vk.com/club152685050 | vk.com/id446425943

Chapter 1: Probabilistic Design

Probabilistic design is an analysis technique for assessing the effect of uncertain input parameters and assumptions on your model.

A probabilistic analysis allows you to determine the extent to which uncertainties in the model affect the results of a finite element analysis. An uncertainty (or random quantity) is a parameter whose value is impossible to determine at a given point in time (if it is time-dependent) or at a given location (if it is location-dependent). An example is ambient temperature; you cannot know precisely what the temperature will be one week from now in a given city.

In a probabilistic analysis, statistical distribution functions (such as the Gaussian or normal distribution, the uniform distribution, etc.) describe uncertain parameters. (See the Modeling and Meshing Guide for a description of parameters.)

The Probabilistic Design System (PDS) allows you to perform a probabilistic design analysis. The following topics are available:

1.1.Understanding Probabilistic Design

1.2.Probabilistic Design Terminology

1.3.Using Probabilistic Design

1.4.Guidelines for Selecting Probabilistic Design Variables

1.5.Probabilistic Design Techniques

1.6.Postprocessing Probabilistic Analysis Results

1.7.Multiple Probabilistic Design Executions

1.8.Example Probabilistic Design Analysis

1.1. Understanding Probabilistic Design

Computer models are expressed and described with specific numerical and deterministic values; material properties are entered using certain values, the geometry of the component is assigned a certain length or width, etc. An analysis based on a given set of specific numbers and values is called a deterministic analysis. Naturally, the results of a deterministic analysis are only as good as the assumptions

and input values used for the analysis. The validity of those results depend on how correct the values were for the component under real life conditions.

In reality, every aspect of an analysis model is subjected to scatter (in other words, is uncertain in some way). Material property values are different if one specimen is compared to the next. This kind of scatter is inherent for materials and varies among different material types and material properties. For example, the scatter of the Young's modulus for many materials can often be described as a Gaussian distribution with standard deviation of ±3 - 5%. Likewise, the geometric properties of components can only be reproduced within certain manufacturing tolerances. The same variation holds true for the loads that are applied to a finite element model. However, in this case the uncertainty is often due to a lack of engineering knowledge. For example, at elevated temperatures the heat transfer coefficients are very important in a thermal analysis, yet it is almost impossible to measure the heat transfer coefficients. This means

that almost all input parameters used in a finite element analysis are inexact, each associated with some degree of uncertainty.

vk.com/club152685050Probabi istic Design | vk.com/id446425943

It is neither physically possible nor financially feasible to eliminate the scatter of input parameters completely. The reduction of scatter is typically associated with higher costs either through better and more precise manufacturing methods and processes or increased efforts in quality control; hence, accepting the existence of scatter and dealing with it rather than trying to eliminate it makes products more affordable and production of those products more cost-effective.

To deal with uncertainties and scatter, you can use the Probabilistic Design System (PDS) to answer the following questions:

•If the input variables of a finite element model are subjected to scatter, how large is the scatter of the output parameters? How robust are the output parameters? Here, output parameters can be any parameter that the program can calculate. Examples are the temperature, stress, strain, or deflection at a node, the maximum temperature, stress, strain, or deflection of the model, etc.

•If the output is subjected to scatter due to the variation of the input variables, then what is the probability that a design criterion given for the output parameters is no longer met? How large is the probability that an unexpected and unwanted event takes place (what is the failure probability)?

•Which input variables contribute the most to the scatter of an output parameter and to the failure probability? What are the sensitivities of the output parameter with respect to the input variables?

Probabilistic design can be used to determine the effect of one or more variables on the outcome of the analysis. In addition to the probabilistic design techniques available, the program offers a set of

strategic tools that can be used to enhance the efficiency of the probabilistic design process. For example, you can graph the effects of one input variable versus an output parameter, and you can easily add

more samples and additional analysis loops to refine your analysis.

1.1.1. Traditional (Deterministic) vs. Probabilistic Design Analysis Methods

In traditional deterministic analyses, uncertainties are either ignored or accounted for by applying conservative assumptions.

Uncertainties are typically ignored if the analyst knows for certain that the input parameter has no effect on the behavior of the component under investigation. In this case, only the mean values or some nominal values are used in the analysis. However, in some situations the influence of uncertainties exists but is still neglected; for example, the Young's modulus mentioned above or the thermal expansion coefficient, for which the scatter is usually ignored. Let's assume you are performing a thermal analysis and you want to evaluate the thermal stresses (thermal stresses are directly proportional to the Young's modulus as well as to the thermal expansion coefficient of the material). The equation is:

σtherm = E α T

If the Young's modulus alone has a Gaussian distribution with a 5% standard deviation, then there is almost a 16% chance that the stresses are more than 5% higher than what you would think they are in a deterministic case. This figure increases if you also take into account that, typically, the thermal expansion coefficient also follows a Gaussian distribution.