Probabilistic Reliability
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PROBABILISTIC ENGINEERING DESIGN
INTRODUCTION
Engineering design relies heavily on complex computer code such as finite element or finite difference analysis. These complex codes can be extremely time-consuming even with the computing power available today. Deterministic design employs the idea of either: (a) running this code with input variables at their worst case values, or (b) running this code with input variables at their nominal values and applying a safety factor to the final result of the output variable. In the generic sense of modeling a typical response like stress, the result of using either of these methods is unknown. Assuming the input distributions are correct, applying worst case scenarios is too conservative. Applying safety factors to a nominal solution can result in either too much or too little conservatism with no method to compute risk or probability of occurrence.
Probabilistic engineering design relies on statistical distributions applied to the input variables to assess reliability, or probability of failure, in the output variable by specifying a design point. Any response value passing beyond this design point (also referred to as the most probable point, or MPP) is considered in the failure region. This method also allows for reverse calculations such that a specific probability of failure can be specified for the response. The MPP is then determined by calculating the response value that yields the specified probability of failure. This concept of designing to reliability instead of designing to nominal is clearly a superior method for engineering design. By choosing a desired reliability from a distribution on the response, a probabilistic risk assessment is built into the design process.
The theoretical improvement of probabilistic design over deterministic design can be seen below in Figures 1 and 2. Figure 1 depicts the general concern when calculating the MPP deterministically using near worst-case scenarios on input variables. The resulting MPP estimate on the output distribution can be greatly biased compared to the true desired probability of failure. Figure 2 shows the theoretical benefit of probabilistic design. In this case, Monte Carlo simulation is portrayed. Instead of using a single combination containing only one extreme point from each input variable as in deterministic design, a series of random combinations are drawn from the assumed input distributions on the parameters. These combinations of input variables are used to calculate a large number of realizations for the output variable for which a distribution is thus generated.
Figure 1. Conceptual Depiction of Deterministic Design.
While Figures 1 and 2 pictorially show the benefit of probabilistic design, it cannot actually be assessed without performing both deterministic and probabilistic designs. The clear concern in deterministic design is “over-design.” In all designs, we must live with some probability of failure, no matter how low, regardless of how many precautionary measures are in place. Calculating the probability of failure based on extreme values from inputs may result in a number much lower than is required. Blind intuition would not express concern over this, but over-design can result in increased costs. Additionally, without having the results of a probabilistic design for comparison, there is no way to measure the potential over-design penalty. The benefit of probabilistic design is clear. The question would then be as to how much effort is needed to carry out a probabilistic design. The same information required for a deterministic design is used in probabilistic design. The analysis process itself will be longer, but the benefits could include validation of deterministic results, calculation of more precise results and potential cost savings.
Figure 2. Conceptual Depiction of Probabilistic Design.
The remainder of this paper will describe the Monte Carlo simulation (MCS) and response surface methodology (RSM) approaches to probabilistic design. The RSM approach will include experimental design selection, model checking based on statistical and engineering criteria, and output analysis. Useful guidelines are given to ascertain statistical validity of the model, and the results.
MONTE CARLO SIMULATION
Once the input variable distributions have been determined, Monte Carlo simulation follows quite easily. A large number of random samples are drawn and run through the complex design code. The resulting realizations create a distribution of the response variable. The distribution fit to the response can be empirical or a fitted parametric distribution can be used. An advantage of fitting a parametric distribution is that specific probabilities of interest requiring extrapolation can be estimated. For instance, if one uses 1000 random samples, it would be impossible to estimate the 0.01th percentile empirically. A parametric distribution would be able to estimate this value. It would be up to the analyst as to whether the extrapolation is too far beyond the range of the data to consider trustworthy. Another advantage of fitting a parametric distribution is for ease of implementation in additional analyses. If a design engineer probabilistically modeled stress as a response and wished to then use stress as an input distribution to calculate life, programming and calculations would be simplified by using a functional distribution. Figure 3 shows the flowchart required for MCS.
Figure 3. Flowchart for MCS Probabilistic Design Method.
While the MCS method is simple, it is also very expensive. As mentioned, the design code can be quite time consuming. It is not uncommon for a single finite element analysis run required to produce a single observation to take as long as 10, 20 or 30 minutes. At ten minutes, one thousand simulated realizations would require just under a full 7-day week to complete. Even at just 30 seconds per run, a simulation of 1000 runs would require more than eight hours to complete. The MCS method will produce a distribution of the response that can essentially be viewed as the truth, but the penalty for demanding many runs is severe.
RESPONSE SURFACE METHODOLOGY
While MCS will produce a true distribution of the response, the cost of using that method is clearly in the number of runs required for the design code. An alternative is to estimate