Probabilistic Design

The term probabilistic design refers to the approach used in systems engineering to facilitate decision-making. Probabilistic tools help identify designs that are least sensitive to random variability in performance of systems. These variability effects are significant when optimiz-ing for quality and reliability and are found to be most useful in fields like manufacturoptimiz-ing and systems engineering. Probabilistic methods incorporate techniques such as robust design, parameter design and design for Six Sigma.

The use of probabilistic methods are found to be most beneficial whenever there is sufficient uncertainty to cause a significant variation in the response. This situation is most common when the design requirements are not well known or are likely going to change throughout the design process. The design of revolutionary vehicles is a prime example where designers would employ probabilistic methods in order to account for changes in mission requirements and uncertainties associated with the maturation of embedded technologies (Mavris, Roth and Elliott, 1998; Roth et al., 2004).

Aircraft engine design benefits from probabilistic methods because it helps to estimate the uncertainties in engine component weights and performance as well as predict the overall performance of the system. The application of robust and probabilistic methods are becom-ing more widespread due to increased demand for safety and reliability and better prediction of technology performance. Studies by Mavris, Macsotai and Roth (1998) have shown how probabilistic design methods are employed throughout engine preliminary design that show the impact of changing engine cycle parameters on the performance of a large commercial aircraft.

Quantifying the uncertainty of customer specifications in engine design is an underlying goal of the this research. Customer requirements are specified as deterministic values that translate into constraints on the engine design space. However, these requirements rarely

stay the same throughout the entire design process. These engine specifications are ulti-mately a function of aerodynamics, structures and other aircraft characteristics that will change whenever the aircraft design changes.

Mission uncertainty faced by the customer results in aircraft performance changes, yield-ing further variation of engine specifications. The vendor observes this variation of engine requirements as a migration of the engine design point. An illustration of this migration scenario is shown in Figure 2.8. Once the Modeling and Simulation Environment has been created, it is possible to challenge the required solution and possible deviations from the original needs. For instance, the solution point contains assumptions about:

1. Structures: If for any reason the initial design empty weight of the vehicle cannot be met when performance and takeoff gross weight are fixed, the engine manufacturer might be asked to lower the engine weight and or fuel consumption to help in reducing the overall vehicle empty weight or block fuel weight. Since there is usually a competing engine manufacturer waiting for an opportunity to enter the game, the company that is willing/capable of providing such a reduction might be the one chosen to power the vehicle. This is illustrated in Figure 2.8 where the maximum allowed engine weight is decreased, shifting the engine design point to (1).

2. Aerodynamics: uncertainty associated with the vehicle aerodynamics may lead to a degradation of the aerodynamic drag coefficients and properties which may increase the needed thrust to maintain the target cruise speed. This situation results in a displacement of the design point to location (2) as shown above.

3. Mission Evolution: Initial user needs usually change over time. These changes include variations in payload and range that regularly occur with time as new versions of this vehicle will be introduced on the market. This migration, due to growth needs, is denoted as (3).

4. Combinations: As shown in the Figure 2.8, it is possible to observe that combinations of these three scenarios can possibly change the design point.

Figure 2.8: Migration of the Design Point with Uncertainty

Uncertainty in engine specifications is modeled by introducing variations in the assump-tions made by the customer. For instance, the airframe manufacturer requires that the performance remain unchanged with a reduction in engine weight. As the aircraft design matures, the empty weight estimation at the outset is no longer valid. The customer needs to maintain range, payload, and performance. To balance this increase in weight, the air-frame manufacturer may pass some of the weight reduction responsibility to the engine manufacturer, requiring a reduction in engine weight to meet the user performance needs.

At this point, the engine manufacturer may no longer guarantee meeting the initial design requirements with full confidence.

Over recent years the field of probabilistic analysis has matured significantly with the emergence of new tools and techniques that have overcome the obstacles of using physics-based sizing codes that are deterministic in nature. The use of a surrogate model, which implies approximating a model with another model, is one of the best ways of building complex models that more accurately represent the physics-based codes. However, these complex models tend to have a large number of inputs and outputs meaning that a surrogate

model usually will take a significant amount of time to construct. There is a tradeoff that is often made between the degree of complexity of the surrogate model to achieve accuracy and amount of time needed to construct that surrogate model. Oftentimes, designers will choose to create simplified models that sufficiently represent the more complex model and accept in return some measure of modeling error.

The biggest advantage of using surrogate models is to alleviate the burden of expending too much time running design codes that can usually take several minutes to run a single simulation. This is particularly common in aerospace design where using aerodynamics codes to perform design optimization and sensitivity analyses would need to be run thousands of times.

The first step in constructing a surrogate model is to identify which input variables are uncertain in the analysis. The creation of an accurate surrogate depends on the number of simulation runs which in turn is a function of the number of input variables used. However, as the number of variables increases more analysis runs will likely be needed which effectively increases the total simulation run time. Determining the most efficient number of runs or experiments needed to create a surrogate model with the minimum amount of effort is a broad field of its own. Experimental designs range from evaluating every possible input variable combination, commonly referred to as full factorial designs, to more efficient designs that leave out specific variable settings, known as composite designs. For a more detailed description and application of these designs, the reader is referred to more in depth literature by Montgomery (2001) and Barros et al. (2004).

After running the experimental designs through the analysis codes the data is regressed to build a linear regression model or a Response Surface Equation (RSE) (Myers and Mont-gomery, 2002). There are many popular methods besides Response Surface Methodology (RSM) such as Kriging, Gaussian Processes, and Neural Networks. Kriging models are advantageous for situations where a non-linear representation is needed (Simpson, 1998).

Neural Networks (NN) are popular non-linear data modeling tools that are based on a mathematical representation of a biological interconnected group of neurons. They are most

commonly found in the study of artificial intelligence and cognitive psychology. A surro-gate model is constructed through a learning process in which Neural Networks must be trained during data-fitting. The statistical software JMP, developed by the SAS Institute Inc. (2007), combined with a surrogate modeling tool- Basic Regression Analysis for Inte-grated Neural Networks (BRAINN) (Johnson and Schutte, 2006) has greatly facilitated the creation of surrogate models and simplified the training process. (More information available in appendix B)

The process for implementing probabilistic analysis is illustrated in Figure 2.9. The representation of the modeling and simulation environment, via polynomials as expressed by the surrogate models, greatly facilitates the use of a Monte Carlo Simulation approach to obtain the probabilistic forecasts. Monte Carlo methods are simulation techniques that use random samplings to observe the statistical output of a mathematical system. They originated with famous physics researchers like Metropolis and Ulam (1949) and have been widely used in many operations research fields. A probability distribution of an appropriate

Figure 2.9: Probabilistic Design Method (Adapted from Bandte (2000))

shape is placed over the ranges set for each input variable. These distribution shapes can be normal, triangular, beta, or any other shape that best represents the perceived confidence and represent the probability of change of any given design metric (aerodynamic, structural, etc.). As the design development matures and the confidence associated with these estimates increases (i.e. reduction in the variability range) these shape functions will also be modified to reflect the change. Since the shape functions are time dependent this formulation may be characterized as a stochastic one.

The distributions defined are then provided as inputs into the computationally expedient surrogate model and a Monte Carlo simulation is performed. At the conclusion of this iter-ative process, probability distribution functions (PDF) or cumuliter-ative distribution functions (CDF) are generated for each of the responses of interest. Once these distributions have been evaluated for each metric of interest, and their corresponding correlation coefficients, ρ, have been determined (using the DoE obtained from observed results), joint probability curves may be introduced by superimposing two or more probability distribution functions (PDF) into the design space. Joint probability distributions (JPD) are useful in disciplines that involve the observation of more than one random variable (Li, 2007). Two probability distribution functions can be analyzed with a bivariate normal distribution in which each of the two random variables (x,y) are normally distributed. The resulting joint probability density function (JPDF) of the bivariate normal distribution is presented in equation 2.1 where “f” is the frequency of the JPDF, “x” and “y” are any parametric axes such as thrust and weight and “a” and “b” are specific values of x and y respectively. Equation 2.1 uses the normal distribution parameters, with mean µ, standard deviation σ, and correlation coefficients ρ of pairs of criteria and is illustrated in Figure 2.10.

f_XY(a,b)= ¹ Joint probability decision-making is a popular method used to analyze various forms of un-certainty in multi-dimensional problems. The reader is referred to a more extensive overview of this field in the work by Bandte (2000) and applications of this method by other authors in this field (Li, 2007; Mavris and Briceno, 2003; DeLaurentis and Mavris, 2000).