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Struct Multidisc Optim (2005) 29: 178–189 DOI 10.1007/s00158-004-0481-1

R E S E A R C H P A P ER

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

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

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

Abstract Multidisciplinary design optimization (MDO) is a concurrent engineering design tool for large-scale, complex systems design that can be affected through the optimal design of several smaller functional units or subsystems. Due to the multiobjective nature of most MDO problems, recent work has focused on formulating the MDO problem to resolve tradeoffs between multiple, conflicting objectives. In this paper, we describe the novel integration of linear physical programming within the collaborative optimization framework, which enables designers to formulate multiple system-level objectives in terms of physically meaningful parameters. The proposed formulation extends our previous multiobjective formulation of collaborative optimization, which uses goal programming at the system and subsystem levels to enable multiple objectives to be considered at both levels during optimization. The proposed framework is demonstrated using a racecar design example that consists of two subsystem level analyses — force and aerodynamics — and incorporates two system-level objectives: (1) minimize lap time and (2) maximize normalized weight distribution. The aerodynamics subsystem also seeks to minimize rearwheel downforce as a secondary objective. The racecar design example is presented in detail to provide a benchmark problem for other researchers. It is solved using the proposed formulation and compared against a traditional formulation without collaborative optimization or linear physical programming. The proposed framework capC.D. McAllister Department of Industrial and Manufacturing Systems Engineering, Louisiana State University, Baton Rouge, LA 70803 T.W. Simpson (B) Departments of Mechanical & Nuclear and Industrial & Manufacturing Engineering, The Pennsylvania State University, University Park, PA 16802 E-mail: [email protected] K. Hacker · K. Lewis Department of Mechanical and Aerospace Engineering, State University of New York, Buffalo, NY 14260 A. Messac Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180

italizes on the disciplinary organization encountered during large-scale systems design. Keywords Collaborative optimization · Multidisciplinary design optimization · Multiobjective optimization · Physical programming

Nomenclature A Ai (x)

Normalized weight distribution Achievement of design metric i in linear physical programming β Vehicle sideslip angle Normalized aero downforce distribution C δ Vehicle wheel steer angle di+ , di− Positive and negative deviation variables in a compromise DSP et Lap time Fx Tractive force Fy Lateral force Fz Normal force IDYaw Yawing moment due to induced drag effects Normalized roll stiffness distribution K Class function for criterion i in linear physical pros¯i gramming tik Range boundary in linear physical programming for design metric i and range k u Vehicle speed x Vector of design variables xc Vector of design variables coupled by multiple subsystems x0c Vector of system-determined target values for coupled design variables Vector of local coupling variables determined by xsc subsystem s YawBal Yaw force balance Z Deviation function in a compromise DSP zi Deviation function at preemptive level i in a compromise DSP


1 Introduction Multidisciplinary design optimization (MDO) is a concurrent engineering design tool for large-scale system design that typically approaches the design problem by decomposing the system into its constituent subsystems. These subsystems are intrinsically linked through design, function, and performance. MDO methods employ individual analyses for each subsystem, which are then aggregated by a systemlevel coordination procedure that ensures compatibility of the subsystems. Reviews of the fundamental approaches to multidisciplinary design optimization can be found in Sobieszczanski–Sobieski and Haftka (1997), Balling and Sobieszczanski–Sobieski (1996), and Cramer et al. (1994). The basic mathematical formulation for multidisciplinary design optimization typically follows a nonlinear programming structure, namely, {Min f(x)|g(x) ≤ 0 & h(x) = 0}, where the values of design variables, x, are determined to minimize an objective, f(x), while satisfying inequality, g(x), and equality, h(x), constraints. In MDO, evaluation of the constraints may require execution of high-fidelity analyses, such as computational fluid dynamics or finite element analyses. These additional routines are often called contributing analyses. 1.1 Single-level formulations The simplest optimization approaches are conducted within a single analysis level. The most common and recognizable form is the traditional or all-at-once (AAO) optimization approach shown in Fig. 1a. Under this approach, we seek to find the values of design variables, x, that minimize an objective function, f(x), while satisfying constraints, g(x) and h(x). Any subsystem dependencies or interactions are addressed through integrated analyses. For a given set of design variables, the integrated analysis returns constraint and objective function values to the optimizer. For largescale design problems, the integrated analysis of all subsystems may be prohibitively complex — both conceptually and computationally. For example, the subsystem analysis codes may reside on different computer platforms (e.g. PC

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vs. Unix), be written in different programming languages (e.g. Fortran vs. C), or be located in different geographic regions (e.g. Pennsylvania vs. California). An approach to overcome these difficulties is illustrated in Fig. 1b, which depicts the simultaneous analysis and design (SAND) framework (Balling and Sobieszczanski– Sobieski 1996; Sobieszczanski–Sobieski 1988). This approach differs from traditional or all-at-once (AAO) optimization, in that the analysis of each subsystem is independently executed, which facilitates MDO analyses in light of the previously discussed computational difficulties associated with traditional optimization. Since the problem has been partitioned into isolated units, the optimizer must ensure compatibility among coupling variables that are common to more than one subsystem. The set of corresponding ¯ c ), is added to the problem. compatibility constraints, h(x The values of the coupling variables, xc , are estimated by the optimizer and passed independently to the contributing analyses (CA j ). The contributing analyses are executed, and the results are returned to the optimizer. The compatibility constraints ensure that the values of xc initially estimated by the optimizer are identically equal to the actual values calculated by the contributing analyses to ensure subsystem and system compatibility. The contributing analyses in SAND simply interact with the optimizer by exchanging design variable estimates and attained values. In many problems (e.g. aircraft design), it may be advantageous to perform optimization at the contributing analysis level. For example, a computational fluid dynamics model of an aircraft wing may be iteratively executed to maximize lift while meeting drag constraints. Concurrent subspace optimization and collaborative optimization are two methods that provide for this optimization structure as discussed next. 1.2 Multi-level formulations Figure2 illustratesconcurrentsubspaceoptimization(CSSO), as proposed by Sobieszczanski–Sobieski (1988), in which there is a system-level coordinator rather than an optimizer. The system-level coordinator only evaluates the compatibility constraints to ensure system-level feasibility of the

Fig. 1 Single-level optimization

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subsystems. Hence, this formulation is only appropriate for problems in which system-level objectives and variables do not exist. The subsystems retain individual, disciplinary objectives and design variables. Applications of CSSO include simple test problems (Balling and Wilkinson 1997), structural design, and aircraft sizing (Wujek et al. 1996). The CSSO implementation significantly reduces the number of contributing analysis calls (Balling and Sobieszczanski–Sobieski 1996), but it faces convergence problems that are detrimental to the identification of optimal solutions (Balling and Wilkinson 1997). Convergence problems are attributed to the lack of a coordination strategy at the system level to arbitrate among discrepancies at the subsystem level. Without an optimization approach at the system level, it is difficult to arbitrate among subsystems. This drawback motivated the development of collaborative optimization (Braun et al. 1996; Braun and Kroo 1997), which is used in this research and is discussed next. Collaborative optimization (CO), developed by Braun and Kroo (1997) is a popular MDO framework (see Fig. 3). Similarly to CSSO, each subsystem consists of a disciplinary level optimizer and disciplinary constraints, but the only objective at the subsystem level in CO is to minimize the violation of the compatibility constraints. In contrast to CSSO, CO uses a system-level optimizer to act on an overall design objective subject to the subsystem compatibility constraints. The lack of system-level optimization in CSSO is a significant drawback to its applicability. In the design of most engineering systems, there are one or more design objectives. For example, an aircraft design problem may be posed to minimize cost and weight, and maximize cargo capacity and range. Furthermore, the system-level optimizer within CO is a method for arbitrating among coupled design variables, xc . If the system-level objective is in terms of one or more of the coupled variables, the corresponding optimal value is selected as the subsystem target for the succeeding iteration. Applications of collaborative optimization include launch vehicle design (Braun et al. 1997), aircraft wing design (Sobieski and Kroo 1996), and undersea vehicle design (McAllister et al. 2000). Implementations of CO are computationally expensive due to the large number of iterations required to satisfy the compatibility constraints at the system level, which ensure equality of shared variables. The selection of collaborative optimization as the basis of this research is motivated by several factors. First, our emphasis is on multi-level frameworks that can accommodate the formulation of design rules and implementation of optimization approaches at the system and subsystem levels. Second, increased computational expense is not considered detrimental in the context of this research; greater importance is placed on the ability to generate an improved solution, regardless of the computational cost required to obtain it. Moreover, we are interested in extending this work to large-scale design problems where the ease of assembling design rules and formulating the multi-level optimization problem provides time savings comparable to the increased solution time. Finally, since collaborative optimization can

C.D. McAllister et al.

be readily implemented using parallel computation, some of the increased solution time can be recovered. Recent extensions of collaborative optimization include multiobjective formulations using the weighted-sum (Tappeta and Renaud 1997) and goal programming (McAllister et al. 2000) approaches, and implementations accounting for uncertainty in design (Gu and Renaud 2001; Lewis and Mistree 1998; McAllister and Simpson 2003). When presented with a multiobjective problem, it is often difficult for a decision maker to specify numeric weights corresponding to relative 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; Messac and Ismail-Yahaya 2002; Messac et al. 2000) is a mathematical construct that has been developed to facilitate multiobjective problem formulation and optimization using parameters that are physically meaningful to the decision maker. Previous applications of physical programming include the design of aircraft structures (Martinez et al. 2000), product family design (Messac et al. 2002), robust propulsion system design (Chen et al. 2000), and rigidified inflatable structures (Messac et al. 2004). These applications use a traditional single-level formulation for multidisciplinary design optimization. While these traditional formulations are valid approaches to engineering design, the premise of this research is that large-scale design scenarios can rarely be posed in a single system domain. For instance, the design of a satellite must be analyzed by taking into account power generation, signal transmission, instrumentation, and maneuverability. A system engineer who completely understands the intricacies of each satellite subsystem is not likely to be found. However, individual satellite subsystem engineers can be identified. Using this disciplinary expertise and establishing mechanisms to resolve conflicts among competing subsystems, the design of a satellite can be conducted. Unlike single-level formulations, collaborative optimization directly preserves the disciplinary organization encountered in large-scale engineering design. Physical programming provides an optimization formulation that allows designers to express their preferences for competing objectives using natural language and parameters that are physically meaningful. The goal in this research is to integrate physical programming within collaborative optimization for multi-level MDO. The primary contribution of this work is the extension of physical programming to multi-level MDO problems, using CO to explore and exploit subsystem interactions. One anticipated practical benefit of this work is a large-scale MDO framework in which problems can be posed and formulated using the disciplinary organization and inherent design language that is unique to each discipline. Our proposed framework is implemented to conduct the conceptual design of a Formula 1 racecar. The remainder of this paper is organized as follows. In the next section, we discuss the mechanics of integrating collaborative optimization and linear physical programming. Section 3 introduces the racecar design analyses and corresponding formulations. Results are given in Sect. 4, and we conclude with a discussion of limitations and future work in Sect. 5.

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port problem (Mistree et al. 1993) approach, proposed by Hernandez et al. (2001) to provide a single-level goal programming formulation with a piecewise linear preference function for each design criterion. Applications of LPP can contain nonlinear constraints and design criteria; only the preference function for each objective is piecewise linear (Messac et al. 1996). Messac (1996) also developed the nonlinear physical programming approach that uses preference functions with continuous first derivatives. Piecewise linear preference functions are sufficient to meet our current research objective, which is to provide a multi-level problem formulation for solving large-scale MDO problems using LPP within the collaborative optimization framework. The fundamental LPP concepts — design metrics, class functions, ranges of desirability, and aggregate objective functions — are described in the following section. 2.1 Overview of linear physical programming 2.1.1 Design metrics A typical optimization problem involves identifying the characteristics of the system, or design, which allow the designer to judge the effectiveness of alternative solutions. These characteristics, or design metrics, are denoted by the vector A = (A1 , . . ., Am ). Design metrics are quantities that the designer wishes to minimize; maximize; take on a certain 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 preferences with respect to each design metric using four different Classes. Each Class comprises two cases, Hard and Soft, referring to the sharpness of the preference. All Soft Class functions become constituent components of the aggregate objective function. Construction of the aggregate objective function is described after the development of additional notation. Figure 4 depicts the qualitative meaning of each Soft Class. The value of the i-th design metric (or objective) under consideration, Ai , is on the horizontal axis, and the function that will be minimized for that objective, s¯i , hereby called the Class function, is on the vertical axis. Class functions provide the means for a designer to express the spectrum of preferences for a given design metric. The desired behavior of a generic design metric is described by one of eight sub-classes: four Soft and four Hard. These classes are characterized as follows.

2 Collaborative optimization with linear physical programming

Soft: Class 1S Class 2S Class 3S Class 4S

Smaller-is-better, i.e. minimization. Larger-is-better, i.e. maximization. Value-is-better. Range-is-better.

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

Hard: Class 1H Class 2H Class 3H Class 4H

Must be smaller, Ai ≤ Ai,max Must be larger, Ai ≥ Ai,min Must be equal, Ai = Ai,val Within range, Ai,min ≤ Ai ≤ Ai,max

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For each of these classes, a Class function is formed; examples for the soft classes 1S and 2S are shown in Fig. 4. Class functions 3S and 4S are omitted without loss of generality because each can be viewed as a combination of the 1S and 2S class functions. 2.1.3 Ranges of desirability Linear physical programming allows a designer to express degrees of desirability within each of the soft sub-classes. For example, the ranges of Class 1S, see Fig. 4, are defined in the order of decreasing preference as follows. + : an acceptable range over which the Ideal range Ai ≤ ti1 improvement that results from further reduction of the performance objective is desired, but is of minimal additional value. + + Desirable range ti1 ≤ Ai ≤ ti2 : an acceptable range that is desirable. + + Tolerable range ti2 ≤ Ai ≤ ti3 : an acceptable, tolerable range. + + Undesirable range ti3 ≤ Ai ≤ ti4 : a range that, while acceptable, is undesirable. + + Highly undesirable range ti4 ≤ Ai ≤ ti5 : a range that, while still acceptable, is highly undesirable. + Unacceptable range Ai ≥ ti5 : the range of values that the generic objective must not take.


The designer defines preference with respect to each design metric by providing numerical values tik corresponding to each of the m design metrics partitioned into k ranges of desirability. In order to define the preference function fully, the de+ , which signer only needs to specify the range boundaries, tik are physically meaningful quantities. For example, in the design of a pressure vessel, we may wish to minimize the material cost with a “desirable range” of $50 to $70, which + + = $50 and ti2 = $70. Definitions of the remainyields ti1 ing ranges would proceed exactly as above according to the preferences of the designer. 2.1.4 Aggregate objective functions Individual design metrics become part of an aggregate objective function (AOF) to be minimized or are treated as inequality or equality constraints. The aggregate objective function is defined as the average class function value of all soft Classes: Aˆ =

n sc 1 s¯i , n sc i=1

where n sc = the number of Soft Classes; s¯i = the value of the i-th Class function. The aggregate objective function is not to be confused with an average or weighted average of the related m design metrics. In Fig. 4, the aggregate objective function is derived from the vertical axis, not the horizontal axis. Linear physical programming is a design language that provides a flexible mechanism to express designer preferences. Hernandez et al. (2001) note that LPP is amenable to being formulated as a modified compromise decision support problem (DSP). The compromise DSP (Mistree et al. 1993) is a multiobjective mathematical programming formulation used to determine the values of the design variables that satisfy a set of constraints, and meet a set of potentially conflicting goals as closely as possible. The compromise DSP has been applied to a variety of single- and multi-objective optimization problems including aircraft design (Lewis et al. 1994), ship design (Smith and Mistree 1994), robust design (Chen et al. 1996), and product family design (Simpson et al. 2001). To formulate a compromise DSP based on linear physical programming, a system goal is + − added for each criterion range parameter tik and tik . The corresponding weight, wi,k , for each design metric required for the preemptive objective function is determined using an iterative procedure (Messac et al. 1996), which enforces class function convexity, identical class function values at range intersections, and requires that the vertical change across any range satisfies the one-versus-others (OVO) rule. OVO is an inter-criteria preference rule that seeks to improve the worst criterion first. 2.2 Collaborative optimization with linear physical programming

Fig. 4 Linear physical programming class function regions

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


form a unified approach for multiobjective analysis in MDO. The realization of this approach includes physical programming to allow designers to formulate the problem using physically meaningful parameters to describe customerspecified requirements. Collaborative optimization is used to cast the hierarchical design problem in a formulation that is reflective of the functional structure of system design problems, and the compromise DSP provides the optimization mechanics. Our original formulation of collaborative optimization using goal programming and the compromise DSP is documented in McAllister et al. (2000). The architecture of the optimization problem is presented mathematically in Figs. 5 and 6 for the system and subsystem levels, respectively. The system analysis (see Fig. 5) determines the target values of the coupled design variables, xc0 j , to minimize the preemptive objective function, Z, subject to system constraints, gi4 (x), and interdisciplinary compatibility (xcs − + − xc0 = 0). Range boundaries tik and tik are specified by the designer for the LPP minimization and maximization classes, − respectively. The goal programming deviation variables di,k + and di,k (Mistree et al. 1993) are determined as a consequence of the current achievement, A+ i , for 1S classes and Ai for 2S classes for criterion i and the corresponding range + boundary tik at priority level k. The preemptive objective function, Z, is constructed by seeking to improve design criteria across the highly undesirable range (k = 4) and then to successively more preferred ranges. The unacceptable range is directly modeled through variable bounds. At the subsystem level (see Fig. 6), the objective is to determine values of the local shared variables, xcs , that

Fig. 5 Collaborative optimization with linear physical programming at the system level

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Fig. 6 Collaborative optimization with linear physical programming at the subsystem level

minimize the sum of squared deviations from the systemnc 2 xis − xi0 subject to specified design variable targets i=15

subsystem constraints, gi5 (x). Once the targets have been attained, optimization proceeds to any local objectives of interest, e.g. maximize the lift-to-drag ratio of the wing (subsystem) during aircraft (system) design. Preferences for these local objectives may also be specified using linear physical programming, if desired. Final values of the local shared variables are returned to the system level for the evaluation of compatibility constraints to ensure feasibility across all subsystems. To demonstrate the proposed framework, the design of a Formula 1 racecar is described next.

3 Racecar design example As discussed by Kasprzak and Lewis (2001), racecar design provides a rich environment to apply MDO techniques. Racecar design and analysis involves knowledge of aerodynamics, structural mechanics, tire performance, and vehicle dynamics. This information is obtained from disciplinary experts who have different opinions and control over the performance of the vehicle. The range of adjustment on the design variables may be limited during the racing season, and sanctioning bodies limit the amount of ontrack testing. As a result, vehicle simulations must be used to optimize a racecar before it is constructed. Advantages gained through simulation increase the vehicle’s potential, and ultimately lead to an improvement in on-track performance. During a lap on a particular racetrack, a driver is faced with a number of different types of corners and straights. Designing a racecar to perform well across turns of all radii on a single track involves resolution of a set of conflicting tradeoffs. Each segment of the racetrack has its own optimal vehicle characteristics, for example, the optimal racecar configuration for tight cornering is significantly different than that for sweeping, large-radii curves. Kasprzak et al. (2000) and Hacker et al. (2000) use single-level multiobjective optimization formulations to maximize racecar performance across multiple tracks of different radii. Our racecar model is based on the classic bicycle model of Milliken and Milliken (1995), which has been expanded to include four wheels. Equations of motion are written for lateral, longitudinal, and yaw accelerations. The tires, which

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Fig. 7 Sketch of the racecar model

Fig. 8 Racecar dynamics

may be different for front and rear, are modeled using tabular tire data including representations of nonlinearities, such as load sensitivity and slip angle saturation. Wheel loads are calculated based on static load, aerodynamic downforce, and lateral load transfer. Figure 7 illustrates a simplified sketch of the racecar model and shows the three design variables: roll stiffness distribution (K ), weight distribution (A ), and aerodynamic downforce distribution (C ).

Table 2 indicates the fixed racecar and track parameters used in this study. Specifically, we considered a racecar with a wheelbase of 9.67 feet and mass of 41.7 slugs traveling on a 400-ft radius curve. Figure 8 illustrates the relationships between lateral forces and slip angles. As indicated in the figure, the center of gravity (CG) defines the origin of the coordinate system, and clockwise moments are positive. Detailed design equations for the racecar are given in the Appendix.

3.1 Racecar analyses The racecar analysis begins with the calculation of tire parameters and concludes with an iterative analysis to solve for lateral forces given the center of gravity and roll stiffness. Table 1 presents the design variables under consideration for the racecar optimization. All design variables are normalized between 0 and 1 and have lower and upper bounds of 0.3 and 0.6, respectively.

Table 1 Racecar design variables Var.

Description

Init. value

A C K

Weight distribution Aero downforce distribution Roll stiffness distribution

0.4 0.4 0.3

Table 2 Racecar and track parameters Parameter

Value

Description

l mass h tF tR RefArea Radius CD

9.67 ft 41.7 slug 1.167 ft 5.5 ft 5.25 ft 10 ft2 400 ft 2.9

Vehicle wheelbase Vehicle mass Height of CG Front track Rear track Frontal area Skidpad radius Drag coefficient

3.2 Racecar problem formulation Requirements imposed on racecar design by the decision maker are incorporated using physical programming. One design objective is to minimize the lap time (i.e. go as fast as possible). However, lap speed and lap time must be considered in conjunction with pit time, which has a negative contribution to the overall race. Maintaining the center of gravity near the midpoint of the wheelbase provides more consistent wear of the front and rear tires and minimizes the frequency of required tire changes. Therefore, the normalized weight distribution, A , and the lap time, et, have specified ranges of desirability, see Table 3. For instance, it is desired to maximize, Class 2S, the normalized weight distribution with the ideal range being greater than 0.5, and desirable range being 0.5–0.4, and so on. The designer also wishes to minimize the lap time; the ideal range is less than 13 seconds, the desirable range is 13 to 14 seconds, and so on, while a lap time greater than 17 seconds is unacceptable. Equations (1)–(26) in the Appendix are used to establish the traditional optimization formulation for the racecar design problem shown in Fig. 9, which seeks to minimize lap time. Formulating this as a multidisciplinary collaborative optimization design problem with physical programming, we define two disciplinary subspaces: (1) aerodynamics and (2) force analysis as shown in Fig. 10. Incorporated in the aerodynamics analysis are (1)–(9) while the force analysis contains (10)–(24). The system-level optimizer seeks to


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Table 3 Physical programming ranges of desirability for racecar design Variable Class A et

2S 1S

Ideal

Desirable 0.5 13 s

Tolerable 0.4 14 s

Highly undesirable

Undesirable 0.36 15 s

0.32 16 s

Unacceptable 0.3 17 s

Fig. 9 Racecar design by traditional optimization

minimize lap time and maximize weight distribution and establishes 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 the deviation from these established targets to ensure the compatibility dictated by a multi-level formulation. Note that at the subsystem level, the aerodynamics subsystem contains the local objective of minimizing rear downforce, which is set at the 2nd priority level in the goal programming formulation, i.e. it is only improved after system-level compatibility, the first priority, is achieved. Minimization of rear downforce is useful for maximizing the

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

straight-line speed of the racecar and is addressed preemptively only after minimization of the discrepancy between system target values for shared variables and the equivalent local subsystem variables. Inclusion of rear downforce as an objective introduces potential conflict with the force subsystem that must meet the cornering requirement of maintaining contact between all four wheels and the track. The force subsystem designer would gladly accept a larger rear downforce if it is best for cornering.

4 Results

Fig. 10 Racecar design formulation using collaborative optimization with linear physical programming

Tables 4 and 5 present results for six different racecar design optimization formulations, which include traditional and CO formulations with and without LPP for comparison purposes. The reported values are consistent across three randomly selected starting points. “Traditional optimization” refers to the single-level formulation depicted in Fig. 9. “Collaborative optimization” indicates the multi-level formulation that seeks to minimize lap time at the system level and minimize the discrepancy between shared variables at the subsystem level. “Collaborative optimization with aero objective” indicates that minimization of rear downforce has

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Table 4 Racecar design optimization results: single-objective case Parameter

Traditional optimization

Collaborative optimization

CO with aero objective

A C K RDwnfc et

0.514 0.392 0.341 0.0361 slug/ft 14.22 sec

0.538 0.365 0.325 0.0378 slug/ft 14.22 sec

0.566 0.344 0.313 0.0390 slug/ft 14.09 sec

Table 5 Racecar design optimization results: multiobjective case Parameter

Traditional with LPP

CO with LPP

CO with LPP & aero objective

A C K RDwnfc et

0.512 0.392 0.336 0.0361 slug/ft 14.24 sec

0.513 0.393 0.348 0.0361 slug/ft 14.21 sec

0.505 0.396 0.342 0.0359 slug/ft 14.29 sec

been included preemptively as a secondary objective in the aerodynamics subsystem, as shown in Fig. 10. The formulation modifier “with LPP” indicates that linear physical programming has been used to specify preferences for lap time and normalized weight distribution, using the ranges of desirability listed in Table 3. First referring to the single objective results in Table 4, the traditional and collaborative optimization formulations achieve distinct optimal solutions with a lap time of 14.22 seconds. When collaborative optimization is implemented with the secondary aerodynamics objective of rear downforce minimization, the resulting design improves the lap time by 0.13 seconds. The primary objective is not the minimization of rear downforce because of the cornering requirement of the racecar; hence, minimum lap times do not necessarily correspond to minimum rear downforce values. The design space is identical across all formulations, and we note that (i) the traditional optimization and pure collaborative optimization formulations did not identify the optimum solution and (ii) the use of local subsystem objectives within collaborative optimization successfully provides additional

guidance for the system optimization to proceed to find the optimal racecar configuration. Physical programming modifies the objective function formulation such that a different design point is selected as the optimum in the multiobjective case (see Table 5). Traditional optimization with physical programming indicates a design with an “ideal” 0.512 normalized weight distribution (A ) and a “tolerable” lap time (et) of 14.24 seconds. When physical programming is used in conjunction with collaborative optimization, the reported design maintains the “ideal” normalized weight distribution and “tolerable” lap time. Though further resolution of normalized aerodynamic downforce distribution (C ) and normalized roll stiffness distribution (K ) could improve the lap time, it would not be sufficient to promote lap time to the “ideal” range. Hence,

Fig. 12 Traditional optimization convergence

Fig. 13 Convergence for collaborative optimization with aero objective

Fig. 14 Convergence for collaborative optimization with linear physical programming


the preferences of the decision maker do not strongly distinguish between the two different design configurations. Figure 12 shows the convergence history for the traditional optimization formulation. The design converges to the reported optimum (et = 14.22 sec). The model mandates that all four wheels remain in contact with the track; hence, negative wheel loads are penalized as evidenced by the large vertical spike in Fig. 12. Figure 13 illustrates the system-level convergence history for the collaborative optimization formulation using the secondary aerodynamics subsystem objective to minimize rear downforce. Compared to the traditional optimization convergence shown in Fig. 12, the collaborative optimization convergence is more frequently affected by infeasible designs due to interactions with the aerodynamics subsystem. When the system inadvertently tests a design that leads to negative wheel loads and corresponding infeasibility, the aerodynamics subsystem follows the direction established by the system, and the resulting deadlock is difficult for the system to resolve. Figure 14 presents convergence plots for the CO-LPP formulation. The trend for the normalized weight distribution exhibits a smooth and consistent improvement across linear physical programming’s regions of successively higher preference. After the optimization initially rejects infeasible designs, the lap time settles into a state of small fluctuations that become secondary to the improvement that is achieved in the weight distribution.

5 Closing remarks Linear physical programming has been integrated within collaborative optimization to provide a novel framework for the design and analysis of large-scale, hierarchical MDO problems. The proposed framework retains physical programming’s strength of contributing a flexible mechanism to express preferences among competing design metrics, while collaborative optimization provides the ability to formulate the multi-level MDO problem. The unified framework is demonstrated using the design of a Formula 1 racecar. Results are compared to the traditional, single-level formulations and CO formulations that do not include linear physical programming. We anticipate improved performance with the implementation of nonlinear physical programming within CO rather than linear physical programming; however, integration of nonlinear physical programming within the compromise DSP is not straightforward. We are also investigating the potential advantages of using this approach to solve large-scale MDO problems. Finally, an added drawback is the increased computation time necessary to enforce the equality-constrained system-level compatibility requirement, which needs to be investigated further; however, the collaborative optimization formulation more accurately represents the disciplinary organization encountered in large-scale systems design. Acknowledgement The authors gratefully acknowledge the racecar modeling expertise contributed by Milliken Research Associates, Inc.

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Appendix: Racecar analyses Equations (1)–(2) are used to compute the front and rear lift coefficients, CLF and CLR, respectively, based on the aerodynamic 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 and rear downforce, FDwnfc and RDwnfc, and aerodynamic drag, 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 before proceeding with the iterative analysis to solve for the lateral forces. Equations (7)–(9) determine the aerodynamic forces, where positive quantities indicate downforce. The aerodynamic force acting on the front and rear wheels is represented by AeroFzF and AeroFzR, respectively. AeroFx is an aerodynamic 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 FzRF FzLR FzRR FyF FyR Fy uOld MaxAlphaF MaxAlphaR

Left front wheel load Right front wheel load Left rear wheel load Right rear wheel load Lateral force front axle Lateral force rear axle Lateral force Velocity last iteration Max front slip angle Max rear slip angle

0 0 0 0 0 0 0 0 0 0

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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 each of the four wheels, including the effects of downforce. For instance, FzRF, is the load acting on the right front wheel. 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 due to normal load and slip angle, a quadratic approximation is used to determine maximum slip angles, MaxAlphaF and MaxAlphaR, and lateral forces, FyF and FyR, for the front and rear axles. Equations (17) and (18) check the lateral forces on the rear wheels, FyLR and FyRR, and, if required, reduce these forces due to the friction ellipse effect.  Fx Req  0 > |FyL R|    2   

FyL R = (17)

FyL R Fx Req 2   2  FyL R else −   2   |FyL R|   0      FyRR=

Fx Req > |FyRR| 2

2



FyRR Fx Req   FyRR2 −  else   |FyRR| 2

(18)

Equation (19) calculates the total rear lateral force, FyR, as a sum of lateral forces acting on each of the two rear wheels. FyR = FyLR + FyRR

(19)

Equations (20) and (21) determine the total yaw force, YawBal. 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 balance, YawBal = 0. If YawBal < 0, (22) provides the neces-

sary adjustment, while (23) is used to correct for YawBal > 0. 1 − A × FyR × cos(MaxAlphaR) − IDYaw FyF = A × cos(MaxAlphaF) (22) A × FyF × cos(MaxAlphaF) + IDYaw (23) FyR = B × cos(MaxAlphaR) Equation (24) calculates total lateral force, Fy, as a sum of front and rear lateral forces. Then, (25) and (26) are used to determine the corresponding speed, u, and lap time, et, respectively. Fy = FyF + FyR Fy × Radius u= mass et =

2π × Radius u

(24) (25)

(26)

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

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