Most practical problems in combustion occur under inhomogeneous, transient conditions. Examples include laminar flames, turbulent flames, and combustion in engines. In order to simulate faithfully the physics involved, the chemistry in these applications is typically modeled using a large, detailed chemical mechanism on the order of tens to thousands of species and hundreds to tens of thousands of reactions. This mechanism is then encoded as a source term in a system of coupled, nonlinear PDEs governing the system temperature and the concentration of all species. Due to the large number of nonlinear PDEs and the wide range of active time scales spanning up to ten orders of magnitude, simulations of combustion in practical applications typically have prohibitive computational costs, even on large supercomputers. For this reason, model reduction strategies are used to reduce the computational requirements of these simulations.

One popular category of model reduction techniques is projection-based model reduction [5]. This class of techniques takes the right-hand side of an ODE and determines a projection matrix from this right-hand side and some additional data (typically reference values of state variables and error tolerances). This righthand side is called the original model. The reduced model constructed by the technique is the right-hand side of the ODE, premultiplied by the projection matrix determined by the technique.

However, despite the popularity of projection-based model reduction techniques, for the case of nonlinear ODEs, there exists no practical procedure to bound the approximation error in the solution of the reduced model ODE constructed by one of these techniques relative to the solution of the original model ODE, given comparable initial conditions for both ODEs.

The contribution of this work will be to propose a new projection-based model reduction technique as a proof of concept. This technique will be extended to give the first projection-based model reduction technique that can be used to construct bounds on the approximation error in the solution of the reduced model ODE it constructs, relative to the solution of the original model ODE, given comparable initial conditions for both ODEs.

Given an original model with N state variables, the proposed algorithm takes a basis set of vectors in N- dimensional Euclidean space, a collection of reference values of state variables, and error tolerances as input. From these inputs, a mixed-integer nonlinear program (MINLP) is constructed, then reformulated exactly as a mixed-integer linear program (MILP).

An optimal solution of the MILP formulation contains a projection matrix that is used for model reduction. Relative to the right-hand side of the original model ODE, the approximation error in the right-hand side of the reduced model ODE is guaranteed to be bounded by the user-supplied error tolerances at the reference values of the state variables. Furthermore, there is no orthogonal projector of lesser rank satisfying these error bounds; this projector corresponds to identifying a maximal number of quasi-steady state-like relationships orthogonal to a minimal-dimensional model-reducing manifold.

Model reduction case studies are carried out to demonstrate that this technique is practical by showing that the computations are feasible.

Both the theory and the results show that the proposed technique reliably bounds the approximation error in the right-hand side of the reduced model ODE relative to the right-hand side of the original model ODE at the reference values of the state variables specified by the user.

In the future, the MILP formulation in this work will be extended using interval arithmetic [3] to bound the approximation error in the right-hand side of the reduced model ODE relative to the right-hand side of the original model ODE over a range of reference values of the state variables specified by the user. This extension parallels the extension of point-constrained reaction elimination [1] to range-constrained reaction elimination [4] and the extension of point-constrained simultaneous reaction and species elimination [2] to range-constrained simultaneous reaction and species elimination [6]. Bounding the approximation error of a range of reference values of state variables enables the construction of error bounds on the solution of the reduced model ODE relative to the solution of the original model ODE. By constructing these bounds, it will be possible to reduce the computational costs of combustion simulations for practical applications and guarantee the accuracy these simulation results to within known tolerances.

References:

[1] B. Bhattacharjee, D. Schwer, P. Barton, and W. H. Green. Optimally-reduced kinetic models: reaction elimination in large-scale kinetic mechanisms. Combustion and Flame, 135(3):191-208, 2003.

[2] A. Mitsos, G. M. Oxberry, P. I. Barton, and W. H. Green. Optimal automatic reaction and species elimination in kinetic mechanisms. Combustion and Flame, 155:118–132, 2008.

[3] R. E. Moore. Methods and applications of interval analysis. Society for Industrial Mathematics, Philadelphia, 1979.

[4] O. O. Oluwole, P. I. Barton, and W. H. Green. Obtaining accurate solutions using reduced chemical kinetic models: a new model reduction methods for models rigorously validated over ranges. Combustion Theory and Modelling, 11(1):127–146, 2007.

[5] G. M. Oxberry,W. H. Green, and P. I. Barton. Affine Lumping Formalism for Comparison of Projection-Based Model Reduction Techniques. In 2009 AIChE Annual Meeting, Nashville, Tennessee, 2009.

[6] G. M. Oxberry, A. Mitsos,W. H. Green, and P. I. Barton. Range-Constrained Simultaneous Reaction and Species Elimination in Kinetic Mechanisms. In 2009 AIChE Annual Meeting, Nashville, Tennessee, 2009.

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