Turbine Blade (Patran Plug-in)
It will be shown that using the features of the Patran Plug-in, the uncertainties in the material properties, e.g. young's modulus, can be modeled easily. Most importantly, the spatial fluctuations of these uncertainties are taken into account in this example. For this purpose, the random field modeling capabilities of the plug-in have been utilized. More detailed description of this example can be found in 
Definition of the probabilistic model
As mentioned above, in this example, the spatial fluctuations of the young's modulus are to be taken into account. Hence, random fields have been used to represent this property. It is assumed that the structure is made of three different component, namely the upper part consisting of the blade itself, the middle part which combines the upper and lower parts, and finally the lower part, which provides the integration of the blade to the turbine.
Please see below a screenshot of an arbitrary realization of these random fields considering two different correlation lengths. As will be shown in this example, in modeling random fields, correlation length is the most important parameter, which has a substantial affect on the result. Besides the random field representation, the young's modulus will be also analysed using a single random variable for each region, in order to observe the differences in results with respect to the random field case.
Please refer to the Modeling Uncertain Parameters page to see the details on how to create random fields using the plug-in.
Performing the DMCS analysis
Once the groups are defined within PATRAN, which represent the regions of each random field, the user can proceed with defining the required parameters on the GUI. Upon the completion of this step, the user is asked to generate the samples of the random field using the SIMULATE button. Before starting the simulations, also the quantity of interest to be extracted at the end of each simulation should be determined using the SELECT RESPONSE button.
100 Monte Carlo simulations have been performed with the constructed model, where the von Mises stresses and displacement have been stored for each element at every simulation. Of particular interest in this tutorial is the question, as to whether different assumptions with respect to the spatial fluctuations of the properties have a significant impact on the predicted variability. Therefore, two sets of Monte Carlo simulation runs have been conducted, one in which the Young’s moduli of each of the three portions of the blade model are fully correlated, i.e. without spatial fluctuations (using Random Variable option in the GUI), and one in which the Young’s moduli are modeled as random fields, as described above, with a correlation length of b = 8mm (using Random Field option in the GUI)
Results of the DMCS analysis
In this part, the variation observed in the von Mises stress and displacements due to the uncertainties in the young's modulus will be demonstrated. It should be noted that the CoV of the responses have been plotted on the FE model itself in this case, using the visualization capabilities of PATRAN.
The first figure, where the coefficient of variation for the Von Mises stresses are plotted, reveals the significant difference of representing the uncertain Young’s modulus with the random variable model on one hand, and the random field model on the other hand. Indeed, for this response quantity of interest, the variability is essentially negligible in the random variable case (left part), as it remains below 0.1% throughout the considered domain, with the exception of small portions at the surface of the middle plate. For these localized portions, however, the significance of CoV is actually
greatly diminished, since the figure reveals they correspond regions in which the mean stress is close to zero.
In contrast, if the uncertainty in the Young’s modulus is modeled with random field (right part), then the CoV of the von Mises stress is more than an order of magnitude higher and reaches peaks between 1% and 5%. The physical explanation of this behavior has to do with the fact that a spatially varying Young’s modulus causes stress redistributions, whereas in the case of a fully correlated Young’s modulus the latter does not affect the stress distribution, but only the displacement.
In the second figure, where the variability of the vertical displacement, due to the two different models for the uncertainty in the Young’s modulus is shown. Indeed, for this quantity of interest, the variability (CoV) is constantly about 10% throughout the domain, in the case in which the Young’s modulus is modeled as a random variable (left portion). The variability is therefore significantly larger than in the case in which the Young’s modulus is modeled as a random field. As the right portion of the figure shows, in the latter case, the CoV is around 5% over most of the domain. It should be noted that the small CoV’s in the region near the support have been set artificially; this is acceptable because the significance of the CoV in this
region is very limited, since mean displacement is approaching zero.
Finally a reliability analysis has been also carried out within this example, using limit state function g = 2.3MPa − vM, where vM denotes the von Mises stress at the considered location. The following figure shows the local failure probability throughout the domain. In the case in which the Young’s modulus is modeled as a spatially constant random variable, the failure probability is below the level that can be estimated with the utilized number of samples. In contrast, for the uncertainty model which does include the spatial fluctuations, there are two regions near the support, in which the failure probability assumes values in excess of 50%. This shows again the importance of considering the random spatial fluctuations of the material properties in the utilized uncertainty model, since otherwise stress based reliability estimates may be in gross error on the unsafe side.
- M.F. Pellissetti and G.I. Schuëller, Scalable uncertainty and reliability analysis by integration of advanced Monte Carlo simulation and generic finite element solvers, Computers and Structures, 930-947, 2009