Abstract
The need for reduced models of chemical kinetics is motivated by the fact that the simulation of reactive flows with detailed chemistry is generally computationally expensive. In dissipative dynamical systems different time scales cause an anisotropically contracting phase flow. Most kinetic model reduction approaches explicitly exploit this and separate the dynamics into fast and slow modes. We propose an implicit approach for the approximation of slow attracting manifolds by computing trajectories as solutions of an optimization problem suggesting a variational principle characterizing trajectories near slow attracting manifolds. The objective functional for the identification of suitable trajectories is supposed to represent the extent of relaxation of chemical forces along the trajectories which is proposed to be minimal on the slow manifold. Corresponding geometric criteria are motivated via fundamental concepts from differential geometry and physics. They are compared to each other through three kinetic reaction mechanisms.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Warnatz, J., Maas, U., Dibble, R.W.: Combustion: Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation. Springer, Berlin (2006)
Lebiedz, D., Kammerer, J., Brandt-Pollmann, U.: Automatic network coupling analysis for dynamical systems based on detailed kinetic models. Phys. Rev. E, 72 (2005) 041911
Lebiedz, D.: Computing minimal entropy production trajectories: An approach to model reduction in chemical kinetics. J. Chem. Phys. 120 (2004) 6890–6897
Lebiedz, D., Reinhardt, V., Kammerer, J.: Novel trajectory based concepts for model and complexity reduction in (bio)chemical kinetics. In Gorban, A.N., Kazantzis, N., Kevrekidis I. G., Theodoropoulos C. (eds.): Model reduction and coarse-graining approaches for multi-scale phenomena. Springer, Berlin (2006) 343–364
Reinhardt, V., Winckler, M., Lebiedz, D.: Approximation of Slow Attracting Manifolds in Chemical Kinetics by Trajectory-Based Optimization Approaches. J. Phys. Chem. A 112 (2008) 1712–1718
Lebiedz, D., Reinhardt, V., Siehr, J.: Minimal curvature trajectories: Riemannian geometry concepts for model reduction in chemical kinetics. J. Comput. Phys. 229 (2010) 6512–6533
Gorban, A.N., Karlin, I.V., Zinovyev, A.Y.: Constructive methods of invariant manifolds for kinetic problems. Phys. Rep. 396 (2004) 197–403
Weinhold, F.: Metric geometry of equilibrium thermodynamics. J. Chem. Phys. 63 (1975) 2479–2483
Shahshahani, S.: A new mathematical framework for the study of linkage and selection. Mem. Am. Math. Soc. 17 (1979)
Chiavazzo, E., Karlin, I.V., Gorban, A.N., Boulouchos, K.: Combustion simulation via lattice Boltzmann and reduced chemical kinetics. J. Stat. Mech. (2009) P06013
Powell, M.J.D.: A fast algorithm for nonlinearly constrained optimization calculations. Vol. 630 of Lecture Notes in Mathematics, Springer, Berlin (1978) 144–157
Forsgren, A., Gill, P.E., Wright, M.H.: Interior Methods for Nonlinear Optimization. SIAM Rev. 44 (2002) 525–597
Ascher, U., Petzold, L.: Computer methods for ordinary differential equations and differential-algebraic equations. SIAM, Philadelphia (1998)
Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Prog. 106 (2006) 25–57
HSL: A collection of fortran codes for large-scale scientific computation. See URLhttp://www.hsl.rl.ac.uk(2007)
Bell, B.M., Burke, J.V.: Algorithmic differentiation of implicit functions and optimal values. In Bischof, C.H., Bücker, H.M., Hovland, P.D., Naumann U., Utke, J. (eds.): Advances in Automatic Differentiation. Springer, Berlin (2008) 67–77
Li, J., Zhao, Z., Kazakov, A., Dryer, F.L.: An updated comprehensive kinetic model of hydrogen combustion. Int. J. Chem. Kinet. 36 (2004) 566–575
Ren, Z., Pope, S.B., Vladimirsky, A., Guckenheimer, J.M.: The invariant constrained equilibrium edge preimage curve method for the dimension reduction of chemical kinetics. J. Chem. Phys. 124 (2006) 114111
HegheÅŸ, C.I.: C1-C4 Hydrocarbon Oxidation Mechanism. PhD thesis, University of Heidelberg (2006)
Troe, J.: Theory of Thermal Unimolecular Reactions in the Fall-off Range. I. Strong Collision Rate Constants. Ber. Bunsenges. Phys. Chem. 87 (1983) 161–169
Gilbert, R., Luther, K., Troe, J.: Theory of Thermal Unimolecular Reactions in the Fall-off Range. II. Weak Collision Rate Constants. Ber. Bunsenges. Phys. Chem. 87 (1983) 169–177
Acknowledgements
This work was supported by the German Research Foundation (DFG) through the Collaborative Research Center (SFB) 568.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lebiedz, D., Reinhardt, V., Siehr, J., Unger, J. (2011). Geometric Criteria for Model Reduction in Chemical Kinetics via Optimization of Trajectories. In: Gorban, A., Roose, D. (eds) Coping with Complexity: Model Reduction and Data Analysis. Lecture Notes in Computational Science and Engineering, vol 75. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14941-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-14941-2_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14940-5
Online ISBN: 978-3-642-14941-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)