Discrete optimization of trusses using ant colony metaphor Saurabh Samdani, Vinay Belambe, B.Tech Students, Indian Institute Of Technology Guwahati, Guwahati – India.

Introduction Design of trusses- active area of research in search and optimization Design of trusses- active area of research in search and optimization Various classical techniques have been developed Various classical techniques have been developed Ant colony metaphor relatively new metaheuristic for solving combinatorial optimization problems Ant colony metaphor relatively new metaheuristic for solving combinatorial optimization problems

Truss optimization problem Objectives Objectives 1. Minimize Material cost 2. Ease of fabrication 3. Service life 4. Construction time

Classification on basis of variables 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 formulation Minimize Subject to j=1….m and k=1….n and k=1….n` S j --stress in member j,s a allowable stress and u k --displacement at node k and u a --allowable displacement

Modified objective function Where K is the penalty factor and C is the cumulative constraint violation calculated as

Why ant colony metaphor? Uses discrete variables Uses discrete variables Can avoid local optima easily Can avoid local optima easily Easy to implement Easy to implement Finds good solutions quickly Finds good solutions quickly Gives a number of solutions from which the best solution can be chosen Gives a number of solutions from which the best solution can be chosen

What is ant colony optimization? Introduced by Dorigo et al.

First application to Travelling Salesman First application to Travelling Salesman Problem (TSP). Problem (TSP). TSP -If a traveling salesman must visit a given number of cities, being sure to visit each city only TSP -If a traveling salesman must visit a given number of cities, being sure to visit each city only once, what is the shortest possible path between all cities? Ant colony optimization (ACO)

ACO 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; initia initia

Pheromone increment calculated as Pheromone increment calculated as

Ant colony approach to truss design Ants walking along the members! Ants walking along the members! Imagine multiple paths between two nodes in a truss. Imagine multiple paths between two nodes in a truss. Length of each path corresponds to the volume of the material Length of each path corresponds to the volume of the material Simulated ants would travel via one of the virtual paths. Simulated ants would travel via one of the virtual paths. Complete traverse over the truss gives a design to be evaluated! Complete traverse over the truss gives a design to be evaluated!

Possible virtual paths for a truss

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.

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 W k 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.

The ACO TRUSS algorithm Procedure_ACO_for_truss_optimizaion() rocedure_ACO_for_truss_optimization() Start Input parameters; Initialize design variables; initialize trail; do cycle=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

Examples Example 1 Six Node-Ten Bar Truss Example 1 Six Node-Ten Bar Truss

Data assumed E= kg/cm2. E= kg/cm2. u a =5.08cm,s a =1759 kg/cm 2. u a =5.08cm,s a =1759 kg/cm 2. The control parameters were The control parameters were The number of ants were set as 1000 and the number of cycles were set to 750. The minimum weight found was kg The number of ants were set as 1000 and the number of cycles were set to 750. The minimum weight found was kg Details are in table Details are in table

Table no 1Displacements And Stresses NodeX Y XY MemberArea sq.cmStress kg/cm2MemberArea cm2Stress kg/cm

Example 2:41 bar 18 node truss

Data E= kg/cm2. E= kg/cm2. u a =8 mm,s a =1250 kg/cm 2. u a =8 mm,s a =1250 kg/cm 2. The member section areas are allowed to take values between 2 and 64 cm 2 in step of 2 cm 2. The member section areas are allowed to take values between 2 and 64 cm 2 in step of 2 cm 2. The control parameters were The control parameters were The number of ants were set as 1200 and the number of cycles were set to The number of ants were set as 1200 and the number of cycles were set to The minimum volume found was cm3 The minimum volume found was cm3 Details are in table Details are in table

Table no 2 # A  #A  #A  Dislacements cm Node XY XY XY a00 g m b h n c i o d j p e k q f l r

Summary ACO used for truss design successfully to get intuitively optimal solutions. ACO used for truss design successfully to get intuitively optimal solutions. Discrete variables Discrete variables Hypothetical ant travels along members Hypothetical ant travels along members Objective function:weight of material used Objective function:weight of material used Penalty function approach for constraints Penalty function approach for constraints

Future research Method could be implemented for Topology & configuration optimization Method could be implemented for Topology & configuration optimization The effect of the parameter values on convergence and speed. The effect of the parameter values on convergence and speed. Multiple objectives can be considered. Multiple objectives can be considered. Application to other structural optimization problems Application to other structural optimization problems Comparisons with other methods. Comparisons with other methods.

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

Thank You!

The algorithm

Best tour check Best tour check For each ant calculate the length of the tour. For each ant calculate the length of the tour. If there is an improvement update If there is an improvement update the best tour found so far. the best tour found so far. Update trails Update trails Evaporate a fixed proportion of pheromone from each road Evaporate a fixed proportion of pheromone from each road For each cycle perform pheromone update For each cycle perform pheromone update