The Effects of Variation in CAE

The vast majority of all analysis performed is deterministic in that all the input values are constant so if you run the analysis 100 times you will get the same answer 100 times.  In reality if you manufacture 100 items and subject them to the same test you will get 100 different answers, even if the differences are small. 

Incorporating random variation into your analysis and then running it many times to get an idea of the average and standard deviation of key responses can be described as stochastic simulation. 

Why is this important? Take the example of a car body.  In the simplest terms the noise and vibration characteristics of a car can be tied to the natural frequencies.  These frequencies are a function of stiffness and mass, both of which can vary from car to car.  The mass can vary due to fluctuations in material density, thickness of panels and the quantities of e.g. adhesive used to bond components.

The stiffness can be affected by variation in the nominal thickness of each panel as well as the integrity of the spot welds or fasteners.  If you are working with modal frequencies from your car model that assume all the spot welds are of perfect integrity, then your predicted frequencies will always be too high. 

MSC Software provides two ways in which you can account for this.  MSC Nastran has a brute force method where we can apply a coefficient of variation (COV) to materials, properties and loads as well as randomly dropping a proportion of the fasteners and welds to simulate a failed joint. 

Configured in this way, each time Nastran is run it will generate a different set of results.  The example below is a very simple cantilever beam made of four sheet metal components spot welded together at 60 locations. 

We randomised the material stiffness and density as well as the panel thicknesses with a COV of 5% using a Gaussian distribution. The model was then run 4000 times and the first five natural frequencies recovered.  Performing some basic statistics on the results shows the comparison between the single deterministic result and the potential range of values.

An interesting side result of this can be seen by comparing the COV of the results with that of the inputs.  If we vary the inputs by 5% and the output varies by less than 5% then our result can be considered ‘robust’ and intolerant to errors in input.  If, however, the variation is greater we might want to consider spending more time on refining our inputs or our modelling methodology.  This result is valid and useful even if the COV of the inputs is arbitrary – think of it as a transfer function of error.

There are some issues with this method though.  Because the sampling of the design space is completely random then we must run many hundreds or thousands of versions to explore the design space.  We are also not able to recover the input values so understanding cause and effect through correlation between inputs and outputs is not possible.

The second method that MSC Software offer to address this need is from the Adams portfolio.  MSC Adams Insight is developed primarily as a Design of Experiments and Stochastic tool for use with MSC Adams, but does have interfaces to other MSC products.

Adams Insight gives us a number of advantages over the brute force method in Nastran.

• We can be more selective over what variables to randomise
• We can use more than just a Gaussian distribution
• Combined with Parametric Patran we can vary part dimensions
• The integrated Latin Hypercube selection method allows us to sample a large design space with fewer unique analysis runs
• Tools for statistical analysis are built in

Repeating the same stochastic simulation with Adams Insight we generated a design space using 128 samples.  We get the same simple output of mean/max/min etc:

But we can also look at correlation between inputs and outputs:

A correlation coefficient with modulus greater than 0.5 suggests a significant result, meaning that varying the input has a direct effect on the output – a scatter plot would look like this:

In this simple example the only significant correlations are between material stiffness/density and modal frequency as you would expect.  In a real, more complex, example you can use this technique to understand two important things:

• Which inputs are my key outputs correlated with? If I need to change my key output then these are the only ones worth changing. If I need to reduce the variation in this output then spending money reducing the variation (tolerance) on these inputs is the way forward
• Inputs that are not correlated with my key outputs could be areas where I could save some money by using a material with more relaxed tolerances