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pyJac: analytical Jacobian generator for chemical kinetics

Kyle E. Niemeyer, Nicholas J. Curtis, Chih-Jen Sung

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Abstract

Accurate simulations of combustion phenomena require the use of detailed chemical kinetics in order to capture limit phenomena such as ignition and extinction as well as predict pollutant formation. However, the chemical kinetic models for hydrocarbon fuels of practical interest typically have large numbers of species and reactions and exhibit high levels of mathematical stiffness in the governing differential equations, particularly for larger fuel molecules. In order to integrate the stiff equations governing chemical kinetics, generally reactive-flow simulations rely on implicit algorithms that require frequent Jacobian matrix evaluations. Some in situ and a posteriori computational diagnostics methods also require accurate Jacobian matrices, including computational singular perturbation and chemical explosive mode analysis. Typically, finite differences numerically approximate these, but for larger chemical kinetic models this poses significant computational demands since the number of chemical source term evaluations scales with the square of species count. Furthermore, existing analytical Jacobian tools do not optimize evaluations or support emerging SIMD processors such as GPUs. Here we introduce pyJac, a Python-based open-source program that generates analytical Jacobian matrices for use in chemical kinetics modeling and analysis. As a demonstration, we first establish the correctness of the Jacobian matrices for kinetic models of hydrogen, methane, ethylene, and isopentanol oxidation, then demonstrate the performance achievable on CPUs and GPUs using pyJac via matrix evaluation timing comparisons.

Preprint submitted to Computer Physics Communications February 21, 2017 Keywords: Chemical kinetics, Jacobian, SIMD, GPU

PROGRAM SUMMARY

Manuscript Title: pyJac: analytical Jacobian generator for chemical kinetics

Authors: Kyle E. Niemeyer, Nicholas J. Curtis, Chih-Jen Sung

Program Title: pyJac

Journal Reference:

Catalogue identifier:

Licensing provisions: MIT License

Programming language: Python

Computer: Windows, Ubuntu, and Mac OS computers

Operating system: Any (Linux, Mac OS X, Windows)

RAM: Problem dependent; typically less than 1 GB

Number of processors used: one

Classification: 16.12, 4.3

External routines/libraries: required: NumPy; optional: Cython, Cantera, PyYAML, Adept

Nature of problem: Automatic generation of source code to evaluate Jacobian matrices for chemical kinetic models

Solution method: Chemical kinetic model interpreted from input file(s), and partial derivatives and necessary supporting functions are written to file based on a theoretical derivation.

Additional comments: This paper describes pyJac v1.0.2, available at http://dx.doi.org/10.5281/zenodo.251144. Current version and support available at https://github.com/SLACKHA/pyjac/

Running time: Problem dependent; from seconds to tens of minutes depending on the model size.

Keywords: Chemical kinetics, Jacobian, SIMD, GPU

Introduction

As the need for detailed and accurate chemical kinetic models2 in predictive reactive-flow simulations has become recognized in recent years, such models describing the oxidation of hydrocarbon fuels simultaneously grew orders of magnitude in size and complexity. For example, a recently developed detailed kinetic model for 2-methylalkanes, relevant for jet and diesel fuel surrogates, consists of over 7000 species and 30000 reactions [1]; similarly large surrogate models exist for gasoline [2, 3] and biodiesel [4]. Since in general the computational cost of solving the associated generally stiff systems of equations scales quadratically with the number of species at best—and at worst, cubically—models of such a large size pose challenges even for lower-dimensional analyses, and cannot practically be used directly in multidimensional reactive-flow simulations.

In an effort to reduce the computational demands of using large, detailed kinetic models, a number of techniques have been developed to reduce their size and complexity while retaining predictiveness, as reviewed by Lu and Law [5] as well as Turányi and Tomlin [6]. Major classes of such approaches include skeletal reduction methods that remove unimportant species and reactions [7, 8, 9, 10], lumping of species that share similar properties [11, 12, 13], and time-scale reduction methods that reduce chemical stiffness [14, 15, 16, 17]. Effective reduction strategies either combine multiple methods a priori [18, 19, 20] or apply them dynamically during a simulation to achieve greater local savings [21, 22, 23, 24, 25, 26]. Techniques such as interpolation/tabulation of expensive terms [27] can also reduce computational costs.

In addition to the aforementioned cost reduction methods that modify the chemical kinetic models, improvements to the integration algorithms that actually solve the governing differential equations can also offer significant gains in computational performance. Due to the chemical stiffness exhibited by most kinetic models, solvers typically rely on robust, high-order implicit integration algorithms based on backward differentiation formulae [28, 29, 30, 31]. In order to solve the nonlinear algebraic equations that arise in these methods, the Jacobian matrix must be evaluated and factorized, operations that result in the quadratic and cubic costs mentioned previously. However, by using an analytical formulation for the Jacobian matrix rather than a typical finite difference approximation, the cost of the numerous evaluations can drop from growing with the square of the number of species to a linear dependence [5].

In parallel with the potential improvements in stiff implicit integrators used for chemical kinetics, algorithms tailored for high-performance hardware accelerators offer another route to reducing computational costs. In the past, central processing unit (CPU) clock speeds increased regularly—i.e., Moore’s Law—but power consumption and heat dissipation issues disrupted this trend recently, slowing the pace of increases in CPU clock rates. While multicore parallelism continues to raise CPU performance, single-instruction multiple data (SIMD) processors, e.g., graphics processing units (GPUs), recently emerged as a low-cost, low-power consumption, and massively parallel high-performance computing alternative. GPUs—originally developed for graphics/video processing and display purposes—consist of hundreds to thousands of cores, compared to the tens of cores found on a typical CPU. Recognizing that the SIMD parallelism model fits well with the operator-split chemistry integration that forms the basis of many reactive-flow codes [32], a number of studies in recent years [33, 34, 35, 36, 37, 38] explored the use of SIMD processors to accelerate the integration of chemical kinetics in reactive-flow codes. Niemeyer and Sung [39] reviewed such efforts in greater detail. While explicit methods offer significant improvements in performance for nonstiff and moderately stiff chemical kinetics [38], experiences thus far suggest that stiff chemistry continues to require the use of implicit or similar algorithms. This provides significant motivation to provide the capability of evaluating analytical Jacobian matrices on GPUs as well as CPUs.

Thus, motivated by the potential cost reductions offered by analytical Jacobian matrix formulations, over the past five years a number of research groups developed analytical Jacobian generators for chemical kinetics, although as will be discussed the software package introduced here offers a number of improvements. The TChem toolkit developed by Safta et al. [40] was one of the first software packages developed for calculating the analytical Jacobian matrix, but provides this functionality through an interface rather than generating customized source code for each model. Youssef [41] recognized the importance of using an analytical Jacobian over a numerical approximation to reduce both computational cost and numerical error when performing eigendecomposition of the matrix. Bisetti [42] released a utility for producing analytical Jacobian matrix source code based on isothermal and isobaric conditions, with the state vector comprised of species concentrations rather than mass fractions; while incompatible with many existing reactive-flow formulations, this formulation resulted in a significant increase of Jacobian matrix sparsity—such a strategy should be investigated further. Perini et al. [43] developed an analytical Jacobian matrix formulation for constant-volume combustion, and when used in a multidimensional reactive-flow simulation—combined with tabulation of temperature-dependent properties—reported a performance improvement of around 80 % over finite-difference-based approximations. Recently, Dijkmans et al. [44] used a GPU-based analytical Jacobian combined with tabulation of temperature-dependent functions based on polynomial interpolations to accelerate the integration of chemical kinetics equations, similar to the earlier approach of Shi et al. [34]. Unlike the current work, the approach of Dijkmans et al. [44] used the GPU to calculate the elements of a single Jacobian matrix in parallel, rather than a large number of matrices corresponding to different states.

To our knowledge, currently no open-source analytical chemical Jacobian tool exists that is capable of generating code specifically optimized for SIMD processors. To this end, pyJac is capable of generating subroutines for reaction rates, species production rates, derivative source terms, and analytical chemical Jacobian matrices both for CPU operation via C/C++ and GPU operation via CUDA [45], a widely used programming language for NVIDIA GPUs. Furthermore, pyJac supports newer pressure-dependent reaction formulations (i.e., based on logarithmic or Chebyshev polynomial interpolation).

The rest of the paper is structured as follows. First, in Section 2 we introduce the governing equations for chemical kinetics and then provide the analytical Jacobian matrix formulation in Section 3. Next, Section 4 describes the techniques used to optimize evaluation of the analytical Jacobian on both CPUs and GPUs. Then, in Section 5 we demonstrate the correctness and computational performance of the generated analytical Jacobian matrices for benchmark chemical kinetic models, and discuss the implications of these results. Finally, we summarize our work in Section 6 and outline future research directions.

Theory

This section describes the theoretical background of the analytical Jacobian generator, first in terms of the governing equations and then the components of the Jacobian matrix itself. The literature contains more detailed explanations of the governing equations development [46, 47, 48], but we include the necessary details here for completeness.

Governing equations

The initial value problem to be solved, whether in the context of a single homogeneous reacting system (e.g., autoignition, perfectly stirred reactor) or the chemistry portion of an operator-split multidimensional reactive-flow simulation [32], is described using an ordinary differential equation for the thermochemical composition vector

Φ={T,Y1,Y2,…,YNsp−1}T,(1)\Phi= \left\{ T, Y_1, Y_2, \ldots, Y_{N_{\mathrm{sp}}-1} \right\}^{\mathrm{T}}, \tag*{(1)}

where TT is the temperature, YiY_i are the species mass fractions, and NspN_{\mathrm{sp}} is the number of species. The mass fraction of the final species, YNspY_{N_{\mathrm{sp}}}, is determined through conservation of mass:

YNsp=1−∑k=1Nsp−1Yk(2)Y_{N_{\mathrm{sp}}} = 1 - \sum_{k=1}^{N_{\mathrm{sp}}-1} Y_k \tag*{(2)}

where the most abundant species (e.g., N2\mathrm{N}_2 in air-fed combustion) can be assigned to this role. In multidimensional simulations where the equations for chemical kinetics are coupled to conservation of energy (or enthalpy), temperature can be determined algebraically in a straightforward manner [32]. Completely defining the thermodynamic state also requires either pressure (pp) or density (ρ\rho), related to temperature and the mixture composition through the ideal equation of state

p=ρRWT=RT∑k=1Nsp[Xk](3)p = \rho\frac{\mathcal{R}}{W} T = \mathcal{R}T \sum_{k=1}^{N_{\mathrm{sp}}} [X_k] \tag*{(3)}

where R\mathcal{R} is the universal gas constant, WW is the average molecular weight of the mixture, and [Xk][X_k] is the molar concentration of the kkth species. In the current implementation of pyJac, we assume a constant-pressure system3; thus, we use (3) to determine density rather than including it in the differential system given by (1). The average molecular weight is defined by

W=1∑k=1NspYk/Wk=ρRTp(4)W = \frac{1}{\sum_{k=1}^{N_{\mathrm{sp}}} Y_k/W_k} = \frac{\rho\mathcal{R} T}{p} \tag*{(4)}

and the molar concentrations by

[Xk]=ρYkWk(5)[X_k] = \rho\frac{Y_k}{W_k} \tag*{(5)}

where WkW_k is the molecular weight of the kkth species.

The system of ODEs governing the time change in thermochemical composition corresponding to (1) is then f=∂Φ/∂tf = \partial\Phi/\partial t:

f=∂Φ∂t={∂T∂t,∂Y1∂t,∂Y2∂t,…,∂YNsp−1∂t}T(6)f = \frac{\partial\Phi}{\partial t} = \left\{ \frac{\partial T}{\partial t}, \frac{\partial Y_1}{\partial t}, \frac{\partial Y_2}{\partial t}, \ldots, \frac{\partial Y_{N_{\mathrm{sp}}-1}}{\partial t} \right\}^{\mathrm{T}} \tag*{(6)}

where

∂T∂t=−1ρcp∑k=1NsphkWkω˙k(7)\frac{\partial T}{\partial t} = -\frac{1}{\rho c_p} \sum_{k=1}^{N_{\mathrm{sp}}} h_k W_k \dot{\omega}_k \tag*{(7)}
∂Yk∂t=Wkρω˙kk=1,…,Nsp−1(8)\frac{\partial Y_k}{\partial t} = \frac{W_k}{\rho}\dot{\omega}_k \qquad k = 1,\ldots,N_{\mathrm{sp}}-1 \tag*{(8)}

where cpc_p is the mass-averaged constant-pressure specific heat, hkh_k is the enthalpy of the kkth species in mass units, and ω˙k\dot{\omega}_k is the kkth species overall production rate.

Thermodynamic properties

The standard-state thermodynamic properties (in molar units) for a gaseous species kk is defined using the standard seven-coefficient polynomial of Gordon and McBride [49]:

Cp,k∘R=a0,k+T(a1,k+T(a2,k+T(a3,k+a4,kT)))(9)\frac{C_{p,k}^{\circ}}{R} = a_{0,k} + T\left(a_{1,k} + T\left(a_{2,k} + T\left(a_{3,k} + a_{4,k}T\right)\right)\right) \tag*{(9)}
Hk∘R=T(a0,k+T(a1,k2+T(a2,k3+T(a3,k4+a4,k5T))))+a5,k(10)\frac{H_k^{\circ}}{R} = T\left(a_{0,k} + T\left(\frac{a_{1,k}}{2} + T\left(\frac{a_{2,k}}{3} + T\left(\frac{a_{3,k}}{4} + \frac{a_{4,k}}{5}T\right)\right)\right)\right) + a_{5,k} \tag*{(10)}
Sk∘R=a0,kln⁡T+T(a1,k+T(a2,k2+T(a3,k3+a4,k4T)))+a6,k(11)\frac{S_k^{\circ}}{R} = a_{0,k}\ln T + T\left(a_{1,k} + T\left(\frac{a_{2,k}}{2} + T\left(\frac{a_{3,k}}{3} + \frac{a_{4,k}}{4}T\right)\right)\right) + a_{6,k} \tag*{(11)}

where Cp,kC_{p,k} is the constant-pressure specific heat in molar units, HkH_k is the enthalpy in molar units, SkS_k is the entropy in molar units, and the ∘^{\circ} indicates a standard-state property at one atmosphere (equivalent to the property at any pressure for calorically perfect gases).

The mass-based specific heat and enthalpy are then defined as

cp,k=Cp,kWkandhk=HkWk,(12)c_{p,k} = \frac{C_{p,k}}{W_k} \quad\text{and} \quad h_k = \frac{H_k}{W_k}, \tag*{(12)}

and the mixture-averaged specific heat is

cp=∑k=1NspYkcp,k.(13)c_p = \sum_{k=1}^{N_{\mathrm{sp}}} Y_k c_{p,k}. \tag*{(13)}

Reaction rate expressions

Next, we define the species rates of production and related kinetic terms as

ω˙k=∑i=1Nreacνkiqi,(14)\dot{\omega}_k = \sum_{i=1}^{N_{\mathrm{reac}}} \nu_{ki}q_i, \tag*{(14)}

where NreacN_{\mathrm{reac}} is the number of reactions, νki\nu_{ki} is the overall stoichiometric coefficient for species kk in reaction ii, and qiq_i is the rate-of-progress for reaction ii. These are defined by

νki=νki′′−νki′and(15)\nu_{ki} = \nu_{ki}^{\prime\prime} - \nu_{ki}^{\prime} \quad\text{and} \tag*{(15)}
qi=ciRi,(16)q_i = c_i R_i, \tag*{(16)}

where νki′′\nu_{ki}^{\prime\prime} and νki′\nu_{ki}^{\prime} are the product and reactant stoichiometric coefficients (respectively) of species kk in reaction ii. The base rate-of-progress for the ith reversible reaction RiR_i is given by

Ri=Rf,i−Rr,i,(17)R_i = R_{f,i} - R_{r,i}, \tag*{(17)}

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