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SAE TECHNICALPAPER SERIES 2004-01-1560

An Optimization Study of ManufacturingVariation Effects on Diesel Injector

Design with Emphasis on Emissions

Zhijun Li, Michael Kokkolaras, Dohoy Jung,Panos Y. Papalambros and Dennis N. Assanis

University of Michigan

Reprinted From: Reliability and Robust Design in Automotive Engineering(SP-1844)

2004 SAE World CongressDetroit, MichiganMarch 8-11, 2004

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ABSTRACT

This paper investigates the effects of manufacturing variations in fuel injectors on the engine performance with emphasis on emissions. The variations are taken into consideration within a Reliability-Based Design Optimization (RBDO) framework. A reduced version of Multi-Zone Diesel engine Simulation (MZDS), MZDS-lite, is used to enable the optimization study. The numerical noise of MZDS-lite prohibits the use of gradient-based optimization methods. Therefore, surrogate models are developed to filter out the noise and to reduce computational cost. Three multi-objective optimization problems are formulated, solved and compared: deterministic optimization using MZDS-lite, deterministic optimization using surrogate models and RBDO using surrogate models. The obtained results confirm that manufacturing variation effects must be taken into account in the early product development stages.

INTRODUCTION

Design of powertrain for ground vehicle applications has traditionally focused on performance metrics such as fuel economy, power/torque and emissions. These metrics are related to physical parameters of the powertrain by advanced simulation tools [1].

Geometric dimensions, flow rates and material properties are examples of physical parameters which are the results of manufacturing processes applied to selected materials. The uncertainty existing widely in these processes may bring on property variations of manufactured parts. For instance, materials selection and composition, choice of machining tools, human operations, temperature and humidity in operational environment, can induce change in tolerances, shapes, and surface characteristics.

In the automobile industry, inferior quality and increased costs due to these variations may cause reduced productivity and loss of market share. Thus the

uncertainty in manufacturing process cannot be neglected and has to be considered in the early phases of product development.

One way to address the uncertainty is to obtain a design that is insensitive to manufacturing process variations with some acceptable sacrifice in performance. Traditional approaches account for the variations through the use of empirical safety factors. However, deterministic safety factors do not provide adequate information to achieve optimal use of the available resources to maximize reliability [2]. In Reliability-Based Design Optimization (RBDO), quantities that exhibit variations are treated as stochastic design variables or parameters and probabilistic constraints are formulated. The means of the random variables are then found such that all probabilistic constraints meet a required reliability level.

VARIATIONS IN FUEL INJECTORS

Injection of the fuel into the combustion chamber is one of the most important processes in direct injection diesel engines. Accordingly, injector nozzle design parameters, such as the number of holes and diameter, play an extremely important role in the mixture preparation process. In addition, cavitation phenomena in the injector add another degree of freedom to the complex process of spray breakup and atomization. Hence, variations of injector holes can affect spray and mixing in two ways: variation of main design parameters, such as hole diameter, will result in different flow conditions; while micro variations of the hole surface or variations of the roughness of edges at the hole entrance will lead to different cavitation conditions. The effect of the former might manifest itself as variation in spray penetration, while the latter could have a more significant impact on spray cone angle and atomization.

The flow coefficient or discharge coefficient is one important characteristic of the injectors. The timing and rate of fuel injection into the chamber affect the spray

2004-01-1560

An Optimization Study of Manufacturing Variation Effects on Diesel Injector Design with Emphasis on Emissions

Zhijun Li, Michael Kokkolaras, Dohoy Jung, Panos Y. Papalambros and Dennis N. Assanis

University of Michigan

Copyright © 2004 SAE International

dynamics and combustion characteristics. If the pressure upstream of the injector nozzle can be estimated or measured, and assuming that the flow through each nozzle is quasi-steady, incompressible, and one-dimensional, the mass flow rate of fuel injected through the nozzle is given by [3]

, (1) where Cd is the discharge coefficient, ρ is the density of liquid fuel, ∆P is pressure drop across the injector nozzle and An is the total injection cross-sectional area (TICSA) computed by

. (2) When pressure and density are fixed, the flow rate is affected by the product of the discharge coefficient and the TICSA.

The discharge coefficient represents the efficiency of the nozzle and thus has significant effects on emissions. On the other hand, it is a key variable which reflects surface quality characteristics. The discharge coefficient depends on the manufacturing process, so the effects of its variation are studied in this paper.

Reliability-based design optimization methodology helps to make design decisions such as: how much should the decision-maker improve the surface conditions of a fuel injector nozzle to make the system meet the required reliability level, reduce the emissions and not deteriorate the engine performance or fuel efficiency.

MODEL DEVELOPMENT

The developed physics-based models are capable of capturing the effect of manufacturing quality and surface finish on engine performance, fuel economy and emissions.

MZDS (Multi-Zone Diesel engine Simulation) has been developed based on the multi-zone spray combustion model with quasi-dimensional approach, which has the capability of predicting engine performance and NO/soot emissions [4]. MZDS was intended for use in fuel injector design optimization studies. However, increased computational requirements prevent integration of MZDS with design optimization, which requires a number of iterative executions of engine simulation code to find an optimized solution. Therefore, a reduced version of MZDS (MZDS-lite) was developed by introducing aggregated spray zone concept for heat release rate and emissions models to improve computational efficiency of MZDS. MZDS-lite is much faster than MZDS (in the order of 100 times) while maintaining its capability of predicting engine emissions.

Sub-models used in MZDS include spray penetration, air entrainment, fuel evaporation, combustion and emissions models. Spray penetration is calculated using

empirical correlations and air entrainment is calculated based on a relatively simple momentum conservation concept in conjunction with spray penetration. Therefore, these two sub-models do not cause any significant computational load. Since the concentration of the air in the zones are critically important for accurate calculation of combustion process and NO and soot emissions formations, spray penetration and air entrainment are decided to be calculated for each zone defined as in Figure 1.

Figure 1: Division of fuel spray into parcels and zones

Combustion and emissions formation sub-models are those that causes computational load. Especially NO formations sub-model based on the extended Zeldovich model requires significant amount of computation time during the process of calculating equilibrium concentrations of 14 different species. To reduce the computational time caused by these sub-models, aggregated spray zone concept is introduced and only three zones are considered for the calculation of spray combustion and emissions formation processes. Conservation equations of mass and energy are applied to air zone and aggregated spray zone to calculate thermodynamic properties. The definition of zones for spray combustion and pollutant formations is illustrated in Figure 2.

Figure 2: Definition of zones for spray combustion and pollutant formations

New features were added to MZDS-lite for the flexibility of fuel injection control. With new features, user can choose the way of controlling fuel injection by specifying fuel injection rate profile, injection duration, or injection pressure with fuel amount depending on the problems that the user want to solve.

MZDS-lite is associated with the engine configuration presented in Table 1. The fuel injector used in the experiment is 200×8 (200µm diameter holes with 8 holes in the nozzle).

Table 1: Heavy duty diesel engine specifications

Displacement Volume (liter) 12.7 Bore (mm) 130

Stroke (mm) 160 Compression Ratio 15 : 1

Rating speed/power (rpm/kW) 2100/350

MZDS-lite has a number of input parameters. In this pa-per, we are interested in the injector design variables as-sociated with the geometry and surface properties of the injector nozzle. Therefore, the injector hole diameter D and the number of injector holes have been chosen to be treated as design variables. However, the effect of microscopic variation of the hole surface or the variation of the roughness of edges at the hole entrance cannot be treated as explicit input parameters. Instead, the effect of those microscopic variations is represented by the discharge coefficient Cd.

MZDS-lite can predict the following quantities: Indicated Mean Effective Pressure (IMEP), Brake-Specific Fuel Consumption (BSFC), and two emission types, NO and soot, as shown in Figure 3.

Figure 3: Inputs, parameters and outputs of MZDS-lite

A “7-Mode” simulation is defined to represent a wide range of engine operating conditions. All outputs of interest are computed based on a weighted average of seven modes. Engine speed and load are specified for each mode in Table 2. Different weights are assigned for different modes according to their operating time length in a specific driving cycle.

Table 2: Definition of “7-Mode” simulation parameters

Mode RPM Load (%) Weight (%) 1 900 50 13 2 1200 50 6 3 1500 50 7 4 1800 25 17 5 1800 50 21 6 1800 75 21 7 1800 100 15

SENSITIVITY STUDY

A number of numerical experiments were performed to study the sensitivity of the outputs to the design variables. Specifically, simulations were executed for varying design variable values, and the output values were recorded. Figures 4 - 7 depict variations of engine performance, fuel efficiency and two emission types, respectively, when Cd is 0.8. The figures with other values of Cd are similar. Relative variation ranges are summarized in Table 3. It can be seen that within the range of design variable values, NO and soot emissions exhibit higher variations than engine performance and fuel efficiency.

Table 3: Output variation

OUTPUT IMEP BSFC NO SOOT Variation 35% 37% 90% 90%

Figure 4: IMEP variation

Figure 5: BSFC variation

Figure 6: NO variation

Figure 7: Soot variation

INDIVIDUAL OPTIMIZATION STUDIES

In this section, we formulate and solve four individual op-timization problems to identify tradeoffs between conflict-ing objectives.

There are three design variables: the number of injector holes, injector hole diameter and discharge coefficient.

Table 4: Lower and upper bound of design variables

Design Variable Lower Bound Upper Bound Number of holes 6 10

Diameter(m) 1.5×10−4 2.5×10−4 Cd 0.5 0.8

The appropriate ranges of the first two design variables in Table 4 were determined by the bounds on TICSA. TICSA must lie between 0.1mm2 and 0.4mm2 approxi-mately, so that the total injection duration time or pres-sure does not exceed the appropriate limits: when the TICSA is too small, unusually high injection pressure oc-curs to inject the amount of fuel specified by the fuel in-jection rate. The normal injection pressure is typically in the order of 100MPa, depending on the operation condi-tions. If the injection pressure is being kept fixed, too small TICSA causes too long injection duration: since

the injection pressure cannot be increased, the only way to inject the necessary amount of fuel is to inject for longer period. The injection duration becomes too long in some cases and the combustion cannot be completed. The lower and upper bounds of the discharge coefficient were chosen based on experience.

Individual optimization problems are formulated as:

(3)

where f represents each objective function: IMEP, BSFC, NO emission and soot emission.

The DIvided RECTangles (DIRECT) optimization algorithm was used [5]. DIRECT can solve mixed integer nonlinear programming problems and locate global minima efficiently without derivative information when the number of variables is small. The obtained results are summarized in Table 5. The following observations can be made based on the obtained results:

1) Performance and fuel efficiency can be improved simultaneously. This is an interesting result that shows optimal injector design can improve both engine performance and fuel efficiency. This may be not possible in many other engine subsystem designs, where these two objectives are conflicting.

2) Tradeoff exists between emission types. Both the optimal results and comparison of Figure 6 and Figure 7 show that NO emission cannot be decreased without increasing soot emission, and vice versa.

3) NO emission cannot be decreased without deteriorating engine performance and fuel efficiency. IMEP and BSFC values did not change much while minimizing soot emission. However, at minimal NO emission, IMEP and BSFC both have been deteriorated dramatically.

This last tradeoff was expected. The timing and rate of fuel injection into the chamber affect the spray dynamics and combustion characteristics. When the pressure and the density are fixed, the flow rate is affected by the product of the discharge coefficient and the TICSA. If this product increases, the flow rate increases. The combustion process is enhanced, which results in improved engine performance, fuel efficiency, and lower soot emissions. However, NO emission cannot be minimized.

Table 5: Individual optimization results

Objective Function max IMEP min BSFC min NO min SOOT Number of Holes 10 10 6 10

Hole Diameter (m) 2.37×10−4 2.35×10−4 1.50×10−4 1.87×10−4 Discharge Coefficient 7.85×10−1 7.99×10−1 5.00×10−1 8.00×10−1

IMEP (atm) 14.31 14.31 9.59 13.66 BSFC (g/kWhr) 170.44 170.31 263.59 181.59

NO (g/kWhr) 18.32 17.84 1.65 9.82 SOOT (g/kWhr) 0.019 0.017 1.294 0.013

MULTIOBJECTIVE DETERMINISTIC OPTIMIZATION USING MZDS-LITE

Based on the results obtained by solving the individual optimization problems, it was decided to focus on the emission types and solve the associated multi-objective optimization problem. The goal is to determine the hole diameter, number of holes and discharge coefficient, and improve emissions without deteriorating the engine performance and fuel efficiency. The multi-objective problem is formulated for minimizing both emission types subject to engine performance and fuel efficiency constraints.

(4)

Baseline values for IMEP and BSFC are computed using the baseline values of input design parameters (Table 6) and engine simulation models.

The multi-objective optimization problem is solved by setting constraint bounds for soot emissions and minimizing NO emissions to generate set of Pareto points. Design variable and output values of the obtained Pareto points are presented in Table 7. The maximal relative variations between optimal design variables and outputs are also shown in the table. It can be seen that IMEP and BSFC variation is relatively small for all Pareto points. The associated Pareto curve is depicted in Figure 8.

Figure 8: Pareto curve for emissions using MZDS-lite

Table 6: Baseline values for design variables and outputs

Number of Holes Diameter Cd NO SOOT IMEP BSFC 8 2×10−4 0.7 10.65 0.0427 13.273 187.684

Table 7: Design variable and output values of the obtained Pareto points using MZDS-lite

Number of Holes Diameter Cd NO SOOT IMEP BSFC 10 1.870×10−4 0.799967 9.8167 0.012587 13.660 181.586 10 1.757×10−4 0.799977 7.9971 0.012908 13.465 184.080 10 1.671×10−4 0.799942 6.6874 0.013443 13.557 182.678 10 1.659×10-4 0.799936 6.6340 0.013868 13.470 183.017 10 1.649×10−4 0.799977 6.4720 0.014277 13.402 184.722 10 1.642×10−4 0.799382 6.3631 0.014661 13.291 186.524 10 1.585×10−4 0.799977 5.5365 0.015153 13.314 185.120 10 1.578×10−4 0.799977 5.4229 0.015485 13.314 185.599 10 1.570×10−4 0.799977 5.2799 0.016008 13.332 185.796 10 1.566×10−4 0.799946 5.2255 0.016174 13.278 186.688 10 1.566×10−4 0.799977 5.2218 0.016171 13.273 186.753 10 1.522×10−4 0.799992 4.6509 0.017296 13.273 185.889

Variation 0% 18.6% 0.08% 52.6% 37.4% 2.9% 2.7%

SURROGATE MODELING

MZDS-lite is fast to be used in the design process. But surrogate models still need to be developed because of the numerical noise of MZDS-lite. This noise prohibits the use of gradient-based optimization methods, such as RBDO, since gradients estimated using finite differences can be inaccurate.

Surrogate modeling helps to smooth out the numerical noise and save computational cost as well. Surrogate models can be generated over the whole design space. However, it is preferred to generate the surrogate models in the neighborhood of a candidate design point [6].

This paper uses quadratic response surface modeling tool. For three inputs (x1, x2, x3), the equation of a quadratic response surface is:

(5)

where y is the response value, x terms are the design variables, b terms are unknown polynomial coefficients. Given a set of response values and associated values of design variables, the polynomial coefficients can be determined. Quadratic polynomials are used due to their computational simplicity and their convenient algebraic form. Quadratic response surface is generated by rstool from Matlab Curve Fitting Toolbox.

The data sampling method for quadratic response surface modeling is design of experiments (DOE) developed for use in planning physical experiments. A full factorial design of experiments was performed. The surrogate models are developed in the neighborhood of the optimum design based on Table 7, where the discharge coefficient always goes to the upper bound and hole diameter favors smaller value.

To validate the surrogate models, Root Mean Squared Error (RMSE), also known as the fit standard error and the standard error of the regression, has been used as an absolute criterion and is defined as:

(6)

where n is the number of response values, m is the number of fitted coefficients, iy is the response value,

ˆiy is the predicted response value.

Table 8 summarizes the sample points and area used for two different surrogate models and corresponding RMSE. Surrogate models (A) for soot emission exhibit big error and cannot be trusted. From the sensitivity study, the error may be caused by the sharp increasing value change for soot emission in the area which contains both small hole diameter and small number of holes stated by Figure 7. For deterministic optimization using MZDS-lite the number of holes always goes to the upper bound, so surrogate models (B) are developed with more restricted value range for each design variable. The error of surrogate model for each response has been reduced compared with that of surrogate models (A). Especially for soot emission, its goodness of fit has been improved over 20%. Surrogate models (B) are used for the following optimization studies. The choice of sample points and sample area is very important in the development of surrogate models. It depends on the information from sensitivity studies and optimization studies. The surrogate models are adjusted in order to increase the fidelity and reflect the engine simulation models well enough in the neighborhood of the optima.

Table 8: Surrogate models used for each response

Surrogate Surrogate Models (A) Models (B)

Sample Points 4500 1638

Cd [0.5, 0.8] [0.75, 0.8] Number of

Holes [6, 10] [8, 10]

Hole Diameter (m) [1.5×10−4,2.5×10−4] [1.5×10−4, 2.0×10−4]

BSFC RMSE 2.3731 1.8775 Error(%) 1.3% 1.00%

IMEP RMSE 0.096489 0.076476 Error(%) 0.7% 0.58%

NO RMSE 0.33481 0.052394 Error(%) 3.1% 0.49%

Soot RMSE 0.010467 0.0015704 Error(%) 24.5% 3.68%

MULTI-OBJECTIVE DETERMINISTIC OPTIMIZATION USING SURROGATE MODELS

Figure 9 shows the Pareto curve using surrogate models (B) can approximate that using MZDS-lite within a certain degree of accuracy. The multi-objective optimization problem is solved by weighting method. Table 9 presents the obtained Pareto points. These results are verified by constraints method, which is minimizing NO emission while setting the constraint bound for soot emission.

The main conclusion that can be drawn by solving two deterministic multi-objective optimization problems is that the number of holes variable has reach its upper bound and the hole diameter variable is just about reaching its lower bound. This points to the fact that injectors with many small holes promote fuel air mixing. As expected, the discharge coefficient should be maximized.

Figure 9: Pareto curves for emissions using MZDS-lite and surrogate models (B)

Table 9: Design variable and output values of the obtained Pareto points using surrogate models (B)

w1:w2 (NO:SOOT)

Number of Holes Diameter Cd NO SOOT IMEP BSFC

0:1 10 1.755×10−4 0.7999975 8.044 0.012765 13.474 184.5870.025:0.975 10 1.748×10−4 0.7999975 7.926 0.012771 13.466 184.692

0.05:0.95 10 1.740×10−4 0.7999975 7.803 0.0112790 13.458 184.8020.075:0.925 10 1.732×10−4 0.7999975 7.676 0.012824 13.449 184.920

0.1:0.9 10 1.723×10−4 0.7999975 7.543 0.012875 13.439 185.0460.125:0.875 10 1.714×10−4 0.7999975 7.404 0.012946 13.428 185.181

0.15:0.85 10 1.705×10−4 0.7999975 7.259 0.013039 13.417 185.3260.175:0.825 10 1.695×10−4 0.7999975 7.107 0.013157 13.405 185.481

0.2:0.8 10 1.684×10−4 0.7999924 6.948 0.013304 13.392 185.6490.233:0.767 10 1.670×10−4 0.7999975 6.725 0.013552 13.373 185.8920.267:0.733 10 1.654×10−4 0.7999975 6.487 0.013871 13.352 186.162

0.3:0.7 10 1.637×10−4 0.7999975 6.232 0.014276 13.328 186.4641:0 10 1.599×10−4 0.7999997 5.678 0.015389 13.273 187.166

Variation 0% 6.7% 0.0006% 46.4% 17.1% 1.5% 1.4%

RELIABILITY-BASED DESIGN OPTIMIZATION STUDY

The design variables are the same as those in the deterministic optimization design. In RBDO problem, the number of holes and hole diameter are deterministic design variables. The discharge coefficient is chosen as the random design variable because it reflects the effect of microscopic variation of the hole surface or the variation of the roughness of edges at the hole entrance. It is assumed that the discharge coefficient has normal distribution which can be expressed by two main parameters: mean and standard deviation. The standard deviation and the target reliability have to be assumed reasonably based on the engine performance and manufacturing process. Here, the standard deviation, σ, is 0.01 for the discharge coefficient. The target reliability, βt, is 3. Probability of failure is Pf = Φ(−βt)=0.0013, while Reliability = 1 − Pf = 0.9987.

According to the results obtained by solving the deterministic optimization problems, it was decided to focus on the emission types and solve the associated multi-objective reliability-based design optimization problem. The goal is to determine the hole diameter, number of holes and discharge coefficient, and improve emissions without deteriorating the engine performance and fuel efficiency. The multi-objective RBDO problem is formulated for minimizing both emission types subject to engine performance and fuel efficiency.

(7)

RBDO problem solution involves two loops. In the outer loop, specific values are assigned to each variable according to the optimizer algorithms. Random variables are represented by their mean, while standard deviation is kept constant. Objective function is evaluated at this point while probabilistic constraints are examined by the inner loop, where the most probable point (MPP) is found. Only the value for random variables will be

changed in inner iterations while deterministic design variables are kept constant. The procedure can be summarized as following

(1) Set k =0, given target reliability and distribution in-formation for random variables, select convergence criteria.

(2) The optimizer updates design point in the design space based on the optimizer algorithms.

(3) (Constraints Evaluation Process)

(3-1) Transform all the random variables to equivalent uncorrelated standard normal variables.

(3-2) Probabilistic constraints are evaluated at the MPP found using Hybrid Mean Value (HMV) method [7].

(3-3) Check if both probabilistic and deterministic con-straints are satisfied.

(4) k = k + 1, check convergence criteria, if the conver-gence criteria hold, then stop; otherwise, go to Step 2.

Pareto points are obtained using surrogate models (B). Weighting method is applied to solve the multi-objective RBDO problem. The associated Pareto curve is depicted in Figure 10 and compared with the result of deterministic optimization. Mean of design variables and output values of the obtained Pareto points are presented in Table 10. The maximal relative variations of optimal design points and outputs are also shown in the table. The result was verified by constraints method.

Figure 10: RBDO Pareto curve using surrogate models (B)

Table 10: RBDO design variable and output values of the obtained Pareto points using surrogate models (B)

w1:w2 (NO:SOOT)

Number of Holes Diameter Cd NO SOOT IMEP BSFC

0:1 10 1.778×10−4 0.7666641 8.953 0.016923 13.443 185.0190.025:0.975 10 1.770×10−4 0.7666641 8.828 0.016929 13.435 185.130

0.05:0.95 10 1.762×10−4 0.7666641 8.696 0.016950 13.426 185.2500.075:0.925 10 1.754×10−4 0.7666641 8.561 0.016986 13.417 185.374

0.1:0.9 10 1.745×10−4 0.7666641 8.420 0.017041 13.407 185.5080.125:0.875 10 1.735×10−4 0.7666641 8.272 0.017116 13.396 185.652

0.15:0.85 10 1.726×10−4 0.7666641 8.118 0.017215 13.384 185.8060.175:0.825 10 1.715×10−4 0.7666641 7.956 0.017341 13.371 185.972

1:0 10 1.670×10−4 0.7666658 7.710 0.017576 13.352 186.233 Variation 0% 6.1% 0.0002% 13.9% 3.7% 0.6% 0.7%

It is interesting to see how much the reliability has been enhanced. Points with the same number of holes and diameter were chosen from the Pareto curves of RBDO design and deterministic optimization design. A Monte Carlo study simulated was used to validate the obtained results. According to [8], when Pf =0.0013, 300,000 evaluations are enough to guarantee an error that is less than 5%. The reliability of the baseline design is also investigated.

Table 11 shows the design variable and output values of the chosen Pareto points, and the corresponding probability constraints reliability as computed by the Monte Carlo method. Given the priority that the number of holes and diameter are the same for both optimization studies, the variations in discharge coefficient lead to different design decisions. The probabilistic constraint P{Cd≤0.8} plays an important role. The upper bound for discharge coefficient expresses manufacturing capabilities [9]. Figure 11 illustrates the effect of deburred inlet corner, which is also called as nozzle inlet chamfer radius, on the discharge coefficient. The discharge coefficient can be increased as much as 0.84 from 0.6, which is the case without corner rounding. Since the upper bound in this case study (0.8) is

artificially set by us, it can be substituted by any other appropriate number. The accuracy of surrogate models is substantial for the reliability assessment. If the surrogate models exhibit large estimation errors, it may be meaningless to consider the uncertainties related to manufacturing processes.

Figure 11: The effect of injector nozzle chamfer radius on discharge coefficient

Table 11: Comparison of reliability

Baseline Deterministic RBDO Design Design Design

Number of Holes 8 10 10 Diameter 2×10−4 1.714×10−4 1.715×10−4

Cd 0.7 0.7999975 0.7666641 NO 10.65 7.404 7.956

SOOT 0.0427 0.012946 0.017341 IMEP 13.273 13.428 13.371 BSFC 187.684 185.181 185.972

Probability constraints reliability P{ Cd ≤ 0.8} 1 0.4991 0.9996

P{IMEP ≥ IMEPBaseline} 0.6530 1 1 P{ BSFC ≥BSFCBaseline} 0.8225 1 1

CONCLUSION

The case study on fuel injectors has shown that the effects of manufacturing variations on powertrain design decisions are prominent. The uncertainties in manufacturing processes cannot be neglected and need to be considered in the early phases of product development.

The results obtained using deterministic and probabilistic optimization formulations demonstrated the need for RBDO to improve not only performance but also reliability.

ACKNOWLEDGMENTS

This work has been partially supported by the US Army TACOM and General Motors Corporation through the Dual Use Science and Technology Program. This support is gratefully acknowledged.

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