discrete optimization of trusses using ant colony metaphor saurabh samdani, vinay belambe, b.tech...
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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.
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!
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
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
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.
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
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