Term paper - State of the art in Structural optimisation

8/7/2019 Term paper - State of the art in Structural optimisation

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STRUCTURAL OPTIMISATION OF TRUSSES AN OVERVIEW OF DIFFERENT METHODS

Lt Cdr V S SwaminathanDept of Ocean Engineering & Naval Architecture, IIT Kharagpur

Abstract

This paper presents the over view of structural optimisation strategies used in truss problems, their merits, demerits and the future trends. Comparisons between different methodshas also been discusses keeping the accuracy, efficiency and the convergence rate of each of the methods in mind. Over the period of time many new methods have been developed in thefield of structural optimisation. They are different from the classical methods developed in many ways. Most of these methods are meta-heuristic in nature. The common factor in meta-heuristic algorithms is that they combine rules and randomness to imitate natural phenomena. Thecomplexities of a problem limit the structural designer from using the classical methods for analysis and optimisation. However, one has to have a clear understanding of the new methods,and exercise caution as any error in formulation can lead to wrong results. Even now most of thegoals of a common structural optimisation problem remain the minimisation of weight of the

structure. In this paper one of the classical case will be discussed which has been researched quiet well.

1. Introduction

Optimisation of structure hasgained lots of significance in the lastdecade. Much of this progress wasmade possible by impressivedevelopments in the field of digital

computer technology . The escalationof cost and time has forced thestructural designer to deviate from theclassical methods of design. Even thestructural optimisation has undergoneevolution from the classical methods tothe meta-heuristic methods. Theclassical methods such as linear,nonlinear, and dynamic programmingare efficient for certain range of problems. However, they have

limitations with respect to complexproblems. Some techniques, includingthe penalty-function, augmentedLagrangian, and conjugate gradientmethods, search for a local optimumby moving in a direction related to thelocal gradient. These methods becomeinefficient when searching for the

optimum design of large structuresdue to the large amount of gradientcalculations those are required.

Structural optimisation problemsare generally characterized by a largenumber of design variables, a simple

objective function and an indirect butwell behaved constraint functions. Theconstraint functions are indirect in thesense that they cannot he expressedexplicitly as functions of the designvariables. Attempts to optimizestructures by nonlinear programmingmethods have met with varying degreeof success. These methods areextremely useful in defining the designproblem in proper mathematical terms.

This paper compares most of themethods, classical and meta-heuristicfor their accuracy, merits, de-merits,computational time and ease and therate of convergence. Due to the factthat material cost is one of the major factors, minimisation of weight hasbecome one of the important goals.

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2. Problem formulation

In general the structuraloptimisation problem can becategorised in different groups viz.

shape optimisation, weightoptimisation, topology optimisation,reliability and dynamic characteristicsetc. However, this paper is limited toweight optimisation only. Theoptimisation problem of the structure isdefined as:

Minimise (X)

(X) The objective function

Subjected to the constrains

G i(X) 0 { i = 1,2,3,4 . n}H j(X) = 0 { j = 1,2,3,4 . m}

X The design vector

As far as weight optimisation isconcerned the objective function isdefined as:

Minimise W(X) =1

k

i ii

l A=

ik Total number of members in thetrussl, A, length, area and density of i thmember.

A typical structural optimisationproblem is restricted with a number of

constraints controlling the structuralresponse. Constraints can be stressconstraints, displacement constraints,frequency constraints etc with boundson the design variables.

l i u

l i uAl A i Au

l, u lower and upper bounds

3. Overview of methods

Some of the methods that willbe seen are modified double cutsapproach, sequential quadraticprogramming, genetic algorithm,harmony search method, improvedmove limit method of SLP and strainenergy method.

3.1 Modified double cuts approach

The work of C J Shih & H W Lee[1] is presented here. The general formof a nonlinear programming problem withfuzzy inequality constraints (FNLP) canbe stated as

Find X = [x 1,x2..x n]

Min (X) (1)s.t g i(X) b i , i = 1,2,3m (2)

The fuzzy number b i, i are in thefuzzy region of [b i, b i + p i] with givenfuzzy tolerance of p i. Assume that thefuzzy tolerance p i for the i th fuzzyconstraint is known, then, b i can beequivalent to (b i + p i), i, where isin [0, 1]. Eq. (2) can be expressed asfollowing formulation used for solving a

FNLP problem:

s.t g i(X) b i + (1- )p i i = 1,2,m (3)

Where [0, 1]. Thus, the FNLPproblem containing constraint given inEq. (3) is equivalent to a parametricprogramming while = 1 - . For

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each , there is an optimal solution,therefore, the solution with a grade of membership function is fuzzy. One canconsequently apply the maxminoperator to obtain the optimal decision.

Then, the problem of Eqs. (1) and (2)can be solved by the max- strategy,where = min[ f (X), g1(X), g2(X),., gm (X)]Max (4)s.t f (X) (5) gi(X) i = 1,2,m

(6)

Where f (X) and gi(X) represent themembership function corresponding tothe objective function and the i th fuzzyconstraint, respectively.

When one observes the conventional -cuts method in which the value of g i (X) goes to b i + p i as gi(X) isapproaching to zero; simultaneously,the value of f (X) goes to the direction of f min (the minimum value of objectivefunction) as f (the level cut valuecorresponding to the objective function)goes to one. For maintaining theequilibrium between f and gi in theoptimization process, there is atendency to pull f toward a lower limitand to push gi toward an upper limitso to achieve a balance state. Theresultant optimal design eventuallyconverges to two different compromisevalues of *f and *gi , where each agiin the formulation uses a same level of . This consideration leads us todevelop the double-cuts approach for FNLP, the formulation can be written as

Find [X, f , ]TMin f (X) x f / (7)

s.t f (X) [ f max - f (f max f min)] = 0(for linear f (X)) (8)

f (X) ( f min/ f ) = 0(for non-linear f (X)) (9)gi(X) b i + (1- )p i, i = 1,2m (10)

where f [0,1] and 0.01 1. Eq.(7) for nonlinear f (X) can be viewedas minimising f (X) and minimising f / simultaneously; which in termsimplies that Eq. (7) can be replaced byan another utility function to beminimised, as written in the following:

Min

2

min2

maxmin

minmin

max

( ) f f

f f X f f f f

+

(11)

An alternate function for Eq. (7) in thecase of linear f (X) can be written as

Min

2

2

min

min

( ) f

p

p

f X f f

+

(12)

Where p is an additional designvariable with the range in 10 -4 p 1.The inadequacy in Eq. (7) resulting thelocal optimum because of f numerically goes to the minimum and goes to the maximum. Especially thevariable p in Eq. (12) seems to beuseless due to yielding to its upper

bound. Therefore, Eqs. (11) & (12) aremodified to be Eqs. (13) & (14) for avoiding those computationallydifficulties.

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Min

2

min2

maxmin

minmax min

max

( )1

0.0001

f f f f X f

f f f f

+

.. (13)

Min

2

2

min

max min

0.0001( )1

0.00010.0001

f

f X f f f

+

. (14)

Eq. (13) is used with nonlinear f (X) and Eq. (14) is used with linear f (X). Formulation of Eqs. (13) & (14)are developed from the idea of quadratic normalization in which theparameter a goes to a small value. TheEq. (14) also shows the advantage of neglecting the additional designvariable p.

3.2 Sequential quadratic

programming (SQP)

The work of R. Sedaghati & E.Esmailzadeh [2] is presented here. Theoptimisation problem is defined asminimisation of the weight of thestructure subjected to stress anddisplacement constrains and lower bound on the cross-sectional areas.SQP is a kind of iterative method, atiterate k it needs to solve a QP sub-

problem:

Min g kTd + d TBkd (15)

s.t c i(xk) + c i(xk)T d 0, I = 1,2 , m

where x k is the current iterate,

gk=f (xk) and B k is an nn matrix.Often B k is required to be positivedefinite, which is supposed to be anapproximate Hessian of the Lagrangian

( ), ( ) ( )T

x f x c x = l (16)

At the current iterate x k. A meritfunction, which is normally a penaltyfunction such as the l 1 exact penaltyfunction, is used to carry out linesearches. It has been proved that SQPis globally convergent. One of theresulted difficulties of SQP for large-scale problem is that the memoryrequisite for each QP sub-problem maybe very large if the original problem islarge.

3.3 Genetic algorithm

The work of Kalyanmoy Deb &Surendra Gulati [3] is presented here.In this method, simulated binarycrossover (SBX) and a parameter-based mutation operator are used.

3.3.1 Simulated binary crossover (SBX)

A probability distribution is usedaround parent solutions to create twochildren solutions. In the proposed SBXoperator, this probability distribution isnot chosen arbitrarily. Instead, such aprobability distribution is fist calculatedfor single-point crossover operator inbinary-coded GAs and then adapted for

real-parameter GAs. The chosenprobability distribution is as follows:

2

0.5( 1) 1,( )

0.5( 1) /

c

c

c

c

if P

otherwise

+ + = +

(17)

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where c is a parameter whichcontrols the extent of spread in childrensolutions. A small value of c allowssolutions far away from parents to becreated as children solutions and a

large value restricts only near-parentsolutions to be created as childrensolutions. The procedure of computingchildren solutions y (1) and y (2) from twoparent solutions x (1) and x (2) are asfollows:1) Create a random number u between0 and 1.2) Find a parameter using thepolynomial probability distribution froma schema processing point of view, asfollows:

1/( 1)

1/( 1)

(2 ) , 0.5

(1/ 2(1 ) ,

c

c

u u

u otherwise

+

+

= (18)

3) The children solutions are thencalculated as follows:

(1) (1) (2) (2) (1)

(2) (1) (2) (2) (1)

0.5[( ) ]

0.5[( ) ]

y x x x x

y x x x x

= + = +

The above procedure is used for variables where no lower and upper bounds are specified. Thus, thechildren solutions can lie anywhere inthe real space [- ,] with varyingprobability. For calculating the childrensolutions where lower and upper bounds (x l and x u) of a variable arespecified, Eq. (18) needs to bechanged as follows:

1/( 1)

1/( 1)

( ) , 1 /

(1/ 2(1 ) ,

c

c

u u

u otherwise

+

+

= (19)

where = 2 ( 1)c + and arecalculated as follows:

(1) 1 (2)(2) (1)

21 min[( ), ( )]u x x x x

y y = +

3.3.2 Parameter-based mutationoperator

A polynomial probabilitydistribution is used to create a solutiony in the vicinity of a parent solution x.The following procedure is used for variables where lower and upper boundaries are not specified:

1) Create a random number u between0 and 1

2) Calculate the parameter asfollows:

1/( 1)

1/( 1)

(2 ) 1, 0.5

1 [2(1 )] ,

m

m

u u

u otherwise

+

+

= where m is the distribution index for mutation and takes any non-negativevalue.

3) Calculate the mutated child asfollows:

Y = x + max

Where max is the maximumperturbance allowed in the parentsolution. For variables where lower andupper boundaries (x l and x u) arespecified, above equation may bechanged as follows:

1 1/( 1)

1 1/( 1)

2 (1 2 )(1 ) ] 1, 0.5

1 [2(1 ) 2( 0.5)(1 ) ] ,

m m

m m

u u u

u u otherwis

+ +

+ +

+ = + Thus, in order to get a mutation

effect of 1% perturbance in solutions,we should set m 100. We terminatea GA simulation when a pre-specifiednumber of generations are elapsed.

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3.4 Harmony search method

This section is based on thework by Kang Seok Lee & Zong Woo

Geem [4]. Harmony search method(HMS) was conceptualized using themusical process of searching for aperfect state of harmony. The HSalgorithm does not require initial valuesand uses a random search instead of agradient search, hence derivativeinformation is unnecessary. Thedifferent steps involved in HSM aregiven below:

Step 1 : Initialize the optimization problem and algorithm parameters

First, the optimization problem isdefined as minimisation of weight of atruss with stress and displacementconstrains. The HS algorithmparameters that are required to solvethe optimization problem are alsospecified in this step: harmony memorysize (number of solution vectors, HMS),

harmony memory considering rate(HMCR), pitch adjusting rate (PAR),and termination criterion (maximumnumber of searches). Here, HMCR andPAR are parameters that are used toimprove the solution vector.

Step 2: Initialize the harmony memory (HM)

In this step, the harmony memory

(HM) matrix shown is filled with asmany randomly generated solutionvectors as the size of the HM (i.e.,HMS) and sorted by the values of theobjective function, ( ) f x .

1

2

3

.

.

. HMS

x

x

x

HM

x

=

Step 3: Improvise a new harmony from the HM

A New Harmony vector, x J = (x 1J ,x2J ,.x NJ) is generated from the HMbased on memory considerations, pitchadjustments, and randomization. For instance, the value of the first decisionvariable (x 1J) for the new vector can bechosen from any value in the specifiedHM range (x 11 ~ x 1HMS). Values of theother decision variables (x iJ) can bechosen in the same manner. Here,there is a possibility that the new valuecan be chosen using the HMCRparameter, which varies between 0 and1 as follows:

}{ 1 2, ,...., .. (1 )

J HMS i i i i J

i J

i i

x x x x w pHMCR x

x X w p HMCR

The HMCR sets the rate of choosing one value from the historicvalues stored in the HM, and (1-HMCR)sets the rate of randomly choosing one

value from the possible range of values. For example, a HMCR of 0.95indicates that the HS algorithm willchoose the decision variable value fromhistorically stored values in the HM witha 95% probability and from the entirepossible range with a 5% probability. AHMCR value of 1.0 is not

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recommended, because there is achance that the solution will beimproved by values not stored in theHM. This is similar to the reason whygenetic algorithms use a mutation rate

in the selection process. On the other hand, every component of the NewHarmony vector, x J = (x 1J , x 2J ,.x NJ), isexamined to determine whether itshould be pitch-adjusted. Thisprocedure uses the PAR parameter that sets the rate of adjustment for thepitch chosen from the HM as follows:

Pitch adjusting decision for x iJ

. .. (1 )

Yesw p PAR Now p PAR

The pitch adjusting process isperformed only after a value is chosenfrom the HM. The value (1-PAR) setsthe rate of doing nothing. A PAR of 0.1indicates that the algorithm will choosea neighboring value with 10% HMCRprobability. If the pitch adjustmentdecision for x iJ is yes, and x iJ isassumed to be x i(k), i.e., the k th elementin X i, the pitch-adjusted value of x i(k) is

xiJ xi(k+m) for discrete decisionvariablesxiJ xiJ + for continuous decisionvariables

where m = the neighboringindex, m {,-2, -1,1,2}; = thevalue of bw x u (-1,1); bw = an arbitrarydistance bandwidth for the continuousvariable; and u(-1,1) = a uniformdistribution between -1 and 1. TheHMCR and PAR parameters introducedin the harmony search help thealgorithm find globally and locallyimproved solutions, respectively.

Step 4: Update the HM

If the New Harmony vector isbetter than the worst harmony in the

HM, judged in terms of the objectivefunction value, the New Harmony isincluded in the HM and the existingworst harmony is excluded from theHM. The HM is then sorted by theobjective function value.

Step 5: Repeat Steps 3 and 4

The computations explained areterminated when the termination

criterion is satisfied. If not, Steps 3 and4 are repeated.

3.5 Improved move limit method

This section is presented basedon the work done by KV John, CVRamakrishnan & KG Sharma [5]. Abrief resume of the improved move limitmethod of sequential linear programming at a particular design

point is presented below.

Let P

X be the current designpoint. The linear programming problemreferred above is solved by the simplexalgorithm leading to the point 1 P X

+

. If 1 P

X + is a feasible point, the objective

function is checked for improvement[W( 1 P X

+

) < W( P X )]. The sequence of linear programming is continued from

1 P

X +

if improvement is found, takingcare to compute the constrain valuescorresponding to this point. Otherwisethe new design point is selected byquadratic interpolation between thedesign points 1 P X

+ andP

X .

Thus if

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1 P P

P

S X X

X X S

+

++

= = +

(20)

Where +

is the optimal step lengthalong S direction. The move limit isalso modified as

P M M

++= (21)

If the linear programming solutionenters an unfeasible region, it issteered to the feasible domain bymoving in the gradient direction of themost violated constraint. Thus the newdesign point is given by

1 1( )

P P

q X X g X + + +

= + (22)

Where q g is the gradient of themost violated constraint q and

1

1 1

( )

( ) ( )

P

j P P T

q q

g X

g X g X

+

+ +=

(23)

It may be necessary in someproblems to use the above technique of steering to the feasible regionrepeatedly. After steering the designvector to the feasible region if noimprovement is found in the objectivefunction, the usability of the direction

* * P S X X =

is checked. The quadraticinterpolation is resorted to only if thedirection is usable. Otherwise quadraticinterpolation erroneously leads to

P

X as optimum. In such cases quadraticinterpolation is to be done between

P

X

and 1 P X + and then optimization

continued.

3.6 Strain energy criterion method

Strain energy criterion methodexplained here is based on the work byV B Venkayya [6]. A structure with mstructural elements and specifiedconfiguration is subjected to ageneralized force vector R. Theproblem is to obtain optimum sizes for the elements such that the weight of the structure is a minimum. The designthat has the same average strainenergy density in all its elements is a

lower weight design than the one inwhich this condition is not satisfied.Now the statement of the optimalitycriteria is as follows:

The optimum structure is theone in which the average strain energy density is the same in all its elements.

If the change in potential of theapplied forces is considered as a

measure of the stiffness of the structurethen it can be shown that the structurethat satisfies the above strain energycriteria will also be the stiffest structurefor that loading condition.

The optimality criteria modifiedfor the general case is as follows:

The optimum design is theone in which the strain energy of eachelement bears a constant ratio to its

energy capacity.

The energy capacity is definedas total strain energy stored if the entireelement is stressed to its limitingnormal stress. The limiting normalstress can be different from the actualstress limit as long as it does not

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exceed it. The step size in iteration,based on energy criteria, can be alteredby varying the magnitude of the limitingnormal stress. It should be pointed outthat the definition of energy capacity is

independent of the actual state of stress in the element and depends onlyon the volume and on the limitingnormal stress of the elements.The expression for the energy capacityof the i th element is given by

( ) ( )12

U U i i i iV = (24)

Assuming the material to be linearly

elastic the relation between limitingnormal stress and strain is written as

( ) ( )U U i i i E = (25)

Substitution of equation (25) in (24)gives the expression for energycapacity in the form

( ) 21 ( )2

U i i i il = (26)

Quantity l i is defined as

i ii

i

V E l

= (27)

The scalar is the base parameter for all the elements and i is the relativevalue of the i th design variable. Theactual design variable vector may bewritten as . The vector alone willbe referred to as the relative designvariable vector or normalized vector.The scalar is then the normalizingfactor. The strain energy in the elementis written in terms of its internal forcesand displacements as

'12

t i i iu s v= (28)

Where s i is the generalized force vector of the i th element and v i is the

corresponding relative displacementvector. It is related to the actualdisplacement vector v i by

v i = v i(29)

According to the modified optimalitycriteria the strain energy of eachelement should bear a constant ratio toits energy capacity. Equations (26) and(28) and the optimality criteria yield

'2 2

'iuC

= (30)

Where C is the constant of proportionality and u i and i are givenby

' ' '

' ( ) 2

121

( )2

i i i

U i i i i

u s v

l

=

=(31) & (32)

Multiplying both sides of equation (30)by i2 and taking the square root yields

1' 2

'i

i ii

uC

=

(33)

Where i is the i th design variablewhich is expressed as a function of xi.The form of equation (33) suggests thefollowing recursion relation for determining the design variable in eachcycle.

1' 2

1 '( ) ( )i

i v i vi v

uC

+

= (34)

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Where the subscripts and +1 refer to the cycles of iteration.When the design conditions includemultiple loading case equation (34) will

be modified to read1

' 2max

1 '( ) ( )i

i v i vi v

uC

+

= (35)

Where u imax is a measure of themaximum strain energy of the i th

element due to any of the loadingconditions. In the case of a singleloading condition with stress constraintsonly, the design that satisfies theoptimality criteria is the lowest weightdesign. Iteration using equation (35)produces such a design. When thereare constraints on the sizes of theelements and multiple loadingconditions the design satisfying theoptimality criteria may not be the lowestweight design. In such cases equation(35) will be used to simply generatedesign lines and in conjunction with thescaling procedure a directed search for the lowest weight design can be made.

4. Case study of a classical problem

25 Bar space truss problemhas been considered for the casestudy. The 25-bar transmission tower space truss, shown in Fig. 1, has been

size optimized by many researchers.These include Schmit and Farshi [7],Schmit and Miura [8], Gellatly andBerke [9], Rizzi [10] apart from theauthors already mentioned in section 3.In these studies, the material densitywas 0.1 lb/in 3 and modulus of elasticitywas 10,000 ksi.

This space truss was subjectedto the two loading conditions shown in

Table 1. The structure was required tobe doubly symmetric about the x- andy-axes; this condition grouped the trussmembers as follows: (1) A 1, (2) A 2 ~ A 5,(3) A 6 ~ A 9, (4) A 10 ~ A 11 , (5) A 12 ~ A 13 ,(6) A 14 ~ A 17 , (7) A 18 ~ A 21 & (8) A 22 ~A25 . The truss members were subjectedto the compressive and tensile stresslimitations shown in Table 2. Inaddition, maximum displacementlimitations of 0.35 in. were imposed on

every node in every direction. Theminimum cross-sectional area of allmembers was 0.01 in 2. The comparisonof results are given in Table 3.

Node No.Condition 1 Condition 2

P x P y P z P x P y P z

1 0.0 20.0 -5.0 1.0 10.0 -5.0

2 0.0 -20.0 -5.0 0.0 10.0 -5.0

3 0.0 0.0 0.0 0.5 0.0 0.0

6 0.0 0.0 0.0 0.5 0.0 0.0

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Table 1 Loading conditions for 25 Bar space truss(Loads are in Kips)

Fig 1 25 Bar space truss

Variables MembersCompressive

stress limitations(Ksi)

Tensile stresslimitations (Ksi)

1 A1 35.092 40.0

2 A2 ~ A 5 11.590 40.0

3 A6 ~ A 9 17.305 40.0

4 A10 ~ A 11 25.092 40.0

5 A12 ~ A 13 35.092 40.0

6 A14 ~ A 17 6.759 40.0

7 A18 ~ A 21 6.059 40.0

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8 A22 ~ A 25 11.082 40.0

Table 2 Member stress limitations for 25 Bar space truss

Members

Schmit

& Farshi[7]

Rizzi[10]

K Deb &

S Gulati[3]

K S Lee& Z WGeem

[4]

KV John, CVRamakrishna

n & KGSharma [5]

V B

Venkayya[6]

A1 0.010 0.010 0.060 0.047 0.01 0.028

A2 ~ A 5 1.964 1.988 2.092 2.022 1.604 1.942

A6 ~ A 9 3.033 2.991 2.884 2.950 3.540 3.081

A10 ~ A 11 0.010 0.010 0.010 0.010 0.01 0.01

A12 ~ A 13 0.010 0.010 0.010 0.014 0.01 0.01

A14 ~ A 17 0.670 0.684 0.690 0.688 0.652 0.693

A18 ~ A 21 1.680 1.677 1.640 1.657 1.761 1.678

A22 ~ A 25 2.670 2.663 2.691 2.663 2.446 2.627

Weight(lb) 545.22 545.16 544.984 544.38 548.07 545.49

Table 3 Comparison of results for 25 Bar space truss (Areas in2)

5. Merits and de-merits

Compared to gradient-basedmathematical optimization algorithms,the HS algorithm imposes fewer mathematical requirements to solveoptimization problems and does notrequire initial starting values for thedecision variables. The HS algorithmgenerates a new vector after considering all of the existing vectorsbased on the HMCR and the PAR,rather than considering only two(parents) as in genetic algorithms.These features increase the flexibility of the HS algorithm and produce better solutions. However, this method hasfew shortcomings. Selection of HMCRand PAR plays a vital role in the resultsand the number of iterations required toconverge to the optimum is very high.

The improved modified double-cut approach for solving nonlinear engineering design problems is veryeffective for fuzzy structuraloptimization. The proposed modifiedapproach is the most recommended for its beneficial to simple algorithm andthe neglecting of the redundant designvariable. It also improves the finaldesign as compared with single-cutapproach and sophisticated multiple-cutapproach. The proposed modifieddouble-cuts approach guarantees toobtain the unique compromise finaldesign.

It is found that the optimizationtechnique that is based on the forcemethod is computationally far moreefficient than the displacement one.

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Improved move limit method of sequential linear programming is anapproach which is naturally suited for solution and works very well for all trialproblems. The number of design

iterations for convergence is nearly thesame or even less than that for other well established methods. However,optimisation of complex space trussproblems using this method needs tobe verified.

The strain energy criterionmethod charts an efficient path to theoptimum and this makes it attractive for the optimization of structures with a

large number of variables. However,the validity of the optimality criteriaapproaches have been established onlyfor restrictive cases and need to beextended to more general designconditions.

One of the resulted difficulties of SQP for large-scale problem is that thememory requisite for each QP sub-problem may be very large if the

original problem is large.6. Way ahead & Conclusion

Most of the new methodsdeveloped in the field of structuraloptimisation are deviating from theclassical approach. As the complexitiesof the structure has increased theefficiency of the method play a vital roleas far as rate of convergence andaccuracy of the results are concerned.Most of the gradient based methodsare getting replaced by meta-heuristicmethods which promises to reduce thecomputational difficulties. The commonfactor in meta-heuristic algorithms isthat they combine rules andrandomness to imitate natural

phenomena . However, most of thesemethods are not yet been tested onwide range of structural optimisationproblems.

One has to exercise extremecaution while dealing with suchmethods as errors in formulation canlead to wrong results. An approach withmixture of different methods, exploitingthe advantages of each one of themlooks quiet promising for complexproblems.

References

[1] C.J. Shih, H.W. Lee. Modified double-cuts approach in 25-bar and 72-bar fuzzytruss optimization. Computers andStructures 84 (2006) 21002104

[2] R. Sedaghati, E. Esmailzadeh. Optimumdesign of structures with stress anddisplacement constraints using the forcemethod. International Journal of Mechanical Sciences 45 (2003) 13691389.

[3] Kalyanmoy Deb, Surendra Gulati. Designof truss-structures for minimum weightusing genetic algorithms. Finite Elementsin Analysis and Design 37 (2001)447}465.

[4] Kang Seok Lee, Zong Woo Geem. A newstructural optimization method based onthe harmony search algorithm.Computers and Structures 82 (2004)781798

[5] K V John, C V Ramakrishnan and K GSharma. Minimum weight design of trusses using improved move limitmethod of sequential linear programming.Computers & Structures Vol. 27. No. 5.pp. 583-591. 1971.

[6] V B Venkayya. Design of optimumstructures. Computers & Structures Vol.1, pp. 265-309. 1987.

[7] Schmit Jr LA, Farshi B. Someapproximation concepts for structuralsynthesis. AIAA J 1974;12(5):6929.

[8] Schmit Jr LA, Miura H. Approximationconcepts for efficient structural synthesis.NASA CR-2552, Washington, DC: NASA;1976.

[9] Gellatly RA, Berke L. Optimal structuraldesign. AFFDLTR-70-165, Air ForceFlight Dynamics Lab., Wright-PattersonAFB, OH; 1971.

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[10] Rizzi P. Optimization of multi-constrainedstructures based on optimality criteria.AIAA/ASME/SAE 17th Structures,Structural Dynamics, and MaterialsConference, King of Prussia, PA; 1976.

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Term paper - State of the art in Structural optimisation

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Transcript of Term paper - State of the art in Structural optimisation