# Dispersion reduction of a beam modeled in Ansys

Hello,
I present a use case of a beam modeled in Ansys where the goal is to reduce the variability of an output. The model here is a simple beam subjected to a central force and with limited displacements at the extremity as boundary conditions.

The coupling with Ansys is described in detail in this Persalys news. I will focus the analysis only on the deviation output coming from the 1st block and I changed the parameter names in Persalys for the sake of an easier understanding. The parameters inputs and outputs are shown in the figure below.

In Ansys:

In Persalys:

All inputs are defined as Normal random variable with a coefficient of variation of 10%.

Analysis

First I create of design of experiment LHS of size 100 and I evaluate the points with the coupling model. And then I obtain the table of the inputs and outputs as shown below:

If if look at the summary, I can see that the computing time was more than 9000 s being about 2h30 and also that my deviation has a great coefficient of variation, around 45%. I would like to reduce this variability by reducing the variability of some influential inputs.

In order to find on which inputs I must focus on, letâ€™s have a look on the plot matrix in the rank space (with the relevant variables selected). It shows that the variability of the computed deviation (disp\_P5) seems to mainly come from the height h\_P11 and the length L\_P9 which is confirmed by the Spearman correlation values in the dependence tab.

So now it seems that if I reduce the variability of these 2 inputs, I will reduce the variability of my deviation but I want to estimate what will be this new variability. To do that, I will first create a metamodel using the computed design of experiments. Here I build a sparse polynomial chaos model of degree 3 and I obtain a nice curve fitting graph and a R2 > 0.999 and a Q2 > 0.98, so I am confident to use it as a surrogate model for other analyses.

As I chose the polynomial chaos, I automatically get the Sobol Indices (global sensitivity indices) which confirms that the variability of h\_P11 and L\_P9 are responsible of more than 80% of the deviation variability.

I can now export this metamodel as a real model.

Now I will change the variability of these to inputs in the probabilistic model of the metamodel, dividing by 2 their standard deviation: \sigma_{L\_P9} = 0.6 / 2 and \sigma_{h\_P11} = 0.04 / 2.
I can then run a central tendency analysis with Monte Carlo really quickly because I use the metamodel. I did more than 900 calls in less than 1s and now I see that the coefficient of variation of the deviation is reduced to 30%, hence reducing also the risk of failure of the beam when exceeding an ultimate tensile strength.

3 Likes

Hi Antoine,

The use case is interesting: thank you for sharing it!

I would suggest to edit the dialog boxes and to add a red circle to the coefficient of variation so that we can better see where the information you discuss is presented.

One of the critical components here is the metamodel. It may happen that the quality of the result can be improved or decreased using a different polynomial degree. What happens in this case if you use a higher polynomial degree, say 4 or 5? I would expect that the quality increase when the polynomial degree increases, up to a point where the model selection method does not select any new function in the basis.

I often look at the part of variance involved in the polynomial chaos result. This is presented in the â€śResultsâ€ť tab of the PCE:

The table provides a detailed sensitivity analysis. Moreover, we can see which variables are involved and which is the polynomial degree associated with each variable. This may help to see if the polynomial degree was correctly chosen, neither too small, nor too large. For example, if you select a polynomial degree equal to 3 and that the sparse PCE involves polynomials of degree 3, then I would suggest to increase the polynomial degree because it may improve the quality.

Best regards,

MichaĂ«l