4 -1008.5

5 36 1024.95 19 4 903.66 33 2 572.05

6 46 829.73 20 6 -844.97 34 2 -1071.1

7 42 668.42 21 12 1021.3 35 4 -1158.2

8 44 454.16 22 2 -950.11 36 14 900.08

9 10 -891.44 23 2 593.4 37 2 147.75

10 32 -1120.4 24 4 -995.2 38 28 -1023.3

11 36 -1087.9 25 2 18.69 39 12 924.2

12 36 -1124.1 26 4 1149.47 40 14 -888.11

13 48 -844.55 27 2 -585.45 41 8 -1089.2

14 38 -1031.4 28 14 1161.04

Dislacements cm

NodeX Y Node X Y Node X Y

a 00 00 g 0.3516 -0.524 m 0.3341 -0.7550

b 0.0694 -0.3206 h 0.4070 -0.3401 n 0.2287 -0.6368

c 0.1356 -0.5334 i 0.4715 0 o 0.1397 -0.7518

d 0.2063 -0.7636 j 0.2352 -0.0422 p 0.1434 -0.5925

e 0.2341 -0.7474 k 0.2878 -0.3118 q 0.1938 -0.2863

f 0.2829 -0.7533 l 0.3388 -0.6100 r 0.2298 -0.0518

SummarySummary

ACO used for truss design successfully to ACO used for truss design successfully to get intuitively optimal solutions.get intuitively optimal solutions.

Discrete variables Discrete variables Hypothetical ant travels along membersHypothetical ant travels along members Objective function:weight of material usedObjective function:weight of material used Penalty function approach for constraintsPenalty function approach for constraints

Future researchFuture research

Method could be implemented for Method could be implemented for Topology & configuration optimization Topology & configuration optimization

The effect of the parameter values on The effect of the parameter values on convergence and speed .convergence and speed .

Multiple objectives can be considered.Multiple objectives can be considered. Application to other structural Application to other structural

optimization problems optimization problems Comparisons with other methods.Comparisons with other methods.

AcknowledgementsAcknowledgements

The authors would like to thank some The authors would like to thank some of their seniors who preferred to of their seniors who preferred to remain anonymous.remain anonymous.

Thank You!Thank You!

1

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The algorithmThe algorithm

Best tour checkBest tour check• For each ant calculate the length of the For each ant calculate the length of the

tour . tour . • If there is an improvement updateIf there is an improvement update

the best tour found so far.the best tour found so far.

Update trailsUpdate trails• Evaporate a fixed proportion of Evaporate a fixed proportion of

pheromone from each roadpheromone from each road• For each cycle perform pheromone For each cycle perform pheromone

updateupdate