Source Term Parameterization for PCA Combustion Modeling

Transcription

1 Paper # 7RK-252 Topic: Reaction Kinetics 8 th US ational Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 9-22, 23. Source Term Parameterization for PCA Combustion Modeling Isaac, B.,2 Parente, A. 2 Sutherland, J. Smith, P. Fru, G. 3 Thevenin D. 3 Chemical Engineering Department, University of Utah, Salt Lake City, Utah 2 Aero-Thermo-Mechanique, Universite Libre de Bruxelles, Brussels, Belgium 3 Institute of Fluid Dynamics and Thermodynamics, Otto-von-Guericke University Magdeburg, Magdeburg, Germany Modeling the physics of turbulent combustion systems remains a challenge due to large range of scales, which are important in these systems. Often, detailed chemical kinetic mechanisms are used to fully describe the chemistry involved in the combustion process, yielding highly coupled partial differential equations for each of the chemical species used in the mechanism. Recently, Principal Components Analysis (PCA) has shown promise in its ability to identify a low dimensional manifold describing the reacting system []. Sutherland and Parente demonstrated the formulation of a PCA model [2] where the Principal Components (PCs) of the system are transported. Evaluation of the PCs source-term is expensive (as all chemical species source-terms must be evaluated) and inherits error through the PC approximation. Parameterization methods can be employed to quickly and accurately produce sourceterm values, allowing one to avoid the expensive calculation of the source-terms and avoid the additional error from the state space approximation. The present work demonstrates the ability to parameterize the source-term space of a 2D Direct umerical Simulation of a spherical premixed syn-gas (CO/H 2 ) air flame using non-linear regression, comparing several non-linear regression methods. In addition the ability to parameterize the source-terms while altering the scaling parameters used in PCA is interesting as the scaling greatly effects the behavior and shape of the low-dimensional space being modeled. Introduction The ability to accurately model a turbulent combustion system remains challenging due to the complex nature of combustion systems. A simple fuel such as CH 4 has been accurately described using 53 species and 325 chemical reactions [3]. More complex fuels require increasingly complex chemical mechanisms. Each resolved chemical species requires a conservation equation which is a coupled, highly non-linear partial differential equation. Such systems are only possible to solve under very limited situations at this time due to computational costs. This issue leads to the need of a reduced model, which can adequately describe the chemical reactions. Many methods such as computational singular perturbation (CSP) [4], and Rate Controlled Constrained Equilibrium (RCCE) [5] attempt to reduce the complexity of the mechanism by using equilibrium assumptions for fast chemical processes, and using the computational resources on the more pertinent evolution

2 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics in the reaction process. Indeed in these complex combustion reaction mechanisms many of the species evolve at time scales much smaller than the time scales of interest, allowing for decoupling of fast and slow processes while maintaining accuracy. Low dimensional manifolds exists in these systems which describe well the governing characteristics of the flames. For example, the steady laminar flamelet model [6] uses the mixture fraction and mixture fraction variance to describe the flame as an ensemble of steady laminar diffusion flames undergoing various strain rates, providing a very good representation of the entire system with low number of variables. Principal Components Analysis (PCA) has been shown to identify the low dimensional manifolds [7] for turbulent combustion systems. PCA uses the eigenvalue decomposition of the covariance matrix of the thermodynamic state space to identify the manifold. In previous work by Sutherland [7], a modeling approach was presented which uses conservation equations for q PCs (η) which are calculated from q independent linear combinations of the state space variables. The selection of q depends on the complexity of the system of interest, as well as the desired accuracy of the representation of the system. The transport equations for η are of the following form: t (ρη)+ (ρu i η)= j η + s η () x i x i where j η is the diffusive flux of η and s η is the source-term for η. A major challenge of this modeling strategy is in the evaluation of s η. Introduction of a small amount of error to the state space variables by using a PCA representation, can dramatically effect the chemical species source-term (ω k ) calculation. The error in many cases is exponentially propagated due to the characteristics of the reaction rate equations. The present work investigates the ability to accurately model s η using a high-fidelity data set containing exact or similar physics to the system of interest. on-linear regression is used to create a model for s η as a function of η, where the training values for s η are calculated from a training set of X which contain no error from PCA approximation. The benefit in this approach is that approximation error due to PCA on the system is not propagated into the model for s η. Several well known non-linear regression techniques are investigated for estimation of s η, as well as a novel regression method based on Gaussian kernels filters. In addition the effects of the scaling used in PCA is assessed in terms of the ability to create regressions models for s η. 2 Approach 2. Principal Component Analysis The PCs (η) are calculated from the PCA analysis. PCA is performed on a data set consisting of n observations with k variables organized as an n k matrix (X). The data X is centered to zero by its corresponding means X, and scaled by the diagonal matrix γ containing the scaling value for each of the k variables: X =(X X)γ (2) PCA identifies a basis matrix A which when multiplied with X creates an approximation to η. The accuracy of the approximation of η is dependent on the number of retained columns of A. A is obtained by the eigenvalue decomposition of the covariance matrix of X: 2

3 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics k X X = A ΛA (3) The PCs are then defined by the projection of the basis matrix onto the scaled and centered data η = XA (4) The amount of variance represented by basis matrix columns are ordered from highest to lowest. Accordingly a subset of q columns of the original k columns in A, where q k may yield a good approximation of X. Accordingly a subset of η may be used to approximate the entire state space with minimal error based on the number of retained eigenvectors from A. X ηa T kq (5) The present application of PCA to the turbulent combustion system uses PCA to approximate the k chemical species. 2.2 PCA Scaling Scaling plays a key role in PCA as well as the non-linear regression that follows. After centering X the data needs to be scaled so that the PCA will give equal weights to the independent variables (γ from Equation 2). The following scaling methods where adopted for this study [8]: - auto scaling (), uses the standard deviation s k. Auto scaling leaves all columns of X with a standard deviation of one, and now the data is analyzed on the basis of correlations instead of covariances, γ k = s k. - range scaling (RA), uses the difference between the minimum and the maximum variable value, γ k =max X k X k min Xk X k. - pareto scaling (PAR), adopts the square root of the standard deviation as scaling factor, γ k = s k. - variable stability scaling (), gives an emphasis to variables which do not show strong variation, by using the product between the standard deviation and the coefficient of variation, γ k = s k k X k s. - level scaling (LEV), uses the mean value of the variables γ k = X k. - max scaling (), uses the maximum variable value as the scaling factor, here γ k = max X k X k. The scaling of the data set effects the shape of the low dimensional manifold calculated from PCA, which yields a significant impact on the ability of the non-linear regression (see Section 3). 3

4 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics 2.3 Principal Components Conservation Equation As is discussed in the work by Sutherland [7], the conservation equations for the PCs are derived from the general species transport equation [9]: t (ρy k)+ (ρu i Y k )= Y k ρd k + ω k (6) x i x i x i PCA provides a kq, the scaling vector γ k, and the centering vector Ȳk. One derives the transport equations for η by first centering and scaling the species mass fractions: ρ Y k Ȳk + Y k ρu Ȳk i = t γ k x i γ k x i multiplying by a kq and substituting Equation 4 leaves: t (ρη)+ (ρu i η)= x i s η = x i Substituting the diffusion term for Ficks law yields Equation. D η D k x i Yk Ȳk γ k + ω k ργ k (7) (η) + s η (8) x i ω k ργ k a kq (9) The source-term for this Equation (s η ) is a highly non-linear function of all of the state variables. Although the resolution of this source-term should be straight forward as it is a simply a function of the species mass fractions and the temperature, issues arise due to the approximated state space. The error in the approximation of the state space propagates into s η. A non-linear regression model can be used to model this source-term as a function of η. By training the non-linear function on values of s η that are free from the PCA approximation errors, the regression will provide accurate values for s η even though the state space is approximated. 2.4 Regression Models In this study on-linear Regression models are used to develop a function, f, which estimates the source-terms as a function of the PCs with an associated estimation error. s η = f(η)+error () The function is created on a training data set where η is calculated from Equation 4 and s η is calculated from Equation 9 with ω k being calculated from the real values of X. The function f is then tested on a distinct testing sample from X. The current study analyzes five unique non-linear regression models, a simple linear regression model, the general additive model, multivariate adaptive regression splines, support vector regression, and the response manifold regression which is a new Gaussian-kernel based regression method. A brief mathematical description and explanation of these regression techniques follows. 4

5 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics - Linear Regression Model (LI) The linear model applied in multiple dimensions is of the form s η = ηa + v () Where a is the regression coefficient vector and v is the intercept vector []. - General Additive Model (GAM) The general additive model is a more rigorous concept of the linear model where instead of fitting a regression coefficient vector, functions are fitted in attempt to more accurately model the dependent variable. The general form of the model is where f η are functions dependent on η []. - Multivariate Adaptive Regression Splines (MARS) s η = f η η (2) Multivariate adaptive regression splines use the concept of building up the model from product spline basis functions. This model creates a number of basis functions, and automatically determines knot location and implements splines at knot boundaries. The model is of the form M s η = a m B m (η) (3) m= where B m are the basis functions and a m are the expansion coefficients [2]. - Support Vector Regression (SVR) Support vector regression is a subset of the support vector machine work. The idea behind SVR is again to create a model which predicts s η given η using learning machines which implement the structural risk minimization inductive principle. The basic model form is s η = (αi α i ) K (η,η i ) (4) i= where α i and α i are Lagrange multipliers, and K (η,η i ) is the kernel operator [3]. - Response Manifold (RM) The response manifold approach uses the concept of the adaraya-watson kernel estimation [4, 5]: K (η,η i ) s η,i i= s η = (5) K (η,η i ) i= where the kernel function K provides the highest weights to the neighboring data points giving a local estimation, being dependent on the selected filter width. η is the current value 5

6 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics of the test PCs, η i are the training PCs, s η,i are the training source-terms and K (η,η i ) is the kernel operator evaluated at the current η. The drawback to the adaraya-watson kernel estimation is the expensive evaluation of the kernel function when dealing with large data sets, or domains. The response manifold approach tabulates a manifold for the dependent variables the response based on a grid spanning the independent variable space. During run-time a quick interpolation provides the dependent variables from the tabulated response manifold. A kernel filter width, and manifold grid point spacing is selected so as to avoid over-fitting and provide accurate estimation of source-terms. 2.5 Data Set The training data set, used to create the non-linear regression models and for the PC analysis, must contain several important features. First, the data-set should contain realizations both varying temporally and spatially. Second the training data-set must produce the same low dimensional manifold in PC space as the combustion problem of interest. It has been demonstrated [8] that the low dimensional manifolds, may in fact be invariant under certain conditions, allowing a system to be modeled using the PCA from a similar combustion case. For demonstration of the regression process for use in the PC Transport mode, detailed in Section 2.3, a two-dimensional Direct umerical Simulation (DS) data set of a spherical premixed syn-gas air flame has been selected [6]. The detailed reaction scheme [7], contains 3 chemical species (CO, HCO, CH 2 O, CO 2, H 2 O, O 2, O, H, OH, HO 2, H 2 O 2, H 2, 2 ), and uses 67 chemical reactions. The DS is initialized with a unity equivalence ratio, with the fuel consisting of.5 CO and.5 H 2, air as an oxidizer, and a rms turbulent velocity fluctuation (u ) of m/s. The DS assumes a unity Lewis number, and has grid consisting of 8 by 8 points spanning.2 by.2 meters. The data set consists of 3 time-steps. 3 Results and Discussion Results using the various regression models outlined in Section 2.4 were computed with the statistical computing package R, and Matlab. The R code implementations for LI, GAM, MARS, and SVM were used. The authors RM model was implemented in Matlab. Sample training and testing data sets were taken randomly from the data set, where each sample consisted of, observations. The coefficient of determination (R 2 ) and the normalized root mean square error (RMSE) were used as a means of quantifying the error produced by the models: R 2 = RMSE = (x predicted,i x) 2 i= (6) (x i x) 2 i= (x predicted,i x i ) 2 i= max(σ(x predicted,x)) 6 (7)

7 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics.8.6 RAGE R η Figure : Mean R 2 from the reconstruction of the chemical species mass fractions, as a function of the number of retained η, using scaling. First the ability of PCA to represent the chemical species mass fractions is shown in Figures and 2. Figure shows R 2 is greater than.95 when η 4 (for scaling), and Figure 2 shows a RMSE value less than.85. As mentioned in Section 2.3 the propagation of error from the approximation of the state space is much more severe in the calculation of s η. Figures 3 and 4 show the error in the calculation of s η using Equation 9 directly. It is observed that when η 9 the coefficient of determination is great than.95 and a significant reduction in mean RMSE is observed. In order to avoid the calculation of the chemical species source-terms and to prohibit the large degree of error in the calculation of s η which is seen when using fewer η, the regression methods are now tested. Table shows the resultant error for the estimation of s η when four PCs are retained, regressing the value of s η on only three PCs. Table : R 2 for estimation of s η using scaling with various Regression Methods Regression Method s η, s η,2 s η,3 s η,4 LI GAM MARS SVM RM A dramatic improvement in the estimation of s η is observed. This dramatic improvement is consistent with the work by Biglari [8]. In particular the SVM, and RM regression methods show remarkable accuracy in estimating the source-terms. Another important factor in PCA is the scaling factor γ used in Equation 2. Figures 5 and 6 show the PCA manifold calculated using scaling and RAGE scaling respectively. The color in the Figures represent the value of the first PC source term s η,. Here one observes the difference in scales, shapes, and gradients the different 7

8 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics RMSE RAGE η Figure 2: Mean RMSE from the reconstruction of the chemical species mass fractions, as a function of the number of retained η, using scaling..8.6 RAGE R η Figure 3: Mean R 2 from the reconstruction of the source-terms for the species mass fractions, as a function of the number of retained η, using scaling. 8

9 8th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics 2 RAGE RMSE η Figure 4: Mean RMSE from the reconstruction of sη, as a function of the number of retained η, using scaling. 6 x 2 η η2 4 η 2 Figure 5: PCA manifold created using scaling, here the independent variables are η, η2, and η3 representing the cartesian coordinates, and the dependent variable sη mapped in color to the manifold. scaling methods produce. Tables 2 and 3 show R2 and RMSE error metrics using the various scaling methods presented in Section 2.2 while using the RM regression method. Table 2: R2 for the estimation of sη using the RM regression method with various scaling methods Scaling Method RAGE sη, sη, sη, sη,

10 8th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics 4 x 2 η η2.5 η.5 Figure 6: PCA manifold created using RAGE scaling, here the independent variables are η, η2, and η3 representing the cartesian coordinates, and the dependent variable sη mapped in color to the manifold. Table 3: RMSE for the estimation of sη using the RM regression method with various scal- ing methods Scaling Method RAGE sη, sη, sη, sη, A clear benefit is seen while using the RM regression method with or scaling. This is due to that the fact that these manifolds contain smooth gradients, and a more simplified manifold shape allowing for a more accurate regression. 4 Conclusion The current work has addressed the ability to use non-linear regression methods to estimate sourceterms for a PCA based combustion model. Various non-linear regression methods have been analyzed showing the ability to produce accurate estimation even when using a lower number of η. In particular the SVM and RM methods have shown desired accuracy in estimation of sη. In addition the effect of various PCA scaling methods on the non-linear regression models have been assessed, with excellent results using and scaling. The current work outlines an example of an apriori analysis which provides the best regression and scaling method for a given turbulent combustion data set. Additional work may include an analysis on the invariance of the PC based manifold with respect to flow conditions, specifically by increasing the turbulence intensity, and a demonstration of the non-linear regression methods in conjunction with a simple perfectly stirred reactor system.

11 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics Acknowledgments We are grateful to our sponsor for which part of the present research was funded: The ational uclear Security Administration under the Accelerating Development of Retrofittable CO 2 Capture Technologies through Predictivity program through DOE Cooperative Agreement DE A 74. References [] Alessandro Parente, James C. Sutherland, L. Tognotti, and Philip J. Smith. Proceedings of the Combustion Institute, (29) journal. [2] J. C. Sutherland and A. Parente. Proceedings of the Combustion Institute, 32 (29) [3] G. P. Smith, D. M. Golden, M. Frenklach,. W. Moriarty, B. Eiteneer, M. Goldenberg, C. T. Bowman, R. K. Hanson, S. Song, W. C. Gardiner, V. V. Lissianski, and Z. Qin journal. [4] R. Fox. Computational Models for Turbulent Reacting Flows. Cambridge University Press, 23. [5] W.P. Jones and Rigopoulos Stelios. Combustion and Flame, 42 (25) [6]. Peters. Progress in Energy and Combustion Science, (984) [7] James C. Sutherland and Alessandro Parente. Proceedings of the Combustion Institute, 32 (29) [8] I. T. Jolliffe. Principal Component Analysis. Springer, ew York, Y, 986. [9] T. Poinsot and D. Veynante. Theoretical and umerical Combustion. R.T. Edwards, Inc., 2. [] William S Cleveland, Eric Grosse, and William M Shyu. Statistical models in S, (992) [] Simon Wood. Generalized additive models: an introduction with R, Vol. 66. Chapman & Hall/CRC, 26. [2] Jerome H Friedman. The annals of statistics, (99) 67. [3] Alex J Smola and Bernhard Schölkopf. Statistics and computing, 4 (24) [4] Elizbar A adaraya. Theory of Probability & Its Applications, 9 (964) [5] Geoffrey S Watson. Sankhyā: The Indian Journal of Statistics, Series A, (964) [6] Gordon Fru, Gábor Janiga, and Dominique Thévenin. Flow, turbulence and combustion, 88 (22) [7] Ulrich Maas and Stephen B Pope. Combustion and Flame, 88 (992) [8] Amir Biglari and James C Sutherland. Combustion and Flame, (22) journal.