Optimization Methodologies

The optimization and uncertainty minimization uses the Method of Uncertainty Minimization by Polynomial Chaos Expansion (MUM-PCE) [1,2], which finds a joint probability density function of the rate parameters within the hypercube of their uncertainties. The joint probability distribution can be propagated into a model prediction. A set of fundamental combustion property data are used as targets for rate parameter optimization and reduction of model prediction uncertainties. Sandia ChemKin code [3] is used to solve the initial value problem for shock tube and flow reactor experiments. Computational ignition delay is determined in a manner consistent with the experiment. Premixed flame simulations are performed using Sandia Premix [4] with multi-component transport and thermal diffusion, incorporating changes made to the diffusion coefficients of H and H₂. See the Transport data page for a detailed explanation.

The optimization procedure starts from the selection of active rate parameters, which involves sensitivity analysis of the experimental targets with respect to the reaction rate coefficients. Sensitivity charts can be found in the target description pages. The rate coefficients or third-body efficiencies are then chosen based on the ranked uncertainty impact index (equal to sensitivity multiplied by uncertainty). Each rate parameter is normalized by [5]

$$x_k=\ln(A_k/A_{k,0})/\ln f_k$$ where A_k is the pre-exponential factor or the third body efficiency of the k^th rate coefficient, f_k is its prescribed uncertainty factor, and the subscript "0" represents the nominal rate parameter value. The uncertainty of the rate coefficient is represented by a lognormal distribution function, in which the median value is equal to the nominal rate coefficient A₀ chosen for the trial model. The square distribution assumption was tested also, but results showed no significant difference in the predictive uncertainty of the model while the assumption has a tendency to “push” the rate coefficients to the extremity of their uncertainties. A more involved analysis of the two distribution assumptions was given in an earlier work [1] with similar observations.

An active rate parameter x is then represented by a normal distribution with mean value 0 and ±2σ range of -1 ≤ x ≤ 1. The model is represented by multivariate polynomial response surface for a given target by Monte Carlo sampling and factorial design [5],

$$\eta_r(\mathbf x)=\mathbf a^T \mathbf x + \mathbf x^T \mathbf b \mathbf x $$ Optimum rate parameters are found by solving the least-squares optimization problem $$\Phi (\mathbf x^*) = \min_{\mathbf x} \left\{\sum_{r=1}^M\left(\frac{\eta_r(\mathbf x)-\eta_r^{obs}}{\sigma_r^{obs}}\right)^2+\sum_{k=1}^N4x_k^2 \right\}$$ where x* is the array of optimized parameters, M is the number of targets considered, $\eta_r^{obs}$ is the mean value of the r^th target, and $\sigma_r^{obs}$ is its uncertainty. Expressing the normalized rate parameters as random variables [2], $$\mathbf x=\mathbf x^{(0)}+\mathbf x^{(1)} \mathbf \xi$$ where x⁽⁰⁾ is the mean of x, elements of $\mathbf \xi $ are independent, identically distributed random variables, and x⁽¹⁾ is a transformation matrix such that the covariance matrix may be approximated by linearizing the response surfaces in the vicinity of x^(0)*: $$\Sigma^*=\mathbf x^{(1)*}\mathbf x^{(1)*T}=\left(\sum_{r=1}^M \frac{\mathbf J_r^T \mathbf J_r}{(\sigma_r^{obs})^2}+4I\right)^{-1}$$ where $\mathbf J_r$ is the derivative of the r^th polynomial evaluated at x^(0)*, $$\mathbf J_r=(\mathbf a + 2\mathbf b \mathbf x^{(0)*})^T $$ MUM-PCE checks for inconsistencies among the experimental targets by calculating the contribution of the r^th experiment to the objective function, $$W_r=S_r \times F_r$$ where $$F_r=\frac{\eta_r^{obs}-\eta_r(\mathbf x^{(0)*})}{2\sigma_r^{obs}}$$ $$S_r=\frac{\mathbf J_r \cdot \mathbf x^{(0)*}}{\left\|\mathbf J_r \right\|\left\|\mathbf x^{(0)*}\right\|}$$ If one or more target satisfies $|F_r| \geq 1$, the target set is considered inconsistent. If there is only one inconsistent target, this target will be removed; if there are more than one inconsistent targets, the one with the largest $W_r$ value is removed. For each case, the model is re-optimized until the remaining target set is consistent.

References

[1]  Sheen DA, Wang H. The method of uncertainty quantification and minimization using polynomial chaos expansions. Combust Flame. 2011;158:2358-74.

[2]  Wang H, Sheen, DA. Combustion kinetic model uncertainty quantification, propagation and minimization. Prog. Energy Combust. Sci. 2015;47:1-31.

[3]  Kee RJ, Rupley FM, Miller JA. Chemkin-II: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics. Livermore, CA, USA: Sandia National Laboratories; 1989. Report No. SAND89-8009.

[4]   Kee RJ, Grcar JF, Smooke MD, Miller JA. A FORTRAN Program for Modeling Steady Laminar One-Dimensional Premixed Flames. Livermore, CA, USA: Sandia National Laboratories; 1986. Report No. SAND85-8240.

[5]  Frenklach M, Wang H, Rabinowitz MJ. Optimization and analysis of large chemical kinetic mechanisms using the solution mapping method—combustion of methane. Prog. Energy Combust. Sci. 1992;18:47-73.