Computational Singular Perturbation Analysis Research Articles

CSPlib: A performance portable parallel software toolkit for analyzing complex kinetic mechanisms

Computational singular perturbation (CSP) is a method to analyze dynamical systems. It targets the decoupling of fast and slow dynamics using an alternate linear expansion of the right-hand side of the governing equations based on eigenanalysis of the associated Jacobian matrix. This representation facilitates diagnostic analysis, detection and control of stiffness, and the development of simplified models. We have implemented CSP in a C++ open-source library CSPlib1 using the Kokkos2 parallel programming model to address portability across diverse heterogeneous computing platforms, i.e., multi/many-core CPUs and GPUs. We describe the CSPlib implementation and present its computational performance across different computing platforms using several test problems. Specifically, we test the CSPlib performance for a constant pressure ignition reactor model on different architectures, including IBM Power 9, Intel Xeon Skylake, and NVIDIA V100 GPU. The size of the chemical kinetic mechanism is varied in these tests. As expected, the Jacobian matrix evaluation, the eigensolution of the Jacobian matrix, and matrix inversion are the most expensive computational tasks. When considering the higher throughput characteristic of GPUs, GPUs performs better for small matrices with higher occupancy rate. CPUs gain more advantages from the higher performance of well-tuned and optimized linear algebra libraries such as OpenBLAS. Program summaryProgram Title: CSPlibCPC Library link to program files:https://doi.org/10.17632/p9gb7z54sp.1Developer's repository link:https://github.com/sandialabs/csplibLicensing provisions: BSD 2-clauseProgramming language: C++Nature of problem: Dynamical systems can involve coupled processes with a wide range of time scales. The computational singular perturbation (CSP) method offers a reformulation of these systems which enables the use of dynamically-based diagnostic tools to better comprehend the dynamics by decoupling fast and slow processes. CSPlib is an open-source software library for analyzing general ordinary differential equation (ODE) and differential algebraic equation (DAE) systems, with specialized implementations for detailed chemical kinetic ODE/DAE systems. It relies on CSP for the analysis of these systems. CSPlib has been used in gas kinetic and heterogeneous catalytic kinetic models.Solution method: CSP analysis seeks a set of basis vectors to linearly decompose the right-hand side (RHS) of a dynamical system in a manner that decouples fast and slow processes. The CSP basis vectors are often well approximated with the right eigenvectors of the RHS Jacobian. And the left basis vectors are found by the inversion of the matrix, whose columns are the CSP basis vectors. Accordingly, the right and left CSP basis vectors are orthonormal. CSP defines mode amplitudes as the projections of the left basis vectors on the RHS; the time scales as the reciprocals of the RHS Jacobian eigenvalue magnitudes; and the CSP pointers, which are the element-wise multiplication of the transpose of the right CSP basis vectors with the left CSP basis vectors. For kinetic models that can be cast as the product of a generalized stoichiometric matrix and a rate of progress vector, CSP defines the participation index, which represents the contribution of a chemical reaction to each mode. Further, it defines the slow and fast importance indices, which describe the contribution of a chemical reaction to the slow and fast dynamics of a state variable, respectively. These indices are useful in diagnostic studies of dynamical systems and the construction of simplified models.Additional comments including restrictions and unusual features: CSPlib is a portable library that carries out many CSP analyses in parallel and can be used in modern high-performance platforms.

Using computational singular perturbation as a diagnostic tool in ODE and DAE systems: a case study in heterogeneous catalysis

We have extended the computational singular perturbation (CSP) method to differential algebraic equation (DAE) systems and demonstrated its application in a heterogeneous-catalysis problem. The extended method obtains the CSP basis vectors for DAEs from a reduced Jacobian matrix that takes the algebraic constraints into account. We use a canonical problem in heterogeneous catalysis, the transient continuous stirred tank reactor (T-CSTR), for illustration. The T-CSTR problem is modelled fundamentally as an ordinary differential equation (ODE) system, but it can be transformed to a DAE system if one approximates typically fast surface processes using algebraic constraints for the surface species. We demonstrate the application of CSP analysis for both ODE and DAE constructions of a T-CSTR problem, illustrating the dynamical response of the system in each case. We also highlight the utility of the analysis in commenting on the quality of any particular DAE approximation built using the quasi-steady state approximation (QSSA), relative to the ODE reference case.

Computational singular perturbation analysis of brain lactate metabolism.

Lactate in the brain is considered an important fuel and signalling molecule for neuronal activity, especially during neuronal activation. Whether lactate is shuttled from astrocytes to neurons or from neurons to astrocytes leads to the contradictory Astrocyte to Neuron Lactate Shuttle (ANLS) or Neuron to Astrocyte Lactate Shuttle (NALS) hypotheses, both of which are supported by extensive, but indirect, experimental evidence. This work explores the conditions favouring development of ANLS or NALS phenomenon on the basis of a model that can simulate both by employing the two parameter sets proposed by Simpson et al. (J Cereb. Blood Flow Metab., 27:1766, 2007) and Mangia et al. (J of Neurochemistry, 109:55, 2009). As most mathematical models governing brain metabolism processes, this model is multi-scale in character due to the wide range of time scales characterizing its dynamics. Therefore, we utilize the Computational Singular Perturbation (CSP) algorithm, which has been used extensively in multi-scale systems of reactive flows and biological systems, to identify components of the system that (i) generate the characteristic time scale and the fast/slow dynamics, (ii) participate to the expressions that approximate the surfaces of equilibria that develop in phase space and (iii) control the evolution of the process within the established surfaces of equilibria. It is shown that a decisive factor on whether the ANLS or NALS configuration will develop during neuronal activation is whether the lactate transport between astrocytes and interstitium contributes to the fast dynamics or not. When it does, lactate is mainly generated in astrocytes and the ANLS hypothesis is realised, while when it doesn’t, lactate is mainly generated in neurons and the NALS hypothesis is realised. This scenario was tested in exercise conditions.

Chemical Ignition Characteristics of Ethanol Blending with Primary Reference Fuels

Synergistic octane blending behavior of ethanol with gasoline and its surrogates has been observed by many researchers. The nonlinear octane boosting tendency is observed at mid and high molar blends of ethanol in primary reference fuels. The present work aims to provide chemical insight into this nonlinear blending behavior of ignition processes when ethanol is blended with primary reference fuels. To this end, ignition delay time (IDT) calculations, using a well-validated mechanism, were performed for several fuel blends of iso-octane, n-heptane, and ethanol. Temperature and pressure values were found, correlating experimentally measured octane numbers and simulated homogenous batch reactor IDTs. The temperature and pressure conditions obtained, were then used to study the evolution of heat release and reactivity before the onset of auto-ignition in a homogeneous premixed reactor. Markers of low- and high-temperature reactivity (OH and HO2) were analyzed for various molar blends of n-heptane with ethanol–iso-octane. Ethanol was observed to be better at radical scavenging than iso-octane at a higher mole fraction. A computational singular perturbation analysis was conducted for a selection of blends to clarify the reactions responsible for the synergistic blending behavior of ethanol in n-heptane. The role of the H-abstraction reactions was highlighted during the first ignition stage; reactions related to n-heptane were found to compete with the H-abstraction reactions of iso-octane or ethanol. Notably, the H-abstraction path of ethanol was more favored than that of the iso-octane, as a result of the smaller activation energies of the related reactions in ethanol. The competition of the H-abstraction paths resulted in a smaller radical pool in the n-heptane–iso-octane–air case, and an even smaller pool in n-heptane–ethanol–air. In all the cases considered, the second stage was dominated mainly by hydrogen-related reactions, regardless of the initial mixture, with H2O2 (+M) → 2OH (+M) and H + O2 → O + OH playing the most important roles. This work employed a novel approach to examine specific reactions responsible for auto-ignition in ethanol blends, which can be used for fuel design, primarily around the generation/consumption of radical pool intermediates by interaction with fuel components.

Effects of Langmuir Turbulence on Upper Ocean Carbonate Chemistry

AbstractEffects of wave‐driven Langmuir turbulence on the air‐sea flux of carbon dioxide (CO2) are examined using large eddy simulations featuring actively reacting carbonate chemistry in the ocean mixed layer at small scales. Four strengths of Langmuir turbulence are examined with three types of carbonate chemistry: time‐dependent chemistry, instantaneous equilibrium chemistry, and no reactions. The time‐dependent model is obtained by reducing a detailed eight‐species chemical mechanism using computational singular perturbation analysis, resulting in a quasi steady state approximation for hydrogen ion (H+); that is, fixed pH. The reduced mechanism is then integrated in two half‐time steps before and after the advection solve using a Runge‐Kutta‐Chebyshev scheme that is robust for stiff systems of differential equations. The simulations show that as the strength of Langmuir turbulence increases, CO2 fluxes are enhanced by rapid overturning of the near‐surface layer, which rivals the removal rate of CO2 by time‐dependent reactions. Equilibrium chemistry and nonreactive models are found to bring more and less carbon, respectively, into the ocean as compared to the more realistic time‐dependent model. These results have implications for Earth system models that either neglect Langmuir turbulence or use equilibrium, instead of time‐dependent, chemical mechanisms.

Reduction of a detailed chemical mechanism for a kerosene surrogate via RCCE-CSP

Detailed mechanisms for kerosene surrogate fuels contain hundreds of species and thousands of reactions, indicating a necessity for reduced mechanisms. In this work, we employ a framework that combines Rate-Controlled Constrained Equilibrium (RCCE) with Computational Singular Perturbation (CSP) for systematic reduction based on timescale analysis, to reduce a detailed mechanism for a jet fuel surrogate with n-dodecane, methylcyclohexane and m-xylene. Laminar non-premixed flamelets are utilised for the CSP analysis for different strain rates and therefore different scalar dissipation rate, covering the flammable region of strain rates for the surrogate fuel.Two RCCE-reduced mechanisms are developed via an RCCE-CSP methodology, one with 17 and one with 42 species, and their accuracy is assessed in a range of cases that test the performance of the reduced mechanism under both non-premixed and premixed conditions and its dynamic response. These include non-premixed flamelets with varying strain rate, laminar premixed flames for a range of equivalence ratios and pressures, flamelets ignited by an artificial pilot or by hot air, and unsteady flamelets with time-dependent strain rate.The profiles of both major and minor species, as well as important combustion characteristics such as the ignition strain rate and the laminar flame speed, are investigated. The structure of non-premixed flamelets is very well predicted, while the premixed flames are overall well predicted apart from a few deviations in certain species and an underprediction in the laminar flame speed. Apart from the large reduction in dimensionality, the reduction in computational time is also considerable (up to 19 times). As the detailed mechanism comprises 367 species and 1892 reactions, this paper presents the first application of RCCE to a mechanism of this size, as well as a comprehensive validation in a set of cases that include non-premixed and premixed laminar flames, atmospheric and elevated pressures and steady-state and dynamic response.

Computational singular perturbation analysis of super-knock in SI engines

Pre-ignition engine cycles leading to super-knock were simulated with a 48 species skeletal iso-octane mechanism to identify the dominant reaction pathways that are present in super-knock. To mimic pre-ignition, a deflagration front was generated via a hot spot that is placed over the piston at close proximity to the end-wall. Computational singular perturbation (CSP) was used to analyze the chemical dynamics at various in-cylinder locations: a point at the center of the cylinder where the deflagration front consumes the air/fuel mixture and two points located at 3 mm from the end-wall where super-knock and mild knock occur. The CSP analysis of the point at the center of the cylinder reveals weak two-stage ignition-like dynamics with a short second stage. At the other points, a pronounced two-stage ignition is displayed with a long second stage. A distinct contribution of formaldehyde (CH2O) at the second stage of ignition that adds to fast explosive modes in the super-knock points is not observed in the point at the center. A comparison between knock and super-knock analysis indicates that a similar set of reactions is responsible for the abnormal behavior but the fast explosive time scales are comparatively slower for knock, indicating lower reactivity, which results in the reduced intensity of knock. The analyzed results decoded important reactions responsible for the occurrence of super-knock.

Chemical model reduction under uncertainty

A general strategy for analysis and reduction of uncertain chemical kinetic models is presented, and its utility is illustrated in the context of ignition of hydrocarbon fuel–air mixtures. The strategy is based on a deterministic analysis and reduction method which employs computational singular perturbation analysis to generate simplified kinetic mechanisms, starting from a detailed reference mechanism. We model uncertain quantities in the reference mechanism, namely the Arrhenius rate parameters, as random variables with prescribed uncertainty factors. We propagate this uncertainty to obtain the probability of inclusion of each reaction in the simplified mechanism. We propose probabilistic error measures to compare predictions from the uncertain reference and simplified models, based on the comparison of the uncertain dynamics of the state variables, where the mixture entropy is chosen as progress variable. We employ the construction for the simplification of an uncertain mechanism in an n-butane–air mixture homogeneous ignition case, where a 176-species, 1111-reactions detailed kinetic model for the oxidation of n-butane is used with uncertainty factors assigned to each Arrhenius rate pre-exponential coefficient. This illustration is employed to highlight the utility of the construction, and the performance of a family of simplified models produced depending on chosen thresholds on importance and marginal probabilities of the reactions.

Computational singular perturbation analysis of stochastic chemical systems with stiffness

Computational singular perturbation (CSP) is a useful method for analysis, reduction, and time integration of stiff ordinary differential equation systems. It has found dominant utility, in particular, in chemical reaction systems with a large range of time scales at continuum and deterministic level. On the other hand, CSP is not directly applicable to chemical reaction systems at micro or meso-scale, where stochasticity plays an non-negligible role and thus has to be taken into account. In this work we develop a novel stochastic computational singular perturbation (SCSP) analysis and time integration framework, and associated algorithm, that can be used to not only construct accurately and efficiently the numerical solutions to stiff stochastic chemical reaction systems, but also analyze the dynamics of the reduced stochastic reaction systems. The algorithm is illustrated by an application to a benchmark stochastic differential equation model, and numerical experiments are carried out to demonstrate the effectiveness of the construction.

Modeling, estimation and optimization in coreflooding experiments for coalbed methane production

We extend a previously derived kinetic model for coal bioconversion and couple it with a transport model to simulate coreflooding experiments with packed crushed coal, which are representations of a coalbed methane (CBM) reservoir at the laboratory scale. We apply a tanks-in-series model to simulate plug flow in the core, and the nonlinear model is regressed against experimental data using particle swarm optimization. The validated model is used to analyze CBM production at different operating conditions and subsequently for the optimization of gas production. Model-based experimental design is applied to improve the accuracy of parameter estimation, and computational singular perturbation analysis is applied to develop a better understanding of the important species and reaction at each stage of the coreflooding experiment, and to develop reduced order kinetic models that can be used in process optimization.

Detailed transient numerical simulation of H2/air hetero-/homogeneous combustion in platinum-coated channels with conjugate heat transfer

Transient two-dimensional simulations of fuel-lean H2/air combustion were performed in a 2-mm-height planar channel coated with platinum, using detailed hetero-/homogeneous chemistry and transport as well as heat conduction in the solid wall. The developed model resolved, for the first time, all relevant spatiotemporal scales in a practical channel-flow reactor configuration. A parametric study was carried out to investigate the effects of wall material, inlet velocity, and inlet temperature on the fundamental catalytic and gas-phase combustion processes. Computational singular perturbation (CSP) analysis identified the key catalytic reactions affecting light-off and homogeneous ignition. Homogeneous ignition crucially depended on the OH desorbing fluxes from the catalyst, while flame propagation and stabilization involved time scales of a few milliseconds. During the short duration of the light-off event, the ensuing Stefan velocity appreciably altered the flow field. Predictions of time accurate numerical simulations were further compared against those of a code relying on the quasisteady state assumption, and the specific conditions under which the latter was invalidated were identified. Finally, CSP analysis unraveled the reasons for the high computational cost of the fully transient 2-D simulations. The surface reaction mechanism exhibited a high stiffness with fastest time scales of the order of 10-12s, pertaining to the hydrogen adsorption and to the H(s)+O(s)=OH(s)+Pt(s) reactions. These time scales were in turn six orders of magnitude shorter than the ones associated with gas-phase chemistry or with a simplified single-step catalytic reaction.

Flame dynamics in lean premixed [formula omitted]/air combustion in a mesoscale channel

The dynamics and stabilization of fuel lean premixed CO/H2/air atmospheric pressure flames in meso-scale channels were investigated numerically, using detailed gas phase chemistry and transport. Experiments in a channel flow reactor by means of chemiluminescence detection of the excited OH radical allowed for model validation at steady conditions and identification of the conditions at which unsteady flame dynamics were present. A detailed parametric study of the influence of wall temperature and CO:H2 ratio on the ensuing flame dynamics was performed. The numerical results revealed different flame modes which included oscillatory ignition, random ignition spots, as well as steady weak and V-shaped flames. The wall temperature stability intervals of these modes changed with the CO:H2 ratio. The richest variety was found for molar CO:H2 ratios between 4 and 10, while at lower ratios the random and the weak modes were absent. At higher ratios all the dynamic modes were suppressed. The Computational Singular Perturbation (CSP) method was used to obtain insights into the physicochemical processes responsible for the weak flames, which were found at relatively high inflow velocities compared to previous studies, and V-shaped flames. A kinetic explanation of the phenomena was supported by the CSP analysis.

Analysis of n-heptane auto-ignition characteristics using computational singular perturbation

To provide insights into auto-ignition characteristics in low temperature combustion (LTC) engine conditions, computational singular perturbation (CSP) analysis is used to gain fundamental understanding into classification of ignition regimes in n-heptane air mixtures in the presence of temperature and composition inhomogeneities. Parametric studies are conducted using high-fidelity simulations with detailed chemistry and transport. In particular, a key interest is to understand different ignition behavior of the n-heptane mixture at the negative-temperature coefficient (NTC) versus the non-NTC conditions. The CSP analysis for the reference homogeneous system yields the number of exhausted modes (M) at various stages during the ignition event. In addition, the merging of two explosive modes is observed at the onset of auto-ignition. For the one-dimensional system, the M profiles along with the importance index (IT) measured in the upstream of the ignition front are used to determine whether the front propagation is the spontaneous (IT→0) or deflagrative (IT→1) regime. At the relatively large temperature fluctuation considered herein, the mixture at non-NTC conditions shows initially a deflagration front which is subsequently transitioned into a spontaneous ignition front. For the mixtures at the NTC conditions which exhibit two-stage ignition behavior, the 1st stage ignition front is found to be more likely in the deflagration regime. On the other hand, the 2nd stage ignition front occurs almost always in the spontaneous regime because the upstream mixture contains active radical species produced by the preceding 1st stage ignition front. The effects of differently correlated equivalence ratio stratification are also considered and the results are shown to be consistent with previous findings.

Simplified CSP analysis of a stiff stochastic ODE system

We develop a simplified computational singular perturbation (CSP) analysis of a stochastic dynamical system. We focus on the case of parametric uncertainty, and rely on polynomial chaos (PC) representations to quantify its impact. We restrict our attention to a system that exhibits distinct timescales, and that tends to a deterministic steady state irrespective of the random inputs. A detailed analysis of eigenvalues and eigenvectors of the stochastic system Jacobian is conducted, which provides a relationship between the PC representation of the stochastic Jacobian and the Jacobian of the Galerkin form of the stochastic system. The analysis is then used to guide the application of a simplified CSP formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system. Two approaches are specifically developed with the resulting simplified CSP framework. The first uses the stochastic eigenvectors of the uncertain system as CSP vectors, whereas the second uses the eigenvectors of the nominal system as CSP vectors. Numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis, and illustrate the effectiveness of the simplified CSP algorithms in addressing the stiffness of the system dynamics.

High temperature ignition and combustion enhancement by dimethyl ether addition to methane–air mixtures

The effects of dimethyl ether (DME) addition on the high temperature ignition and burning properties of methane–air mixtures were studied experimentally and numerically. The results showed that for a homogeneous system, a small amount of DME addition to methane resulted in a significant reduction in the high temperature ignition delay. The ignition enhancement effect by DME addition was found to exceed that possible with equivalent amounts of hydrogen addition, and it was investigated by using radical pool growth and computational singular perturbation analysis. For a non-premixed methane–air system, it was found that two different ignition enhancement regimes exist: a kinetic limited regime and a transport limited regime. In contrast to the dramatic ignition enhancement in the kinetic limited regime, the ignition enhancement in the transport limited regime was significantly less effective. Furthermore, laminar flame speeds as well as Markstein lengths were experimentally measured for methane–air flames with DME addition. The results showed that the flame speed increases almost linearly with DME addition. However, the Markstein length and the Lewis number of the binary fuel change dramatically at small DME concentrations. Moreover, comparison between experiments and numerical simulations showed that only the most recent DME mechanism well reproduced the flame speeds of both DME–air and CH 4–air flames.

An automatic procedure for the simplification of chemical kinetic mechanisms based on CSP

An algorithm is developed to generate simplified (skeletal) kinetic mechanisms from a given detailed one. The algorithm is able to replicate the dynamics of a user-specified set of species (chosen from the original set) when a finite set of sampling points, D, in the chemistry configuration space is given. The simplification procedure involves discarding elementary reactions and species that are deemed unimportant to the fast and slow dynamics of a set of specific scalars. The criteria used in deciding which elementary reactions or species to discard are based on the computational singular perturbation (CSP) method. The procedure involves applying the CSP analysis to each point in D and an algorithm to assemble the simplified mechanism, the validity of which extends to all points in D and is tailored for the set of specified scalars. This algorithm provides a convenient way to construct comprehensive simplified mechanisms, applicable over a wide range of parameters and combustion processes. The effectiveness of this new algorithm is demonstrated by constructing simplified mechanisms for three methane/air reactive systems: autoignition in a constant-pressure reactor, a premixed flame, and a counterflow diffusion flame.

Computational Singular Perturbation Analysis of Two-Stage Ignition of Large Hydrocarbons

Computational singular perturbation (CSP) analysis has been used to gain understanding of the complex kinetic behavior associated with two-stage ignition of large hydrocarbon molecules. To this end, available detailed and reduced chemical kinetics models commonly used in numerical simulations of n-heptane oxidation phenomena are directly analyzed to interpret the underlying fundamental steps leading to two-stage ignition. Unlike previous implementations of the CSP methodology, temperature is included as one of the state variables so that factors controlling ignition can be unambiguously determined. The analyzed models show differences in the factors contributing to the initial development and shutdown of the first ignition stage. However, during the second stage, both models show the importance of the degenerate branching decomposition of hydrogen peroxide, which contradicts some previous interpretations of this phenomenon.

Reduced chemical mechanisms for atmospheric pollution using Computational Singular Perturbation analysis

This paper explores the derivation of reduced chemical mechanisms for atmospheric pollution using the Computational Singular Perturbation technique. The technique is based on an algorithmic selection of “fast-” and “slow-” reacting species, which can then guide the identification of appropriate steady-state approximations to reduce the number of variables that have to be solved. The chemical mechanism analyzed here is the Carbon-Bond Mechanism (CBM-IV), which is already a skeletal form of the detailed kinetics of the atmosphere. A box-model using CBM-IV and a set of measurements of mixing ratios to reflect realistic urban and rural conditions were used as initial conditions. It was found that the selection of steady-state species depends on the integration period, i.e. whether it is over a diurnal period, or during day only, or during night only. Reduced mechanisms with seven, 11 and 15 steady-state species (out of 28 species present) were tested and it was found that the predictions of CBM-IV under different scenarios were reproduced satisfactorily for most species, with inaccuracies increasing as the reduction level increases. The method selects the appropriate steady-state approximations and hence is suitable for analyzing more complex atmospheric pollution chemical mechanisms.

Reduced chemical mechanisms for atmospheric pollution using Computational Singular Perturbation analysis

Reduced chemical mechanisms for atmospheric pollution using Computational Singular Perturbation analysis