Paper # 0701C-0326 Topic: Internal Combustion and gas turbine engines 8 th U. S. National Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 19-22, 2013 Towards Accommodating Comprehensive Chemical Reaction Mechanisms in Practical Internal Combustion Engine Simulations Mandhapati Raju 1 Mingjie Wang 1 Meizhong Dai 1 Shaoping Quan 1 Peter Kelly Senecal 1 Sibendu Som 2 Matt McNenly 3 Daniel L Flowers 3 1 Convergent Science Inc., Middleton, WI 53562, USA 2 Argonne National Laboratory, Argonne, IL 60439, USA 3 Lawrence Livermore National Laboratory, Livermore, CA 94551, USA The use of detailed chemical mechanisms, although preferable, poses an enormous challenge for practical engine simulations. To accommodate the use of comprehensive detailed mechanisms various acceleration strategies have been implemented into the CONVERGE CFD solver. The different strategies involve (a) dynamic mechanism reduction, (b) adaptive zoning, (c) use of a preconditioned iterative ODE solver, and (d) strategies for reducing the CPU time needed for species transport. The feasibility of using comprehensive detailed reaction mechanisms in practical engine simulations is tested. As a test case, a single-cylinder Diesel engine is simulated using a detailed mechanism for n-heptane developed by Lawrence Livermore National Laboratory (LLNL). The reduction in CPU time for each of the reduction strategies along with the accuracy of the simulated results is reported. 1. Introduction Using detailed chemical reactions mechanisms for engine and gas turbine applications is formidable. Typically mechanisms consisting of less than 50 species or so are used to simulate within a reasonable amount of time. However, the sizes of the detailed chemical reaction mechanism are quite large and impractical for three dimensional simulations [Lu and Law, 2009]. Different strategies are being applied by researchers to reduce the computational time significantly. The different strategies for reducing the simulation time are (i) reducing the number of combustion calls using an adaptive zoning approach [Aceves et al. 2000; Babajimopoulos et al., 2005; Raju et al., 2012a] (ii) reduce the size of system using a pre-reduced mechanism [Raju et al., 2012b] or by using dynamic mechanism reduction [Raju et al., 2013] (iii) reduce the computational time of each combustion call using methods like analytical evaluation of Jacobian, exploiting the sparsity of the Jacobian matrix and using special techniques like preconditioned iterative solver (as will be demonstrated in this paper). When large mechanism is involved, the transport of all the species could also significantly increase the computational time. New strategies have to be developed to reduce the time for transport of all the species. This paper demonstrates the combination of all the above strategies for engine simulations giving the capability to run full detailed chemical reaction mechanisms within reasonable computational times. Some of the strategies like adaptive zoning and dynamic mechanism reduction were already demonstrated elsewhere. The use of preconditioned iterative solver is a novel contribution and hence the advantage of using it is first demonstrated for a HCCI case. This is followed by the results for a diesel engine sector simulation using a detailed n-heptane mechanism from Lawrence Livermore National Laboratory. 2. Acceleration strategies (a) Adaptive zoning Adaptive zoning is used to accelerate the chemistry solver by reducing the number of calls to the chemistry solver. Instead of invoking chemistry at each and every cell, cells of similar physical and thermodynamic properties are grouped into zones. At each discrete time t, the cells are grouped in zones based on the physical and thermodynamic state of the
cell. In a two-dimensional zoning strategy, the zoning is done based on two variables, the temperature and the equivalence ratio of the cells. For each zone, the average temperature, pressure and the composition are determined to specify the thermodynamic state of the mixture in that zone. The SAGE chemistry solver is invoked on each zone. The zonal solution is then remapped onto the individual cells. The zoning strategy and remapping strategy are discussed in detail in Raju et al. 2012. (b) Dynamic Mechanism Reduction Dynamic mechanism reduction involves the reduction of detailed chemical reaction mechanism during run time for each cell. The state of the art Directed relation graph with error propagation (DRGEP) reduction technique [Pepiot and Pitsch, 2008; Niemeyer et al., 2010] is used to generate locally reduced mechanisms for each cell and for each of the CFD time step. Since the local mechanisms are much smaller in size in comparison to the full size detailed mechanism, significant savings in computational time is achieved. Locally reduced mechanisms are created at each cell/zone at each time step. Chemistry solver is invoked in based on the locally reduced mechanism. The production/consumption rate of the unimportant species is taken as zero. There is some overhead for generating the reduced mechanism. The details of the implementation are presented in Raju et al. 2013. (c) Preconditioned Iterative solver The chemistry solver uses CVODES solver which is part of the public domain Sundials package developed by Lawrence Livermore National Laboratory [Hindmarsh et al., 2005; Serban and Hindmarsh, 2005]. When a mechanism consists of large number of species, the size of the Jacobian becomes very large. The factorization of the Jacobian matrix in a direct solver becomes very expensive and often leads to a high CPU time for solving the system of ODE equations. CVODES solver provides an option to use iterative solver. However, iterative solver fails miserably without a suitable preconditioner. In this work an incomplete LU (ILU) factorization is used as a preconditioner for the original Jacobian matrix. Advantage is taken of the sparsity of the original Jacobian and the preconditioned matrix. The following section describes briefly the methodology of ODE integration using iterative solver in CVODES. The chemistry solver solves the species and the energy equation at each and every cell/zone. The system of ODE equations can be represented as y f t, y, y t0 y0. (1) For stiff systems, the CVODES solver uses the backward differentiation formula (BDF) method, which results in a system of non-linear equations at each time step represented as n n G( yn) y hn n,0 f tn, y an 0 (2) nm 1 H y y nm G y nm (3) H I J, hn n,0 (4) f J y (5) where hn tn tn 1 is step size, a n and n,0 are the coefficients. The assembly of the jacobian matrix J is analytical evaluation of the derivatives which results in a significant computational savings as compared to the conventional evaluation of derivative using numerical finite difference approximation. The results system of linear equations can be solved either using a direct solver or an iterative solver. As mentioned earlier, the use of iterative solver necessitates provision of a good preconditioner. The preconditioned system of linear equations can be written as 1 M H y M 1 G (6) The preconditioner M is approximate H in some sense. This preconditioner is called a left preconditioner as the preconditioning is applied left of the LHS matrix. The preconditioning M is obtained by dropping some of the elements of the original H matrix which are below a certain tolerance level ε. This results in increasing the sparsity of the M matrix. Both the matrix H and the matrix M are stored in sparse format to take advantage of the sparsity. In-house sparse Gaussian elimination routine is used to factorize the matrix M. Being sparser than the original matrix H, Matrix M would take lesser time for factorization and hence is suitable as a good preconditioner. GMRES is used as the iterative solver for solving the system of equations. The value of tolerance level ε is hardcoded as 0.1. Further study is required for determining the proper choice of value for ε. 2
(d) Strategies for reducing the time for transport of the species The above strategies (a-c) deal with the reduction of the chemistry computational time. Typically, the chemistry time is the bottleneck for the internal combustion (IC) engine simulations, especially for large size mechanisms. When the chemistry time is significantly reduced by the above strategies, the time for the transport of the species becomes significant. For all the above strategies, the transport of all the species is inevitable. Even though the mechanism is reduced locally, all the species need to be transported. Earlier researchers [Ren et al., 2011] have proposed RCCI type algorithms to reduce the number of species to be transported. In this paper, only a simple strategy is used to reduce the number of transported species. Transport of a species is skipped if the maximum mass fraction of the species in the entire domain is less than a tolerance level (typically 1.e-14). The species mass fractions are renormalized at every time step to avoid losing any mass due to the above approximation. This approximation gives slight reduction in the time required for transport of the species. Ongoing research is still being conducted to add in more robust algorithm to reduce the time for transport of the species. 3. Results and Discussion The strategies (a-c) accelerate the combustion simulations and strategy (d) accelerates the calculation of the transport of the species. The efficiency of strategy (a) has been demonstrated in an earlier study [Raju et al. 2012] and that of strategy (b) has been demonstrated in another study [Raju et al. 2013]. The primary purpose of this study is to first demonstrate the efficiency of the iterative preconditioned solver and then to demonstrate the combination of all the strategies for a real engine simulation. To test the computational efficiency of the iterative preconditioner solver, a HCCI simulation is performed for n- heptane combustion using a full LLNL n-heptane mechanism consisting of 654 species. Two cases are run, one with a dense solver and the other using an iterative solver. Adaptive zoning is used for both the cases as adaptive zoning significantly reduces the CPU time for combustion. There is a negligible difference in the solutions between the two cases (not shown here) as would be expected. Table 1 shows the CPU time comparison for both the cases. For the given case, the combustion time is only a small portion of the total CPU time. Due to the presence of large number of species, the transport time dominates the simulation. A speed up of 13 times is obtained for the combustion simulation using an iterative preconditioned solver. Table 1: Comparison of CPU Times for Dense Solver and Iterative Preconditioned Solver (654 species) Case CPU time for combustion (sec) Total CPU time (sec) Dense solver 9633 39467 Iterative preconditioned solver 755 30196 The efficiency of the iterative preconditioned solver using a large size mechanism has been demonstrated using a HCCI case. In a real engine simulation, the time for combustion dominates the total simulation time. To demonstrate the use of a large size mechanism for engine simulation, a diesel sector simulation using detailed n-heptane mechanism is performed. Table 2 shows the engine parameters. For large size mechanisms (> 500 species or so), the use of SAGE (chemistry on each cell) is prohibitively expensive and is not attempted. Adaptive zoning is used by default. Three different simulations are performed (i) only strategy (a), referred as AZ, (ii) strategies (a) and (b), referred as AZ-IP, (iii) strategies (a),(b), (c) and (d), referred as AZ-IP-DMR. For dynamic mechanism reduction, the DRGEP tolerance value is chosen as 0.005. The chosen target species are n-c7h16, OH and N2. Figure 1 shows the comparison of the average cylinder pressure for cases (ii) and (iii). The results for case (i) and case (ii) are exactly the same and hence case (i) is not shown in the figure. Figure 1 shows the pressure profiles for both case (ii) and case (iii) are close to each other. The peak pressure for case (iii) is slightly less than that for case (ii). The reason for the differences is due to the approximations involved in mechanism reduction on the fly. Table 2 shows the comparison of the computational time for cases (i), (ii) and (iii) on 12 cores. Table 2 shows a huge savings in the computational time using the iterative preconditioned solver (speed up of ~7.5 times). A further speed up of 4 times is obtained by using dynamic mechanism reduction. The overall computational time is reduced from 2 months to 2 days using all the above mentioned strategies. It is to be noted that without the use of adaptive zoning strategy, the simulation is prohibitively expensive. It is projected that it would take around 6 months and so is not attempted here. The use of all 3
the acceleration strategies has significantly reduced the total computational time that it is now possible to use full size mechanism with a little loss in accuracy. 12 11 10 Average Pressure [MPa] 9 8 7 6 5 4 3 AZ-IP AZ-IP-DMR 2 1 0-100 -50 0 50 100 CA Figure 1: Comparison of average cylinder pressure as a function of crank angle for case (ii) and case (iii) 1400 1200 Heat release rate [J/m 3 s] 1000 800 600 400 AZ-IP AZ-IP-DMR 200 0 0 50 100 CA Figure 2: Comparison of heat release rate as a function of crank angle for case (ii) and case (iii) 4
Table 2: Comparison of the total CPU time for the three different cases Case Set up Total CPU time AZ AZ-IP AZ-IP-DMR Over 60 days 8 days 2 days 4. Conclusions Different strategies for accelerating combustion simulations towards accommodating large size chemical mechanisms for engine simulations have been proposed. The first strategy is to use adaptive zoning to reduce the number of calls to the combustion calculations. The second strategy is to use dynamic mechanism reduction, in which each cell/zone performs combustion simulations on a locally reduced chemical reaction mechanism. The third strategy is to use iterative preconditioned based ODE solver which takes advantage of the sparsity of the jacobian matrix and incomplete LU based preconditioner to accelerate the ODE solver. The fourth strategy is to skip the transport of insignificant species. It is found that for large size mechanism, the use of iterative preconditioned solver itself gives a significant speedup. For a HCCI simulation with 654 species mechanism, it is found that the time for combustion simulation is 13 times faster. With the use of all the above mentioned strategies, it is found that for an engine sector simulation with 654 species, the simulation can be completed within a couple of days, making it possible to use large size mechanisms for practical engine simulations within a reasonable amount of simulation time. Acknowledgements The authors acknowledge the support of Convergent Science Inc. References Aceves, S. M. Flowers, D. L., Westbrook, C., Smith, J. et al., "A Multi-Zone Model for Prediction of HCCI Combustion and Emissions," SAE Technical Paper 2000-01-0327. Babajimopoulos, A., Assanis, D, N., Flowers, D. L., Aceves, S. M., Hessel, R. P, "A fully coupled computational fluid dynamics and multi-zone model with detailed chemical kinetics for the simulation of premixed charge compression ignition engines", International Journal of Engine Research, 6(5), pp. 497-512, 2005. Hindmarsh, A. C., Brown, P. N., Grant, K. E., Lee, S. L., Serban, R., Shumaker, D. E. and Woodward, C. S. "SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers," ACM Transactions on Mathematical Software, 31(3), pp. 363-396, 2005. Lu, T., Law, C.K., Toward accommodating realistic fuel chemistry in large-scale computations, Progress in Energy and Combustion Science, 35(2), pp. 192-215, 2005. K.E. Niemeyer, C.-J. Sung, M.P. Raju, Skeletal mechanism generation for surrogate fuels using directed relation graph with error propagation and sensitivity analysis, Combustion and Flame, 157(9), pp. 1760 1770, 2010. Pepiot-Desjardins, P. and Pitsch, H., An efficient error propagation based reduction method for large chemical kinetics mechanisms, Combustion and Flame, 154(1-2), pp. 67-81, 2010. Raju, M., Wang, M., Dai, M., Piggott, W., Flowers, D. Acceleration of Detailed Chemical Kinetics Using Multi-zone Modeling for CFD in Internal Combustion Engine Simulations, SAE 2012-04-16, 2012a. 5
Raju, M., Wang, M., Senecal, P.K., Som, S., Longman, D. E., "A Reduced Diesel Surrogate Mechanism For Compression Ignition Engine Applications," Proceedings of the ASME 2012 Internal Combustion Engine Division Fall Technical Conference, ICEF2012, September 23-26, Vancouver, BC, Canada, 2012b. Raju, M., Wang, M., Senecal, P.K., "Dynamic Chemical Mechanism Reduction for Internal Combustion Engine Simulations" SAE 2013-01-1110, 2013. Ren Z., Goldin, G.M., Hiremath, V., Pope, S. B., "Reduced description of reactive flows with tabulation of chemistry," Combustion Theory and Modelling, 15(6), pp.827-848, 2011. Serban, R., and Hindmarsh, A. C., "CVODES: the Sensitivity-Enabled ODE Solver in SUNDIALS," Proceedings of IDETC/CIE 2005, Sept. 2005, Long Beach, CA. 6