COUPLED SYSTEMS DESIGN IN PROBABILISTIC ENVIRONMENTS

COUPLED SYSTEMS DESIGN IN PROBABILISTIC ENVIRONMENTS Tom Halecki NASA Graduate Researcher, Student Member AIAA Department of Mechanical and Aerospace Engineering University at Buffalo, SUNY Buffalo, NY Kemper Lewis Associate Professor, Member AIAA Department of Mechanical and Aerospace Engineering University at Buffalo, SUNY Buffalo, NY 1. ABSTRACT In traditional robust design techniques, one is concerned with minimizing the variation in the system output due to uncertainty or variation in external conditions without eliminating the source of the variation or uncertainty. Recent applications of robust design have dealt with modeling the interfaces between designers in highly coupled systems where the uncertainty is due to the lack of knowledge of non-local design decisions. In this paper, we extend robust design principles into probabilistic design problems. Non-local information is modeled as uncertainty in a probabilistic sense and decisions are made using simulation and optimization. The sensitivity of these probabilistic decisions to the distribution of decisions of other designers is also modeled and approximated. Validation is offered in the form of a comparison with probabilistic theory prediction using the design of a pressure vessel.. INTRODUCTION As technological advances continue to be achieved, the design processes that are involved in developing these products increase in complexity as well. In order to combat this growing complexity, organizations have been breaking up the design into smaller components or subsystems 1. For instance, an airplane design may be broken up into an aerodynamics design team, aero structures design team, controls design team, etc. Each design group has their own objective to extremize and constraints to satisfy. While breaking up the design responsibilities among several or even hundreds of individual designers/design teams may reduce the scope that one designer or group needs to concentrate upon, a problem arises when these subsystems are coupled with one another. If the subsystems are coupled, information, which may include design variable values, FEA analysis, material selection, needs to be passed between the subsystems. In a perfect design process this information would be exchanged flawlessly and on demand but this is rarely the case. Many organizations will assign subsystems to different vendors. One vendor may need information from another, but information may not be exchanged because of concerns with trade secrets. Also, geographic constraints may not allow the designers to actively exchange information in a timely manner. Design information, such as FEA code or CAD drawings, may not be in a compatible form from design team to design team. These problems can drag a design process to a halt, which can cost a company a share of the market or they may lose out on creating a new market in an increasingly competitive global economy. Thus, a major issue being addressed is how can a designer(s) model and mitigate the coupling of subsystems such that design decisions may be made independently of other subsystems. Moreover, by incorporating a probabilistic model for design decisions, this paper focuses on modeling and predicting the sensitivities of probabilistic decisions (means and variances) to other coupled probabilistic decisions. There are a number of ways in which a designer might handle the uncertainty due to unknown nonlocal variables : 1. Ignore Uncertainty: A guess is made for the uncertain linking variables.. Seek out perfect information: The local designers determine the nonlocal variables by collaborating with the other designers. 3. Represent and Manipulate the Uncertainty: Techniques are used to model the uncertainty and make the solution robust to the uncertainty in the nonlocal variables. 1

Intuition tells us that collaboration between the designers should yield the best results, but in many cases there may be barriers, which make full cooperation difficult or impossible 3. Communication between the interacting disciplinary subsystems might be hampered by geographical separation (i.e. teams may not be collocated) or by the fact that the design teams are part of different departments within the same company or even different companies. Guessing at the unknown design information is always an option, especially when based on experience, but if the guess is inaccurate it can lead to degradation in performance and expensive and timeconsuming design iterations. Modeling the interaction between the designers and making the solution robust to the uncertainty may be the preferred strategy in many cases. In this work a method that incorporates robust design, optimization, and Monte Carlo simulation is used to combat the problems that arise due to certain types of uncertainty in a design process. Fundamentally, robust design is concerned with minimizing the effects of uncertainty or variation in design parameters on a design without eliminating the source of the variation 4. The combination of numerical techniques of optimization with the principles of robust design lends itself to the idea of robust optimization. In robust optimization, the focus is in finding a feasible combination of design variables, which not only optimize the function value but also minimize the sensitivity to variations of design variables and parameters. Depending on one s definition of uncertainty or the cause of the variation in the design parameters, the way in which robust design is applied to design applications can change. In traditional applications, uncertainty and variation were defined by manufacturing tolerances or small changes in design variables. In an adaptation of robust design, Chen 5 introduces two broad categories of problems associated with robust design: Type I - minimizing variations in performance caused by variations in noise factors Type II - minimizing variations in performance caused by variations in control factors. In previous work, a procedure is developed for Robust Design including both types 6. This procedure allows the designer to consider minimizing variations in the performance caused by both noise factors and control factors. It is also possible to apply the theories of Type I and Type II robust design to a multiple designer protocol as developed in other work 7. In this application to multiple designer problems, it has been assumed that the design process follows a sequential or leader/follower protocol 8. In the leader/follower problem, the leader solves their problem or model and then passes this information to the follower who thus solves their design problem using information given from the leader. Taking into account those considerations, Type I and Type II robust design are applied to multiple designer problems to help alleviate the coupling between designers. In this work 8, the notion of noise factors was broadened to include deterministic decisions made by other designers. However, the uncertainty due to the coupling is determined using simple approximations and not in a probabilistic sense. Type II robust design is also applicable to the multiple designer environment. Type II robust design is applied to these problems to allow for more flexibility into the design. The previous approach 8 allows for the definition of a robust solution range. A range is chosen rather than a single point solution. This range provides a stable and satisfactory subsystem performance, although possibly a suboptimal solution. The range of solutions is then given to the other designer allowing for flexibility in the sequential design process. A sensitivity-based robust optimization method has been implemented, which makes use of sensitivity information, i.e. gradient information for the objective function and constraints 9. The sensitivity information is used to approximate variations in the objective function and constraints due to variation in the design variables. The sensitivity-based robust optimization method is modified by implementing a worst-case scenario 1. The worst-case estimation of propagated uncertainty is developed and applied as an alternative means to the estimate function variations. This uncertainty is created from the uncertainty associated with disciplinary design tools, such as FEA or CFD, and variations in the design variables. In this formulation, both the robust objective function and the robust constraints consist of two parts, the original function and an estimate of the variation of the function, which is obtained from the worst-case estimation and verified using Monte Carlo simulation. When dealing with probabilistic design problems, simulation must be used 11. There are several different ways to physically implement the simulation such as system simulation models or Monte Carlo simulation. In this paper, Monte Carlo simulation is used. Monte Carlo simulation is a sampling experiment whose purpose is to estimate the distribution of an outcome variable that depends on several probabilistic variables 11. The basic principle behind Monte Carlo simulation is that behavior of a statistic can be assessed by the empirical process of drawing many random samples

from a population and observing its behavior 1. This is accomplished by creating an artificial model of the system being simulated. This model resembles the real physical system in all important characteristics. These artificial models are usually probabilistic in nature. By using the tools of simulation, one is able to model uncertainties and explore potential outcomes. From a design viewpoint, simulation can be used to explore the potential outcomes that may occur due to the uncertainty in the design process caused by non-local, unknown design information. In this paper, the focus is on two fundamental issues. The first is how to model uncertain parameters in a coupled design problem when each subsystem is determining values for their own local design variables. The second issue is how to model the sensitivity of local decisions on nonlocal decisions in a probabilistic sense. That is how do the means and variances of local decisions depend upon the means and variances of nonlocal decisions? The approach presented in this paper couples simulation and optimization techniques to model the uncertain parameters. It then uses approximation techniques to construct sensitivity information using partial derivatives of probabilistic parameters. In the following section, more background about the approach is presented. Then, a case study is used to illustrate the approach in more detail and to validate the results against theoretical predictions from probability theory. experience with the unknown information. The distributions are defined by appropriate statisitical parameters. For instance, with the normal distributions, a mean and standard deviation are defined. Step 4: A point is randomly sampled from the nonlocal distributions and the local design problem is solved using the given nonlocal data point. This process of sampling and optimizing is continued until a maximum number of points is reached. It is important to note that this process is an augmented Monte Carlo approach using an optimization routine to determine the output instead of a more common analysis routine. Local Design X local known & Define: F(X local, X nonlocal ) g j (X local, X nonlocal ) h k (X local, X nonlocal ) Optimize Local Non-Local Design X nonlocal unknown Simulate Using Probabilistic Function Calculate Output & Sensitivity 3. ROBUST DESIGN APPROACH DISCUSSION The approach to robust design simulation (RDS) discussed here consists of a set of steps that combine robust design methods, simulation, optimization, and sensitivity analysis. These steps are discussed briefly in this section and are illustrated in the following section using an example. These steps are also illustrated in Figure. Step 1: A designer defines their own problem and the criteria of interest. Step : A designer identifies unknown information. This information is controlled by another nonlocal designer but needed by a local designer. Step 3: Since uncertainty is probabilistic and random in nature, a designer must make assumptions about how to represent the unknown design information in a probabilistic sense. This typically is done using some sort of probabilistic distribution based on the knowledge or experience a designer has with a problem. Normal distributions are used in this paper, but any other distribution could be used. Uniform distributions represent the worse case scenario where a designer has no understanding or prior design 3 The sensitivity analysis is the cornerstone of the robust nature of this algorithm. In essence, the sensitivity of the local design distributions to the nonlocal design distributions used in the simulation are predicted. This sensitivity can be defined by the following partial derivatives: objective objective Figure. Illustration of RDS objective objective local _ dv local _ dv local _ dv local _ dv Where the subscripts are defined as: non_local_dv non-local design variable local_dv local design variables objective local designer s goal or objective function

and µ is the mean of a given probabilistic model while σ is the standard deviation of the probabilistic model. As shown in these partial derivatives, the designer needs to know information about the effects on the local design variables due to changes in the non-local design variable s simulated means and variances. In doing so the designer is able to determine the effects that these changes have on the local design. The changes in the non-local design information are modeled as noise factors from a robust design perspective. Therefore the local designer is looking for a design that is insensitive to changes from the non-local design variables. In terms of the sensitivity parameters (the partial derivatives), the local designer wants to find a design where these parameters are close to zero and the objective function(s) is close to the target value. Since the unknown parameters are being treated as being probabilistic, there are two characteristics that must be accounted for: the mean and standard deviation. Each is now investigated in more detail. 3.1 Modeling Unknown Parameters: Means If the designer is interested in determining the effects of a change in the mean of non-local design variable Y 1 (e.g., objective / non_local_dv ) then the RSD algorithm is performed keeping all other non-local probabilistic parameters constant, incrementing only the values of the mean of Y 1,. Since is it assumed that the designer knows the range for the non-local design information, the sensitivity analysis begins at the lower end of the range. This initial value is approximated to be (Y 1LOWER BOUND + 3σ Y1 ), as illustrated in Figure 3. RDS is then performed until enough points have been simulated. Next the mean is incremented by some predetermined µ non-local and RDS is again performed. This continues until the appropriate range of means is investigated. Figure 3 illustrates the sensitivity analysis of a non-local mean. As is shown, the range of the non-local design variable is broken up into smaller ranges,. Monte Carlo simulation is performed for each new value of the mean. Once this is completed, the designer now has statistical information for an optimized local objective function and design variables at several values of the nonlocal design variable mean. Thus if the designer is interested in the change in mean of the local objective function with respect to the change in the mean of Y 1, then the designer uses the mean of the objective function found for each value of and fits a function through these points. Once complete, the derivative of this function is taken and the designer can now view how the change in the mean of nonlocal design variable Y 1 affects the mean of the local µobjective function objective function, as shown in Figure 4. From a robust design perspective, regions of interest are those that cause the least amount of variation in the optimized objective function mean. The same rationale is implemented to determine the sensitivity of local design parameters with respect to the change in the variance of the non-local probabilistic models. In this case all probabilistic parameters are kept constant except for the variance. 3σ Y1 1 Y 1lower bound 3 i Y 1upper bound Figure 3. Sensitivity Analysis of Nonlocal Parameter Mean a) µ obj.func. versus b) obj.func. / Y1 versus Figure 4. Illustrative Sensitivities of Objective Function with Respect to Mean of Nonlocal Variable 3. Modeling Unknown Parameters: Variances In the sensitivity analysis of the change in variance of the non-local design information, the variance is incremented from 5% to 15% of its original value, by σ, while keeping all other parameters constant, as shown in Figure 5. For each different value of variance RDS is completed. Again the designer is able to find local objective function and variable sensitivities for each value of the nonlocal variable variance, similar to Figure 4. From a robust design viewpoint the variation in the variance of the non-local design information is modeled as a noise factor(s). The designer is thus interested in finding an optimal solution where the optimum is close to a target value and the variation caused by the changing values of the non-local design variable variance is reduced to a minimum. obj.func/y1 4

By applying simulation in application to robust design, the designer is eliminating the need for complex analytical expressions to represent the nonlocal design information, and also the designer is avoiding the errors that may arise through erroneous single point assumptions of the non-local design information. 1.5σ Y1 Probability.5σ Y1 σ Y1 that these two subsystems are working independently of each other. Under this assumption, information is not exchanged from one subsystem to another. Table 1: Nomenclature of the Pressure Vessel Design Problem Variable Description Units W V Weight of the pressure vessel Volume of the pressure vessel lbs. in. 3 R Radius in. Figure 5. Sensitivity Analysis of Non-local Parameter Variance 4. CASE STUDY To verify the RDS algorithm, a pressure vessel design problem is used 13-15. The pressure vessel is thin-walled with hemispherical ends, as shown in Figure 6. Nomenclature of this problem is shown in Table 1. Section A-A T Section A-A L Figure 6: Cross-Section of a Thin-Walled Pressure Vessel 4.1 Problem Set-up The design variables of the design problem are the, R, the, L, and the, T. The pressure vessel is to withstand an internal pressure P and the material of the vessel has already been defined. In this design problem there are two system level objectives. The first objective is to maximize the volume, V, of the pressure vessel and the second objective is to minimize the, W, of the vessel. This design problem is broken up into two subsystems: a Weight subsystem and a Volume subsystem. It is recognized that one designer using multi-objective optimization techniques can typically be used to solve this problem, but to illustrate the use of RDS, the problem is broken up such that there are two designers working on the problem. It is assumed R 5 L Length in. T Thickness in. P Pressure inside the cylinder Klbs. S t ρ Allowable tensile strength of cylinder material Density of the cylinder material Klbs. lbs./in. 3 σ circ Circumferential stress lbs./in. The Weight subsystem design problem is defined in Table. In this subsystem design formulation it is important to note several factors. The only local design variable, or design variable under the control of the Weight subsystem is the of the pressure vessel. The other two design variables are under the control of the Volume subsystem. Thus, the, R, and the, L, are defined as nonlocal design variables for subsystem Volume. Minimize: Subject to: Bounds: Table : Weight Subsystem 4 W ( R, T, L) = ρ π L 3 3 g g 1 PR = σ circ = St T = 5T R g 3 = R + T 4 3 4 3 π ( R + T ) + πl( R + T ) πr + R g 4 = L + R + T 15 T T T L U Note that the design variables R and L are not under the control of the Weight Subsystem and thus are termed non-local design variables. This subspace controls design variable T only, and thus is considered a local design variable. The Volume subsystem design problem is defined in Table 3.

Maximize: Subject to: Bounds: Table 3: Volume Subsystem 4 3 V ( R, T ) = π R + πr 3 PR g1 = σ circ = St T g = 5T R g = R + T 4 3 g 4 = L + R + T 15 R R R L L L L L U Note that the design variable T is not under the control of the Weight Subsystem and thus is termed a non-local design variable. This subspace controls design variables R and L, and thus are considered local design variables. Again, one must note the important features of the Volume subsystem design problem. In this subsystem, the design variables that are controlled are the, R, and the, L. Thus these variables are termed local design variables. The, T, is controlled by another subsystem, and thus this design variable is termed a non-local design variable for subsystem Volume. The design parameters for this case study are listed in Table 4. Table 4: Design Parameters P 3.89 Klbs. S t 35. Klbs. ρ.83 lbs./in 3 L L L U R L R U T L T U U.1 in. 14. in..1 in. 36. in..5 in. 6. in. While full results for both subsystems and all variables are reported in other work 16, only the Weight subsystem will be discussed in this paper. In addition, only the effects of one nonlocal variable, the, and its distributions are studied here. Under the assumption of subsystem independence, the designer must decide how to deal with the coupled non-local design variables that are needed to optimize the local objective,, while satisfying all constraints. RDS is applied to this problem to allow the design to view how assumptions of the non-local design information will change the design locally. In other words, how L 6 do changes in the and the distributions affect the optimization of the distributions of the and of the pressure vessel? A stepby-step implementation of RDS is presented in the next section. 4. Implementation of Robust Design Simulation Approach The application of the robust design simulation approach to this problem is discussed in the context of the steps presented in Section 3. Step 1: Definition of the subsystem has been provided in the previous subsection. The designer s goal is to minimize the of the pressure vessel while satisfying all constraints. Step : The designer has defined all local and non-local design information. As stated in the previous section, the subsystem s local design variable is the, T. The non-local design variables are the and, R and L respectively. Using a P-Diagram from robust design 4, shown in Figure 7, the and are being modeled as noise factors and the local design variable is thus defined as the control factor. System Model Signal Factor Radius, Length Thickness Noise Factors PRESSURE VESSEL Control Factors Weight Response Figure 7. P-Diagram of Pressure Vessel: Weight Subsystem Step 3: The designer now must make assumptions about the non-local design information. In this problem it is assumed that the designer has had enough prior experience with the problem in order to choose normal distributions to represent both of the non-local design variables. Choosing a normal distribution allows the designer to simulate non-local design variables about a central point, or mean, with a specified variance away from the mean. If the designer did not have a good understanding of the problem or prior design experience, the designer could then choose a uniform distribution to represent the nonlocal design information or some other type of distribution that appropriately represents the tendency of the non-local design variables. The

Frequency mean of the, µ, is chosen to be inches with a variance, σ of 5 inches. The mean of the, µ, is chosen to be 75 inches with a corresponding variance, σ, of 1 inches. Again, these choices are made due to prior experience with this design problem. Figure 8 illustrates these distributions. Figure 8. Distributions of the Non-Local Variables Step 4: A point, X R, is randomly sampled from the distribution and a point, X L, is randomly sampled from the distribution and then the design problem is optimized. For each simulated point, the designer has an optimized objective function, W * and a corresponding value for the, T * with respect to the simulated non-local design variables, R and L. The process proceeds until 5 simulated values of R and L and optimized values of T and Weight have been defined. Note that the number of points needed to create the subsystem distributions is left to the designer to decide. An appropriately large enough number must be selected to accurately model the non-local design variable assumptions. The distribution of the optimized values and the optimized values are shown in Figure 9. Frequency Radius Frequency Frequency Length and σ =1 inches. With these values used to define the probabilistic models of the non-local design variables, the objective function is found to have a mean of µ =1,18 lbs. and a corresponding variance of, σ =6.967*1 6 lbs. In addition it is found that the has a mean of, µ thick =.1 inches and a corresponding variance of σ thick =.6 inches. This data gives the designer an understanding of what to expect due to uncertainties caused by non-local design information, giving a designer more assurance than picking a single point value for the non-local design information. The simulation of the non-local design variables allows the designer to more accurately define the uncertainty and the corresponding effect on the local subsystem s objectives and design variables. 4.3 Validation of Simulation-Based Approach In order to provide some verification to the values of µ and σ in the previous section an analytical approach is used to modeling the nonlocal probability distributions. In order to accomplish this, the mean and variance must be defined analytically. The analytical equations for the mean and variance are defined as follows: µ = xf ( x) dx (1) σ = x f ( x) dx µ () where f(x) is defined as the normal probability density function and x is an independent random variable 17. It is assumed that the output for the from the RDS algorithm is a normal distribution with the corresponding mean and variance. This formulation is slightly modified to accommodate a multiple random variable problem and thus has the form: µ = 14 36.1.1 W ( R, L, T ) f ( R, L) drdl (3) Weight (1*1 4 lbs.) Thickness (in.) σ = 14 36 W ( R, L, T ) f ( R, L) drdl µ (4).1.1 Figure 9: Output Summary from RDS From Figure 9, the designer is able to visually explore how the assumptions he or she have made impact the optimum value of the of the pressure vessel and the corresponding values for the. In this case study, it is assumed that µ = inches, σ =5 inches, µ =75 inches In this form, f(r,l) is the joint probability density function, and W(R,T,L) is the objective function. The limits of the integrals are based on the upper and lower bounds of R and L defined in the problem setup (Table 4). The is not included in the joint probability density function because it is dependent upon the values of the and therefore is not considered an 7

independent random variable. Due to the complexity of the function being integrated, the joint probability density function, f(r,l), had to be calculated using numerical techniques. After performing the integration in Eqn. 3 using a computational mathematics program, MATLAB, it was found that µ = 1,87 lbs. Comparing this answer to the simulated mean of 1,18 lbs., it can be seen that both the simulated mean and the analytical mean are very close. The error of the mean is approximately 7.4%. This error is largely due to the approximated solution and the assumption that the output follows a normal distribution exactly. Solving Eqn. 4, the analytical solution of the standard deviation is 15 lbs, and the simulated value of the variance is 64 lbs, giving an error of 18.54%. While this seems large, there are several sources of error in this calculation. First, the output from the simulation is not exactly a normal distribution. Since the equations for solving the analytical mean and variance are based on a normal distribution, some error will arise in the analysis. Secondly, the joint probability density function is approximated. This also adds error into the calculation and since terms are being square error propagates through the analysis. While this analysis predicts expected values for local objective functions and variables, it is senstivity information that is truly valuable in an MDO environment. Namely, the following sensitivities are approximated using the RDS and investigated in the next section. Influence of on Weight: Influence of on Weight: Influence of on : Influence of on : 4.4 Sensitivity Due to Changes in Radius: Changes in Mean To determine the effect of a change in µ, a range of means must be determined. As previously discussed, the smallest mean value used in the analysis is dependent upon the variance of the nonlocal design variable distribution. In this case study a 8 variance of 5 inches is assumed for the. Thus, the first mean will be defined as the lower bound of the plus three times σ. µ = 3 (5) _ initial RL + σ The final mean to be explored by the sensitivity analysis is defined in a similar manner: µ = 3 (6) _ final RU σ The initial mean, µ _initial, is incremented by µ, where µ is defined as: 1 µ = ( µ _ final µ _ initial ) (7) 1 Thus the range is broken up into 11 distinct points at which a 5-point simulation is performed at each point. The values of µ are found in Table 5. Table 5: Values of the Radius Mean (inches) µ _1 6.8351 µ _ 9.781 µ _3 11.311 µ _4 13.564 µ _5 15.87 µ _6 18.5 µ _7.93 µ _8.536 µ _9 4.7789 µ _1 7.19 µ _11 9.649 At each value of the mean shown in Table 5, the RDS algorithm is used in the same manner as used before with all other non-local design information remaining constant. Therefore in the first step of the sensitivity analysis, the mean of the is 6.8351 inches with a variance of 5 inches and constant values of µ =75 inches and σ =1 inches. The RDS algorithm is performed until 5 design points have been simulated. The process continues, incrementing the values and keeping the distribution constant. Figures 11 through 14 illustrate the values of the means of the and, as well as the variances of the and with respect to the corresponding values of the mean. From Figure 11, one can see that the of the pressure vessel increases exponentially as the becomes larger. This is the logical reaction to the increasing because by increasing the the designer is in essence increasing the surface area

of the pressure vessel. This in turn leads to a higher value for the. Also, in Figure 13, one can see that the of the pressure vessel increases linearly with an increasing value of the. This is also a logical reaction since as the increases the stress in the vessel increases. In order to satisfy the stress constraint, the must be increased. From Figures 1 and 14 the variance of the local design information with respect to the non-local design variable,, is shown. In a robust sense, the designer wants to find a solution that is very close to the target value, which in this case is zero pounds, and yet provides little variance in the output. From Figure 1 and Figure 14, the standard deviation, σ, of the output is plotted versus the mean. It can be seen that as the mean increases so too does σ. Ideally the should be as small as possible to minimize σ. Figure 14 illustrates the fact that the changing mean has relatively no effect on the variance of the mean until the mean becomes very large. This can be due to several factors. When the begins to reach the upper limits of its bounds the stress constraint becomes very sensitive to values. Any changes in R could affect the optimized value of T so that the stress constraint is satisfied. Also as R reaches larger values, feasible solutions may be hard to find and thus the solutions may vary to a greater extent than when µ is closer to its lower bound. µ σ 3 5 15 1 5 8 7 6 5 4 3 1 5 1 15 5 3 35 µ Figure 11: µ versus µ 1 3 µ Figure 1: σ versus µ µ σ 3.5 3..5. 1.5 1..5. 5 1 15 5 3 35.8.6.4.. µ Figure 13: µ versus µ 1 3 4 µ Figure 14: σ versus µ A function is fit through the design points and the partial derivatives of these functions with respect to µ are taken. This provides the designer the sensitivity of the local design to the change in the mean. These sensitivity parameters are plotted versus the mean in Figure 15 through Figure 18. From these figures the designer is given an indication as to how the mean affects the local design when changes are made in the non-local design variable s mean. Figure 16 and Figure 18 show a low level of correlation between the variance of the and the and the mean. On the other hand, in Figure 17, the mean changes at a constant rate with respect to the mean, and the increases rapidly with the increasing values of the mean in Figure 15. This information allows a designer to determine how the non-local design parameters directly affect the local design. In this problem the designer would want to have the partial derivatives as close to zero as possible. This would indicate that a change in the non-local design information has no effect on the local design at that point. Only the standard deviation of the shows no effect at small values. The other significant sensitivities give an indication to the designer that the plays a key role in the design decisions and must be taken into careful consideration. In the nest section the sensitivity of the local design information as a function of the variance is explored. 9

δ(µ)/δ(µ) δ(σ)/δ(µ) 18 16 14 1 1 8 6 4 5 15 1 5-5 5 1 15 5 3 35 µ Figure 15: δ(µ )/δ(µ ) vs. µ 5 1 15 5 3 35 µ Figure 16: δ(σ )/δ(µ ) vs. µ 4.5 Sensitivity Due to Changes in Radius: Changes in Variance To determine the effects of changes in the variance of the, namely, a range for σ is needed. As discussed in Section 3., the variance is broken into ten ranges. The variance is varied between 5% to 15% of its original value. The number of ranges is left up to the designer as are the ranges of the variance that the designer wants to explore. The original value of the variance of the is 5 inches. Therefore the upper and lower bounds are.5 inches and 7.5 inches. Figure 19 illustrates how this change affects the probability distributions of the. The values for σ are shown in Table 6. Probability.3.5..15.1.5 σ =.5 σ = 7.5 δ(µ)/ δ(σ)/δ(µ).1.1.8.6.4...5.4.3..1. -.1 5 1 15 5 3 35 µ Figure 17: δ(µ )/δ(µ ) vs. µ 5 1 15 5 3 35 µ Figure 18: δ(σ )/δ(µ ) vs. µ 1 65 7 75 8 85 Radius (in.) Figure 19: Probability Distribution for Variances of.5 and 7.5 inches Table 6: Values of σ (inches) σ _1 1.15 σ _ 1.347 σ _3 1.5715 σ _4 1.796 σ _5.5 σ _6.45 σ _7.4695 σ _8.694 σ _9.9185 σ _1 3.1431 σ _11 3.3676 At each value of σ shown in Table 6, the RDS algorithm is run with all other non-local design information remaining constant. Therefore in the first step of the sensitivity analysis, σ is 1.15 inches with a mean of inches. The mean of the

remains the same from the original assumptions made, which are µ =75 inches and σ =5 inches. The RDS algorithm is performed until 5 design points have been found. Next, σ is incremented by σ to 1.347 inches with a mean of inches. This is done until the final value of σ has been simulated. Figures through Figure 3 illustrate the means of the and, as well as the variances of the and with respect to the corresponding values of σ. In these figures, the designer gains some insight to how σ affects the local design. In Figure, the change in σ generally has little affect on the. While it may look like the varies greatly, the actual change in µ is very small and on the order of.1 inches. Also it can be noted that the amount of change in the variance of the is very small as well. The statistical parameters of the both increase with an increase in σ as shown in Figures and 1. This is logical since when σ is large, there is a greater possibility of achieving a large simulated value of the and in turn a larger value for the. Functions are fit through these points and the corresponding partial derivatives are taken with respect to σ. The resulting sensitivities are shown in Figure 4 through Figure 7. µ µ 135 13 15 1 115 11 15 1 995 99 985 98 45 4 35 3 5 15 1 5 1 3 4 σ Figure : µ versus σ..5 1. 1.5..5 3. 3.5 4. σ µ σ.38.36.34.3.3.8.6.4. 1 3 4 5.4.35.3.5..15.1.5 σ Figure : µ versus σ. 1 3 4 σ Figure 3: σ versus σ As discussed in previous sections the designer wants to find areas that have very small changes in the variance and mean of the local design information. The closer to zero the partial derivatives are the less sensitive the local design information is to the non-local design information. From Figure 4, when σ is approximately 1.3-1.6 inches and around 3. inches the change in µ are at a minimum. These points are critical to the designer as it is advantageous to find areas where the changes of non-local design variables minimally affect the local design. The designer may also note the constant values of the change in local variances with respect to the variance, as shown in Figure 5 and Figure 7. This indicates that changes in the local design s variations are constant, and thus unaffected by the changes in σ. This constant value is due to the fact that the local variances are linearly proportionate to the non-local variance, σ, as shown in Figure 1 and 3. As σ increases, the will also vary to a greater extent. This is caused by the only active constraint in this problem, the stress constraint, which is a function of R and T. Any changes in R will impact the optimal solution greatly since this constraint is the driving force behind finding a feasible optimal design. Figure 1: σ versus σ 11

δ(µ)/δ(σ) 45 4 35 3 5 15 1 5.5 1 1.5.5 3 3.5 4 σ δ(µ)/δ(σ) δ(µ)/δ(σ) δ(σ)/δ(σ) Figure 4: δ(µ )/δ(σ ) vs. σ 14 1 1 8 6 4..5 1. 1.5..5 3. 3.5 4. σ Figure 5: δ(σ )/δ(σ ) vs. σ..1 -.1 1 3 4 5 -. -.3 -.4 -.5 -.6 -.7 σ Figure 6: δ(µ )/δ(σ ) vs. σ.1.1.8.6.4.. 1 3 4 5 σ Figure 7: δ(σ )/δ(σ ) vs. σ 6. CONCLUSIONS The idea behind the RDS approach presented in this paper is to give the designer valuable information enabling him or her to make design decisions in an MDO environment marked by uncertainty. Ideally these decisions will allow the local designer to make their solution robust, or insensitive, to the non-local design information. In defining the sensitivity parameters, the local designer is able to view how the local design reacts to changes based on assumptions regarding uncertainty in non-local variables. When these senstivities are equal to zero, or very close to it, the designer knows that changes in non-local design information, such as a non-local design variables mean or variance, will not affect the local solution. If the senstivities are relatively large, the designer knows that any change in the uncertain non-local design information will greatly affect the local design s performance. This method is unique in its ability to couple simulation, optimization, and robust design in an MDO environment. Typically robust design is concerned with design variable values, whereas in this paper robust design has been extended to consider how changes in probabilistic models of uncertainty affect a local design. Changes in probabilistic models or uncertainty are important features of this design process. The mean of a probabilistic function is analogous to design variable values, and the variance of the function is analogous to the controllability of a design variable. This approach to design gives the designer a way to determine robust regions in a probabilistic sense. Simulation plays an important role in the RDS algorithm, allowing the local designer to make probabilistic assumptions of the non-local design information. In doing so the designer has an active way to represent the non-local design variables that may be needed to perform an optimization routine on the local design space. The simulations take advantage of an optimization routine to predict the effect of uncertain noise on optimized design variables and objective functions. While the work presented here investigated distribution changes one at a time (i.e., means and variances were studied independently), current work is focused on varying each simulaneously to uncover any correlation between the parameters. In addition, the computational cost of the method is relatively high. While the problems studied thus far have not posed a computing problem, plans for more complex problems will necessitate an investigation of techniques to make the method more efficient. 1

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