Skeletal model reduction with forced optimally time dependent modes

Abstract

Skeletal model reduction based on local sensitivity analysis of time dependent systems is presented in which sensitivities are modeled by forced optimally time dependent (f-OTD) modes. The f-OTD factorizes the sensitivity coefficient matrix into a compressed format as the product of two skinny matrices, i.e. f-OTD modes and f-OTD coefficients. The modes create a low-dimensional, time dependent, orthonormal basis which capture the directions of the phase space associated with most dominant sensitivities. These directions highlight the instantaneous active species, and reaction paths. Evolution equations for the f-OTD modes and coefficients are derived, and the implementation of f-OTD for skeletal reduction is described. For demonstration, skeletal reduction is conducted of the constant pressure ethylene-air burning in a zero-dimensional reactor, and new reduced models are generated. The laminar flame speed, the ignition delay, and the extinction curve as predicted by the models are compared against some existing skeletal models in literature for the same detailed model. The results demonstrate the capability of f-OTD to eliminate unimportant reactions and species in a systematic, efficient and accurate manner.

Introduction

Detailed reaction models for C${}_1$-C${}_4$ 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. $S\left(t\right)\in {\mathbb{R}}^{n_{eq}\times n_r}$ is modeled on-the-fly as the multiplication of two skinny matrices $U\left(t\right)=[{\mathit{u}}_1\left(t\right),{\mathit{u}}_2\left(t\right),\cdots ,{\mathit{u}}_r\left(t\right)]\in {\mathbb{R}}^{n_{\mathit{eq}}\times r}$, and $Y\left(t\right)=[{\mathit{y}}_1\left(t\right),{\mathit{y}}_2\left(t\right),\cdots ,{\mathit{y}}_r\left(t\right)]\in {\mathbb{R}}^{n_r\times r}$ which contain the f-OTD modes and f-OTD coefficients, respectively, where $n_{eq}$ is the number of equations (or outputs), $n_r$ is the number of independent parameters, $r\ll$ min{$n_s,n_r$} is the reduction size, and $S\left(t\right)\approx U\left(t\right)Y^T\left(t\right)$. The key characteristic of f-OTD is that both $U\left(t\right)$ and $Y\left(t\right)$ 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 $U$ and $Y$ instead of storing and solving the full sensitivity matrix $S$, 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.