Tolerance Study of a Diesel Fuel Injector Model using Sensitivity Analysis and Variability Analysis in GT-SUITE
This example demonstrates a tolerance study using GT-SUITE. The application is a detailed diesel injector model, but the concepts and tools described here can apply to a tolerance study applied to any other model or engineering domain. It demonstrates ranking of influential factors, identification and removal of negligible factors, evaluation of probability distributions of output variables, and execution of what-if studies to predict improvements to the output variable distribution. These objectives are accomplished by running a couple different Designs of Experiments (DOE) and performing various analyses in GT-SUITE’s DOE post-processor (DOE-POST).
We consider a detailed diesel injector model where the model map layout and component schematic is shown here:
Although the application details are not the primary focus of this writing, more details about diesel fuel injectors are included here for readers that are interested. In a diesel injector the fuel flow to the cylinder is governed by a large number of parameters. In this type of injector, a solenoid actuates the control valve, which connects the high and low pressure sides of the fuel system. The resulting flow passes through the control chamber, for which an inlet and outlet orifice diameter significantly influences the static pressure drop. Because the injector’s needle, which is close to the cylinder, is still subject to higher pressure, an upward motion of the control rod and needle releases fuel to the cylinder. The transient mass flow, or rate shape, is an important characteristic for the control of the combustion processes quality and is dependent on a large number or parameters. Aside from the actuation duration of the solenoid, mechanical parameters such as spring characteristics and clearances, as well as flow parameters such as restriction diameters at various locations, are important factors and can interact with each other. If the control piston and needle never reach their designed upper stop, the injector is referred to as being operated in the ballistic regime. Because of the floating nature of the needle, this regime is most sensitive to manufacturing tolerances.
For this study 11 known sources of variability are investigated. A single operating point is studied at 1300 bar rail pressure and 0.5 ms energizing time, which puts the injectors in the ballistic regime.
The 11 input variables, or factors, are listed in the table below, along with their known tolerances. It is assumed that all variations conform to normal distributions such that 95% of the variance falls within the tolerances listed, and therefore the standard deviation is half of the tolerance value. For example, the standard deviation for peak current duration is 0.025 ms, such that the real operating duration is between 0.095 and 0.105 ms 95% of the time (+/- 2 standard deviations).
The main output, or response, of interest is the injected fuel mass. Another key output is the cumulative flow through the control valve for each injection event, which represents fuel mass for which energy is expended to push through the control valve, but which doesn’t get injected. Ideally this mass would be minimized. The objectives of this study are the following:
- Use factor screening to rank the 11 factors with respect to their influence on the two responses.
- In accordance with the first objective, identify negligible factors that can be omitted from further analysis.
- Evaluate the expected probability distributions of the two responses caused by the variation in the factors.
- Predict the improvement in the injected mass probability distribution if the two most influential factors can be adjusted.
The third objective requires evaluating the model with Monte Carlo sampling, where the factor values are sampled from their normal distributions. The second objective is important for achieving the third, because as the number of factors increases, the number of Monte Carlo model evaluations will need to increase to obtain confidence in the results. If the number of factors can be decreased, the Monte Carlo model evaluations can either decrease or provide more resolution for the results.
Identifying Important and Unimportant Factors
We approach the first two objectives by performing the Morris Method on the model. The Morris Method is a global sensitivity analysis that utilizes one-at-a-time step changes in factor values. It calculates k different elementary effects in different areas of the factor domain, where an elementary effect, EE, is the change in response, R, that occurs with a step change in a factor, F.
It is computationally efficient and only requires k(F+1) model evaluations, where k is typically 15-30. The results of the analysis get conveniently summarized in a single plot.
For a given factor, the mean and standard deviation of the multiple elementary effects are calculated and placed on a plot of “EE standard deviation” vs. “EE mean”, with one point per factor. Factors whose points lie near the origin (0, 0) have negligible effect on the response. Factors whose points have large “EE mean” have large main effect on the response, and factors whose points have large “EE standard deviation” contain higher-order effects or interaction effects. However, the Morris Method cannot distinguish between higher-order effects and interaction effects, and it cannot identify specific interaction terms.
For this study, k was chosen to be 20, resulting in 20(11+1) = 240 model evaluations, and the lower and upper values in the table were used as variable bounds. The Morris sampling was configured in the DOE Setup (within Case Setup) as shown here:
The Morris sampling is a reduced version of a 4-level full-factorial scheme which includes interior points and not just the lower and upper bounds of the factor ranges. The parallel coordinates plot shown below, which is available in DOE-POST’s Select Experiments page, also illustrates the spread of the sampling for the 11 factors. In contrast, a 2-level full-factorial sampling scheme would result in 2^11 = 2048 points, but would not contain any interior points. A 3-level full-factorial sampling scheme would result in 3^11 = 177,147. As a result, Morris sampling has a computational advantage over 2-level and 3-level full-factorial methods.
After running the DOE simulations, the resulting Morris Method plots are provided in the Analyze Experiments page of the DOE post-processor and are shown below, where the analysis uses standardization and normalization such that the maximum allowed mean and standard deviation are 1.0. Using a maximum y-axis scaling of 1.0, the first observation from these plots is that the standard deviation is very low and negligible for all factors, meaning there are no higher-order or interaction effects within the relatively small ranges over which they were varied.
For easier readability, the y-axis is scaled to better show the individual points.
The plots show that, according to x-axis magnitude which represents the first-order effect, the OutletOrificeDiameter has the largest effect on the injected mass, followed by InletOrificeDiameter and CtrlValvePreload. For the control valve mass loss response, the OutletOrificeDiameter has the largest effect, followed by CtrlValvePreload, CtrlValveLift, and SolenoidCrossSectionalArea.
The full set of tabulated Morris results are provided in this table.
The Morris results are most useful for determining which factors can be excluded from further analysis. One might apply a threshold of 0.05 or 0.1 to the mean and standard deviations of the elementary effects to determine which factors are worth keeping. A threshold of 0.05 will be used for this example. The goal is to exclude factors which have EE Mean less than 0.05 for both responses. Those consist of CtrlValveStiffness, NozzleDiam, and NeedleSpringStiffness. The example will proceed with the remaining 8 factors.
Predict the Probability Distributions of the Responses
To determine the probability distributions of the responses, a new Monte Carlo DOE with 2000 experiments is configured in DOE Setup in GT-ISE. This screenshot shows the configuration which applies normal distributions to the 8 factors using the syntax normdist(mean, standard deviation).
After running the DOE simulations, the Variability Analysis page of the DOE post-processor is used to analyze the results. The resulting probability distributions for the two output distributions of interest are shown below, where they appear to conform to normal distributions. For the injected mass, 90% of the values fall between 10.85 and 14.24 mg, representing a relative spread of 2.9/12.6 = 27%, where 12.6 mg is the baseline injected mass. Relatively little variation is present in the control valve mass loss, where 90% of the values fall between 6.0 and 6.3 mg, with a relative spread of 0.3/6.14 = 4.9%, and as a result, the injected mass will take the remainder of the focus of this study.
The relatively large variation in injected mass is typical of an injector in the ballistic phase, where the needle is not at the upper stop. Injectors in the ballistic phase tend to be dynamic and unstable, as minimal force differences can cause substantial variation in injected mass. This behavior often causes difficulty in calibrating an injector model to measurement data, because these slight variations in input variables can cause a wide range of injected mass values. This variability can discourage the modeler by unnecessarily making them think that either something is wrong with the model or something is wrong with the measurement.
The modeler or design engineer might want or need to determine some worst-case scenarios for the injector. For example, it’s clear that the injected mass profile has a bell-shaped probability distribution that makes it possible for some injectors to inject as little as 8.5 mg or as much as 16.5 mg at the operating point of interest, even though none of the 2000 experiments resulted in these two values. This injector might be installed in tens of thousands of vehicles, and each vehicle would have multiple injectors, one for each cylinder. As a result, hundreds of thousands of this injector might be produced, making it likely for these more extreme injected mass values to occur.
To explore these worst-case scenarios, it is necessary to fit the injected mass data to an ideal distribution. This utility is included in the Variability Analysis page of the DOE post-processor. By enabling this feature, a distribution line representing the best fit is overlayed with the distribution plot, and a table appears that shows the metrics of different distributions. Changing the checkbox selection updates the plot so that the user can see how each distribution appears to fit the data. A few different distribution selections are shown below. The lower the “Error” metric in the table, the better the fit. The table also provides the parameters necessary to reproduce the distribution in the right-most column. The normal and log-normal distributions clearly fit the data best, but there is very little visual distinction between the two. Because of its lower error, the normal distribution will be used.
With the desired distribution selected, a calculator utility can be opened to determine worst-case scenarios. For example, the probability of the injected mass being as little as 8.5 mg is 0.000040, meaning it is likely to occur 40 times out of a million. The probability of the injected mass being as high as 16.5 mg is 1 – 0.999939 = 0.000061, meaning it is likely to occur 61 times out of a million.
Alternatively, the calculator utility can be used to enter cumulative probability values to calculate corresponding response values. For example, it might be desirable to determine the response values for the 0.1% and 99.9% cumulative probabilities. These are 9.4 and 15.7 mg, respectively.
Given the wide range of injected mass values that can be expected at this operating point, it might be necessary to ensure a narrower range of values in the context of injector design. For demonstration, we’ll assume that lower injected mass values are more problematic and that 99% of the time, at least 11 mg of fuel needs to be delivered. In the Variability Analysis page, the injected mass slider bars serve as specification limits. The lower bar is positioned at 11 mg, and the upper-right corner of the page displays the percentage of experiments that are out-of-spec according to these specification limits. 6.6% of the experiments yield injected mass less than 11 mg. The right-side table also provides process capability metrics, and the common metric Cpk is 0.57.
Predicting Improvements to the Response Distribution
The Variability Analysis page within the DOE post-processor serves as a powerful analysis tool for performing fast what-if studies to predict changes in response distributions according to changes in the factor variances. To experiment with changing the factor distributions, one should understand which factors should be focused on. We have already seen from the Main Effects Ranking plot that the OutletOrificeDiameter, InletOrificeDiameter, and CtrlValvePreload have the largest first-order effects on the injected mass. As a result, these three factors would be most appropriate for experimenting with adjusting the response distribution. These Main Effects rankings are indicative of using all the data, or in other words the entire factor domain. However, in some situations, a different factor can have a larger effect on a response in a smaller, specific region of the response. It is worth checking, since we are particularly interested in determining which factors mainly affect injected mass at the lower values.
To accomplish that task, which is known as Monte Carlo Filtering, we view the factor CDF plots in the Variability Analysis page. These plots are provided below, where the in-spec and out-of-spec CDF is provided for each factor. In summary, they show that higher values of InletOrificeDiameter and lower values of OutletOrificeDiameter cause the most significant discrepancy between the injected mass being in-spec and out-of-spec. SolenoidCrossSectionalArea and CtrlValvePrelaod also have a statistically significant effect on this discrepancy, but we will focus on the two diameters since their effects are so much larger than that of the other two.
Because the higher values of InletOrificeDiameter cause the injected mass to violate the specification limit, it is necessary to adjust the mean or nominal value, and attempting to improve its variance will not be sufficient. The same is observed for OuletOrificeDiameter. To test the effect of adjusting their means, we run a new Monte Carlo analysis, but we can do so directly within the DOE post-processor; it is not necessary to run additional DOE experiments from GT-ISE. To run them from the DOE post-processor, a metamodel is needed. It has already been shown that a linear metamodel without interaction effects can be used for the range of the factors being studied. The regression plot showing how well the metamodel’s predicted points align with the original data points is shown here.
After creating a linear metamodel in the Create Metamodels page, a new Monte Carlo experiment set is configured in the Variability Analysis page, where we experiment with adjusting these two diameters by 5 micron. In addition, we use a larger number of 5000 Monte Carlo experiments for each of these case studies, since the metamodel evaluations are very fast and practically free. We test three modifications to the input variables:
- Decreasing the InletOrificeDiameter mean by 0.005 mm.
- Increasing the OutletOrificeDiameter mean by 0.005mm.
- Making modifications 1 and 2 simultaneously.
The plots below compare the original injected mass distribution with those of the three tests, and the table below the plots summarizes the effects of the modifications. Making a modification to either the InletOrificeDiameter or OutletOrificeDiameter is enough to decrease the probability of having the injected mass be less than 11 mg. Using these results, the design engineer can assess making one of these changes in the product.
This study demonstrated using sensitivity analysis and variability analysis to analyze and improve an injector design in GT-SUITE. A sensitivity analysis utilizing the Morris method identified the most influential factors on the two responses of interest, and three factors were found to be negligibly important for both responses and were therefore omitted from further analysis. Then normal distributions were applied to the remaining eight factors using a 2000-point Monte Carlo DOE. After running the simulations, the results were analyzed in DOE-POST, where the response distributions were plotted and observed. The response data was fit to a normal distribution so that worst-case scenarios could be calculated based on cumulative probabilities. Finally, a regional sensitivity analysis was conducted to determine how best to avoid injected mass values at the lower end of the distribution, and three fast what-if studies were conducted to predict improvements in the injected mass distribution.
By Ryan Dudgeon