Command(s): PDMETH,RSM,USER PDUSER

GUI: Main Menu> Prob Design> Prob Method> Response Surface

By using this option, you have complete control over the sampling data. You are required to give the file name and path.

1.6. Postprocessing Probabilistic Analysis Results

There are two groups of postprocessing functions in the probabilistic design system: statistical and trend.

A statistical analysis is an evaluation function performed on a single probabilistic design variable; for example, a histogram plot of a random output parameter. The ANSYS PDS allows statistical evaluation of either random output parameters or random input variables.

A trend analysis typically involves two or more probabilistic design variables; for example, a scatter plot of one probabilistic design variable versus another.

Probabilistic postprocessing is generally based on result sets. A result set is either a solution set generated by a probabilistic analysis run (PDEXE) or a response surface set (RSFIT). With a few exceptions, you

can perform the same type of probabilistic postprocessing operations regardless of whether you postprocess the results of a solution set or the results of a response surface set.

1.6.1. Statistical Postprocessing

Statistical postprocessing allows you several options for reviewing your results.

1.6.1.1. Sample History

The most fundamental form of postprocessing is directly reviewing the simulation loop results as a function for the number of simulation loops. Here, you can review the simulation values (for either response surface or Monte Carlo methods), or for Monte Carlo Simulations only, the mean, minimum, or maximum values, or the standard deviations.

It is most helpful to review the mean values and standard deviation history for Monte Carlo Simulation results if you want to decide if the number of simulation loops was sufficient. If the number of simulation loops was sufficient, the mean values and standard deviations for all random output parameters should have converged. Convergence is achieved if the curve shown in the respective plots approaches a plateau. If the curve shown in the diagram still has a significant and visible trend with increasing number of simulation loops then you should perform more simulation loops. In addition, the confidence bounds plotted for the requested history curves can be interpreted as the accuracy of the requested curve. With more simulation loops, the width of the confidence bounds is reduced.

Typically, postprocessing Monte Carlo results based on response surfaces is based on a very large number of simulation loops using the response surface equations. Therefore, the accuracy of the results is not a function of the number of simulation loops, but rather the goodness-of-fit measures of the individual response surfaces. As one example, suppose the goodness-of-fit measures indicate a very poor quality of a response surface fit, but the mean value and standard deviation history indicate that the results have converged (because of the large number of loops) and the accuracy is excellent (again because confidence bounds shrink with increasing number of loops). This could lead you to an incorrect conclusion. This is why the PDS only allows you to visualize the sample value history directly, but not the mean value history and so on.

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To review the simulation loop results:

Command(s): PDSHIS

GUI: Main Menu> Prob Design> Prob Results> Statistics> Sampl History

You need to select the results set label and the design variable, select a plot type, and set the confidence level.

1.6.1.2. Histogram

A histogram plot is most commonly used to visualize the scatter of a probabilistic design variable. A histogram is derived by dividing the range between the minimum value and the maximum value into intervals of equal size. Then the PDS determines how many samples fall within each interval, that is, how many "hits" landed in the intervals.

Most likely, you will use histograms to visualize the scatter of your random output parameters. The ANSYS PDS also allows you to plot histograms of your random input variables so you can double check that the sampling process generated the samples according to the distribution function you specified. For random input variables, the PDS not only plots the histogram bars, but also a curve for values derived from the distribution function you specified. Visualizing histograms of the random input variables is another way to make sure that enough simulation loops have been performed. If the number of simulation loops is sufficient, the histogram bars will:

•Be close to the curve that is derived from the distribution function

•Be "smooth" (without large “steps”)

•Not have major gaps

A major gap is given if you have no hits in an interval where neighboring intervals have many hits. However, if the probability density function is flattening out at the far ends of a distribution (for example, the exponential distribution flattens out for large values of the random input variable) then there might logically be gaps. Hits are counted only as positive integer numbers and as these numbers gradually

get smaller, a zero hit can happen in an interval.

To plot histograms:

Command(s): PDHIST

GUI: Main Menu> Prob Design> Prob Results> Statistics> Histogram

1.6.1.3. Cumulative Distribution Function

The cumulative distribution function is a primary review tool if you want to assess the reliability or the failure probability of your component or product. Reliability is defined as the probability that no failure occurs. Hence, in a mathematical sense reliability and failure probability are two sides of the same coin and numerically they complement each other (are additive to 1.0). The cumulative distribution function value at any given point expresses the probability that the respective parameter value will remain below that point. The figure below shows the cumulative distribution function of the random property X:

Figure 1.11: Cumulative Distribution Function ofX

The value of the cumulative distribution function at the location x0 is the probability that the values of X stay below x0. Whether this probability represents the failure probability or the reliability of your component depends on how you define failure; for example, if you design a component such that a certain deflection should not exceed a certain admissible limit then a failure event occurs if the critical deflection exceeds this limit. Thus for this example, the cumulative distribution function is interpreted as the reliability curve of the component. On the other hand, if you design a component such that the eigenfrequencies are beyond a certain admissible limit then a failure event occurs if an eigenfrequency drops below this limit. Thus for this example, the cumulative distribution function is interpreted as the failure probability curve of the component.

The cumulative distribution function also lets you visualize what the reliability or failure probability would be if you chose to change the admissible limits of your design.

Often you are interested in visualizing low probabilities and you want to assess the more extreme ends of the distribution curve. In this case plotting the cumulative distribution function in one of the following ways is more appropriate:

•As a Gauss plot (also called a "normal plot"). If the probabilistic design variable follows a Gaussian distribution then the cumulative distribution function is displayed as a straight line in this type of plot.

•As a lognormal plot. If the probabilistic design variable follows a lognormal distribution then the cumulative distribution function is displayed as a straight line in this type of plot

•As a Weibull plot. If the probabilistic design variable follows a Weibull distribution then the cumulative distribution function is displayed as a straight line in this type of plot.

The advantage of these plots is that the probability axis is scaled in a nonlinear fashion such that the extreme ends of the distribution function are emphasized and more visible.

To plot the cumulative distribution function:

Command(s): PDCDF

GUI: Main Menu> Prob Design> Statistics> Prob Results> CumulativeDF