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Probabilistic Design Terminology |
Young's modulus (Gaussian dis- |
~16% |
~2.3% |
tribution with 5% standard devi- |
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ation) |
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Young's modulus and thermal |
~22% |
~8% |
expansion coefficient (each with |
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Gaussian distribution with 5% |
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standard deviation) |
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When a conservative assumption is used, this actually tells you that uncertainty or randomness is involved. Conservative assumptions are usually expressed in terms of safety factors. Sometimes regulatory bodies demand safety factors in certain procedural codes. If you are not faced with such restrictions or demands, then using conservative assumptions and safety factors can lead to inefficient and costly over-design.
You can avoid over-design by using probabilistic methods while still ensuring the safety of the component.
Probabilistic methods even enable you to quantify the safety of the component by providing a probability that the component will survive operating conditions. Quantifying a goal is the necessary first step toward achieving it. Probabilistic methods can tell you how to achieve your goal.
1.1.2. Reliability and Quality Issues
Use probabilistic design when issues of reliability and quality are paramount.
Reliability is usually a concern because product or component failures have significant financial consequences (costs of repair, replacement, warranty, or penalties); worse, a failure can result in injury or loss of life. Although perfection is neither physically possible nor financially feasible, probabilistic design helps you to design safe and reliable products while avoiding costly over-design and conserve manufacturing resources (machining accuracy, efforts in quality control, and so on).
Quality is the perception by a customer that the product performs as expected or better. In a quality product, the customer rarely receives unexpected and unpleasant events where the product or one of its components fails to perform as expected. By nature, those rare "failure" events are driven by uncertainties in the design. Here, probabilistic design methods help you to assess how often "failure" events may happen. In turn, you can improve the design for those cases where the "failure" event rate is above your customers' tolerance limit.
1.2. Probabilistic Design Terminology
To understand the terminology involved in probabilistic design, consider the following problem.
Find the scatter of the maximum deflection of a beam under a random load of snow. The snow should have a linear distribution along the length of the beam with a height of H1 at one end and a height of H2 at the other. The beam material is described by various parameters including the Young's modulus, for which a mean value and standard deviation have been derived.
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Figure 1.1: A Beam Under a Snow Load
H1 
H2
E
PDS Term
random input variables (RVs)
correlation
random output parameters (RPs)
Description
Quantities that influence the result of an analysis.
In probabilistic design, RVs are often called "drivers" because they drive the result of an analysis. You must specify the type of statistical distribution the RVs follow and the parameter values of their distribution functions.
For the beam example, the heights H1 and H2 and the Young's modulus E are clearly the random input variables. Naturally, the heights H1 and H2 cannot be negative and more often there will be little snow and only a few times there will be a lot of snow. Therefore, it might be appropriate to model the height of the snow as an exponential or a lognormal distribution, both of which have the bulk of the data at lower values.
Two (or more) RVs which are statistically dependent upon each other.
In the beam example, it is unlikely that one side of the roof (beam) supports a great deal of snow while almost no snow exists on the other. It is not necessary that H1 and H2 are exactly identical, but with a lot of snow then H1 and H2 both likely have larger values and with little snowfall then both would be lower. Therefore, H1 and H2 are correlated, although the correlation must not be mistaken for a direct mathematical dependency. In the beam example, no numerical dependency exists but rather a certain trend between the two heights; with this particular H1 and H2 it is unlikely that their values will be drastically different at any given point in time.
Note
Mathematical dependencies have some numerical dependence on each other. For example, a true correlation exists if, when one parameter value doubles, another parameter value also doubles.
The results of a finite element analysis.
The RPs are typically a function of the RVs; that is, changing the values of the random input variables should change the value of the random
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Probabilistic Design Terminology |
PDS Term |
Description |
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output parameters. In our beam example, the maximum beam deflec- |
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tion is a random output parameter. |
probabilistic |
The random input variables (RVs) and random output parameters (RPs) |
design vari- |
are collectively known as probabilistic design variables. |
ables |
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In the Probabilistic Design System (PDS), you must identify which |
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parameters in the model are RVs and which are RPs. |
sample |
A unique set of parameter values that represents a particular model |
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configuration. |
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A sample is characterized by random input variable values. If you |
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measure the Young's modulus of a given beam, and the snow height |
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on a given day at each end of that beam, then the three values for E, |
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H1, and H2 would constitute one sample. |
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Think of a sample as one virtual prototype. Every component manu- |
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factured represents one sample, because you can measure its particular |
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properties (material, geometry, etc.) and obtain specific values for |
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each. |
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In statistics, however, sample also has a wider and more general use. |
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For example, any single measured value of any physical property is |
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considered to be one sample. Because a probabilistic analysis is based |
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on a statistical evaluation of the random output parameters (RPs), the |
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values of the RPs are also called samples. |
simulation |
The collection of all samples that are required or that you request for |
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a given probabilistic analysis. |
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The simulation contains the information used to determine how the |
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component would behave under real-life conditions (with all the ex- |
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isting uncertainties and scatter); therefore, all samples represent the |
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simulation of the behavior. |
analysis file |
An input file containing a complete analysis sequence (preprocessing, |
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solution, and postprocessing). |
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The file must contain a parametrically defined model using parameters |
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to represent all inputs and outputs to be used as RVs and RPs. |
loop |
A single pass through the analysis file. |
or |
In each loop, the PDS uses the values of the RVs from one sample and |
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executes the user-specified analysis. The PDS collects the values for |
simulation |
the RPs following each loop. |
loop |
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loop file |
The probabilistic design loop file (Jobname.LOOP), created automat- |
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ically by via the analysis file. |
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The PDS uses the loop file to perform analysis loops. |
probabilistic |
The combination of definitions and specifications for the deterministic |
model |
model (in the form of the analysis file). The model has these compon- |
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ents: |
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PDS Term |
Description |
•Random input variables (RVs)
•Correlations
•Random output parameters (RPs)
•The selected settings for probabilistic method and its parameters.
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If you change any part of the probabilistic model, then you will gen- |
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erate different results for the probabilistic analysis (that is, different |
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results values and/or a different number of results). For example, |
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modifying the analysis file may affect the results file. If you add or |
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take away an RV or change its distribution function, you solve a differ- |
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ent probabilistic problem (which again leads to different results). If |
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you add an RP, you will still solve the same probabilistic problem, but |
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more results are generated. |
probabilistic |
The database containing the current probabilistic design environment, |
design data- |
which includes: |
base (PDS |
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database) |
• Random input variables (RVs) |
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• Correlations between RVs |
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• Random output parameters (RPs) |
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• Settings for probabilistic methods |
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• Which probabilistic analyses have been performed and in which files |
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the results are stored |
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• Which output parameters of which probabilistic analyses have been |
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used for a response surface fit, the regression model that has been used |
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for the fitting procedure, and the results of that fitting procedure. |
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The database can be saved (to Jobname.PDS) or resumed at any |
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time. The results of a probabilistic analysis are not stored in the data- |
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base but in separate files. The samples generated for a fitted response |
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surface are in neither the database nor in files, because they can be |
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regenerated very quickly. (Files consume disk space, and reading the |
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files requires as much time as regenerating the sample data.) |
mean value |
A measure of location often used to describe the general location of |
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the bulk of the scattering data of a random output parameter or of a |
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statistical distribution function. |
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Mathematically, the mean value is the arithmetic average of the data. |
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The mean value also represents the center of gravity of the data points. |
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Another name for the mean value is the expected value. |
median |
The statistical point where 50% of the data is below the median value |
value |
and the 50% is above. |
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Probabilistic Design Terminology |
PDS Term |
Description |
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For symmetrical distribution functions (Gaussian, uniform, etc.) the |
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median value and the mean value are identical, while for nonsymmet- |
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rical distributions they are different. |
standard |
A measure of variability (dispersion or spread) about the arithmetic |
deviation |
mean value, often used to describe the width of the scatter of a ran- |
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dom output parameter or of a statistical distribution function. |
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The larger the standard deviation, the wider the scatter and the more |
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likely it is that there are data values further apart from the mean value. |
solution set |
The collection of results derived from the simulation loops performed |
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for a given probabilistic analysis. |
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The solution set includes the values of all random input variables and |
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all random output parameters for all simulation loops of a probabilistic |
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analysis. A unique label identifies each solution set. |
response |
The collection of response surfaces derived from a fitting procedure |
surface set |
(regression analysis) and the results derived from using the response |
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surfaces for a probabilistic analysis. |
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A response surface set is identified by a unique response surface label. |
remote host |
A computer in your local area network used to execute a probabilistic |
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analysis in parallel mode. |
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A remote host can have more than one CPU. In parallel processing, |
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you can use multiple CPUs on remote hosts. |
The following figure shows the flow of information during a probabilistic design analysis. The analysis file must exist as a separate entity, and that the probabilistic design database is not part of the model database.
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