Table 2 Design parameters. Fig. 1 A snapshop of a CFD-simulation at the time of injection of the main burst of fuel. Illustrated is the significance of the four spray parameters that are related to the geometry. Tip protrusion is the distance the nozzle in inserted into the geometry, span angle is the angle at which the fuel is injected, and swirl number relates the swirling flow to the radial flow in the combustion chamber at the time of injection. The nozzle hole diameter is, naturally, the hole of the nozzle injecting the fuel. Fig. 2 Inlet mass flow of fuel as a function of crank angle degree. Fig. 3 Mesh of the 72 degree cut of the cylinder. The sections in different colors correspond to the parts of the domain solved one each processor. Table 3 Optimal value of 7 as determined by cross validation for different data sets. difficult to interpolate which reflects in the larger values of 7 for those objec- tives. There is a large jump in the magnitude of the 7 parameter for IMEP Cross validation can be used to estimate the parameter 7 in the RBF expan- sion, which determines the level of approximation. Table 3 list the optimal values of 7 found for the approximations. Cross validations were evaluated at the end of the optimization, 25 iterations before the end and 50 iterations before the end. The general trend is that NO, and particularly soot is more Fig. 4 Visualization of the Pareto Front with constraints on IMEP. The two marked curves correspond to the cases plotted in Figures 11 and 12, respectively. Looking at the results in Table 3 it is clear that we should expect interpo- lation to perform well with IMEP. This can be used to check the CFD data for possibly corrupted data. For instance, a rapid variation of the 1? value of the objective will indicate that something is wrong with the interpolation. Figure 14 shows the result of a validation of the CFD data using the described cross validation technique. Noteworthy is the dip in the /? line for IMEP, suggesting that the cases 64 and 67 may be incorrect, which they indeed were. Fig. 5 IMEP as function of the hole diameter and span angle. The other design variables were set to their middle value. The thicker lines in the contour corresponds to the complete data set, and the other two to all but the last 42 and all but the last 84, respectively. Fig. 6 NOz as function of the hole diameter and span angle. The other design variables were set to their middle value. The thicker lines in the contour corresponds to the complete data set, and the other two to all but the last 42 and all but the last 84, respectively. Fig. 7 Soot as function of the hole diameter and span angle. The other design variables were set to their middle value. The thicker lines in the contour corresponds to the complete data set, and the other two to all but the last 42 and all but the last 84, respectively. Fig. 8 CFD image showing temperature field, fuel droplets and fluid streamlines for a simulation case on the Pareto front where IMEP and soot are bad, but NOz is good. Fig. 9 CFD image showing temperature field, fuel droplets and fluid streamlines for a simulation case on the Pareto front where IMEP and soot are good, but NO; is bad. Note that there are less visible fuel droplets and generally a higher temperature in this case than in the previously shown case. This is due to the more complete combustion in this case. Fig. 10 Evolution of the Pareto curves with constraints on IMEP. Full line: full data set. Dashed line: All but the last 25 simulations. Dotted line: All but the last 50 simulations. Surrogate models based on interpolation. Fig. 11 Design parameters along the selected path of IMEP > 10. Fig. 12 Design parameters along the selected path of IMEP > 12. Fig. 13 Evolution of the Pareto curves with constraints on IMEP. Full line: full data set. Dashed line: All but the last 25 simulations. Dotted line: All but the last 50 simulations. Surrogate models built with the value of 7 taken from table 3. Fig. 14 Results from a two-point cross validation of simulation data for combustion engines.
