StrucOpt manuscript No.(will be inserted by the editor)

Concurrent Performance and Manufacturing CostOptimization of Structural Components?

W. D. Nadir, I. Y. Kim?, and O. L. de Weck

Abstract This paper presents a structural shape opti-mization method that considers not only structural per-formance but also manufacturing cost. Typical structuraldesign optimization involves the optimization of impor-tant structural performance metrics such as stress, mass,deformation, or natural frequencies. Often factors suchas manufacturing cost are not considered in structuraloptimization. In this paper, manufacturing cost is an im-portant performance metric along with typical structuralperformance metrics. The trade-off between manufactur-ing cost and structural performance is observed in twoexamples using the manufacturing process of abrasivewaterjet (AWJ) cutting.

Key words Cost modeling, multi-objective optimiza-tion, abrasive waterjet

Accepted: July 29, 2005

W. D. Nadir, I. Y. Kim, and O. L. de Weck

Department of Aeronautics & Astronautics, Room 33-409,Massachusetts Institute of Technology, Cambridge, Massa-chusetts 02139, USAe-mail: bnadir@alum.mit.edu, e-mail: iykim@me.queensu.caand e-mail: deweck@mit.edu

Send offprint requests to: William D. Nadir

? Presented as paper AIAA–2004–4593 at the 10th AIAA-ISSMO Multidisciplinary Analysis and Optimization Confer-ence, Albany, New York, August 30-September 1, 2004

? Present address: Dept. of Mechanical Engineering,Queen’s University, Kingston, Ontario, K7L 3N6, Canada

Nomenclature

C = Abrasive waterjet (AWJ) cutting speedestimation constant

Cman = Total manufacturing cost, USDdm = Mixing tube diameter of the AWJ

cutting machine, indo = AWJ cutter orifice diameter, inE = AWJ cutter error limitfa = Abrasive factor for abrasive used in

AWJ cutterh = Thickness of material machined by

AWJ, cmJ = Objective functionLj = Step length for jth step along cut curvem = Number of curves being optimized in

the structureM = Mass, kgMa = AWJ abrasive flow rate, lb/minNm = Machinability NumberOC = Overhead cost for machine shop, $/hrPw = AWJ water pressure, ksiq = AWJ cutting qualityR = Arc section cut radius for AWJ cutter, insi = Total number of steps along ith cutting

curveumax = AWJ maximum linear cutting speed

approximation, in/minx = X-coordinate design variable vectory = Y-coordinate design variable vectorα = Objective function weighting factorδ = Deflection, mmσ = Stress, Pa

1Introduction and Literature Review

Typical structural design optimization involves the op-timization of important structural performance metricssuch as stress, mass, deformation, or natural frequencies.This structural design method often does not consideran important factor in structural design: manufacturingcost. In this research, manufacturing cost is an important

2

performance metric in addition to typical structural per-formance metrics. The weighted sum method by Zadeh(1963), is used to observe the trade-off between manu-facturing cost and structural performance. Two exam-ples are presented which exhibit this trade-off. These ex-amples involve optimization of two-dimensional metallicstructural parts: a generic part and a bicycle frame-likepart.

While it is not possible to construct a manufacturingcost model that represents all manufacturing processes,the scope of this research has been limited to one man-ufacturing process: rapid prototyping using an abrasivewater jet (AWJ) cutter. Although AWJ cutting is theonly manufacturing process considered, this framework isgeneralizable to other manufacturing processes providedthat realistic parametric cost models of the manufactur-ing process can be created and verified.

1.1Literature review

The aim of structural optimization is to determine thevalues of structural design variables which minimize anobjective function chosen by the designer for a struc-ture while satisfying given constraints. Structural op-timization may be subdivided into shape optimizationand topology optimization. For shape optimization, thetheory of shape design sensitivity analysis was estab-lished by Zolesio (1981) and Haug (1986). Bendsøe andKikuchi (1988) and Suzuki and Kikuchi (1991) proposedthe homogenization method for structural topology op-timization by introducing microstructures and appliedit to a variety of problems. Yang and Chuang (1994)proposed artificial material and used mathematical pro-gramming for topology optimization. Kim and Kwak (2002)first proposed design space optimization, in which thenumber of design variables and layout change during thecourse of optimization. Kim and de Weck (2005) de-veloped a genetic algorithm in which the chromosomelength changes as optimization progresses and appliedthe method to structural topology optimization prob-lems.

Structural shape optimization has been performedalong with an estimation of manufacturing cost by Changand Tang (2001). This work involved optimization of athree-dimensional part in order to reduce mass and man-ufacturing cost for the special application of the fabrica-tion of a mold or die. However, manufacturing cost wasnot included in either the objective or constraint func-tion, as is done in this paper. Park (2004) performedoptimization of composite structural design consideringmechanical performance and manufacturing cost. Thiswork focused on the optimal stacking sequence of com-posite layers as well as the optimal injection gate loca-tion to be used in the composite material manufacturingprocess, but also did not perform multidisciplinary opti-mization including manufacturing cost. Martinez (2001)

and Curran (2005) performed optimization consideringstructural performance and manufacturing cost. Theircost models are based on empirical data, not a theo-retical model as in this paper. In addition, their designvariables consisted of component sizes and section prop-erties while we change the structural shape using splinecurves, allowing for significantly more design freedom.

The weighted sum method is a popular method forhandling multi-objective problems. Zadeh (1963) per-formed early work on the weighted sum method. In ad-dition, Koski (1988) used the weighted sum method forthe application of multicriteria truss optimization.

The standard method for determining manufacturingcost for the AWJ manufacturing process is presented byZeng (1993) as well as Singh and Munoz (1993). Toestimate manufacturing cost, Zeng (1993) use the cut-ting speed of the water jet cutter to estimate cost via therequired cutting length and layout.

AWJ cutting speed prediction models have been pre-sented by Zeng (1999). Zeng and Kim developed a widelyaccepted AWJ cutting speed prediction model. In addi-tion, Zeng (1992) developed the theory behind AWJcutting process. Zeng et. al. (1992) conducted an ex-perimental study to determine the machinability num-bers of engineering materials used in water jet machiningprocesses.

For the purposes of this paper, the AWJ cutting speedmodel presented by Zeng and Kim is used. The Zeng andKim model has been used by Singh and Munoz to predictAWJ cutting speed and is also used in Omax water jetCAM software.

While other researchers have performed structuralshape optimization and investigated manufacturing cost,a lack of research exists for true concurrent structuralperformance and manufacturing cost optimization of struc-tural components with the use of spline curves for in-creased design freedom.

2Optimization Framework

This section presents the optimization framework usedto obtain an optimal structural design which meets thegiven design requirements. The modeling assumptions,optimization problem statement, optimization algorithm,and details of the software modules used in the simula-tion are presented.

2.1Modeling assumptions

Several assumptions are made in the models for simpli-fication. These are:

– The cuts made by the abrasive waterjet cutter forstructural optimization example are closed curves.

3

– The cuts can not disappear or join together.– The cuts can not intersect each other or the structural

part boundary unless they define the part boundary.

These models were developed to investigate the trade-off between structural performance and manufacturingcost by incorporating a manufacturing cost model into amulti-objective optimization framework. These assump-tions allowed for an exploration of the design space withina reasonable amount of time. More advanced models canbe developed to allow for hole generation or merging, asdone by Lee (2004).

2.2Design objectives

Using the weighted sum method, the two considered de-sign objectives are combined into a single objective func-tion to minimize. The first design objective is structuralperformance defined as mass. The second is manufactur-ing cost.

J(xij , y

ij) = αM + (1− α)Cman (1)

The objective function used for these simulations isshown in (1). In (1), J is the objective function, M is thestructural mass, Cman is the total manufacturing cost ofthe structural component, xi

j and yij are the design vec-

tors composed of the X and Y-coordinates of the jth

control point for the ith Non-uniform rational b-spline(NURBS) curve, respectively, and α is the weighting fac-tor for the two objectives.

NURBS, defined in Piegl and Tiller (1997), are usedto describe the cut curves in the part. NURBS curves arechosen for their ability to control the shape of a curveon a local level by each of the defined control points, orknots. A complex shape can be represented with littledata in the form of several of these control points.

2.3Design variables

The design variables for the simulation are the X andY coordinates of the control points defining the curvesalong which the abrasive waterjet cuts are made. There-fore, two design variables are required for each controlpoint to define cuts in the component being optimized.The total number of design variables depends on thenumber of cutting curves and the number of control pointsused for each curve.

x ≡({x1

1}, {x12}, . . . , {x1

n1}, . . . , {xm

1 }, {xm2 }, . . . , {xm

nm})

(2)

y ≡({y1

1}, {y12}, . . . , {y1

n1}, . . . , {ym

1 }, {ym2 }, . . . , {ym

nm})

(3)

In (2) and (3), ni is the total number of control pointsfor the ith curve and m is the total number of curvesbeing optimized in the structure.

2.4Design constraints

The constraints imposed on this problem statement areside constraints of the design variables and maximumvon-Mises stress in the structure. These constraints aredefined in the following equations.

σmax ≤ σc (4)

xij,LB ≤ xi

j ≤ j, xiUB (5)

yij,LB ≤ yi

j ≤ yij,UB (6)

In (4), (5), and (6), σmax is the maximum von-Misesstress in the structure, σc is the maximum von-Misesstress constraint, and xi

j,LB , xij,UB , yi

j,LB , and yij,UB are

the lower (LB) and upper bound (UB) side constraintsfor the design vector variables controlling the jth controlpoint for the ith NURBS curve. These side constraintsare different for each design variable given the nature ofthe problems being optimized.

2.5Flow chart

The optimal structural design for the given range of de-sign requirements is determined using an optimizationapproach shown in Figure 1. A gradient-based optimizeris combined with a finite element analysis software mod-ule and an abrasive waterjet manufacturing cost estima-tion module to determine the optimal design solution.

The initial design, defined from X, Y coordinatesand geometrical parameters, is input to the system andthe objective function is evaluated using finite-elementanalysis with ANSYS 8.1 and the manufacturing costestimation model. Rather than perform structural opti-mization and then off-line manufacturing cost evaluation,manufacturing cost and structural performance are bothcalculated simultaneously for each design from the opti-mizer. These designs are then evaluated based on theirrespective objective function values.

2.6Gradient-based shape optimization

The optimization procedure used to optimize the shapeof the cutting curves is performed using a gradient-basedoptimization algorithm, a MATLAB sequential quadraticprogramming optimization function, fmincon.

4

Fig. 1 Shape optimization flow chart.

2.7Manufacturing cost estimation

This module is used to determine the manufacturingcost for performing abrasive waterjet manufacturing forstructural components. The manufacturing process ofabrasive waterjet cutting uses a powerful jet of a mixtureof water and abrasive and a sophisticated control systemcombined with computer-aided machining (CAM) soft-ware. This provides for accurate movement of the cuttingnozzle. The result is a machined part with tolerancesranging from ±0.001 to ±0.005 inches. It is possible forAWJ cutting machines to cut a wide range of materialsincluding metals and plastics (Zeng et. al. (1992)).

The inputs to the AWJ manufacturing cost estima-tion module include design variables and parameters suchas material properties, material thickness, and abrasivewaterjet settings. The output of this module is the AWJmanufacturing cost and time to manufacture the struc-tural design.

Based on the material thickness and material prop-erties, a maximum cutting speed is determined for theAWJ cutting machine. While the cutting speed of thewaterjet cutter is constant throughout most of the cut-ting operation for a sufficiently large cutting path radiusof curvature, the cutting speed of waterjet slows if anysharp corners or curves with small arc radii lie alongthe cutting path. (7) is used to determine the maximumlinear cutting speed of the AWJ cutter, umax. The over-head cost associated with using the AWJ cutting ma-chine, OC, is shown in (8). This cost factor is providedas an estimate of the manufacturing cost overhead forthe MIT Department of Aeronautics and Astronauticsmachine shop.

umax =(

faNmP 1.594w d1.374

o M0.343a

Cqhd0.618m

)1.15

(7)

OC = $75/hr (8)

In (7) and (8), fa is an abrasive factor, Nm is themachinability number of the material being machined,Pw is the water pressure, do is the orifice diameter, Ma

is the abrasive flow rate, q is the user-specified cuttingquality, h is the material thickness, dm is the mixing tubediameter, and C is a system constant that varies depend-ing on whether metric or Imperial units are used (Zeng(1993)). The AWJ settings used for this simulation areshown in Table 1.

AWJ Setting ValueAbrasive factor, fa 1Machinability number, Nm 87.6Water pressure, Pw 40Orifice diameter, do 0.014Abrasive flow rate, Ma 0.71Cutting quality (1 = min, 5 = max), q 5Mixing tube diameter, dm 0.030Constant, C 163

Table 1 Abrasive waterjet machining settings used in costmodel.

The cutting path in a typical abrasive waterjet man-ufacturing job is not linear. This issue requires a modi-fication to the linear cutting speed estimation equationin order to estimate the cutting speed along cut curveswith an arc section radius, uas. This involves a modifi-cation to (7) using (9) to replace the quality factor, q.This modification takes into account the radius of cur-vature of the cut path, R. The resulting cutting speedestimation is shown in (10).

q =0.182h

(R + E)2 −R2(9)

umax =

faNmP 1.594w d1.374

o M0.343a

[(R + E)2 −R2

]0.182Ch2d0.618

m

1.15

(10)

In (9) and (10), E is the error limit. When the water-jet traverses a curve or executes a sharp corner, the lagcauses an error in following the true line because the exitpoint at the bottom of the material is not above the entrypoint at the top of the material. As the traverse speedis lowered, the lag and the associated error is reduced.The user-defined error limit is related to the quality levelfor the surface of the cut. From Olsen (1996), the ap-propriate error limit of 0.001 is used with respect to thedesired cutting quality.

Total manufacturing cost is estimated using (11).

Cman = OC

m∑i=1

si∑j=1

Lj

u(i,j)

(11)

5

Fig. 2 Omax output screenshot for short cantileveredbeam example.

Fig. 3 MATLAB AWJ cost model output for shortcantilevered beam example.

In (11), Lj is the length of the jth step along thecutting curve, u is the AWJ cutting speed for the ith

step along the jth curve, either arc section or maximumlinear cutting speed, m is the maximum number of closedcurves, and si is the total number of steps along thecutting curve for the ith curve.

In order to validate the manufacturing cost estima-tion model, results from the model are compared to Omaxresults for a short cantilevered beam manufacturing sce-nario. Omax contains an accurate manufacturing costestimator and is a good benchmarking tool for this ap-plication. A screenshot of the Omax result is shown inFigure 2. Figure 3 is the output of the MATLAB AWJcost estimation model.

The results of the software validation shown in Ta-ble 2 show the MATLAB manufacturing cost estima-tion software accurately estimates manufacturing cost forabrasive waterjet cutting.

Omax Cost ModelManufacturing Time (min) 1.69 1.71Manufacturing Cost $2.14 $2.11

Table 2 Manufacturing cost estimation module validationresults.

2.8Structural analysis module

The structural analysis software is linked to MATLABfor the optimization process. Required inputs to thismodule are the material properties, geometrical defini-tions for the structure, degree of freedom constraints forthe structure, and load vectors applied to the structure.Outputs obtained from the module are the maximumvon-Mises stress and the structural volume. These out-puts are used to evaluate the objective function and de-termine if the structural design satisfies the constraints.

3Example 1: Generic Part Optimization

The first example presented is mass versus manufactur-ing cost optimization for a simple structural part. Visu-alization of the design variable side constraints for theshape control points is shown in Figure 4. The grey ar-eas denote zones in which the three sets of four shapecontrol points are free to move.

Fig. 4 Side constraints of the shape control points for genericstructural part optimization example.

It can be seen in Figure 4 that the side constraintsrestrict the simulated abrasive waterjet cuts to be inter-nal to the part. The side constraints for this example arerestricted to the zones shown in order to prevent NURBS

6

curves from intersecting each other or with the boundaryof the part. If any of these intersections were to occur,the ANSYS structural analysis module would not be ableto mesh the part and compute a solution.

3.1

Optimization procedure

MATLAB modules were created to perform the struc-tural optimization for manufacturing cost and structuralperformance for this example. These routines include amain software module, an AWJ manufacturing cost es-timation module (Section 2.7), and a structural analysismodule (Section 2.8). Important parameters and initial-ization techniques associated with each software modulefor this design example are presented in this section.

3.1.1

Simulation parameters

The important parameters set in this module are thegeometry of the structural component, the number ofinitial designs to consider, objective function weightingfactors, material properties of the truss structure ele-ments, and abrasive waterjet settings. For this structuraldesign example, the geometry defining the boundary ofthe part is defined. These properties are presented inSection 2.4. Three different initial designs were selectedfor the simulations. This is explained in more detail inSection 3.1.2. The material properties are defined in thismodule as well. The material selected is A36 Steel witha Young’s modulus of 200 GPa, a Poisson’s ratio of 0.26,and a yield strength of 250 MPa. The abrasive waterjetsettings used are defined in Section 2.7.

The material thickness of the part is assumed to be1 centimeter. The boundary conditions of the part aredesigned such that the part is fixed in all directions atthe base as shown in Figure 5. The evenly-distributedpressure across the top of the part, also shown in Figure5, is 3.7x107N/m2. A factor of safety of 1.5 is assumedfor this example.

Three holes are cut in the metallic part and the shapesof these holes are controlled by four control points each.These control points are illustrated in Figure 6. The cut-ting path created by the control points is determinedusing NURBS curves created in ANSYS using the splinecommand.

In this example, objective function weighting factorsof 0.2, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, and 0.95 areused. The criteria used for selecting the weighting factorsis explained in Section 3.2.3.

Fig. 5 Generic structural part design including loading andboundary conditions.

3.1.2

Initialization

This design optimization example is performed by start-ing the optimization algorithm at three different initialdesigns. Optimization is performed by first defining aninitial structural solution guess. These three designs areselected to attempt to broadly search the design spacewith the goal of finding solutions close to the global op-timum. The initial designs for the example, shown inFigure 6, include small, medium, and large holes cut inthe blank metallic part.

Fig. 6 Initial designs for the generic structural part shapeoptimization example.

7

The goal of starting the optimization with many dif-ferent initial guesses is to attempt to find a near-globaloptimal solution. A gradient-based optimization method,used for the outer loop of this optimization framework,has a tendency to get trapped at a local optimal solu-tion. By starting the optimization routine from severaldifferent locations in the design space, there is a greaterpotential for finding a near-optimal solution.

3.2Results

Structural component shape optimization considering bothperformance and manufacturing cost is performed for ageneric metallic structural part shown in Figure 5.

3.2.1Objective space results

Pareto frontier results for shape optimization for this ex-ample are shown in Figure 7. The maximum von-Misesstress constraint is active for all designs along the Paretofrontier except the results for weighting factors of 0.2 and0.6.

Fig. 7 Objective space results for generic part optimizationwith objective function weighting factor, α, labeled for eachdesign.

An evenly distributed Pareto frontier is not foundin this multiobjective optimization. This phenomenon islikely caused by the fact that the objectives being min-imized are highly nonlinear in terms of the weightingfactor, α, and an even distribution of weighting factorsis not the best method to find the Pareto front. The useof the adaptive weighted-sum (AWS) method by Kimand de Weck (2005) may alleviate this problem and will

be implemented in future work. In order to attempt toovercome this difficulty, a select set of weighting factorsis chosen to attempt to obtain a well-distributed Paretofrontier. As can be seen in Figure 7, even this select setof weighting factors does not yield such a Pareto frontier.

Although the Pareto frontier is not well-distributed,the trade-off between mass and manufacturing cost canbe seen. A pseudo-Pareto frontier is denoted by connect-ing all the non-dominated design solution points becausethe actual Pareto frontier is not known given the designsolutions obtained.

3.2.2Design space results

Selected structural designs from the set from Figure 7are shown in Figures 8, 9, and 10.

Fig. 8 Weighting factor of 0.6.

Fig. 9 Weighting factor of 0.7.

8

Fig. 10 Weighting factor of 0.9.

The structural design results demonstrate the trade-off between cost and mass. When manufacturing cost isweighted more heavily, the cut-outs in the metallic partare small. However, when mass is weighted more heavily,the cut-outs in the part are significantly larger and oneor more of the holes are at or near the side constraintboundaries. This means the optimization algorithm isattempting to remove material to minimize structuralmass, as expected.

3.2.3Objective space results discussion

It is observed that the weighted sum design solutions arenot in the expected order. The solution from the weight-ing factor of 0.2 should have lower cost and greater massthan the solution for the weighting factor of 0.6, yet thisis not the case. There are two likely causes for this prob-lem. First, it is possible that too few initial designs areinvestigated in order to find a near-global optimal designsolution. The design solutions found are likely local op-tima and not global optimal solutions. However, a morelikely cause of this problem is that manufacturing cost isnot only a function of cutting curve length but also theradius of curvature of the cutting path. As mentionedpreviously, in the manufacturing cost model, a specificcutting path radius of curvature limit exists at whichcuts with radii greater than the limit are assumed tobe at the maximum cutting speed. As shown in Figure11, below this radius of curvature limit, cutting speed isslower and not constant and therefore the cost per unitlength of material increases.

Figure 11 illustrates this radius of curvature limitfor manufacturing cost minimization. The example usedto illustrate this phenomenon is a comparison of closedcircular cuts with varying radii. Figure 11 shows theminimum manufacturing cost with respect to radius ofcurvature. A clear minimum manufacturing cost can be

Fig. 11 Manufacturing cost vs. radius of curvature for cir-cular cuts.

seen at the limit of the maximum linear cutting speed.This minimum was obtained from observations of the ra-dius of curvature limit at which Omax software assumedthe maximum linear waterjet cutting speed was used forvarious cutting qualities. Two important trends can beseen in Figure 11. First, when the radius of curvature isless than the minimum cost radius of curvature, cuttingspeed dominates the manufacturing cost. This results ina dramatic rise in manufacturing cost for small reduc-tions in radius of curvature. For radii of curvature largerthan the minimum cost radius, cost is dominated by cut-ting length. This leads to an increase in manufacturingcost with a linear relationship to radius of curvature.

3.2.4Convergence information

The convergence histories for the optimizations performedfor each weighting factor are shown in Figure 12. Theconvergence rate is dependent on the objective functionweighting factor and ranges from two to thirty-eight iter-ations. As shown in the figure, starting the optimizationfrom an infeasible region does not prevent the optimizerfrom finding feasible designs and converging. In Figure12, for infeasible initial designs, the change to the feasibledesign region is noted.

4Example 2: Bicycle Frame Optimization

This section includes the same optimization algorithmapplied to a more complex structural component designexample. This component is a two-dimensional bicycleframe-like structure. The design objectives for this ex-ample are the same as those for the generic part opti-

9

Fig. 12 Convergence histories for the generic part structuraloptimization example.

mization example (see (1)). The design variables for thesimulation for this example are identical to those pre-sented in Section 2.3.

4.1Design constraints

The constraints imposed on this problem statement areX, Y location shape constraints of the design variablesand maximum von-Mises stress in the structure. Theseconstraints are defined in (4), (5), and (6). Shape designvariable constraints are shown in Figure 13.

Fig. 13 Shape design constraints of the control points forbicycle frame design optimization example.

Figure 13 illustrates how the simulated abrasive wa-terjet cuts form large portions of the part boundary. Thecontrol point shape constraints are restricted to the zones

shown in order to prevent any of the resulting NURBScurves from intersecting each other.

4.2Optimization procedure

The same optimization procedure presented for Example1 is used for this design example. Differences in the designexample problem setup are presented in this section.

4.2.1Parameters

For this example, the boundaries of the portions of thestructure not being optimized are predefined. These prop-erties are presented in Section 4.1. Three different ini-tial designs were selected for the simulations. This isexplained in more detail in the following Initializationsection. The material properties and abrasive waterjetsettings for this design example are the same as Exam-ple 1.

For this example, an evenly distributed set of elevenweighting factors between 0 and 1 are used. The criteriaused for selecting the weighting factors is explained inSection 4.3.2.

4.2.2Initialization

Design optimization is performed by starting the opti-mization algorithm at three different initial designs aswas done for Example 1. These initial designs are shownin Figures 14, 15, and 16. The bicycle frame structureshave thin, medium, and thick-sized structural members.A near-global optimum design is found by selecting thebest design of the three optimized solutions resultingfrom the three different initial designs. These best de-sign solutions are used to create the Pareto frontier.

Fig. 14 Thin structural member initial design mesh and con-trol points for bicycle frame structural optimization example.

10

Fig. 15 Medium thickness structural member initial designmesh and control points for bicycle frame structural optimiza-tion example.

Fig. 16 Thickest structural member initial design mesh andcontrol points for bicycle frame structural optimization ex-ample.

4.3Results

Shape optimization considering both structural perfor-mance and manufacturing cost is performed for a bicy-cle frame-like part shown in Figure 17. The size of thestructure is roughly 20 cm width by 10 cm height.

4.3.1Simulation parameters

The material thickness of the part is 1 centimeter. Theloads and restraints are shown in Figure 17. A factor ofsafety of 1.5 is assumed.

Ten curves controlled by three control points each areused to determine the shape of the structure while thestructural shape at the vertices of the structure remainunchanged. The relationship of the control points to thecurves are shown in Figures 14, 15, and 16.

4.3.2Objective space results

The Pareto frontier shown in Figure 18 demonstratesthe trade-off between manufacturing cost and mass. Themagnitude of improvement in manufacturing cost alongthe Pareto frontier is not large. For this example, a manu-facturing cost savings of approximately 1.6% is observed

Fig. 17 Structural part design with loading and boundaryconditions shown.

when comparing the two anchor points of the Paretofrontier. However, a small improvement in manufacturingcost applied to a product being mass produced can resultin a large cost savings for a manufacturer. In addition,the observed trade-off between cost and mass would bemore significant if the shapes of the bicycle frame jointsare included in the design space. Since large portions ofthe structure are fixed, the design space and thereforethe cost and mass trade-off is restricted.

Fig. 18 Pareto frontier for bicycle frame structural optimiza-tion with weighting factor, α, labeled for each design.

The maximum von-Mises stress constraint is not ac-tive for any of the structural designs included in thePareto frontier. This is a result of the control point X,Y constraints being restrictive. Design freedom is lim-ited by these constraints in order to prevent part edgecurves from intersecting each other, resulting in designsfor which structural analysis cannot be performed.

Abrasive waterjet cutting speeds for all designs forthis example are determined to be at the maximum lin-

11

Fig. 19 Structural design solution for weighting factor of0.1.

Fig. 20 Structural design solution for weighting factor of0.6.

ear cutting speed of the AWJ cutter for the selected ex-ample. This results in better objective space results thanare obtained for the generic structural part example pre-sented in Section 3.2.1.

4.3.3Design space results

Selected structural designs from the Pareto set are shownin Figures 19 and 20. The trade-off between objectivescan be seen by comparing structural designs for theseweighting factors. The design for which the weighting fac-tor is 0.1 results in a structure with nearly straight edgesfor minimum manufacturing cost. However, the designfor a weighting factor of 0.6 results in a design with nar-row structural members in order to minimize structuralmass. This results in low mass but higher manufacturingcost as a result.

5Conclusions and Future Work

While the area of structural shape optimization is fairlymature, we introduce in this paper the considerationof manufacturing cost in the optimization process. Al-though a two-dimensional manufacturing process, abra-sive waterjet cutting, is selected for this paper, othermore complicated manufacturing processes can be usedas well. Two examples are used to exemplify the appli-cation of this procedure for multiobjective structural op-timization problems.

The trade-off between structural performance and man-ufacturing cost is shown with Pareto frontiers for two ex-ample structural components. Mass is used as the met-ric for structural performance and maximum von-Misesstress is the constraint.

Future work will include implementing the adaptiveweighted sum (AWS) method developed by Kim and deWeck (2005) for the generic structural part example.This method may allow for the generation of a well-distributed Pareto frontier for the example. The bicy-cle frame example results will be improved by includingthe bicycle frame joints in the design space by allowingtheir shapes to be optimized. Additional future work willinclude performing topology optimization in which thenumber of curves are considered as design variables andthe creation and merging of holes is allowed. Finally, themethod will be applied to more complex structures and anew manufacturing cost model will be implemented. Po-tential manufacturing process cost models could includemilling and stamping.

References

Bendsøe, M. O.; Kikuchi, N. 1988: Generating opti-mal topologies in structural design using a homogenizationmethod. Comput. Methods Appl. Mesh. Eng., 71, 197-224.

Chang, K. H.; Tang, P. S. 2001: Integration of design andmanufacturing for structural shape optimization. Adv. Eng.Softw., 32, 555-567.

Curran, R. et. al. 2005: Integrating Manufacturing Costand Structural Requirements in a Systems EngineeringApproach to Aircraft Design. Proceedings of the 46th

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dy-namics & Materials Conference, Austin, Texas.

Haug, E. J. et. al. 1986: Design sensitivity analysis of struc-tural systems, Academic Press, San Diego, California.

Kim, I. Y.; Kwak, B. M. 2002: Design Space OptimizationUsing a Numerical Design Continuation Method. Int. J. Nu-mer. Meth. Eng., 53, 1979-2002.

Kim, I. Y.; de Weck, O. L. 2005: Adaptive Weighted SumMethod for Bi-objective Optimization: Pareto front genera-tion. Struct. Multidisc. Optim., 29, 149-158.

Kim, I. Y.; de Weck, O. L. 2005: Variable chomosome lengthgenetic algorithm for progressive refinement in topology op-timization. Struct. Multidisc. Optim., 29, 445-456.

Koski, J. 1988: Multicriteria truss optimization. In: Stadler,W. (ed.), Multicriteria Optimization in Engineering and inthe Sciences, Plenum Press, New York, New York.

Lee, S. B. et. al. 2004: Continuum Topology Optimiza-tion. Proceedings of the 10th AIAA/ISSMO MultidisciplinaryAnalysis and Optimization Conference, Albany, New York.

Martinez, M. P. et. al. 2001: Manufacturability Based Opti-mization of Aircraft Structures Using Physical Programming.AIAA Journal, 39, 517-525.

Olsen, J. 1996: Motion Control with Precomputation, U.S.Patent 5,508,596, Assignee: Omax Corporation.

12

Park, C. et. al. 2004: Simultaneous optimization of compositestructures considering mechanical performance and manufac-turing cost. Compos. Struct., 65, 117-127.

Piegl, L.; Tiller, W. 1997: The NURBS Book, Springer Verlag,Heidelberg, Germany.

Singh, P.; Munoz, J. 1993: Cost Optimization of AbrasiveWaterjet Cutting Systems. Proceedings of the 7th AmericanWater Jet Conference, Seattle, Washington, 191-204.

Suzuki, K.; Kikuchi, N. 1991: A homogenization method forshape and topology optimization. Comput. Methods Appl.Mech. Eng., 93, 291-318.

Yang, R. J.; Chuang, C. H. 1994: Optimal topology designusing linear programming. Comput. Struct., 52, 265-275.

Zadeh, L. 1963: Optimality and Non-Scalar-Valued Perfor-mance Criteria. IEEE Trans. Automat. Contr., 8, 59-60.

Zeng, J. 1992: Mechanisms of Brittle Material Erosion Asso-ciated with High-pressure Abrasive Waterjet Processing, PhDthesis, University of Rhode Island, Department of MechanicalEngineering and Applied Mechanics.

Zeng, J. et. al. 1992: Quantitative Evaluation of Machinabilityin Abrasive Waterjet Machining. Precision Machining: Tech-nology and Machine Development and Improvement at theWinter Annual Meeting of The American Society of Mechan-ical Engineers, Anaheim, California, 169-179.

Zeng, J.; Kim, T. 1993: Parameter Prediction and CostAnalysis in Abrasive Waterjet Cutting Operations. Proceed-ings of the 7th American Water Jet Conference, Seattle,Washington, 175-189.

Zeng, J. et. al. 1999: The Abrasive Water Jet as a PrecisionMetal Cutting Tool. Proceedings of the 10th American WaterJet Conference, Houston, Texas, 829-843.

Zolesio, J. P. 1981: Multicriteria truss optimization. In: Opti-mization of Distributed Parameter Structures, Sijthoff & No-ordhof, The Netherlands.