research paper c.d. mcallister · t.w. simpson · k. hacker

DOI 10.1007/s00158-004-0481-1

RESEARCH PAPER

Struct Multidisc Optim (2005) 29: 178–189

C.D. McAllister · T.W. Simpson · K. Hacker · K. Lewis · A. Messac

Integrating linear physical programming within collaborativeoptimization for multiobjective multidisciplinary designoptimization

Received: 12 September 2003 / Revised manuscript received: 21 July 2004 / Published online: 26 October 2004 Springer-Verlag 2005

Abstract Multidisciplinary design optimization (MDO) isa concurrent engineering design tool for large-scale, com-plex systems design that can be affected through the optimaldesign of several smaller functional units or subsystems.Due to the multiobjective nature of most MDO problems,recent work has focused on formulating the MDO problemto resolve tradeoffs between multiple, conflicting objectives.In this paper, we describe the novel integration of linearphysical programming within the collaborative optimizationframework, which enables designers to formulate multiplesystem-level objectives in terms of physically meaningfulparameters. The proposed formulation extends our previousmultiobjective formulation of collaborative optimization,which uses goal programming at the system and subsys-tem levels to enable multiple objectives to be consideredat both levels during optimization. The proposed frame-work is demonstrated using a racecar design example thatconsists of two subsystem level analyses — force and aero-dynamics — and incorporates two system-level objectives:(1) minimize lap time and (2) maximize normalized weightdistribution. The aerodynamics subsystem also seeks to min-imize rearwheel downforce as a secondary objective. Theracecar design example is presented in detail to providea benchmark problem for other researchers. It is solvedusing the proposed formulation and compared against a tra-ditional formulation without collaborative optimization orlinear physical programming. The proposed framework cap-

C.D. McAllisterDepartment of Industrial and Manufacturing Systems Engineering,Louisiana State University, Baton Rouge, LA 70803

T.W. Simpson (B)Departments of Mechanical & Nuclear and Industrial & Manufactur-ing Engineering, The Pennsylvania State University, University Park,PA 16802E-mail: [email protected]

K. Hacker · K. LewisDepartment of Mechanical and Aerospace Engineering, State Univer-sity of New York, Buffalo, NY 14260

A. MessacDepartment of Mechanical, Aerospace, and Nuclear Engineering,Rensselaer Polytechnic Institute, Troy, NY 12180

italizes on the disciplinary organization encountered duringlarge-scale systems design.

Keywords Collaborative optimization · Multidisciplinarydesign optimization · Multiobjective optimization · Physicalprogramming

Nomenclature

A′ Normalized weight distributionAi(x) Achievement of design metric i in linear physical

programmingβ Vehicle sideslip angleC′ Normalized aero downforce distributionδ Vehicle wheel steer angled+

i , d−i Positive and negative deviation variables in a com-

promise DSPet Lap timeFx Tractive forceFy Lateral forceFz Normal forceIDYaw Yawing moment due to induced drag effectsK ′ Normalized roll stiffness distributionsi Class function for criterion i in linear physical pro-

grammingtik Range boundary in linear physical programming

for design metric i and range ku Vehicle speedx Vector of design variablesxc Vector of design variables coupled by multiple sub-

systemsx0

c Vector of system-determined target values forcoupled design variables

xsc Vector of local coupling variables determined by

subsystem sYawBal Yaw force balanceZ Deviation function in a compromise DSPzi Deviation function at preemptive level i in a com-

promise DSP

Integrating linear physical programming within collaborative optimization for multiobjective multidisciplinary design optimization 179

1 Introduction

Multidisciplinary design optimization (MDO) is a concur-rent engineering design tool for large-scale system designthat typically approaches the design problem by decompos-ing the system into its constituent subsystems. These subsys-tems are intrinsically linked through design, function, andperformance. MDO methods employ individual analyses foreach subsystem, which are then aggregated by a system-level coordination procedure that ensures compatibility ofthe subsystems. Reviews of the fundamental approachesto multidisciplinary design optimization can be found inSobieszczanski–Sobieski and Haftka (1997), Balling andSobieszczanski–Sobieski (1996), and Cramer et al. (1994).The basic mathematical formulation for multidisciplinarydesign optimization typically follows a nonlinear program-ming structure, namely, {Min f(x)|g(x) ≤ 0 & h(x) = 0},where the values of design variables, x, are determinedto minimize an objective, f(x), while satisfying inequality,g(x), and equality, h(x), constraints. In MDO, evaluation ofthe constraints may require execution of high-fidelity ana-lyses, such as computational fluid dynamics or finite elementanalyses. These additional routines are often called con-tributing analyses.

1.1 Single-level formulations

The simplest optimization approaches are conducted withina single analysis level. The most common and recognizableform is the traditional or all-at-once (AAO) optimizationapproach shown in Fig. 1a. Under this approach, we seekto find the values of design variables, x, that minimize anobjective function, f(x), while satisfying constraints, g(x)and h(x). Any subsystem dependencies or interactions areaddressed through integrated analyses. For a given set ofdesign variables, the integrated analysis returns constraintand objective function values to the optimizer. For large-scale design problems, the integrated analysis of all subsys-tems may be prohibitively complex — both conceptuallyand computationally. For example, the subsystem analysiscodes may reside on different computer platforms (e.g. PC

Fig. 1 Single-level optimization

vs. Unix), be written in different programming languages(e.g. Fortran vs. C), or be located in different geographicregions (e.g. Pennsylvania vs. California).

An approach to overcome these difficulties is illustratedin Fig. 1b, which depicts the simultaneous analysis anddesign (SAND) framework (Balling and Sobieszczanski–Sobieski 1996; Sobieszczanski–Sobieski 1988). This ap-proach differs from traditional or all-at-once (AAO) opti-mization, in that the analysis of each subsystem is indepen-dently executed, which facilitates MDO analyses in lightof the previously discussed computational difficulties asso-ciated with traditional optimization. Since the problem hasbeen partitioned into isolated units, the optimizer must en-sure compatibility among coupling variables that are com-mon to more than one subsystem. The set of correspondingcompatibility constraints, h(xc), is added to the problem.The values of the coupling variables, xc, are estimated bythe optimizer and passed independently to the contributinganalyses (CAj). The contributing analyses are executed, andthe results are returned to the optimizer. The compatibilityconstraints ensure that the values of xc initially estimated bythe optimizer are identically equal to the actual values calcu-lated by the contributing analyses to ensure subsystem andsystem compatibility.

The contributing analyses in SAND simply interact withthe optimizer by exchanging design variable estimates andattained values. In many problems (e.g. aircraft design), itmay be advantageous to perform optimization at the con-tributing analysis level. For example, a computational fluiddynamics model of an aircraft wing may be iteratively exe-cuted to maximize lift while meeting drag constraints. Con-current subspace optimization and collaborative optimiza-tion are two methods that provide for this optimization struc-ture as discussed next.

1.2 Multi-level formulations

Figure2 illustratesconcurrentsubspaceoptimization(CSSO),as proposed by Sobieszczanski–Sobieski (1988), in whichthere is a system-level coordinator rather than an optimizer.The system-level coordinator only evaluates the compati-bility constraints to ensure system-level feasibility of the

180 C.D. McAllister et al.

subsystems. Hence, this formulation is only appropriate forproblems in which system-level objectives and variables donot exist. The subsystems retain individual, disciplinary ob-jectives and design variables.

Applications of CSSO include simple test problems(Balling and Wilkinson 1997), structural design, and aircraftsizing (Wujek et al. 1996). The CSSO implementation sig-nificantly reduces the number of contributing analysis calls(Balling and Sobieszczanski–Sobieski 1996), but it facesconvergence problems that are detrimental to the identifi-cation of optimal solutions (Balling and Wilkinson 1997).Convergence problems are attributed to the lack of a co-ordination strategy at the system level to arbitrate amongdiscrepancies at the subsystem level. Without an optimiza-tion approach at the system level, it is difficult to arbitrateamong subsystems. This drawback motivated the develop-ment of collaborative optimization (Braun et al. 1996; Braunand Kroo 1997), which is used in this research and is dis-cussed next.

Collaborative optimization (CO), developed by Braunand Kroo (1997) is a popular MDO framework (see Fig. 3).Similarly to CSSO, each subsystem consists of a disci-plinary level optimizer and disciplinary constraints, but theonly objective at the subsystem level in CO is to minimizethe violation of the compatibility constraints. In contrast toCSSO, CO uses a system-level optimizer to act on an over-all design objective subject to the subsystem compatibilityconstraints. The lack of system-level optimization in CSSOis a significant drawback to its applicability. In the designof most engineering systems, there are one or more designobjectives. For example, an aircraft design problem may beposed to minimize cost and weight, and maximize cargocapacity and range. Furthermore, the system-level optimizerwithin CO is a method for arbitrating among coupled designvariables, xc. If the system-level objective is in terms of oneor more of the coupled variables, the corresponding optimalvalue is selected as the subsystem target for the succeedingiteration.

Applications of collaborative optimization include launchvehicle design (Braun et al. 1997), aircraft wing design(Sobieski and Kroo 1996), and undersea vehicle design(McAllister et al. 2000). Implementations of CO are com-putationally expensive due to the large number of iterationsrequired to satisfy the compatibility constraints at the systemlevel, which ensure equality of shared variables.

The selection of collaborative optimization as the basisof this research is motivated by several factors. First, our em-phasis is on multi-level frameworks that can accommodatethe formulation of design rules and implementation of op-timization approaches at the system and subsystem levels.Second, increased computational expense is not considereddetrimental in the context of this research; greater impor-tance is placed on the ability to generate an improved solu-tion, regardless of the computational cost required to obtainit. Moreover, we are interested in extending this work tolarge-scale design problems where the ease of assemblingdesign rules and formulating the multi-level optimizationproblem provides time savings comparable to the increasedsolution time. Finally, since collaborative optimization can

be readily implemented using parallel computation, some ofthe increased solution time can be recovered.

Recent extensions of collaborative optimization includemultiobjective formulations using the weighted-sum (Tap-peta and Renaud 1997) and goal programming (McAllisteret al. 2000) approaches, and implementations accounting foruncertainty in design (Gu and Renaud 2001; Lewis and Mis-tree 1998; McAllister and Simpson 2003). When presentedwith a multiobjective problem, it is often difficult for a de-cision maker to specify numeric weights corresponding torelative preferences among the individual design objectives

Fig. 2 Concurrent subspace optimization

Fig. 3 Collaborative optimization


(Keeney and Raiffa 1993; Sadagopan and Ravindran 1986;Traintaphyllou 2002; Zeleny 1982). Physical programming(Messac 1996, 1998, 2000; Messac and Chen 2000; Messacand Ismail-Yahaya 2002; Messac et al. 2000) is a mathemat-ical construct that has been developed to facilitate multiob-jective problem formulation and optimization using param-eters that are physically meaningful to the decision maker.Previous applications of physical programming include thedesign of aircraft structures (Martinez et al. 2000), prod-uct family design (Messac et al. 2002), robust propulsionsystem design (Chen et al. 2000), and rigidified inflatablestructures (Messac et al. 2004). These applications use a tra-ditional single-level formulation for multidisciplinary de-sign optimization. While these traditional formulations arevalid approaches to engineering design, the premise of thisresearch is that large-scale design scenarios can rarely beposed in a single system domain. For instance, the de-sign of a satellite must be analyzed by taking into accountpower generation, signal transmission, instrumentation, andmaneuverability. A system engineer who completely un-derstands the intricacies of each satellite subsystem is notlikely to be found. However, individual satellite subsystemengineers can be identified. Using this disciplinary exper-tise and establishing mechanisms to resolve conflicts amongcompeting subsystems, the design of a satellite can be con-ducted. Unlike single-level formulations, collaborative op-timization directly preserves the disciplinary organizationencountered in large-scale engineering design. Physical pro-gramming provides an optimization formulation that allowsdesigners to express their preferences for competing objec-tives using natural language and parameters that are phys-ically meaningful. The goal in this research is to integratephysical programming within collaborative optimization formulti-level MDO. The primary contribution of this work isthe extension of physical programming to multi-level MDOproblems, using CO to explore and exploit subsystem in-teractions. One anticipated practical benefit of this work isa large-scale MDO framework in which problems can beposed and formulated using the disciplinary organizationand inherent design language that is unique to each disci-pline. Our proposed framework is implemented to conductthe conceptual design of a Formula 1 racecar.

The remainder of this paper is organized as follows. Inthe next section, we discuss the mechanics of integratingcollaborative optimization and linear physical programming.Section 3 introduces the racecar design analyses and corres-ponding formulations. Results are given in Sect. 4, and weconclude with a discussion of limitations and future work inSect. 5.

2 Collaborative optimization with linear physicalprogramming

This work extends our previous implementation of collabo-rative optimization using goal programming to handle multi-criteria system and subsystem-level objectives (McAllisteret al. 2000). We first utilize the linear physical programming(LPP) adaptation, based on the compromise decision sup-

port problem (Mistree et al. 1993) approach, proposed byHernandez et al. (2001) to provide a single-level goal pro-gramming formulation with a piecewise linear preferencefunction for each design criterion. Applications of LPP cancontain nonlinear constraints and design criteria; only thepreference function for each objective is piecewise linear(Messac et al. 1996). Messac (1996) also developed the non-linear physical programming approach that uses preferencefunctions with continuous first derivatives. Piecewise linearpreference functions are sufficient to meet our current re-search objective, which is to provide a multi-level problemformulation for solving large-scale MDO problems usingLPP within the collaborative optimization framework. Thefundamental LPP concepts — design metrics, class func-tions, ranges of desirability, and aggregate objective func-tions — are described in the following section.

2.1 Overview of linear physical programming

2.1.1 Design metrics

A typical optimization problem involves identifying thecharacteristics of the system, or design, which allow thedesigner to judge the effectiveness of alternative solutions.These characteristics, or design metrics, are denoted by thevector A = (A1, . . ., Am). Design metrics are quantities thatthe designer wishes to minimize; maximize; take on a cer-tain value (goal); fall in a particular range; or be less than,greater than, or equal to particular values.

2.1.2 Class functions

Within the LPP method, the designer expresses preferenceswith respect to each design metric using four differentClasses. Each Class comprises two cases, Hard and Soft,referring to the sharpness of the preference. All Soft Classfunctions become constituent components of the aggregateobjective function. Construction of the aggregate objectivefunction is described after the development of additionalnotation. Figure 4 depicts the qualitative meaning of eachSoft Class. The value of the i-th design metric (or objec-tive) under consideration, Ai , is on the horizontal axis, andthe function that will be minimized for that objective, si ,hereby called the Class function, is on the vertical axis. Classfunctions provide the means for a designer to express thespectrum of preferences for a given design metric. The de-sired behavior of a generic design metric is described by oneof eight sub-classes: four Soft and four Hard. These classesare characterized as follows.

Soft:Class 1S Smaller-is-better, i.e. minimization.Class 2S Larger-is-better, i.e. maximization.Class 3S Value-is-better.Class 4S Range-is-better.

Hard:Class 1H Must be smaller, Ai ≤ Ai,maxClass 2H Must be larger, Ai ≥ Ai,minClass 3H Must be equal, Ai = Ai,valClass 4H Within range, Ai,min ≤ Ai ≤ Ai,max


For each of these classes, a Class function is formed; ex-amples for the soft classes 1S and 2S are shown in Fig. 4.Class functions 3S and 4S are omitted without loss of gen-erality because each can be viewed as a combination of the1S and 2S class functions.

2.1.3 Ranges of desirability

Linear physical programming allows a designer to expressdegrees of desirability within each of the soft sub-classes.For example, the ranges of Class 1S, see Fig. 4, are definedin the order of decreasing preference as follows.

Ideal range(Ai ≤ t+i1

): an acceptable range over which the

improvement that results from further reduction of theperformance objective is desired, but is of minimal ad-ditional value.

Desirable range(t+i1 ≤ Ai ≤ t+i2

): an acceptable range that is

desirable.Tolerable range

(t+i2 ≤ Ai ≤ t+i3

): an acceptable, tolerable

range.Undesirable range

(t+i3 ≤ Ai ≤ t+i4

): a range that, while ac-

ceptable, is undesirable.Highly undesirable range

(t+i4 ≤ Ai ≤ t+i5

): a range that,

while still acceptable, is highly undesirable.Unacceptable range

(Ai ≥ t+i5

): the range of values that the

generic objective must not take.

Fig. 4 Linear physical programming class function regions

The designer defines preference with respect to each de-sign metric by providing numerical values tik correspondingto each of the m design metrics partitioned into k ranges ofdesirability.

In order to define the preference function fully, the de-signer only needs to specify the range boundaries, t+ik , whichare physically meaningful quantities. For example, in thedesign of a pressure vessel, we may wish to minimize thematerial cost with a “desirable range” of $50 to $70, whichyields t+i1 = $50 and t+i2 = $70. Definitions of the remain-ing ranges would proceed exactly as above according to thepreferences of the designer.

2.1.4 Aggregate objective functions

Individual design metrics become part of an aggregate ob-jective function (AOF) to be minimized or are treated asinequality or equality constraints. The aggregate objectivefunction is defined as the average class function value of allsoft Classes:

A = 1

nsc

nsc∑

i=1

si ,

where nsc = the number of Soft Classes; si = the value of thei-th Class function. The aggregate objective function is notto be confused with an average or weighted average of therelated m design metrics. In Fig. 4, the aggregate objectivefunction is derived from the vertical axis, not the horizontalaxis.

Linear physical programming is a design language thatprovides a flexible mechanism to express designer prefer-ences. Hernandez et al. (2001) note that LPP is amenable tobeing formulated as a modified compromise decision sup-port problem (DSP). The compromise DSP (Mistree et al.1993) is a multiobjective mathematical programming for-mulation used to determine the values of the design vari-ables that satisfy a set of constraints, and meet a set ofpotentially conflicting goals as closely as possible. The com-promise DSP has been applied to a variety of single- andmulti-objective optimization problems including aircraft de-sign (Lewis et al. 1994), ship design (Smith and Mistree1994), robust design (Chen et al. 1996), and product fam-ily design (Simpson et al. 2001). To formulate a compromiseDSP based on linear physical programming, a system goal isadded for each criterion range parameter t+ik and t−ik . The cor-responding weight, wi,k, for each design metric required forthe preemptive objective function is determined using an it-erative procedure (Messac et al. 1996), which enforces classfunction convexity, identical class function values at rangeintersections, and requires that the vertical change acrossany range satisfies the one-versus-others (OVO) rule. OVOis an inter-criteria preference rule that seeks to improve theworst criterion first.

2.2 Collaborative optimization with linear physicalprogramming

Collaborative optimization and linear physical programmingare integrated using the compromise DSP framework to


form a unified approach for multiobjective analysis in MDO.The realization of this approach includes physical program-ming to allow designers to formulate the problem usingphysically meaningful parameters to describe customer-specified requirements. Collaborative optimization is used tocast the hierarchical design problem in a formulation that isreflective of the functional structure of system design prob-lems, and the compromise DSP provides the optimizationmechanics. Our original formulation of collaborative opti-mization using goal programming and the compromise DSPis documented in McAllister et al. (2000). The architectureof the optimization problem is presented mathematicallyin Figs. 5 and 6 for the system and subsystem levels, re-spectively. The system analysis (see Fig. 5) determines thetarget values of the coupled design variables, x0

c j , to mini-mize the preemptive objective function, Z , subject to systemconstraints, gi4(x), and interdisciplinary compatibility (xs

c −x0

c = 0). Range boundaries t+ik and t−ik are specified by the de-signer for the LPP minimization and maximization classes,respectively. The goal programming deviation variables d−

i,kand d+

i,k (Mistree et al. 1993) are determined as a conse-quence of the current achievement, A+

i , for 1S classes andAi for 2S classes for criterion i and the corresponding rangeboundary t+ik at priority level k. The preemptive objectivefunction, Z , is constructed by seeking to improve design cri-teria across the highly undesirable range (k = 4) and then tosuccessively more preferred ranges. The unacceptable rangeis directly modeled through variable bounds.

At the subsystem level (see Fig. 6), the objective isto determine values of the local shared variables, xs

c, that

Fig. 5 Collaborative optimization with linear physical programmingat the system level

Fig. 6 Collaborative optimization with linear physical programmingat the subsystem level

minimize the sum of squared deviations from the system-

specified design variable targetsnc∑

i=15

(xs

i − x0i

)2subject to

subsystem constraints, gi5(x). Once the targets have beenattained, optimization proceeds to any local objectives ofinterest, e.g. maximize the lift-to-drag ratio of the wing(subsystem) during aircraft (system) design. Preferences forthese local objectives may also be specified using linearphysical programming, if desired. Final values of the localshared variables are returned to the system level for theevaluation of compatibility constraints to ensure feasibilityacross all subsystems. To demonstrate the proposed frame-work, the design of a Formula 1 racecar is described next.

3 Racecar design example

As discussed by Kasprzak and Lewis (2001), racecar de-sign provides a rich environment to apply MDO techniques.Racecar design and analysis involves knowledge of aero-dynamics, structural mechanics, tire performance, and ve-hicle dynamics. This information is obtained from disci-plinary experts who have different opinions and control overthe performance of the vehicle. The range of adjustmenton the design variables may be limited during the racingseason, and sanctioning bodies limit the amount of on-track testing. As a result, vehicle simulations must be usedto optimize a racecar before it is constructed. Advantagesgained through simulation increase the vehicle’s potential,and ultimately lead to an improvement in on-track perform-ance.

During a lap on a particular racetrack, a driver is facedwith a number of different types of corners and straights.Designing a racecar to perform well across turns of all radiion a single track involves resolution of a set of conflictingtradeoffs. Each segment of the racetrack has its own optimalvehicle characteristics, for example, the optimal racecar con-figuration for tight cornering is significantly different thanthat for sweeping, large-radii curves. Kasprzak et al. (2000)and Hacker et al. (2000) use single-level multiobjective op-timization formulations to maximize racecar performanceacross multiple tracks of different radii.

Our racecar model is based on the classic bicycle modelof Milliken and Milliken (1995), which has been expandedto include four wheels. Equations of motion are written forlateral, longitudinal, and yaw accelerations. The tires, which


Fig. 7 Sketch of the racecar model

may be different for front and rear, are modeled using tabu-lar tire data including representations of nonlinearities, suchas load sensitivity and slip angle saturation. Wheel loads arecalculated based on static load, aerodynamic downforce, andlateral load transfer. Figure 7 illustrates a simplified sketchof the racecar model and shows the three design variables:roll stiffness distribution (K ′), weight distribution (A′), andaerodynamic downforce distribution (C′).

3.1 Racecar analyses

The racecar analysis begins with the calculation of tire pa-rameters and concludes with an iterative analysis to solve forlateral forces given the center of gravity and roll stiffness.Table 1 presents the design variables under consideration forthe racecar optimization. All design variables are normal-ized between 0 and 1 and have lower and upper bounds of0.3 and 0.6, respectively.

Table 1 Racecar design variables

Var. Description Init. value

A′ Weight distribution 0.4C′ Aero downforce distribution 0.4K ′ Roll stiffness distribution 0.3

Table 2 Racecar and track parameters

Parameter Value Description

l 9.67 ft Vehicle wheelbasemass 41.7 slug Vehicle massh 1.167 ft Height of CGtF 5.5 ft Front tracktR 5.25 ft Rear trackRefArea 10 ft2 Frontal areaRadius 400 ft Skidpad radiusCD 2.9 Drag coefficient

Fig. 8 Racecar dynamics

Table 2 indicates the fixed racecar and track parametersused in this study. Specifically, we considered a racecar witha wheelbase of 9.67 feet and mass of 41.7 slugs traveling ona 400-ft radius curve.

Figure 8 illustrates the relationships between lateralforces and slip angles. As indicated in the figure, the centerof gravity (CG) defines the origin of the coordinate sys-tem, and clockwise moments are positive. Detailed designequations for the racecar are given in the Appendix.

3.2 Racecar problem formulation

Requirements imposed on racecar design by the decisionmaker are incorporated using physical programming. Onedesign objective is to minimize the lap time (i.e. go as fastas possible). However, lap speed and lap time must be con-sidered in conjunction with pit time, which has a negativecontribution to the overall race. Maintaining the center ofgravity near the midpoint of the wheelbase provides moreconsistent wear of the front and rear tires and minimizesthe frequency of required tire changes. Therefore, the nor-malized weight distribution, A′, and the lap time, et, havespecified ranges of desirability, see Table 3. For instance, itis desired to maximize, Class 2S, the normalized weight dis-tribution with the ideal range being greater than 0.5, anddesirable range being 0.5–0.4, and so on. The designer alsowishes to minimize the lap time; the ideal range is less than13 seconds, the desirable range is 13 to 14 seconds, and soon, while a lap time greater than 17 seconds is unacceptable.

Equations (1)–(26) in the Appendix are used to establishthe traditional optimization formulation for the racecar de-sign problem shown in Fig. 9, which seeks to minimize laptime.

Formulating this as a multidisciplinary collaborative op-timization design problem with physical programming, wedefine two disciplinary subspaces: (1) aerodynamics and(2) force analysis as shown in Fig. 10. Incorporated in theaerodynamics analysis are (1)–(9) while the force analy-sis contains (10)–(24). The system-level optimizer seeks to


Table 3 Physical programming ranges of desirability for racecar design

HighlyVariable Class Ideal Desirable Tolerable Undesirable undesirable Unacceptable

A′ 2S 0.5 0.4 0.36 0.32 0.3et 1S 13 s 14 s 15 s 16 s 17 s

Fig. 9 Racecar design by traditional optimization

minimize lap time and maximize weight distribution and es-tablishes subsystem-level targets for design variables A′, C′,and K ′ and coupled variables AeroFzF, AeroFzR, and FxReq(see Fig. 11). The goal of each subsystem is to minimize thedeviation from these established targets to ensure the com-patibility dictated by a multi-level formulation.

Note that at the subsystem level, the aerodynamics sub-system contains the local objective of minimizing reardownforce, which is set at the 2nd priority level in the goalprogramming formulation, i.e. it is only improved aftersystem-level compatibility, the first priority, is achieved.Minimization of rear downforce is useful for maximizing the

Fig. 10 Racecar design formulation using collaborative optimizationwith linear physical programming

Fig. 11 Compromise DSP for system level collaborative optimizationusing linear physical programming for racecar design example

straight-line speed of the racecar and is addressed preemp-tively only after minimization of the discrepancy betweensystem target values for shared variables and the equivalentlocal subsystem variables. Inclusion of rear downforce as anobjective introduces potential conflict with the force subsys-tem that must meet the cornering requirement of maintainingcontact between all four wheels and the track. The force sub-system designer would gladly accept a larger rear downforceif it is best for cornering.

4 Results

Tables 4 and 5 present results for six different racecar de-sign optimization formulations, which include traditionaland CO formulations with and without LPP for compari-son purposes. The reported values are consistent across threerandomly selected starting points. “Traditional optimiza-tion” refers to the single-level formulation depicted in Fig. 9.“Collaborative optimization” indicates the multi-level for-mulation that seeks to minimize lap time at the system leveland minimize the discrepancy between shared variables atthe subsystem level. “Collaborative optimization with aeroobjective” indicates that minimization of rear downforce has


Table 4 Racecar design optimization results: single-objective case

Parameter Traditional Collaborative CO with aerooptimization optimization objective

A′ 0.514 0.538 0.566C′ 0.392 0.365 0.344K ′ 0.341 0.325 0.313RDwnfc 0.0361 slug/ft 0.0378 slug/ft 0.0390 slug/ftet 14.22 sec 14.22 sec 14.09 sec

Table 5 Racecar design optimization results: multiobjective case

Parameter Traditional CO CO with LPP &with LPP with LPP aero objective

A′ 0.512 0.513 0.505C′ 0.392 0.393 0.396K ′ 0.336 0.348 0.342RDwnfc 0.0361 slug/ft 0.0361 slug/ft 0.0359 slug/ftet 14.24 sec 14.21 sec 14.29 sec

been included preemptively as a secondary objective in theaerodynamics subsystem, as shown in Fig. 10. The formu-lation modifier “with LPP” indicates that linear physicalprogramming has been used to specify preferences for laptime and normalized weight distribution, using the ranges ofdesirability listed in Table 3.

First referring to the single objective results in Table 4,the traditional and collaborative optimization formulationsachieve distinct optimal solutions with a lap time of 14.22seconds. When collaborative optimization is implementedwith the secondary aerodynamics objective of rear down-force minimization, the resulting design improves the laptime by 0.13 seconds. The primary objective is not the min-imization of rear downforce because of the cornering re-quirement of the racecar; hence, minimum lap times do notnecessarily correspond to minimum rear downforce values.The design space is identical across all formulations, and wenote that (i) the traditional optimization and pure collabora-tive optimization formulations did not identify the optimumsolution and (ii) the use of local subsystem objectives withincollaborative optimization successfully provides additional

Fig. 14 Convergence for collaborative optimization with linear physical programming

guidance for the system optimization to proceed to find theoptimal racecar configuration.

Physical programming modifies the objective functionformulation such that a different design point is selected asthe optimum in the multiobjective case (see Table 5). Tra-ditional optimization with physical programming indicatesa design with an “ideal” 0.512 normalized weight distribu-tion (A′) and a “tolerable” lap time (et) of 14.24 seconds.When physical programming is used in conjunction withcollaborative optimization, the reported design maintains the“ideal” normalized weight distribution and “tolerable” laptime. Though further resolution of normalized aerodynamicdownforce distribution (C′) and normalized roll stiffnessdistribution (K ′) could improve the lap time, it would not besufficient to promote lap time to the “ideal” range. Hence,

Fig. 12 Traditional optimization convergence

Fig. 13 Convergence for collaborative optimization with aero objec-tive


the preferences of the decision maker do not strongly distin-guish between the two different design configurations.

Figure 12 shows the convergence history for the tradi-tional optimization formulation. The design converges tothe reported optimum (et = 14.22 sec). The model mandatesthat all four wheels remain in contact with the track; hence,negative wheel loads are penalized as evidenced by the largevertical spike in Fig. 12.

Figure 13 illustrates the system-level convergence his-tory for the collaborative optimization formulation using thesecondary aerodynamics subsystem objective to minimizerear downforce. Compared to the traditional optimizationconvergence shown in Fig. 12, the collaborative optimiza-tion convergence is more frequently affected by infeasibledesigns due to interactions with the aerodynamics subsys-tem. When the system inadvertently tests a design that leadsto negative wheel loads and corresponding infeasibility, theaerodynamics subsystem follows the direction establishedby the system, and the resulting deadlock is difficult for thesystem to resolve.

Figure 14 presents convergence plots for the CO-LPPformulation. The trend for the normalized weight dis-tribution exhibits a smooth and consistent improvementacross linear physical programming’s regions of succes-sively higher preference. After the optimization initiallyrejects infeasible designs, the lap time settles into a state ofsmall fluctuations that become secondary to the improve-ment that is achieved in the weight distribution.

5 Closing remarks

Linear physical programming has been integrated withincollaborative optimization to provide a novel framework forthe design and analysis of large-scale, hierarchical MDOproblems. The proposed framework retains physical pro-gramming’s strength of contributing a flexible mechanism toexpress preferences among competing design metrics, whilecollaborative optimization provides the ability to formu-late the multi-level MDO problem. The unified frameworkis demonstrated using the design of a Formula 1 racecar.Results are compared to the traditional, single-level for-mulations and CO formulations that do not include linearphysical programming. We anticipate improved perform-ance with the implementation of nonlinear physical pro-gramming within CO rather than linear physical program-ming; however, integration of nonlinear physical program-ming within the compromise DSP is not straightforward. Weare also investigating the potential advantages of using thisapproach to solve large-scale MDO problems. Finally, anadded drawback is the increased computation time neces-sary to enforce the equality-constrained system-level com-patibility requirement, which needs to be investigated fur-ther; however, the collaborative optimization formulationmore accurately represents the disciplinary organization en-countered in large-scale systems design.

Acknowledgement The authors gratefully acknowledge the racecarmodeling expertise contributed by Milliken Research Associates, Inc.

Appendix: Racecar analyses

Equations (1)–(2) are used to compute the front and rear liftcoefficients, CLF and CLR, respectively, based on the aero-dynamic downforce distribution, C′.

CLF = −0.5×C′ (1)

CLR = −1× (5+ (−5×C′)) (2)

Equation (3) calculates half the weight of the car, halfwt,where g is the acceleration due to gravity.

halfwt = mass× g/2 (3)

Equations (4)–(6) determine the coefficients for front andrear downforce, FDwnfc and RDwnfc, and aerodynamicdrag, Dragc, where Den is the atmospheric density.

FDwnfc = −(Den×CLF×RefArea)/2 (4)

RDwnfc = −(Den×CLR×RefArea)/2 (5)

Dragc = (0.2×FDwnfc)+ (0.4×RDwnfc) (6)

Table 6 indicates the parameters that must be initialized be-fore proceeding with the iterative analysis to solve for thelateral forces.

Equations (7)–(9) determine the aerodynamic forces,where positive quantities indicate downforce. The aerody-namic force acting on the front and rear wheels is repre-sented by AeroFzF and AeroFzR, respectively. AeroFx is anaerodynamic force that opposes forward motion.

AeroFzF = FDwnfc×uOld2 (7)

AeroFzR = RDwnfc×uOld2 (8)

AeroFx = Dragc×uOld2 (9)

Equation (10) indicates the required tractive effort, FxReq,which is always positive.

FxReq = AeroFx+|FyF× sin(MaxAlphaF)|+ |FyR× sin(MaxAlphaR)| (10)

Table 6 Initialization of lateral force loop

Parameter Description Init. value

FzLF Left front wheel load 0FzRF Right front wheel load 0FzLR Left rear wheel load 0FzRR Right rear wheel load 0FyF Lateral force front axle 0FyR Lateral force rear axle 0Fy Lateral force 0uOld Velocity last iteration 0MaxAlphaF Max front slip angle 0MaxAlphaR Max rear slip angle 0


Front and rear wheel load transfers due to centrifugal force,FLT and RLT , are given by (11)–(12).

FLT = (Fy×h/tF)× K ′ (11)

RLT = (Fy×h/tR)× (1− K ′) (12)

Equations (13)–(16) determine the wheel loads on eachof the four wheels, including the effects of downforce.For instance, FzRF, is the load acting on the right frontwheel.

FzLF = (1− A′)×halfwt+FLT+AeroFzF/2 (13)

FzRF = (1− A′)×halfwt−FLT+AeroFzF/2 (14)

FzLR = A′ ×halfwt+RLT+AeroFzR/2 (15)

FzRR = A′ ×halfwt−RLT+AeroFzR/2 (16)

Based on the installed tires with tabulated lateral forces dueto normal load and slip angle, a quadratic approximation isused to determine maximum slip angles, MaxAlphaF andMaxAlphaR, and lateral forces, FyF and FyR, for the frontand rear axles.

Equations (17) and (18) check the lateral forces on therear wheels, FyLR and FyRR, and, if required, reduce theseforces due to the friction ellipse effect.

FyL R =

0FxReq

2> |FyL R|

FyL R

|FyL R|

√√√√∣∣∣∣∣FyL R2 −

(FxReq

2

)2∣∣∣∣∣

else(17)

FyRR=

0FxReq

2> |FyRR|

FyRR

|FyRR|

√√√√∣∣∣∣∣FyRR2 −

(FxReq

2

)2∣∣∣∣∣

else

(18)

Equation (19) calculates the total rear lateral force, FyR,as a sum of lateral forces acting on each of the two rearwheels.

FyR = FyLR+FyRR (19)

Equations (20) and (21) determine the total yaw force, Yaw-Bal.

IDYaw = (FyRF−FyLF)× tF× sin(MaxAlphaF)

+ (FyRR−FyLR)× tR× sin(MaxAlphaR) (20)

YawBal = [A′ ×FyF× cos(MaxAlphaF)]− [B′ ×FyR× cos(MaxAlphaR)]+ IDYaw (21)

Equations (22) and (23) are used to enforce yaw bal-ance, YawBal = 0. If YawBal < 0, (22) provides the neces-

sary adjustment, while (23) is used to correct for YawBal> 0.

FyF =((

1− A′)× FyR × cos(MaxAlphaR))− IDYaw

A′ × cos(MaxAlphaF)(22)

FyR =(A′ × FyF × cos(MaxAlphaF)

)+ IDYaw

B′ × cos(MaxAlphaR)(23)

Equation (24) calculates total lateral force, Fy, as a sum offront and rear lateral forces. Then, (25) and (26) are usedto determine the corresponding speed, u, and lap time, et,respectively.

Fy = FyF+FyR (24)

u =√∣∣∣∣

Fy × Radius

mass

∣∣∣∣ (25)

et = 2π × Radius

u(26)

The analysis has converged if the difference in lap time be-tween successive iterations does not exceed a small number,| etOld-et | ≤ 0.005. Otherwise, time and velocity estimatesare updated, uOld = u and etOld = et, and the analysis loopreturns to (7).

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