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Contents Abstract

Introduction

Differential evolution

Algorithm

Mutation operator

Crossover operator

Selection operator

Adaptive multi-population differential evolution

Multi-population mechanism

Self- adaptive strategy

Adaptive multi-population differential evolution

for truss optimization

Design variables

Objective function

Constraints

Application of Differential Evol

Case 1 Size optimization

Case 2 Size and topolog

Case 3 Size, shape and T

optimization

Conclusion

Optimization through a MATLA

Reference

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Abstract

This paper applies multi-population differential evolution (Mpenalty-based, self-adaptive strategythe adaptive mudifferential evolution (AMPDE)to solve truss optimization p

design constraints. The self-adaptive strategy is a new adaptive approach that

control parameters of MPDE by monitoring the number solutions generated during the evolution process.

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Abstract (contd.)

Self-adaptive strategy can improve the performance of MPDE in con

optimization problems, especially in the case of simultaneous optim

size, topology, and shape of truss structures.

Multiple different minimum weight optimization problems of the tsubjected to allowable stress, deflection, and kinematic stability cused to demonstrate that the proposed algorithm is an efficien

finding the best solution for truss optimization problems.

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Introduction

Due to its direct applicability to the design of structures truss problem has received increasing attention.

Optimization problems can be classified into three categoriesand topology (Krisch 1989).

Optimizatio

n category

Design variables Constants

Sizeoptimization

Cross sectional areas of trussmembers

Coordinates of thenodes and connectivity

Shapeoptimization

Coordinates of the nodes Cross-sectional areasand connectivity

Topology

optimization

Connectivity of a member in a truss structure (objective funcdetermined)

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Introduction (contd.)

Gradient based approaches like sensitivity analysis or aconcepts, can offer high effectiveness in terms of solving truss problems.

The disadvantages of the gradient approach are:

1. Not efficient for the simultaneous optimization of size and topstructures, because the connectivity of the truss members is a discretnot easy to represent (Deb and Gulati 2001).

2. It cannot provide sufficient diversity in terms of locating a global optimized size, shape, and topology of truss structures.

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Introduction (contd.)

Apart from gradient based approach, several heuristic algorithmused to tackle truss optimization problems.

1. Genetic algorithms (GA)

2. Particle swarm optimization (PSO)3. Applied artificial immune system (AIS)4. Harmony search (HS)5. Combination Ant algorithm and the API algorithm

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solving trussstructure

optimization

One stage Two

cross-sectional area(size), shape, and

topology

Differe

Time consumingdue to larger search

space

Approaprovid

opti

Single stage

Types of Truss Optimization

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Differential Evolution

Differential evolution (DE) is a simple and efficient popumethod invented by Storn and Price (1995) to optimize the crarea of truss members

The initial population of DE is generated randomly between t

upper bounds for each design variable. DE uses a multi-dimensional real-value vectorfor the design

three control parameters:

1. Crossover rate (CR),2. Scaling factor (F), and3. Population size (NP).

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Differential Evolution (contd.)

There are three main DE steps:

1. Mutation,2. Crossover, and3. Selection.

The mutation operator of DE is executed by adding a weightvector between two individuals to a third individual.

After crossover and selection, the mutated individuals produce

The whole search process is iterated until a convergence state

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Differential Evolution(contd.)

Differential evolution flowchart

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Algorithm

1. Choose target vectorand base vector.

2. Random choice of twopopulation members

3. Compute weighted

difference vector4. Add to base vector5. Selection operator to

form new population

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Algorithm

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Differential Evolution (contMutation ope

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Mutation operator

The mutation operator of DE is executed by adding a weighted difbetween two individuals to a third individual.

This operator generates a new mutated vector according to the follequation:

G = index of the current generation

r1 and r2 = random numbers generated between 1 and NP

xr1,G and xr2,G = two random target vectors chosen in the population of the curren

xbest, G and vi,G+1 = the best target vector and mutated vector of the current gene

F = Scaling factor is a real value between 0 and 1 that will enlarge or reduce the dbetween the two random vectors

,+1 = , 1, 2, ,

i= 1,2,3.. NP

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Mutation operator

Various other mutation mechanisms

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Differential Evolution (contCrossover Ope

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Crossover operator

The crossover

mechanism of differential

Evolution

R >CR

Variable

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Differential Evolution (contSelection Ope

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Selection operator

The trial vector ui,G+1 is then compared with the target vectorindividuals for the next generation.

The selection operator is as follows :

Objectiv

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Selection operator (contd.)

Next

Generation

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Adaptive multi-population differen

evolution

Benefit: Increasethe diversity of the

search design

space

Divides a singlepopulation into

multiplesubpopulations

Multi-populationmechanism

Two methods

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Adaptive multi-population differen

evolution

Self adstra

Benefit: Increasethe diversity of the

search design

space

Divides a singlepopulation into

multiplesubpopulations

Multi-populationmechanism

Redunuminfesolu

Increaseffic

Two methods

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Adaptive multi-populationdifferential evolution (contd.)

Adaptive multi-population

differential evolution flowchart

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Adaptive multi-population differentevolution (contMulti-population mecha

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Multi-population Mechanism

Migration rate

Migrationmechanism

Migration in

Three parameters

Migration policy

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Adaptive multi-population differentevolution (cont

Self-Adaptive Stra

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Self-adaptive Strategy

In this paper, we develop a penalty-based self-adaptive strateenhance the performance of multi-population differential esolving truss optimization problems with constraints.

It increases the search diversity and convergence speed.

The adjustment method of the proposed self-adaptive dependent on the number of infeasible trial vectors Ninf andvectors Nf.

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Self-adaptive Strategy (contd.)

Ninf

> Nf < Nf

F valuedecreased bymultiplying

F = F R

if Ninf> Nf

and F > Fmin

F = F R

if Ninf< Nf

and

Penalty-based self-adaptive strategy

Original value Rrange 0.0 to 1.0

withby Nextiteration

Nextiteration

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Self-adaptive Strategy

The counters of the accumulated infeasible and feasible solutito zero after the F value is adjusted.

When using the self-adaptive strategy within the mul

framework, different F values are assigned to subpopulatiovalues are adjusted independently according to the search patsubpopulation.

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Adaptive multi-population differential evofor truss optimization

Three truss optimization cases selected from the literature:

1. Size optimization 3D, 25 member truss

2. Size and topology optimization simultaneously 39 member, 12 node3. Size, shape and topology simultaneously 2-tier, 39 members, 12 nod

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Adaptive multi-population differential evofor truss optimization (contd.)

Comparisons among DE, MPDE and AMPDE were used to effects of the proposed self-adaptive strategy in terms of enhaperformance across different types of truss optimization proble

The parameters were determined through numerical expermultiple simulation runs, and based on the suggestions of Sto(1997).

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Adaptive multi-population differential evofor truss optimization (contd.)

Parameter setting used in

AMPDE for truss structureoptimization

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Design variables

1. Ai represents the cross-sectional areas for each truss member;2. j stands for n real-valued coordinates of all non-basic nodes p

truss;

3. i is the material density of a truss member; and4. Li is the length of each truss member.

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Objective function

The formulation of the truss optimization problem is described

Minimize: =

Where

Ai represents the cross-sectional areas for each truss member;

i is the material density of a truss member; and

Li is the length of each truss member.

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Constraints

The constraints considered in truss optimization problems usuallydisplacement, kinetic stability, and nodal connectivity of truss member

Constraint violations may easily appear when simultaneously optimiztopology of truss structures.

The constraints the objective function is subjected to are:

1. G1 Truss structure is acceptable to the user2. G2 Truss is kinetically stable.3. G3 iallow, i= 1, 2 . . . . . .m4. G4 jallow, j= 1, 2 . . . . . . n5. G5 Ai, i= 1, 2 . . . . . .m

6. G6 j, j= 1, 2 . . . . . .n

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Constraints (contd.)

Constraints G5 and G6

The design variables should be bound by pre-specified values

The parameters allow and allow indicate the allowable strength of the mallowable deflection of the node as defined by the designer, respectivel

The existence or absence of a truss member in AMPDE follows the pro

and Gulati (2001) and in the ground structure is determined through a ccross-sectional area with a pre-defined, small critical cross-sectional are

If the cross-sectional area is smaller than the critical area , the memberremoved from the truss structure. Otherwise, the truss member is retastructure within the cross-sectional area.

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The cross-sectional area must be between the lower and upp

,

, and the value of must be the same as the n

of(Deb and Gulati 2001).

In the current study, these definitions of critical cross-sectionnegative area are employed in cases exploring the topology optruss structures.

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The penalty function of the constraint violations used in this below and are applied as in GA by Deb and Gulati (2001).

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Application of Differential EvSiz

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Case 1 Size Optimization

The geometry model and loading conditions ofis a three-dimensional truss ground structurewith 25 members are as shown

The material properties and design constraintsare:

Modulus of elasticity E = 1 104 ksi

Density= 0.1 lb/in3

Maximum allowable stress a = 25 ksi

Allowable displacement allow = 2.0 in

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Case 1 Size Optimization (contd

The max. allow is 40 ks

The max allow is 0.3node one and node tw

The structure was re

symmetric about the simultaneously.

The area of each crossfall between 0.01 and 3

The loading conditions

Case 1 Size Optimization (contd )

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Case 1 Size Optimization (contd.)

The results of the size optimization

Case 1 Size Optimization (contd )

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Case 1 Size Optimization (contd.)

Performance comparison of case 1 using DE, MPDE an

DE

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Application of Differential EvSize and Topolog

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Case 2 Size and Topology Optimiza

A two-dimensional 39 members, 12 node truss is asshown.

This model is symmetrical about the middlevertical member; hence the number of designvariables is reduced from 39 to 21.

The material properties and design constraints are:Modulus of elasticity E = 1 104 ksi

Density= 0.1 lb/in3


Allowable displacement allow = 2.0 in

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(contd.)

The allowable strength is 20 ksi.

The range of cross-sectional area varies between 2.25 and 2.25

The critical area is 0.05 in2. for each truss member.

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Case 2 Size and Topology Optimization

Best solutions for sizing and topology

Genetic Algorithm Ant algor

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Case 2 Size and Topology Optimization

The results for the Sizing and

topology optimization for case 2

GA

Case 2 Size and Topology Optimization (co

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p gy p (


AMPDE

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(contd.)

The number of the objective function calculation for AMPDE is(100 individuals with 323 generations); it apparently outperfoalgorithm (303,600) in terms of convergence speed.

Further, the population size of AMPDE is 100, much smaller (840).

In general, AMPDE obtains better optimum solutions for truswith fewer objective function evaluations than those obtained and Gulati (2001) and Luh and Lin (2008).

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Application of Differential EvSize, Shape and Topolog

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Case 3 Size, Shape and Topolog

Optimization

The geometry model of a two-tier, 39 member, 12node ground structure is as shown.

In addition to the cross-sectional areas and thetopologies of members in the truss structure, thecoordinates of the non-basic nodes aresimultaneously kept as decision variables.

The material properties and design constraints are:

Modulus of elasticity E = 1 104 ksi

Density= 0.1 lb/in3


Allowable displacement allow = 2.0 in Paramete

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Optimization (contd.)

These extra decision variables are assumed to vary within [120

Further, symmetry about the middle vertical member is used tonumber of variables, as in Deb and Gulati (2001) and Luh and Li

The total population size of MPDE and AMPDE is 200, with 40 ieach of the five subpopulations.

The population size of DE is the same as those of MPDE and AM

Case 3 Size, Shape and Topology Optim

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3 , p p gy p(contd.)

Best solutions for sizing and topology

Genetic Algorithm

Ant algo

Case 3 Size, Shape and Topology Optimizatio

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The results of the size optimizationGA Ant

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Optimization (contd.)

AMPDE requires 137,200 function evaluations, far less than the 453,60

algorithm (Luh and Lin 2008) and 504,000 for the GA (Deb and Gulati

The population size of AMPDE is 200 (40 individuals in each of five su

populations), which is also smaller than that of Deb and Gulati (1680

the GA).

Based on these numerical results, AMPDE offers better performance

approaches used in previous research (Deb and Gulati 2001; Luh and

Case 3 Size, Shape and Topology Optimization

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Conclusion

This paper applies multi-population differential evolution (MPDE) augpenalty-based self-adaptive strategy called adaptive multi-populatevolution (AMPDE) to solve truss optimization problems with constraints.

The self-adaptive strategy proposed in this study can enhance the effective

Several examples of truss optimization problems are used to illustrate the

AMPDE. In terms of size optimization, AMPDE can find truss structures superior to th

the literature without incurring constraint violations (Haug and Arora 19Gurdal 1992; Lee and Geem 2004; Perez and Behdinan 2007).

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Conclusion (contd.)

While simultaneously optimizing two (size and topology) and thre

and topology) types of truss structure problems, it is clear tha

approach as a one stage approach performs with greater efficienc

finding better or similar solutions with fewer function evaluations) t

stage and two stage approaches (Deb and Gulati 2001; Luh and Lin 20 It is also observed that the developed self-adaptive strategy can fu

the performance of MPDE across different types of truss optimizatio

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Optimization through a MATLAB c

C:\Users\Aaron\Desktop\DeMat\Rundeopt.m
http://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.mhttp://c/Users/Aaron/Desktop/DeMat/Rundeopt.m

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