pareto frontier for simultaneous structural ... - pasini...

Struct Multidisc Optim (2010) 40:497–511DOI 10.1007/s00158-009-0366-4

INDUSTRIAL APPLICATION

Pareto frontier for simultaneous structuraland manufacturing optimization of a composite part

H. Ghiasi · D. Pasini · L. Lessard

Received: 2 October 2007 / Revised: 30 January 2009 / Accepted: 6 February 2009 / Published online: 7 March 2009© Springer-Verlag 2009

Abstract Optimized design of composite structures re-quires simultaneous optimization of structural perfor-mance and manufacturing process. Such a challengecalls for a multi-objective optimization. Here, a gen-erating multi-objective optimization method callednormalized normal constraint method, which attainsa set of optimal solutions and allows the designer toexplore design alternatives before making the finaldecision, is coupled with a local-global search calledconstrained globalized bounded Nelder–Mead method.The proposed approach is applied to the design of aZ-shaped composite bracket for optimization of struc-tural and manufacturing objectives. Comparison of theresults with non-dominated sorting genetic algorithm(NSGA-II) shows that when only a small number offunction evaluations are possible and a few Paretooptima are desired, the proposed method outperformsNSGA-II in terms of convergence to the true Paretofrontier. The results are validated by an enumerationsearch and by exploring the neighbourhood of the finalsolutions.

Keywords Multi-objective optimization ·Composite materials · Pareto frontier ·Global optimization · Simultaneous optimization

H. Ghiasi (B) · D. Pasini · L. LessardDepartment of Mechanical Engineering,McGill University, MacDonald Eng. Bldg,817 Sherbrooke St. West, Montreal,Quebec, Canada, H3A 2K6e-mail: [email protected]

Nomenclature

n Number of design variablesm Number of objectives�n Design space of a problem with n design

variablesS Feasible region of the design space

(S ⊂ �n)Z Criterion space

(Z = �f (S ) ⊂ �m

)

�x∗ A Pareto optimum solution in the designspace

X∗ All Pareto optimal points in the designspace

�f ∗ = �f (x∗) A Pareto optimum in criterion space

f̄ Normalized objectiveφ(x) Probability of sampling a point x ∈ Sr Number of non-dominated solutions ob-

tained during the optimization process

1 Introduction

Because of their excellent mechanical properties, lami-nated fibrous composite materials are successfully usedin a wide range of structural applications. However, dueto the large number of design variables and objectives,the design of composite materials is more complexthan the design of uniform isotropic materials. Thisis not only due to the anisotropic material propertiesbut also because of the strong interconnection betweendesign and manufacturing issues. Thus, a common wayto simplify the composite design problem is to separatestructural design from manufacturing design (Le Riche

498 H. Ghiasi et al.

et al. 2003) and performing the latter just after theformer is completed. However, researches by Wangand Costin (1992), Costin and Wang (1993), Hendersonet al. (1999), Wang et al. (2002), Le Riche et al. (2003),and a series of papers by Park et al. (2001, 2003, 2004,2005) have shown that a better overall performanceis obtainable if structural and manufacturing objec-tives are simultaneously optimized. This task requiressolving a multi-objective optimization (MOO) prob-lem, which is generally more challenging than solvinga single-objective optimization problem and involvesmore computational effort.

This paper proposes using a method called nor-malized normal constraint method (NNCM) to simul-taneously optimize several objectives in a compositedesign problem. To obtain each of the numerous pos-sible solutions to this problem, NNCM performs asingle-objective optimization. For the single-objectiveoptimization, a local-global search called constrainedglobalized Nelder–Mead (CGBNM) is suggested. Theproposed method is applied to the design of a Z-shapedcomposite bracket, the goal of which is to find thefiber orientations and the geometrical parameters thatmaximize the strength and minimize the weight and thespring-in after demoulding. The results are comparedto the ones obtained by one of the most successfulmulti-objective evolutionary optimization methods inthe literature, called non-dominated sorting geneticalgorithm (NSGA-II; Deb et al. 2002).

In the following sections, the main terminology anddefinitions used in MOO are provided before reviewingthe applications of these methods in design of compos-ite materials. The proposed MOO method, its internalsingle-objective method, and the two parameters usedto measure the performance of a MOO method areexplained in Section 3. Section 4 applies the proposedmethod to the design of a Z-shaped composite bracketand compares the results with NSGA-II. Validationof the numerical results, discussion, and concludingremarks are given at the end.

2 MOO and design of composite materials

In many applications, such as design with compositematerials, usually there is more than one objective in-volved; this type of optimization problems is generallycalled multi-objective optimization (MOO) problems.This section introduces the terminology used in MOOand reviews its applications in design of compositematerials.

2.1 Multi-objective optimization

The process of systematically and simultaneously opti-mizing several objectives is called multi-objective opti-mization (MOO). A MOO problem is mathematicallyexpressed as:

minimize �f (x) = {fi (x) : �n �→ �;i = 1, ..., m; m > 1

}

subject to

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

g j (x) ≥ 0 ; {g j (x) : �n �→ �;j = 1, ..., J; J ≥ 0

}

hk (x) = 0 ; {hk (x) : �n �→ �;k = 1, ..., K; K ≥ 0

}

(1)

The objectives in this problem are in contrast to eachother, and there is no unique solution to a problemwith more than one conflicting objective. There exista number of solutions which all are optimum. Thesesolutions are called Pareto optimum solutions (Pareto1906). The definition of Pareto optimality is based ondomination that follows.

Definition 1 (domination) Considering the optimiza-tion problem formulation in (1), a solution x1 ∈ S domi-nates a solution x2 ∈ S, if x1 is smaller than x2 in at leastone objective and is not bigger with respect to the otherobjectives.

x1 dominate x2 ⇔{∀i : 1 ≤ i ≤ m ⇒ fi (x1) ≤ fi (x2)

∃ j : 1 ≤ j ≤ m : f j (x1) < f j (x2)∧

(2)

Definition 2 (Pareto optimal) Considering the prob-lem formulation (1), a solution x∗ ∈ S is Pareto optimalif and only if it is not dominated by any other solutionin S. Collection of all Pareto solutions is called Paretofrontier, represented by X∗.

x∗ ∈ X∗ ⇔ � ∃x ∈ S : x dominate x∗ (3)

There are numerous methods for multi-objectiveoptimization, and they can be classified in many waysaccording to different criteria. A comprehensive reviewof these methods can be found in Miettinen (1999), Deb(2001), and Marler and Arora (2004). For the purposeof this paper, we categorize these methods into twodistinct groups:

1. Non-generating methods that find only one Paretosolution for a given MOO problem. The Pareto so-lution may be selected with or without consideringthe user’s preferences. If the user preferences are

Pareto frontier for simultaneous structural and manufacturing optimization of a composite part 499

required, they may be obtained at the onset of theoptimization process (method with priori prefer-ences) or interactively as the optimization processgoes on (methods with progressive preferences).

2. Generating methods that generate a set of Paretooptimal solutions among which the user choosesthe final solution according to his/her preferences.These methods are also called methods with poste-riori preferences.

This section reviews the application of differentMOO methods to the design of composite materials,which is the focus of this paper.

2.2 Multi-objective design of composite materials

Current literature on MOO of composite materialsreveals mainly the use of non-generating methods.Although these are preferred for their simplicity andfor being less time consuming; there are attempts ofusing generating methods to design a composite part.Both methods are reviewed in this section.

Method of weighted-sum is the most commonMOO method used in composite design. In this non-generating method, the objectives are combined intoa single-objective problem using user-defined weight-ing factors. The resulted single-objective optimizationproblem may be solved by any optimization technique.Examples of application of this method in stackingsequence design of composite materials can be foundin Adali (1983), Walker et al. (1997), Manne and Tsai(1998), Walker and Smith (2003), Deka et al. (2005),and Mohan Rao and Arvind (2005), and Abouhamzeand Shakeri (2007). Kere et al. (2003) and Kere andKoski (2002) also used the weighted-sum, but only toreduce their multi-objective optimization problem toa bi-criterion problem. The most sensitive objectivewas used as the first criterion, while the others werecollected in a weighted sum as the second criterion.The resulted bi-criterion problem was solved by using alayer-wise approach, in which all possible permutationsof adding a new layer to the laminate were examined,and the one that made the maximum improvement ineither of the two criteria was accepted.

Another strategy, also classified among non-generating methods, is to optimize one criterion whileconstraining the others to user-defined limits. Thesemethods are generally known as “ε-constraintmethods” and are commonly used in composite designoptimization. Wang and Costin (1992) and Costin andWang (1993) found the minimum weight design of acomposite shell by applying constraints on manu-facturing objectives. Henderson et al. (1999) integrated

manufacturing considerations as constraints into thedesign optimization of a blade stiffened panel. Parket al. (2001) applied constraints on processing time andpanel stiffness to minimize the weight of a plate madeby RTM. Wang et al. (2002) optimized the numberof ribs and spars in an aerospace composite structureusing weight as the primary design drive and thecost parameter as a constraint. (Le Riche et al. 2003)used the same method but frequently exchanged theobjective and constraints.

Reference point method is another non-generatingMOO method that has been used in composite designapplications (Saravanos and Chamis 1992; Kere andLento 2005). This method minimizes an achievementfunction based on a reference point. The referencepoint, defined by the designer, is a feasible or infea-sible point in the criterion space that is reasonable ordesirable to the designer. The achievement functionmay be the Euclidian distance to the reference point orany other user-defined measure. Appropriate selectionof the reference point has a major effect on the finalsolution obtained by this method.

There are some non-generating methods with nouser-defined preferences; an example is the min-maxstrategy, in which only the critical objective is opti-mized. For instance, if the goal of a problem is tomaximize the strength of several components withinan assembly, only the part with the minimum strengthis considered in each step of the optimization process.The limitation is that this method requires the objec-tives to have comparable values. It is particularly usefulfor stress or stiffness minimization (Suresh et al. 2007).

Non-generating methods generally require an in-sight into the problem because the preference parame-ters (e.g. weighting factors, constraint values, referencepoint, etc.) must be set by the designer. These meth-ods find only one optimal solution; however, they canalso be used to generate a set of Pareto optimal solu-tions by varying the user-defined preference parame-ters (Watkins and Morris 1987; Adali et al. 1996; MohanRao and Arvind 2005), but the resulted solutions maynot be uniformly spread along the Pareto frontier. Incontrast, generating methods have the advantage ofnot requiring any user-defined preferences and gener-ating a set of optimum solutions. The penalty, on theother hand, is that they usually need a great deal ofcomputation. Non-dominated sorting genetic algorithm(NSGA) and particle swarm are examples of methodsfalling in this group that are used for optimization ofcomposite materials.

Non-dominated sorting genetic algorithm (NSGA)was proposed by Srinivas and Deb (1995) as an evo-lutionary method based on genetic algorithm (GA).


NSGA differs from simple GA only in the waythe selection operator works. Before selection, indi-viduals are ranked according to the level of non-domination. Each solution assigned a fitness equal toits non-domination level (first is the best level), thusminimization of the fitness promotes non-dominatedsolutions and eventually reaches the Pareto frontier.The original NSGA has no control on the spread ofthe obtained solutions. Deb et al. (2002) overcomethis problem by adding a crowded control parameterto NSGA. The new method, called NSGA-II, is anelitist non-dominated sorting GA that provides a setof non-dominated solutions uniformly distributed onPareto frontier. Using mathematical test functions, Debet al. (2002) reported that NSGA-II outperforms othercontemporary multi-objective evolutionary methodsin terms of convergence to the true Pareto fron-tier and achieving a uniform distribution of solutions.An integer-coded version of NSGA-II was used byPelletier and Vel (2006) to find Pareto-optimal designsfor a composite laminates.

Particle Swarm Optimization (PSO) is anotherpopulation-based, stochastic optimization method usedin design of composite materials. Inspired by the flock-ing behaviour of the birds, each solution in this methodis called a “particle” and resembles a bird amongothers. Each bird adjusts its position according to itsown flying experience (best solution in its individualhistory) and the flying experience of the others (thebest solution among all particles). There are severalmethodologies using PSO to handle problems with mul-tiple objectives, among them Vector Evaluated PSO(VEPSO), a multi-swarm variant of PSO, was used byOmkar et al. (2008) to minimize weight and cost oflaminated composite components. VEPSO considers mswarms, each consisting of n particles. Each swarm isexclusively evaluated according to one of the objectivefunctions. The adjustment process takes place accord-ing to the flying experience of the particle itself, andthe particles in one of the other swarms. Althoughits performance is reported to be “satisfactory”; nocomparison with other methods has been found by theauthors.

A population-based generating MOO, such asNSGA-II or VEPSO, is computationally time consum-ing because it needs numerous function evaluations.Composite design problems have the particularity ofhaving a time consuming function evaluation process,which usually involves performing several finite ele-ment analysis. Therefore, in many cases, it may notbe possible or desirable to perform as many func-tion evaluations as is required for a population-basedmethod to converge. The other property of population-

based methods is that they usually return a set of non-dominated solutions almost as large as their populationsize. To facilitate the selection of the final solution, adesigner may prefer to have only a few solutions witha low computational cost rather than a large numberof alternative solutions with a high computational cost.For these situations, this paper proposes a combinationof a local single-objective optimization technique anda MOO approach called NNCM. The next section ex-plains the proposed method, the single-objective opti-mization method and how the latter is embedded in theformer.

3 Optimization procedure

Among generating methods, a survey of which can befound in Marler and Arora (2004), Miettinen (1999),and Deb (2001), normalized normal constraint method(NNCM; Messac et al. 2003) is implemented in thiswork. This method normalizes the design space andintroduces new constraints. Considering the new con-straints, optimization of only one of the objectives re-turns a non-dominated solution. When several of thesesingle-objective optimization problems are solved, sev-eral non-dominated solutions are obtained. The differ-ence between this method and varying user preferencesin a non-generating method is that here the set ofconstraints are introduced to spread the final solutionsuniformly in the criterion space. This section explainsthe NNCM method, the single-objective optimizationmethod and their integration. Finally, two parametersare presented to measure the performance of a MOOmethod.

3.1 Normalized normal constraint method

Normalized normal constraint method (NNCM) is analgorithm for generating a set of evenly spaced solu-tions on a Pareto frontier (Messac et al. 2003). Thismethod yields Pareto optimal solutions, and its per-formance is independent of the scale of the objectivefunctions. NNCM method and some related definitionsare presented in this section.

Definition 3 (utopia point) Considering optimizationproblem (1), a point �f o ∈ Z in the criterion space iscalled a utopia point if and only if:

f oi = min

x

{fi (x)| x ∈ S

}; i = 1, ..., m. (4)

Because of contradicting objectives, the utopia point isunattainable.


Definition 4 (anchor point) A non-dominated point�f ∗∗ ∈ Z is an anchor point if and only if it is Pareto

optimal and at least for one i = 1, 2, ..., m; f ∗∗i =

minx

{ fi (x)| x ∈ S}.

The first step in NNCM is to normalize the de-sign space. For this purpose, the utopia and theanchor points are required. These points are found byoptimizing only one of the objectives at a time. Afterfinding these points, the criterion space is normalizedusing the following transformation.

f̄i = fi − f oi

f maxi − f o

i(5)

f maxi = max

{fi (x) ; x /∈ X∗} (6)

The normalization process locates the utopia pointat the origin and the anchor points at the unit coordi-nates. Figure 1a shows the original criterion space andthe Pareto frontier of a generic bi-criterion problem.Figure 1b represents the Pareto frontier of the sameproblem after normalization.

The next step is to form the utopia hyperplane, whichis a hyperplane with vertices located at the anchorpoints. For a bi-criterion problem, the utopia hyper-plane is a line as shown in Fig. 1c. Next, a grid of evenlydistributed points on the utopia hyperplane is gener-ated. The number of points in this grid is defined by theuser. Figure 1c shows, for example, a grid of six pointson the utopia line. If these points are projected ontothe Pareto frontier, several Pareto optimum solutionsare obtained. To find the Pareto optimum solution cor-responding to each point in this grid, a single-objectiveoptimization problem must be solved. This problem

entails minimizing one of the normalized objectiveswith an additional inequality constraint. For example,the Pareto optimum solution corresponding to pointP in Fig. 1c can be found by minimizing f̄2 while thefeasible region is cut by the line passing through thispoint and perpendicular to the utopia line. The feasibleregion of this single objective optimization problem isshown in Fig. 1c. The solution of this problem, f̄ ∗,is a Pareto optimum solution for the original multi-objective problem. Other Pareto optimal points can befound by repeating the same procedure for other pointson the utopia line.

If the objective functions have many local optimaor the Pareto frontier is discontinuous, it is possible tohave some dominated solutions among the final solu-tions. Composite stacking sequence design problem iswell known for having many local optima; therefore,dominated solutions are expected. Messac et al. (2003)proposed a filter that removes all dominated solutions,after all the single-objective optimizations are com-pleted. This filter requires a pair wise comparison ofall the solutions. Since this algorithm aims at findinga small number of solutions on the Pareto frontier(e.g. around ten points) the performance of the filteringalgorithm is of minor concern and not discussed here.

3.2 CGBNM for single-objective optimization

In order to find each Pareto optimum solution, NNCMrequires solving a single-objective optimization prob-lem. Since this algorithm is proposed for solving acomposite design problem, in which the gradients ofthe objectives are not available, a direct optimizationmethod is required. On the other hand, consideringthe time consuming process of structural and manu-facturing analysis of a composite part, an evolutionary

Feasible Region

1f

2f

1

1O

**2f

**1

f

Of

(c)

P*f Utopia Line

Feasible Region

Pareto Frontier

1f

2f

1

1O

**2

f

**1

f

Of

(b)

Feasible Region

Pareto Frontier

Of

2f

1f

**1f

**2f

(a)

Fig. 1 a A typical bi-criterion space, b normalized criterion space, c a normal constraint introduced by NNCM and the feasible regionof the resulted single-objective problem (min f̄2)


algorithm may not be a good choice due to the lowrate of convergence. Therefore, a local-global searchcalled CGBNM (Ghiasi et al. 2008) is used for this pur-pose. CGBNM, which stands for constrained globalizedbounded Nelder–Mead, is an improved version of thealgorithm introduced by Luersen and Le Riche (2004).This method is using several restarts of a simplex opti-mization method called Nelder–Mead (N–M) method(Nelder and Mead 1965). The restart procedure usesa probability distribution function that initiates N–Mlocal search from new points far from previously sam-pled ones. CGBNCM is capable of finding several localoptima for a multi-modal function. It has been shownthat CGBNM is more efficient than an evolutionaryalgorithm, when a small number of function evaluationsare possible (Ghiasi et al. 2008).

The flowchart of CGBNM is shown in Fig. 2. Themain blocks marked by the gray pattern and thebold border, are: “Nelder–Mead local optimizer” and“Restart procedure”. The first finds a local optimum,while the second restarts the local search to confirmits convergence to a true optimum or to help findinganother local solution. The maximum number of itera-tions for each restarts and the total number of functionevaluations are defined by the user at the onset of theprocess.

Nelder–Mead (N–M) sequential optimization meth-od, proposed by Nelder and Mead (1965), is amongthe most popular direct methods for local optimizationof unconstrained problems. N–M method compares theobjective values at a set of n + 1 points called a simplex.The simplex is moved toward the optimum solutionby four operators: reflection, expansion, contraction, orshrinkage. Reflection operator mirrors the worst pointin the simplex with respect to the other points in thatsimplex. This is the basic step that moves the simplextoward a better solution. If the new point is betterthan the old point, the move is expanded by usingthe expansion operator, else it is contracted by usingthe contraction operator. If neither of these operatorscould find a better point, the simplex is shrunk towardits best point. The process is terminated when thesimplex converges to an optimum. It has been shownthat this method is effective in practice by producinga rapid initial decrease in function values (Lagariaset al. 1998), but the local optimum found is dependenton the initial simplex (Humphrey and Wilson 2000).Therefore, in CGBNM a probability function is usedto restart the local search from new points far frompreviously inspected regions in the design space.

The restart procedure re-initializes N–M in order toensure that the solution found in the last try is a local

Fig. 2 Restart andconvergence tests linkingin CGBNM

Initialize a random Polyhedron with size “a”

Solution of the Nelder-Mead local optimizer

T1: Maximum number of analyses is reached. T2: This point is already known as a local optimum. T3: (Small or Large test and return to

the same point) OR (not Small test and point not on the bound and small simplex) OR (flat simplex)

T4: Large test or Probabilistic and not return to the same point and point on the bound

T5: Small test and not return to the same point and not on the bounds

T6: N-M stopped by maximum number of iterations allowed for each N-M local optimizer.

T3

T4

T5

Save possible local optima

Save possible local optima

Save local optima

Set Restart= Probabilistic

Set Restart = Small test

Set Restart= Large test

T6Save possible local optima

Set Restart = Small test

Set Restart= Large test

Restart

Uses initial points and local optima to calculate the probability function (either by Luersen or VVP method)

T2

T1

Y

Y

Y

Y

Y

Y End

Save Initial Points

N

N

N

N

N

N


optimum (small test or large test) or to find a newlocal optimum (probabilistic restart). The small test andlarge test restart the N–M from the last solution ob-tained with a simplex of size as or al, user-defined para-meters. If the new small or large simplex returns to thesame solution, the result is saved as a local optimum,and the N–M is initialized with a probabilistic restart.The probabilistic restart initiates the local search froma simplex located in the region that has not been pre-viously explored. This strategy eases the finding of anew local optimum. The probabilistic restart procedureuses a one-dimensional adaptive probability functioncalled variable variance probability (VVP). Using VVP,probability of sampling point x is calculated as follow:

φ (x) = 1√2πσ

(1 − e−(d2

min

/2σ 2)

)(7)

dmin = mini=1,...,I

⎧⎨

⎩di =

√∑n

k=1

(xk,i − xk

xku − xkl

)2⎫⎬

⎭(8)

xi s in this equation are the points in the design spacethat are previously sampled. dmin is the minimum non-dimensional Euclidian distance between point x andone of these points. The variance of the normal proba-bility density, σ , is updated in each restart by using thefollowing equation:

σ = 1

3 n√

m(9)

Then a selection pool is created, in which each pointhas a number of copies proportional to its probabilityvalue. Therefore, points far from previously sampledpoints have more chance to be selected as an initialpoint for the next local optimization.

N–M method in its original form is unable to dealwith nonlinear constraints; however, a composite de-sign problem is often constrained by several nonlinearconstraints, such as failure criteria, and others that mayemerge by using NNCM. Therefore, it is important thatthe selected optimization method be able to handlethese constraints. In CGBNM, this goal is achieved byusing a repair procedure that brings the infeasible so-lutions into the feasible region. This procedure consistsof a backtracking scheme; when a new point generatedby reflection or expansion violates one of the nonlinearconstraints, the new point is moved toward the originalfeasible point such that the distance between these twopoints is reduced by a factor α ∈ (0,1). The procedureis terminated when either a feasible point is found or

a predetermined number of trials is reached. If theprocedure fails to find a new feasible point, the originalpoint is kept, and the simplex is shrunk towards its bestvertex.

3.3 Performance measures

Two performance metrics used by Deb et al. (2002) tomeasure the performance of a non-generating MOOmethod are presented in this section. These two para-meters are used later to compare the proposed methodwith NSGA-II. The first metric, γ , measures the ex-tent of convergence to a known set of Pareto-optimalsolutions, while the second, �, measures the extentof spread achieved among obtained solutions. Bothmetrics are positive numbers and are desired to be assmall as possible.

Convergence of a set of solutions to a known Paretofrontier is measured by the average of the minimumdistance of all the solutions from the Pareto frontier. Tofind the minimum distance from the Pareto frontier, agrid of uniformly distributed points on Pareto frontierare generated. The minimum distance of a solutionfrom one of the points in this grid is used as theminimum distance to Pareto frontier. The convergencemetric, γ , is mathematically defined as below:

γ = 1p

p∑

i=1

di; di = minj

∥∥∥ �fi − �f ∗

j

∥∥∥ (10)

In this equation, ‖ ‖ shows the Euclidian distancebetween the two points in the criterion space. Figure 3aillustrates how this metric is calculated for a bi-criterionproblem.

The second metric, �, provides information aboutthe extent of spread achieved among the obtained so-lutions. It is desired that a set of solutions obtainedby a generating MOO spans the entire Pareto-optimalregion and is uniformly distributed along the Paretofrontier. The following equation is used to calculate thismetric for a bi-criterion problem:

� = l0 + l p + ∑p−1i=1

(li − l̄

)

l0 + l p + (p − 1) l̄(11)

l̄ = 1p−1

∑p−1

i=1li (12)

As shown in Fig. 3b, l0 and l p of the above equa-tion are the Euclidian distances between the extremesolutions and the anchor points. li is the Euclidian dis-tance between two solutions. This metric is zero if the


Fig. 3 a Convergence metricγ , b diversity metric,� (Deb et al. 2002) Feasible Region

Anchor points

1f

2f

(b)

**2

f

**1f

il

1f

if

pf

Feasible Region

1f

2f

(a)

*jf

*1f

*2f

�

��

�

�

�

�

�

�

�

�

1f

2f

if id

Points on Pareto frontier

solutions are equally spaced and includes both anchorpoints.

4 Composite test case and numerical results

In this section, NNCM is applied to the simultane-ous structural and manufacturing optimization of aZ-shaped composite bracket shown in Fig. 4. Thebracket is made of 16-ply balanced symmetric laminateof graphite/epoxy (AS4/8552) with fiber orientation[±θ1/±θ2±θ3/±θ4

]s. The goal is to find geometries and

lamination sequences that minimize weight and spring-in and maximize strength. The part should not fail ordelaminate anywhere and should satisfy a safety factorof 1.5 against failure and 2 against delamination. De-lamination is calculated in the curved regions where theangle shape causes high interlaminar normal stresses.The vertical deflection of less than 1 mm and the spring-in of less than 0.5◦ are strictly required for an acceptabledesign. The semi-analytical models of first-ply-failure,delamination, deflection, and spring-in used for theoptimization process are described in Appendix 1.

4.1 Optimization set up

The MOO method described in section three is usedto solve this composite design problem. The objectivefunctions are divided into two groups of manufacturingobjectives and structural objectives. Considering thesetwo group of objectives, NNCM is used to find thePareto solutions.

The first group of objectives deals with the structuralperformance including the weight minimization andstrength maximization. The two objectives are groupedinto one weighted-sum as:

min fs = W(g)

5 g− R

1.5(13)

where W is the weight, and R is the load factor. Thesecond group of objectives deals with the manufactur-ing aspect and includes only minimization of spring-inafter demoulding.

min fm = |�θ | (14)

Fig. 4 Geometrical variablesand applied loads on thecomposite bracket

N1=500 N/m

N3=100 N/m

M1=50 N.m/m

τ 12=300 N/m


By incorporating the set of inequality constraints de-scribed in the problem definition, the optimizationproblem is formulated as follow:

min{

fs (x) , fm (x)}; x = {

θ1, θ2, θ3, θ4, e, sr, r}

Subject to

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{R (x) ≥ 1.5 ∧ D (x) ≥ 2.0

∧ Sr (x) ≥ 10 mm ∧ |δ (x) | ≤ 1 mm

∧ |�θ (x)| ≤ 0.5◦}

θi ∈ [−90◦, 90◦] ; i = 1, ..., 4;e ∈ [0, 0.15] (m) ;sr ∈ [2, 5] (cm) ;r ∈ [2, 20] (mm) ;

(15)

Where D(x) in this equation is the delamination factor.Sr, e, and r are the geometrical parameters shown inFig. 4. δ(x) is the maximum vertical deflection, andfinally �θ(x) is the spring-in.

The first step of NNCM is to find the utopia andthe anchor points. To this end, two single-objectiveoptimization problems must be solved; one optimizesonly the structural objectives while the manufacturingobjective is set free to take any value. The secondproblem minimizes only the manufacturing objectiveand the structural ones are ignored. We refer to thesolution of the first problem as structural-only solu-tion

(f ∗∗s

), and the solution of the second problem as

manufacturing-only solution(

f ∗∗m

).

f ∗∗s = (0, |−0.149|) = (0, 0.149) ;

f ∗∗m =

(9.47 g

5 g− 1.51

1.5, 0

)= (0.886, 0) ; (16)

The utopia point does not correspond to any physicaldesign and is calculated as follow.

�f O = (f Os , f O

m

) ; f Os = 6.99 g

5 g− 2.15

1.5= −0.037;

f Om = |−0.0048| = 0.0048. (17)

Figure 5 shows the two anchor points, f ∗∗s and f ∗∗

m ,in the normalized criterion space. The correspondinglamination sequences and 2D cross-section shapes arealso represented beside each point. Crossed-line di-agrams in this figure and Fig. 6 represent the fiberorientations in the bracket. Vertical lines represent thefibers running along the length of the bracket, whereas

Sf

mf

O

Utopia Line

1

**sf

**mf C

B

A

Pareto Frontier

W=7.0gr R=2.2

Δθ=-0.15o

W=9.5gr R=1.5

Δθ=0.00o

1 W=9.5gr R=1.5

Δθ=0.00o

Fig. 5 Utopia point, two anchor points, utopia line and the user-selected points for composite bracket design problem

horizontal ones show the fibers running along the widthof the bracket (normal to the cross-section).

Regarding the objective values at utopia and anchorpoints the normalization is performed using the follow-ing transformation.

f̄s = fs (x) + 0.037

0.923; f̄m = fm (x) − 0.005

0.144(18)

4.2 Numerical results

NNCM requires solving a set of single-objective min-imization problems with an additional nonlinear con-straint. The additional nonlinear constraint makes the

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

Anchor pt minStr minMfg

mf

Sf

C

B

A

1

2

Fig. 6 Graphical representation of 2D cross-section and lami-nation sequences of the solutions found on the Pareto frontier.Crossed-line diagrams represent the fiber orientations


feasible region tighter, thus the solution to the single-objective minimization problem would be differentthan the anchor points. Here, two different scenariosare considered. In the first scenario, called minStr, thestructural objective is minimized. In contrast, the sec-ond scenario, called minMfg, minimizes the manufac-turing objective. Each scenario uses three values for theconstraints, seeking three points on the Pareto frontier(i.e. A, B, and C in Fig. 5).

The optimization problem in the first scenario, min-Str, is formulated as:

min f̄s (x) ; x = {θ1, θ2, θ3, θ4, e, sr, r}

S.T. :

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

R (x) ≥ 1.5

D (x) ≥ 2.0

Sr (x) ≥ 10 mm

|δ (x) |≤ 1 mm

|�θ (x)| ≤ 0.5◦

f̄s − f̄m + a ≥ 0 (19)

The last constraint in (19) is added by NNCM, inwhich the constant a in is set to −0.5, 0, and 0.5 to obtainpoints A, B, and C in Fig. 5. The optimization problemfor the second scenario, minMfg, is identical to the firstone but involving minimization of f̄m and replacing thelast constraint with f̄m − f̄s + a ≥ 0.

Figure 6 shows the results obtained in two scenar-ios, minStr and minMfg, each including 2,000 functionevaluations. As expected, the points obtained in thesetwo scenarios are similar regarding the objective values.Points A, B, and C in this figure are shown by theirnormalized objective values. The optimized geometry(left) and lamination sequence (right) of the corre-sponding bracket are also shown for each data point.

Table 1 summarizes the objective values of the op-timal solutions. As the design point moves from thestructural-only design (point 1) toward manufacturing-only design (point 2), (a) the geometry graduallychanges from small brackets to elongated ones and (b)the fiber orientations are changed from laminates inwhich fibers are distributed in all directions to laminateswhere fibers are mostly aligned at 0◦.

The Pareto curve helps the designer better under-stand the trade-off between the conflicting objectives.

Table 1 The performance criteria at the two anchor points andat three other points on the Pareto frontier

Point 1 A B C 2

Weight (g) 7.0 7.0 6.9 7.0 9.5Deflection (mm) 0.046 0.073 0.099 0.166 0.133Load factor 2.2 2.1 2.0 1.8 1.5Spring-in (degree) −0.15 −0.14 −0.07 −0.02 −0.00

For instance, the results have shown that starting fromthe best manufacturing design, penalizing only 10%the manufacturing objective (e.g. moving from a pointwith f̄m = 0 to f̄m = 0.1), the structural objective willimprove more than 75% (e.g. is reduced from f̄s = 1to f̄s = 0.25). However, this is not valid at the pointwith f̄s = 0.25, where 10% penalty on manufacturingobjectives results in almost equivalent improvement instructural performance.

4.3 Validation and interpretation of the results

In this section, the optimization procedure is validatedby comparing the results to the ones obtained by arough enumeration search over the design domain anda non-dominated sorting genetic algorithm (NSGA).The local optimality of the results is also demon-strated by analysing the neighbourhood of the optimalsolutions.

This problem, similar to all other composite designproblems, has many local optima and the global op-tima are unknown. Thus, to judge the validity of theresults we resort to an enumeration search. Obtainingall possible fiber orientations and geometries requiresnumerous function evaluations (i.e. 1010 function evalu-ations for this problem). Here, we decide to examine allpossible designs with fibers oriented at 0◦, ±45◦, and 90◦each with 64 different geometries, which still requiresmore than 5,000 function evaluations.

Figure 7 shows the results of the enumeration searchin the normalized criterion space beside the solutionsfound by NNCM. The solutions obtained by NNCMare located at the boundary of the feasible region. Theunexplored region, right of these solutions, is resulted

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

Anchor pt minStr minMfg Enumeration Search

mF

A

B

C

SF

Fig. 7 Enumeration search and the points on Pareto frontier thatobtained by NNCM


Table 2 Performance parameters, γ and �, and number of non-dominated solutions, r ′, obtained by NNCM and NSGA-II

γ � r′

NNCM 0.2415 0.9077 7NSGA-II: p20g100 0.3110 0.8193 20NSGA-II: p50g40 4.9185 1.4724 48NSGA-II: p100g20 0.4449 1.0788 70

by the choice of having examined fiber orientationsonly at 0◦, ±45◦, and 90◦. A valuable result is that thesolutions found by NNCM dominate the optimum solu-tions found by the enumeration search, and it capturesthe overall shape of the boundary of the feasible region.For instance, at the lower-right corner of this figure,the feasible region is tangent to the horizontal axes.This property is also reflected by plotting a line passingthrough solutions found by NNCM.

4.3.1 Comparison with NSGA

Evolutionary algorithms are commonly used in designof composite materials and also to solve MOO prob-lems. Thus, here the result obtained by NNCM is com-pared to those obtained by NSGA-II. The comparisonis made with respect to the performance parametersdescribed in Section 3.3. Both methods are applied tothe composite design problem with maximum 2,000function evaluations.

Performance of NSGA-II is dependent on the pop-ulation size and the number of generations. Three dif-ferent cases were examined and the results are shownin Table 2. Each case is labelled with the populationsize and number of generations; for instance the caselabelled as p50n40 has the population size of 50 chro-

mosomes and the optimization process was continuedtill 40 generations. Number of generations and popula-tion sizes are selected such that, in each case, the totalnumber of 2,000 function evaluations is performed. Theoptimal solutions obtained are presented in Fig. 8a.This figure clearly shows that the solutions obtainedby NNCM dominate all NSGA-II’s solutions. Table 2compares the same results in terms of the performancemetrics introduced in Section 3.3. Since the real Paretofrontier of the composite design problem is unknown,the origin and the two anchor points are used to calcu-late �. This table shows that the NNCM can achievea better convergence to the Pareto frontier than allNSGA-II runs with almost similar level of spread to thebest results obtained by NSGA-II.

One of the advantages of the improved GBNM,which makes it comparable to population-based meth-ods, is that it not only returns one global solution, butalso provides a set of local optima called a local set.Non-dominated points in the local set can be used togain more insight into the shape of the Pareto frontier.Figure 8b shows the local optima found during theoptimization of the problem at hand. The local setprovides additional non-dominated solutions.

4.3.2 Analysis of the neighbourhoodof the optimal solutions

Verifying the global optimality of the solutions is avery difficult task; however it is possible to investigatethe neighbourhood of the solutions to verify their localoptimality. The neighbourhood of a solution is specifiedby all possible solutions that can be obtained whenonly one of the design variables is varied. As exam-ples, this is applied to points A, B, C. Figure 9 shows

Fig. 8 a Comparison of thePareto frontier found byNNCM and NSGA,b all local solutions obtainedby NNCM coupled withGBNM

(a) (b)

0 0.25 0.5 0.75 1

mF

A

B

C

2

1

Feasible Region

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

NSGA-II: p20g100

NSGA-II: p100g20

NSGA-II: p50g40

NNCM

SF SF

mF


Point A in minStr scenario

0.5

0.75

1

1.25

1.5

0.0 0.1 0.2 0.3 0.4 0.5

Point B in minStr scenario

0

0.25

0.5

0.75

1

0.0 0.1 0.2 0.3 0.4 0.5

Point C in minStr scenario

0

0.25

0.5

0.75

0.0 0.2 0.4 0.6

Fig. 9 Neighbourhood of points A, B, and C in normalized criterion space (FS vs. = Fm)

the neighbourhood of the optimal solutions when alldesign variables are changed one at a time. Appar-ently, all the solutions found are non-dominated in thatneighbourhood.

In this section, the design of a composite bracket hasbeen simultaneously optimized for structural and man-ufacturing objectives. The results were validated bycomparing them with an enumeration search and anevolutionary optimization method and also by explor-ing the neighbourhood of the solutions. The result notonly attests to the ability of the algorithm to grab thegeneral form of the Pareto frontier but also showsthat the proposed MOO is able to find better resultsthan those obtained with an evolutionary MOO (i.e.NSGA-II). The trade-off between structural and manu-facturing objective also proves the necessity of a multi-objective approach for composite material design.

4.4 Discussion

The overall performance of a method is controlledby the parameters defined by the user. This sectiondiscusses how user-defined parameters may affect theperformance of NNCM and NSGA-II.

If NSGA-II is used with a limit on the total numberof function evaluations (N), the only parameter leftfor the designer to select is the population size (p), orthe number of generations (g) considering that N = pg.In NSGA-II, the user does not have control over thenumber of non-dominated solutions, q, which is usuallyknown, by experience, to be close to the populationsize, q ≈ p (see Table 2 for examples). If only a smallnumber of non-dominated solutions are required, asmall population should be utilized. This choice is at therisk of losing the diversity of the population and con-verging to non-optimal solutions. On the other hand,using a large population size reduces the number ofgenerations, but it may not converge to a promising

set of solutions. Similarly, the total number of functionevaluations (N) performed in NNCM is

N = q (2 lr) , (20)

q Number of non-dominated solutions desired by theuser;

l Number of N–M iterations for each local optimiza-tion process;

r Number of restarts in CGBNM

If q is small, which is the case we are interested in,there would be enough number of function evaluationsfor each single objective optimization to find a pointclose to the Pareto frontier. This explains why in thenumerical example solved in this paper NNCM outper-forms NSGA-II. If more non-dominated solutions arerequired, NSGA-II will be a better choice.

Regarding the number of function evaluations, N,and number of non-dominated points required, q, it issuggested to use NNCM when q and N are small. Incontrast, when N and q are large numbers, NSGA-II isrecommended. The threshold between small and largevalues for N and q is problem dependent and mustbe found by experience. Figure 10 plots these regions.The top left quarter corresponds to the case where asmall number of function evaluations is required toobtain a large number of non-dominated solutions. Inthis region, none of the considered algorithms are able

N

q

NNCM NNCM NSGA-II

NSGA-II NSGA-II

Fig. 10 Suggested optimization method regarding the maxi-mum number of function evaluations (N) and number of non-dominated solutions required (q)


to obtain a good solution, but comparatively NSGA-IIhas been observed to yield better results. In contrast,at the bottom right quarter of the chart, both methodsare expected to have a similar performance, althoughNSGA-II might be selected due to its simplicity.

For a function with a large number of local optima,the local search embedded in NNCM may converge toa non-optimal solution. This drawback amplifies evenmore when non-linear constraints are imposed by theproblem. Figure 8b shows that, among all the localsolutions found by CGBNM, many are located on theboundary of the feasible region (the imaginary linespassing through points A, B, and C in Fig. 8b). To makea more efficient single-objective optimization search byCGBNM, a more efficient constraint handling methodis required.

5 Conclusion

In order to find a set of Non-dominated solutions for acomposite design problem, the normalized normal con-straint method (NNCM) and a local-global optimiza-tion method called constrained globalized boundedNelder–Mead (CGBNM) are coupled in this paper. Theproposed method is compared to an evolutionary multi-objective optimization method by solving the simulta-neous structural and manufacturing optimization of aZ-shaped composite bracket. This type of problem hastwo following characteristics: (1) function evaluation istime consuming, thus performing only a small numberof function evaluation is possible, (2) the designer doesnot need a large number of non-dominated solutions.For such problems, the NNCM is found to be moreefficient than a population-based method, consideringthe proximity to the true Pareto frontier and the uni-form spread of the solutions. Comparing the resultswith an enumeration search and investigation of theneighbourhood of the optimal solutions also confirmsthat the proposed method is able to accurately and

efficiently find non-dominated solutions very close tothe Pareto frontier. The solutions found by the pro-posed algorithm clearly dominate the ones obtained byNSGA-II. The corresponding geometries, fiber orien-tations, and performance criteria of the optimal solu-tions are plotted, showing that the trade-off betweenstructural and manufacturing objectives exist and mustbe considered during the design process. Having a setof non-dominated solution helps the designer to betterunderstand this trade-off and have some alternativedesigns before making the final decision.

Appendix 1

Modeling and simulation of the composite bracket

To evaluate the objectives of a composite bracketdesign problem, an appropriate processing simulationand structural analysis are required. For this purpose,a semi-analytical model is developed in MATLAB©.The model should be able to evaluate objectives andconstraints including the load factor (R), the verticaldeflection (δ), the spring-in (�θ), and the delaminationfactor (D).

The load factor is calculated using first-ply-failureof the classical lamination theory and Hashin stress-based failure criterion (Tsai 1992). Vertical deflection iscalculated by numerical integration and energy method(Megson 1999). The spring-in, which is the angulardeformation of a part after demoulding, is a function ofcure shrinkage and thermal expansion, and is calculatedusing the following equation:

�θ = θ

[((αl − αt) �T

1 + αt�T

)+

(φl − φt

1 + φt

)](21)

where �θ is spring-in, θ is the angle of the bracket,�T is the temperature difference between the cure

Fig. 11 Approximatedistribution of interlaminarnormal stress at a free edge


Fig. 12 Interlaminar normalstresses in the curved regionof an angle-shaped flange

–

temperature and the room temperature. φ and α arecoefficients of shrinkage and thermal expansion, sub-scripts l and t, respectively, stand for longitudinal andthrough thickness directions.

Delamination is a critical mode of failure in compos-ite materials, and it is due to the interlaminar stressesbetween subsequent laminates. In a flat plate, interlam-inar stresses are created only by the free-edge effect(Pagano and Pipes 1989), but in a curved part, the 3Dstress field also creates significant interlaminar stressesthat may cause delamination at the curved regions. Inthis paper, we adopt a convenient, albeit crude, modelof free edge interlaminar stresses by Pagano and Pipes(1989). In this model the interlaminar normal stress isestimated as shown in Fig. 11.

Interlaminar normal stresses created by the angle-shape effect are shown in Fig. 12. Sequentially solvingthe equilibrium equations for all layers, starting formthe innermost layer results in an interlaminar normalstress between layers n and n + 1:

σz,n−(n+1) =n∑

k=1

σx,ktk/

R +n∑

k=1

tk (22)

where, ti shows the thickness of ith layer, and R isthe inner radius of the curved part. Interlaminar shearstresses are of minor importance with respect to inter-laminar normal stresses (Tsai 1992), thus an approxi-mation of shear stress in a prismatic member under atransverse load is used.

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