A reliability analysis with a Monte Carlo approach using modeFRONTIER

Introduction
The following article shows how to use in a profitable way the Design Of Experiments tools available in modeFRONTIER to perform a reliability analysis.
It is clear that, knowing the quality level of a product or a process in advance, allows the designer to eventually modify the actual design, looking for an optimized configuration to warrant the desired quality level. Obviously, it is mandatory to have a reliable and accurate virtual model in order to predict the response of the system and check the quality level that it can reach.
The exposed methodology can be used, for example, when dealing with a DFSS project (Design For Six Sigma) to design high quality products, or, more generally, whenever a new design has to be performed using a probabilistic approach.
Let us imagine that the design of a new washing machine requires checking the reliability of a sealing gasket. The designer suspects that the actual design is not able to reach a six sigma quality standard because the probability that the water drops out or the gasket brakes is too high.
The possible reasons of this undesired situation have been found in bad material properties and in a too high water pressure. In order to check if this statement is true or not and explore in detail the problem, it is necessary to simulate the gasket in operating conditions, measure the deformed shape and the stress state and verify if they satisfy a Six Sigma quality level.
Obviously, this could be done with an experimental campaign, but it is faster and cheaper to reproduce this investigation virtually, by means of some numerical simulations.
For sake of simplicity, the gasket has been modeled as an infinite circular holed cylinder made of a linear and elastic material, subjected to a uniform internal pressure. For this problem there is the analytical solution, which can be easily computed in terms of geometrical quantities, material properties and loads.
The gasket has been supposed to brake when the maximum von Mises stress reaches an ultimate value, which simply is a material characteristic.
The following equations report the maximum radial displacement (ur) and the maximum non-zero stress components (σr and σt) and the corresponding von Mises stress (σvM) inside the cylinder. The internal ad external radius are a and b respectively, E and n are the elastic engineering material properties while p is the internal pressure.

An excel file collects all these equations and it is used to simulate the gasket response.
It is clear that this simple model could be substituted, without any variation in the approach to the problem, with a more realistic one, where the actual geometry of the gasket and the loads are modeled.
The gasket material properties (Young modulus, Poisson coefficient and ultimate stress) have been supposed to be normally distributed, with the following means and standard deviations.
These values could be known thanks to an experimental investigation or simply given, for example, by the material supplier.
Also the internal pressure is normal distributed, with mean of 5 [MPa] and standard deviation 1.1E-1. On the contrary, the internal and external radius have been supposed to have a deterministic dimensions of 30 [mm] and 40 [mm] respectively.
In order to work properly (that is, to avoid water drops and premature failures) the gasket has to exhibit a radial deformation less than 1.5 [mm] and, obviously, a von Mises stress less than the material ultimate stress. In a Six Sigma context, these values could be seen as the Upper Specification Limits (USL) for the deformed shape and for the stress state.

Table 1: The Gaussian probabilistic characterization of the material properties.

The simulation should clarify if the gasket is able to satisfy a Six Sigma quality level, with reference to the above mentioned USLs, taking into account the material and load variations.
In this example there are not any lower specification limits, but of course this does not yield to any loss of generality.
Obviously, more complex systems can be considered, taking advantage of a wide variety of direct connection nodes to third-parties simulation software available in modeFRONTIER.

Reliability analysis
The workflow depicted in the following figure collects all the ingredients needed to simulate the gasket response. On the left the input variables (Young, Poisson, stress and Pressure), while on the right the output variables (Deformation, Stress and Result) have been placed. They have been connected to the Excel file which computes the response of the gasket and, finally, the workflow has been completed with the DOE and scheduler nodes.

Figure 1: The modeFRONTIER workflow ready to run and a detail of the DOE panel.

The output variable Result assumes the 0 value if at least one of the two checks (deformation and stress) are not satisfied and it allows us to monitor the number of failures out of the total number of designs.

Using the DOE node it is possible to generate a certain number of designs using the Latin Hypercube or Monte Carlo technique and run the simulation.
Once the run has finished, it is interesting to plot a histogram of the Deformation and Stress output variables starting from the design table. The aim is to understand how these two quantities are distributed and check the reliability of the gasket.

Table 2: The limits that the gasket has to satisfy to meet a Six Sigma quality level.

If we analyze the Deformation distribution in a Six Sigma context, it is possible to compute the upper acceptable limit (as mean value plus six times the standard deviation) to obtain 1.485 [mm].
It is clear that this value is less than the specified limit of 1.5 [mm].
On the contrary, if the stress on the gasket and ultimate stress distributions are compared, it is possible to understand that the desired reliability is not guaranteed. The upper limit for the stress distribution (23.554) is larger than the lower limit of the ultimate stress (21.066); therefore the probability to have a failure of the gasket is higher than 3.4 DPMO, corresponding to a Six Sigma quality level.

Some statistical analyses
With a scatter matrix plot we can find out if there is any linear relation between variables. It is clear that, in this case, the Pressure, the Deformation and the Stress have very strong positive relations. This was expected, in view of the nature of the equations describing the system behavior. As it can be seen, only linear (positive or negative) relations can be detected with the scatter matrix, but this does not exclude that other non-linear relations are present between variables. To understand this aspect, note that there is a clear hyperbolic relation between the radial displacement and the material Young modulus, which could be never identified only by means of a scatter matrix.
The correlation coefficient is, as expected, closed to zero, when couples of input variables are considered; this means that the DOE generation algorithm has worked properly, generating independent designs (from a linear point of view, at least).

Figure 2: Comparison between the distributions of the stress inside the gasket and the material ultimate stress.

Figure 3: The correlation matrix highlights the linear relations between the variables.

As shown before, the gasket does not satisfy the Six Sigma quality level because the maximum stress is too high. It is necessary to identify the variables which influence the response of the system in terms of stress and try to control their values (or distributions), in order to improve the gasket quality.
As the t-Student analyses suggest, together with the Scatter matrix, the input variable which seems to more influence the stress (the largest size effect) is the pressure. The significance of the test is extremely small (practically zero) and this means that the size effect can be statistically accepted as a real effect (and not due to random causes) with a high level of confidence.

For this reason, the designer should try to understand if the operating pressure can be reduced or better controlled (with a reduction of the standard deviation) without any loss of performance; obviously, one other possible solution is also to improve the material ultimate stress, increasing its mean value or reducing the dispersion in the attempt to satisfy the Six Sigma requisites.
It is interesting to note that the result output variable could always assume the value 1 (both checks are satisfied) if a low number of runs is performed. This could lead the designer to the wrong conclusion that a high quality level is assured by the gasket; obviously, this is not true, as shown before. When dealing with Six Sigma, a very low number of discards is tolerated (3.4 over a million) and, therefore, a very large number of simulations should be run to really check if the above mentioned quality level is actually really reached by the system.

Figure 4: The t-Student analysis allows identifying the input variables which most influence the outputs.

Conclusions
This article shows how to plan a virtual campaign of experiments to check if a system is able to provide sufficient reliability and quality warranties, in a Six Sigma context.
A very simple example has been proposed, to describe the procedure which should be used in these cases. Obviously, more complex and closer-to-reality problems could be solved using the same technique.
The statistical tools available in modeFRONTIER can be employed to improve the knowledge of the system under exam, achieving a better design.