Skeletal model reduction with forced optimally time dependent modes☆
Introduction
Detailed reaction models for C-C hydrocarbons usually contain over 100 species in about 1000 elementary reactions [1], [2], [3], [4], [5], [6]. Direct application of such models is limited only to simple, canonical combustion simulations because of their tremendous computational cost. Various reduction techniques have been developed to accommodate realistic fuel chemistry simulations, and to capture intricacies of chemical kinetics in complex multi-dimensional combustion systems. As the first step in developing model reduction, it is important to extract a subset of the detailed reaction model, skeletal model, by eliminating unimportant species and reactions [7], [8]. Local sensitivity analysis (SA), reaction flux analysis [9], [10], [11], and directed relation graph (DRG) and its variants [12], [13], [14], [15] have often been utilized for skeletal model reduction. Local SA, which is the subject of the present work, explores the response of model output to a small change of a parameter from its nominal value [16] while global sensitivity analysis is useful for studying uncertainty of kinetic parameters (i.e. collision frequencies and activation energies) which propagate through model and non-linear coupling effects [2], [17], [18], [19], [20], [21], [22], [23], [24].
Model reduction with local SA contains methods such as PCA [1], [25], [26], [27], [28], [29], [30], [31], and construction of a species ranking [32]. In local SA, the sensitivities are commonly computed either by finite difference (FD) discretizations, directly solving a sensitivity equation (SE), or by an adjoint equation (AE) [33]. The computational cost of using FD or SE, which are forward in time methods, scales linearly with the number of parameters making them impracticable when sensitivities with respect to a large number of parameters are needed. On the other hand, computing sensitivities with AE requires a forward-backward workflow, but the computational cost is independent of the number of parameters as it requires solving a single ordinary/partial differential equation (ODE/PDE) [34], [35], [36]. The AE solution is tied to the objective function, and for cases where multiple objective functions are of interest, the same number of AEs must be solved. Regardless of the method of computing sensitivities, the output of FD, SE, and AE at each time instance is the full sensitivity coefficient matrix, which can be extremely large for systems with large number of parameters.
Recently, the forced optimally time dependent (f-OTD) decomposition method was introduced for computing sensitivities in evolutionary systems using a model driven low-rank approximation [33]. This methodology is the extension of OTD decomposition in which a mathematical framework is laid out for the extraction of the low-rank subspace associated with transient instability of the dynamical system [37]. The OTD approximates sensitivities with respect to initial conditions, while f-OTD approximates sensitivities with respect to external parameters, e.g., forcing. As a consequence, in the formulation of f-OTD there is a two–way coupling between the evolution of the f-OTD modes and the f-OTD coefficients, whereas in OTD formulation, the evolution of the modes is independent of the coefficients. In forward workflow of f-OTD, the sensitivity matrix i.e. is modeled on-the-fly as the multiplication of two skinny matrices , and which contain the f-OTD modes and f-OTD coefficients, respectively, where is the number of equations (or outputs), is the number of independent parameters, min{} is the reduction size, and . The key characteristic of f-OTD is that both and are time-dependent and they evolve based on closed form evolution equations extracted from the model, and are able to capture sudden transitions associated with the largest finite time Lyapunov exponents [38]. The time-dependent bases have also been used for stochastic reduced order modeling [39], [40], [41], [42], [43], and recently for on-the-fly reduced order modeling of reactive species transport equation [44]. In a nutshell, f-OTD workflow i) is forward in time unlike AE, ii) bypasses the computational cost of solving FD and SE, or other data-driven reduction techniques, and iii) stores the modeled sensitivities in a compressed format, i.e. we only store and solve for two skinny matrices and instead of storing and solving the full sensitivity matrix , as in FD, SE and AE.
The major advantage of PCA in skeletal reduction is to combine the sensitivity coefficients for a wide range of operating conditions (e.g. equivalence ratio and pressure) [1]. The PCA finds the low-dimensional subspace of data gathered from different (temporal or spatial) locations by applying a minimization algorithm over the whole data at once. Therefore, PCA is a low-rank approximation in a time-averaged sense and may fail to capture highly transient finite-time events (e.g. ignition). In order to resolve this issue, one needs to pre-recognize the locations of such events and use the data mainly from these locations. This requires extensive knowledge and/or expertise. Moreover, PCA modes are time invariant, and the process of selecting sufficient eigenvalues/eigenmodes to capture the essence of all observed phenomena (e.g. ignition, flame propagation), is crucial but is usually done by trial and error. References [1], [45] show that for certain problems, a skeletal model built solely upon the information conveyed by that first reaction group (first eigenmode) from PCA can fail to accurately reproduce the detailed model over the entire domain of interest. Therefore, one needs to deal with several eigenmodes with close eigenvalues and choose essential reaction groups among them [1].
In order to resolve the drawbacks of current SA methods and PCA for skeletal model reduction, we use f-OTD methodology for both SA and skeletal model reduction. The applicability of our approach is demonstrated for ethylene-air burning with the University of Southern California (USC) chemistry model [2] as the detailed model. Adiabatic, constant pressure, spatially homogeneous ignition is the canonical problem; and the generated skeletal models with f-OTD are compared against detailed and several skeletal models.
The remainder of this paper is organized as follows. The theoretical description of PCA and f-OTD and their mathematical derivations for SA are presented in Section 2. Model reduction with f-OTD is first described in Section 3, with a simple reaction model for hydrogen-oxygen combustion, followed by skeletal model reduction with f-OTD for the more complex ethylene-air system in Section 4. The paper ends with conclusions in Section 5. All the generated models are supplied in supplementary materials section.
Section snippets
Formulation
Consider a chemical system of species reacting through irreversible reactions, where is a symbol for species , and and are the molar stoichiometric coefficients of species in reaction . Changes of mass fractions and temperature in an adiabatic, constant pressure , and spatially homogeneous reaction system of ideal gases can be described by the following initial value problems (IVPs) [46]
Model reduction with f-OTD: application for hydrogen-oxygen combustion
In this section, the process of eliminating unimportant reactions and species from a detailed kinetic model with f-OTD is described, and its differences with PCA are highlighted. The Burke model [49] for hydrogen-oxygen system which contains =10 species,1 and =54 irreversible (27 reversible) reactions is considered as the detailed model. The
Skeletal reduction: application for ethylene-air burning
Several detailed kinetic models for ethylene-air burning are available in literature, and are developed at the University of California, San Diego (UCSD) [4], the University of Southern California (USC - a subset of JetSurf) [2], the KAUST (AramcoMech2) [50], and the Politecnico of Milan (CRECK) [5]. Figure 6 indicates that the ignition delays as predicted by all these models are in a reasonable agreement with each other. Moreover, it is shown in Ref. [51] that USC ignition delays are closer to
Conclusions
Instantaneous sensitivity analysis with f-OTD is described and implemented for a systematic skeletal model reduction. A key feature of the f-OTD approach is that it factorizes the sensitivity matrix into a multiplication of two low-ranked time-dependent matrices which evolve based on evolution equations derived from the governing equations of the system. Modeled sensitivities are then normalized and the most important reactions and species of a detailed model are ranked in a systematic manner
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work has been co-authored by an employee of Triad National Security, LLC which operates Los Alamos National Laboratory under Contract No. 89233218CNA000001 with the U.S. Department of Energy/National Nuclear Security Administration. The work of PG was supported by Los Alamos National Laboratory, under Contract 614709. Additional support for the work at Pitt with H.B. as the PI is provided by NASA Transformational Tools and Technologies (TTT) Project Grant 80NSSC18M0150, and by NSF under
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