**6. Computational Aspects**

With respect to the SI engine campaign (see model settings in Table 8), a single SRM cycle simulation employing the 475 species chemical mechanism using the online chemistry solver takes approximately 19 min to complete on 24 parallel cores (Intel Xeon E5-2687W v4 @ 3.00GHz processors from the year 2016). The simulation of thirty consecutive cycles results in a total CPU time per operating point of 9.4 h. Although these figures are a small fraction of the CPU cost of a RANS 3-D CFD multi-cycle simulation, typical 0-D/1-D simulation frameworks (i.e., based on a multi-zone Vibe combustion model) usually require a few tenths of a second to run and, in some cases (i.e., Mean Value Engine Models MVEMs), real-time simulation capability is easily reached. In addition, considering that the turbulence model calibration procedure [49,58] relies on a few thousands of Genetic Algorithm (GA) driven SRM simulations to find the optimal constants (see Tables 5 and 9), the application of the online chemistry solver with large mechanisms (i.e., more than 150 species) becomes unfeasible

for engine development studies such as driving cycle simulations or engine performance mapping. With respect to the present simulation campaigns, a summary of the CPU times obtained with both solvers and the reported model settings (see Tables 4 and 8) are reported in Table 10.

**Table 10.** Computational performance summary assuming SRM model settings listed in Tables 4 and 8, on an Intel Xeon E5-2687W v4 @ 3.00GHz CPU from the year 2016.


Considering that the SRM with CPV tabulated chemistry solver can be easily run on a single core, as opposed to the online chemistry solver that requires multiple cores per run, one can conclude that the present solver delivers a speed-up of at least three orders of magnitude. The size of the auto-ignition table valid for a wide range of typical engine relevant conditions including EGR variations (between 0 and 40%) requires about 1.0 GB of RAM memory. These level resource requirements allow usage of the SRM with CPV not only on dedicated high-performance computing (HPC) systems but also on modern industry grade laptops. Moreover, given the high degree of physical and chemistry models included in its formulation, engine parameter optimization campaigns can be performed within feasible engineering times. To put the computational results shown in Table 10 in a broader prospective, in Figure 11 are shown the extrapolated computational costs of two relevant engine development simulation campaigns: an engine performance mapping and a WLTP cycle.

**Figure 11.** Comparison between online and tabulated chemistry solver computational performances for (**a**) a full engine performance mapping simulation campaign comprising a total of 38 operating conditions and (**b**) the simulation of the full WLTP cycle (30 min) in terms of combustion and emission simulations only.

Both results have been extrapolated considering only the CPU time needed by in-cylinder combustion model. Additional system components (i.e., intake and exhaust air paths, aftertreatment systems) and their contribution to the total simulation time are not considered. Nevertheless, it can be stated that the tabulated chemistry allows to include detailed chemistry e ffects in a number of applications that are, in most cases, unfeasible for the online chemistry solver.

The developed CPV tabulated chemistry solver was recently applied in two publications for the engine and fuel co-optimization of Diesel and gasoline engines. In the work of Franken et al. [58] a heavy-duty Diesel engine was optimized to find the best set of engine parameters to reduce fuel consumption and NOx emissions at di fferent speeds and loads. The authors reported optimization times of 20 h to 40 h for one operating point. For a single-cylinder, research on gasoline engine optimization campaign was published by Franken and co-authors [48,61]. A dual fuel tabulated

chemistry approach, based on CPV, was used to find the best set of engine parameters in terms of water/fuel-ratios to reduce the knock tendency at a high load operating point and improve the engine efficiency. The optimization times reported in [48,61] are within 10h for one operating point using 4 cores (Intel i7-7820HQ @ 2.90 GHz, from the year 2017) while an equivalent run with the online chemistry solver would have taken several days.
