What’s driving your problem?
As design engineers, one of the most frequent questions we need to answer is: what are the most important parameters that drive our problem? This is a very important question, most engineering systems are defined by dozens or hundreds of parameters, if we were to design or optimise a system considering all of them at the same time, we would embark on an epic task that would require massive resources, would take too long to complete and would likely end in a poorly understanding of the behaviour and the sensitivities of the system.
A better approach is to first identify what are the real drivers of our problem, and then focus mainly on those to design a system with better performances in less time. I will show you how this can be easily achieved in Nexus.
Correlation and Sensitivity Analyses
One of my favourite features introduced with Nexus 3.1 is the new Analysis View within the Design Space Explorer: Correlation, Sensitivity and Robustness analyses can now be performed in Nexus with few clicks of the mouse.
To show some of these features, consider this very simple problem: an I-section cantilever beam loaded with a vertical force at the free end. Assuming a fixed geometry for beam, we want to focus on 2 design variables: the thickness of the web and the thickness of the two flanges (assumed to be the same for the upper and lower flange). Our design task is to size these thicknesses so that we get the minimum weight while constraining the displacement at the tip. The problem is illustrated below
|Flange Width||10.0 mm|
|Initial Flange Thickness||1.0 mm|
|Web Height||10.0 mm|
|Initial Web Thickness||1 mm|
|Vertical Load||-200 N|
Table 1: Initial dimensions of the I section of the cantilever beam
This is of course just a simple problem, even if we considered both design variables for our optimisation, Nexus would be able to identify the optimal solution in just few iterations. I have chosen this problem for two main reasons: it is a very common problem in structural mechanics and there is a well-known solution provided by the elastic beam theory. In fact, as you probably remember from your Continuum Mechanics course, in such a problem the bending load is carried almost entirely by the flanges. But pretend for now we don’t know how this structure works – this is typically the case for many engineering problems anyway, where we don’t have the luxury of an analytical solution to help us identify what parameters are driving the problem. I am going to show you how you can prepare a virtual experiment to obtain this information using Nexus and its Design Space Explorer.
In this example, we are going to use CalculiX to calculate the tip displacement, while we use a simple expression to calculate the mass of the structure based on the two thicknesses that represent our design variables. Notice that the Y axis of my model is directed upwards, so we will be dealing with negative displacements in this problem. For the Design of Experiment, I am going to use a Latin Square allocation algorithm: this is a very efficient algorithm that allows us to explore the design space with a limited number of design points and is my go-to algorithm most of the times. The Figure below shows the WorkFlow and TaskFlow I have prepared to run the numerical campaign (I am using 100 points in my Latin Square). After running the DoE, we get a table with all the evaluated design points. Let me now show you how to use Nexus Design Space Explore to identify the driving parameter of our problem.
Let’s start with measuring the correlation between our design variables (thickness of the web and thickness of the flanges) and our responses (mass and vertical displacement at the tip). From the Analyses View of the Design Space Explorer, select Correlation Analysis. In the Analysis Wizard, you can select inputs and outputs for the correlation study and a table containing the data source. I am going to use the output context of the Latin Square allocation as my data source for the correlation analysis. The procedure is illustrated below.
Nexus calculates three different correlation coefficients for each variable-output pair: Kendall, Pearson and Spearman. Each of these coefficients answers a different question that might be more or less relevant for a given problem. Regardless of the actual value of each coefficient, however, we can use the results of this analysis to measure the relative strengths of association between the variables and the direction of the relationship. The value of the correlation coefficient varies between +1 and -1.
When the value of the correlation coefficient lies around ± 1, then it is said to be a perfect degree of association between the two variables. To make sense of these charts, remember that we have a negative displacement at the tip, so the correlation coefficient is positive since an increment in thickness result in an increment in the displacement signed value (but a reduction in its absolute value!). As the correlation coefficient value goes towards 0, the relationship between the two variables will be weaker. As you can see from Figure 5, it’s easy to see that both the vertical displacement and the mass of the beam are more strongly correlated to the thickness of the flanges than to the thickness of the web. No surprise here!
Let’s now get a more significant measure of this relationship by calculating the sensitivity of each output with respect to the design variables. Similar to what we did earlier, let’s select the Sensitivity Analysis from the Analyses View. In the Sensitivity Wizard (Figure 6), we have three options on how to run this type of analysis. We can select one from the Eval. Method drop-down menu:
- Full Direct Evaluation: the gradient around the selected point is evaluated running the Workflow (so, in this case, running CalculiX) to calculate the finite difference according to the selected scheme. The baseline point can be selected using the Highlight Data Point from the Nexus main toolbar. If no point is selected (as in this case), default values for the design variables are used. This provides the most accurate evaluation of the sensitivity, but it requires more time, especially when we are dealing with heavy computations in the Workflow.
- Averaging from Tabulated Data: reuses the data already available in a table (for instance from the evaluation table of a previous optimisation) so that Nexus does not have to run a full evaluation of the Workflow. While this option allows to obtain a sensitivity measure really quickly, the result is only an approximation of the sensitivity, and the accuracy depends on the number of points in the table near the point of interest. Use this only when you’re happy with a very rough, preliminary estimation.
- K-Nearest RS from Tabulated Data: uses the points from the selected table to build a response surface using the K-Nearest algorithm and uses the response surface to evaluate the sensitivity. As in the previous case the accuracy of this method depends on the number of available points around the point of interest, but in general it should give better results than the simple averaging while still not requiring a full evaluation. You can see this option as a compromise between the two previous ones.
Since in this case I am dealing with very quick simulations (CalculiX runs in few seconds on my machine), I am going to use a Full Direct Evaluation. You can see in Figure 6 that this result confirms what we saw in the correlation analysis and is in line with beam theory: the vertical displacement is driven almost exclusively by the thickness of the flanges (their contribution is almost 8 times the one of the web). The mass is also more sensitive to the thickness of the flanges, but in this case only by a factor of 2. This means that if we want to optimise this structure, we should focus almost entirely on the thickness of the flanges: it is more efficient to control the vertical displacement at the tip by increasing the thickness of the flanges because, for a moderate increment in mass, we get a significant reduction of the displacement. By the way, this is the design philosophy of the IPE beams: notice that the thickness of the flanges of an IPE beam is, on average, 50% bigger than the thickness of the web. This is the reason.
Using the new analyses panels of the Design Space Explorer you can easily identify the driving parameters of your problem up-front, so you focus only on what really matters. This can give you a significant advantage in your design work: think about the huge benefits of reducing of 50% or more the design parameters you need to consider. Next time you know how!