An Optimization Approach to Kinetic Model Reduction for Combustion Chemistry

Model reduction methods in chemical kinetics are used for simplification of models which involve a number of different time scales. Slow invariant manifolds in chemical composition space are supposed to be identified. A selection of state variables serve for parametrization of these manifolds. Species reconstruction methods are used to compute the values of the remaining variables in dependence of the parameters. We discuss theoretical results and numerical methods for an application of a model reduction method that is developed by D. Lebiedz based on optimization of trajectories. The main focus of this work is an application of the model reduction method to models of chemical combustion. The existence of a solution of the semi-infinite optimization problem, which has to be solved to obtain a local approximation of the slow manifold, is proven. A finite optimization problem for the same purpose is presented which can be solved with a generalized Gauss-Newton method. This method is used with an active set strategy. A filter framework and iterations with second order correction are employed for globalization of convergence. Families of neighboring optimization problems can be solved efficiently in a predictor corrector continuation scheme. The tangent space of the slow manifold can be computed by evaluation of sensitivity equations for the parametric optimization problem. A step size strategy is applied in the continuation scheme for efficient progress along the homotopy path. Results of an application of the presented method are shown and discussed. The test models range from simple test examples to realistic models of syngas combustion in air.

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.KeywordsChemical KineticModel ReductionSlow ManifoldProgress VariableForward ModeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Minimal curvature trajectories: Riemannian geometry concepts for slow manifold computation in chemical kinetics

This paper presents a model reduction and control approach for a class of active flow control problem. Computational fluid dynamics (CFD) has been used extensively to study fluid dynamic systems. It is a powerful tool to gain insight and understanding, but it is also extremely computationally intensive and thus unsuitable for control design and iteration. Various model reduction schemes have been proposed in the past to approximate the Navier-Stokes equation with a low-dimensional model. There are essentially two approaches: input/output model identification and modal decomposition based on snapshots of high-dimensional states such as proper orthogonal decomposition (POD). The former captures mostly the local behavior near a steady state and the latter is highly dependent on the snapshots of the flow state used to extract the projection. This paper presents a novel model reduction approach that attempts to combine the attractive features of the two approaches. The local behavior is first obtained through the identification of a linear (or semi-linear) time invariant model using the input/output data. To capture the large nonlinear structure, we project the difference between CFD response and the linear identified model response onto a set of POD basis. This trajectory is fit to a nonlinear dynamical model to augment the input/output linear model. The resulting linear+nonlinear model is used to design a feedback control law. The difference between the predicted trajectory and CFD simulated trajectory under this control law is further used to refine the POD basis, until a good match is obtained. The proposed method is applied to 2D compressible flow example with outlet flow rate control for a contraction section. Preliminary results for lift control of a 2D airfoil is also included.

ABSTRACT(Chemical) kinetic models in terms of ordinary differential equations correspond to finite-dimensional dissipative dynamical systems involving a multiple time scale structure. Most dimension reduction approaches aimed at a slow mode description of the full system compute approximations of low-dimensional attracting slow invariant manifolds and parameterize these manifolds in terms of a subset of chosen chemical species, the so called reaction progress variables. The invariance property suggests a slow invariant manifold to be constructed as (a bundle of) solution trajectories of suitable ordinary differential equation initial or boundary value problems. The focus of this work is on a discussion and exploitation for deeper insight of unifying geometric and analytical issues of various approaches to trajectory-based numerical approximation techniques of slow invariant manifolds that are in practical use for model reduction in chemical kinetics. Two basic concepts are pointed out reducing various model reduction approaches to a common denominator. In particular, we discuss our recent trajectory optimization approach in the light of these two concepts. We newly relate both of them within our variational boundary value viewpoint, propose a Hamiltonian formulation and conjecture its relation to conservation laws, (partial) integrability and symmetry issues as novel viewpoints on dimension reduction approaches that might open up new deep perspectives and approaches to the problem from dynamical systems theory.

A model reduction approach aimed at systematically reducing computational complexity of multiple scale homogenization for nonlinear history-dependent problems is developed. The so-called N-scale model reduction (NMR) approach is based on reformulating a sequence of nonlinear unit cell problems at multiple scales in terms of inelastic deformation modes that a priori satisfy equilibrium equations at multiple scales and thus eliminating the need for costly solution of discretized nonlinear equilibrium. The N-scale model reduction approach is validated on tube crush tests.

1 Methods for Model Reduction.- 2 Multiplicative Approximation.- 3 Low Order Controller Design.- 4 Model and Controller Reduction Based On Coprime Factorizations.

This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics. Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects. This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.

Considering the speed information as a prior constraint, constrained cubature kalman filter (CCKF) tracking filtering method based on model reduction (MR) method and perfect measurement (PM) method is studied and compared for the tracking problem of fixed single observer passive positioning using only angle pair constant speed targets. The simulation results show that the fixed single observer passive positioning system can track the moving target with constant speed by measuring angle only under the condition of speed prior information, and the perfect measurement method is more robust to the initial state of the filter than the model reduction method.

This chapter proposes a clustered model reduction method for interconnected linear systems evolving over bidirectional networks. This model reduction method belongs to a kind of structured model reduction methods, where network clustering, namely clustering of subsystems, is performed according to a notion of uncontrollability of local states. We refer to this notion of uncontrollability as cluster reducibility, which can be captured by a coordinate transformation called positive tridiagonal transformation in an algebraic manner. In this chapter, it is shown that the aggregation of the reducible clusters retains the stability of the original system as well as an interconnection topology among clustered subsystems. Furthermore, an \(\fancyscript{H}_{\infty }\)-error bound is derived for the state discrepancy due to the cluster aggregation. The efficiency of the clustered model reduction is shown through an example of large-scale complex networks.KeywordsClustered model reductionInterconnected linear SystemsLarge-scale complex networks

We consider the dynamics of the general Maxwell–Bloch equations modified by a linear restoring term. We use the slowly varying amplitude approximation to eliminate second-order derivatives in the polarization, but we keep the second-order derivatives in the electric field. This model governs a two-level laser in a ring cavity. For some class of lasers, this system of equations has two separated time scales and forms a singularly perturbed, hyperbolic system with three distinct characteristics. We extend Menon–Haller’s infinite-dimensional geometric singular perturbation theory [G. Menon and G. Haller, Siam J. Math. Anal. Society for Industrial and Applied Mathematics Vol. 33, No. 2, pp. 315–346 c (2001)] to the general Maxwell–Bloch equations by proving the persistence of an arbitrarily smooth, infinite-dimensional, slow manifold, under an unbounded perturbation. The attractor of the system is contained in this manifold. We use a modified graph transform method to prove the existence of this manifold and a cone condition to define the graph transform. The fact that the Maxwell–Bloch and the modified general Maxwell–Bloch equations have the same asymptotic behavior suggests a justification of the slowly varying amplitude approximation.KeywordsMaxwell–Bloch systemInvariant manifolds theoryGeometric singular perturbation

: In this project, methods of model reduction that integrate feedback active flow control with applications to nonlinear convection and turbulent flows governed by Navier-Stokes equations are developed. A new methodology which extracts boundary conditions in reduced order proper orthogonal decomposition (POD) and finite difference models is developed. A new model reduction method based on empirical data and balanced truncation was developed and applied to nonlinear Galerkin models. Based on this method a new empirical Hankel norm model reduction algorithm is proposed. These methods are applied to a prototype nonlinear convective problem governed by the two-dimensional (2D) Burgers' equation. The reduced models are used in the design of robust boundary controllers that achieve tracking, and implemented on the full order Computational Fluid Dynamics (CFD) models.

A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing such manifolds for kinetic models using trajectory optimization. The corresponding objective functional reflects a variational principle that characterizes trajectories on, respectively near, slow invariant manifolds. For a two-dimensional linear system and a common nonlinear test problem we show analytically that the variational approach asymptotically identifies the exact slow invariant manifold in the limit of both an infinite time horizon of the variational problem with fixed spectral gap and infinite spectral gap with a fixed finite time horizon. Numerical results for the linear and nonlinear model problems as well as a more realistic higher-dimensional chemical reaction mechanism are presented.

This paper provides three model reduction methods for linear time-invariant systems in the view of the Riemannian Newton method and the Jacobi-Davidson method. First, the computation of Hankel singular values is converted into the linear eigenproblem by the similarity transformation. The Riemannian Newton method is used to establish the model reduction method. Besides, we introduce the Jacobi-Davidson method with the block version for the linear eigenproblem and present the corresponding model reduction method, which can be seen as an acceleration of the former method. Both the resulting reduced systems can be equivalent to the reduced system originating from a balancing transformation. Then, the computation of Hankel singular values is transformed into the generalized eigenproblem. The Jacobi-Davidson method is employed to establish the model reduction method, which can also lead to the reduced system equivalent to that resulting from a balancing transformation. This method can also be regarded as an acceleration of a Riemannian Newton method. Moreover, the application for model reduction of nonlinear systems with inhomogeneous conditions is also investigated.

Large scale finite element models are routinely used in design and optimization for complex engineering systems. However, the high model order prevents efficient exploration of the design space. Many model reduction methods have been proposed in the literature on approximating the high dimensional model with a lower order model. These methods typically replace a fine scale model with a coarser scale model in schemes such as coarse graining, macro-modeling, domain decomposition and homogenization. This paper takes a systems perspective by stating the model reduction objective in terms of the approximation of the mapping between specified input and output variables. Methods from linear systems theory, including balance truncation and optimal Hankel norm approximation, are reviewed and compared with the standard modal truncation. For high order systems, computational load, numerical stability, and memory storage become key considerations. We discuss several computationally more attractive iterative schemes that generate the approximate gramian matrices needed in the model reduction procedure. A numerical example is also included to illustrate the model reduction algorithms discussed in the paper. We envision that these systems oriented model reduction methods complementing the existing methods to produce low order models suitable for design, optimization, and control.