An improved algorithm for in situ adaptive tabulation

doi:10.1016/j.jcp.2008.09.015

Journal of Computational Physics

Volume 228, Issue 2, 1 February 2009, Pages 361-386

https://doi.org/10.1016/j.jcp.2008.09.015 Get rights and content

Abstract

In situ adaptive tabulation (ISAT) is a proven storage/retrieval method which efficiently provides accurate approximations to high-dimensional functions which are computationally expensive to evaluate. Previous applications of ISAT to computations of turbulent combustion have resulted in speed-ups of up to a thousand. In this paper, improvements to the original ISAT algorithm are described and demonstrated using two test problems. The principal improvements are in the table-searching strategies and the addition of an error checking and correction algorithm. Compared to an earlier version of ISAT, reductions in CPU time and storage requirements by factors of 2 and 5, respectively, are observed for the most challenging, 54-dimensional test problem.

Introduction

In computer simulations in different disciplines and applications, it is often required to evaluate a function (denoted by $f (x)$ ) very many times, for different values of the input, $x$ . For example, if time-evolving partial differential equations are being solved by a finite-difference method, then some function $f$ may need to be evaluated for each grid node on each time step based on the current values of the dependent variables at the nodes. If the function $f$ is computationally expensive to evaluate, then these evaluations can dominate the overall computation cost, and it is natural to seek less expensive ways to evaluate $f$ , or to obtain an acceptably accurate approximation.

The above problem of the efficient approximation of a function $f (x$ ) is extremely general and can be quite varied depending on: the dimensionality, $n_{x}$ , of the input $x$ ; the dimensionality, $n_{f}$ , of the output, $f$ ; the distribution of the values of $x$ ; the smoothness of the function $f$ ; the number of evaluations required; the accuracy required; and the available computer memory. For low-dimensional problems (e.g. $n_{x} ⩽ 3$ ), often an effective strategy is to use structured tabulation. In a pre-processing stage, values of $f (x)$ are tabulated; and then, during the simulation, approximate values of $f$ are obtained by interpolation in the table. Both the work and the storage necessary to construct the table increase exponentially with $n_{x}$ , which is why this approach is restricted to low-dimensional problems.

Since its introduction in 1997 [1] the in situ adaptive tabulation (ISAT) algorithm has proved very effective for a class of higher-dimensional problems (e.g. $n_{x} ⩽ 50$ ). The original application of ISAT was in the particle method used to solve the probability density function (PDF) equations for turbulent combustion [2]. In that case, $x$ consists of the thermochemical state of a particle at the beginning of the time step (of duration $Δ t$ ); and $f$ represents the composition at the end of the step (resulting from adiabatic, isobaric reaction). Evaluating $f (x)$ involves solving a stiff set of $n_{f}$ ordinary differential equations (ODEs) for a time $Δ t$ . Typically this requires of order $10^{4} μ s$ of CPU time, whereas ISAT yields an accurate approximation in of order $10 μ s$ , resulting in a thousandfold speed-up. Based on these timings, if $f (x)$ needs to be evaluated for each of $10^{6}$ particles on each of $10^{3}$ time steps, then integrating the ODEs requires over 100 days of CPU time, whereas the same task is accomplished by ISAT in under 3 hours.

The ISAT algorithm continues to be heavily used for combustion problems both in the authors’ group (e.g. [3], [4], [5], [6]) and by others (e.g. [7], [8], [9]); and the algorithm has been incorporated into the ANSYS/FLUENT software [3], [10]. ISAT has also been used in chemical engineering [11], [12], [13], in solid mechanics [14], control [15] and the study of thin films and surface reaction [16], [17]. Several variants of ISAT have been proposed and investigated [18], [19], [20], [21], [22], [23], [24]. Other methods that have been used to address the same problem include: artificial neural networks [25], [26], [27]; high-dimensional model representation [28]; PRISM [29]; orthogonal polynomials [30]; and kriging [31].

The purpose of this paper is to describe and demonstrate several new additions to the original ISAT algorithm, which significantly enhance its performance. The next section provides an overview of the ISAT algorithm, and then the new components are described in Sections 3 Retrieve search, 5 Adding. In Section 6, quantitative testing of the algorithm is performed to demonstrate the efficacy of the new methods and to compare the computational performance with a previous implementation of ISAT. (Further tests are reported in [32], [33].)

The ISAT algorithm makes extensive use of ellipsoids and hyper-ellipsoids (which, for simplicity, are referred to as ellipsoids, regardless of the dimension of the space). All of the algorithms used in ISAT involving ellipsoids are described in [34].

Section snippets

Overview of the ISAT algorithm

The purpose of ISAT is to tabulate a function $f (x$ ), where $x$ and $f$ are of dimension $n_{x}$ and $n_{f}$ , respectively. Given a query, $x^{q}$ , ISAT return $f^{a} (x^{q})$ – an approximation to $f (x^{q})$ . It is assumed that $f$ and $x$ are appropriately scaled, with variations of order unity, so that the two-norm $ε = ∥ f^{a} (x^{q}) - f (x^{q}) ∥,$ is an appropriate measure of the approximation error. (There is no difficulty in using a more general definition of the error.)

An essential aspect of ISAT is that the table is built up, not in a

Retrieve search

Given a query $x^{q}$ , the objective of the retrieve search is to identify any EOA which covers $x^{q}$ . There may be zero, one or more such EOAs. If at least one EOA covers $x^{q}$ , then the search process is deemed to be complete if it is guaranteed to identify one such EOA: otherwise it is incomplete. If the search identifies a covering EOA it is deemed to be successful. Any search strategy inevitably involves testing to determine whether or not a particular EOA covers the query point $x^{q}$ . This testing is

EOA growing

If the retrieve attempt fails, then some leaves are selected as candidates for having their EOAs grown. We describe here the “grow search” performed to identify candidate leaves, and the subsequent EOA growth (and other modifications to the leaf). The treatment used is heuristic, and is the result of a good deal of experimentation with alternative implementations and variants. The fundamental reason for the need for heuristics is that the region of accuracy (ROA) is not in all circumstances

Adding

On a given query, if the retrieve and grow attempts fail, and if the table is not full, then a new entry is added to the table. (The treatment used when the table is full is discussed at the end of this section.) The primary operations involved in an “add” are the initialization of the new leaf (EOA, EOI, etc.) and its insertion into the various data structures. These operations are now outlined.

Performance of ISAT

Comprehensive testing of the ISAT algorithm has been performed for two test cases of a partially-stirred reactor (PaSR). The PaSR test cases are described in the first sub-section. Then test results are presented on the performance of ISAT, including the incurred error, CPU time per query, and number of table entries. Different variants of the ISAT implementation, and different values of various parameters are investigated, with the objectives of assessing the efficacy of different components

Conclusions

In situ adaptive tabulation (ISAT) has proved to be an effective method to reduce the computer time required to evaluate (approximately) high-dimensional functions, $f (x)$ . In this paper, several new algorithms for ISAT are described and demonstrated. These include: new search strategies (MRU, MFU, EBT); the use of low-dimensional affine spaces to reduce the cost of searching; ellipsoids of inaccuracy (EOIs) to identify candidates for growing; and error checking and correction (ECC).

The

Acknowledgments

For technical discussions and suggestions, the authors are grateful to several colleagues at Cornell, especially to Biswanath Panda, Zhuyin Ren and Paul Chew. This research was supported in part by the National Science Foundation through Grant CBET-0426787. This research was conducted using the resources of the Cornell University Center for Advanced Computing, which receives funding from Cornell University, New York State, the National Science Foundation, and other leading public agencies,

References (43)

S.B. Pope
PDF methods for turbulent reactive flows
Prog. Energy Combust. Sci.
(1985)
M.A. Singer et al.
Modeling unsteady reacting flow with operator-splitting and ISAT
Combust. Flame
(2006)
R. Cao et al.
The effect of mixing models in PDF calculations of piloted jet flames
Proc. Combust. Inst.
(2007)
S. James et al.
Large eddy simulations of turbulent flames using the filtered density function model
Proc. Combust. Inst.
(2007)
A. Arsenlis et al.
Generalized in situ adaptive tabulation for constitutive model evaluation in plasticity
Comp. Methods Appl. Mech. Eng.
(2006)
J.D. Hedengren et al.
Approximate nonlinear model predictive control with in situ adaptive tabulation
Comput. Chem. Eng.
(2008)
A. Varshney et al.
Multiscale optimization using hybrid PDE/kMC process systems with application to thin film growth
Chem. Eng. Sci.
(2005)
S. Mazumder
Adaptation of the in situ adaptive tabulation (ISAT) procedure for efficient computation of surface reactions
Comput. Chem. Eng.
(2005)
J.-Y. Chen et al.
An economical strategy for storage of chemical kinetics: fitting in situ adaptive tabulation with artificial neural networks
Proc. Combust. Inst.
(2000)
F.C. Christo et al.
An integrated PDF/neural network approach for simulating turbulent reacting systems
Proc. Combust. Inst.
(1996)

F.C. Christo et al.

Artificial neural network implementation of chemistry with PDF simulation of $H_{2} / {CO}_{2}$ flames

Combust. Flame

(1996)

J.A. Blasco et al.

Modelling the temporal evolution of a reduced combustion chemical system with an artificial neural network

Combust. Flame

(1998)

T. Turányi

Parameterization of reaction mechanisms using orthogonal polynomials

Comput. Chem.

(1994)

N.R. Barton et al.

Embedded polycrystal plasticity and adaptive sampling

Int. J. Plasticity

(2008)

S.B. Pope

The probability approach to the modeling of turbulent reacting flows

Combust. Flame

(1976)

S.M. Correa

Turbulence-chemistry interactions in the intermediate regime of premixed combustion

Combust. Flame

(1993)

B. Yang et al.

An investigation of the accuracy of manifold methods and splitting schemes in the computational implementation of combustion chemistry

Combust. Flame

(1998)

Z. Ren et al.

An investigation of the performance of turbulent mixing models

Combust. Flame

(2004)

V. Saxena et al.

PDF calculations of major and minor species in a turbulent piloted jet flame

Proc. Combust. Inst.

(1998)

M. Caracotsios et al.

Sensitivity analysis of initial value problems with mixed ODEs and algebraic equations

Comput. Chem. Eng.

(1985)

S.B. Pope

Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation

Combust. Theory Modell.

(1997)

Cited by (141)

In situ adaptive tabulation of vapor-liquid equilibrium solutions for multi-component high-pressure transcritical flows with phase change
2024, Journal of Computational Physics
Vapor-liquid equilibrium (VLE) is a family of first-principled thermodynamic models for transcritical multiphase flows, which can accurately capture the phase transitions at high-pressure conditions that are difficult to deal with using other models. However, VLE-based computational fluid dynamics (CFD) simulation is computationally very expensive for multi-component systems, which severely limits its applications to real-world systems. In this work, we developed a new ISAT-VLE method based on the in situ adaptive tabulation (ISAT) method to improve the computational efficiency of VLE-based CFD simulation with reduced memory usage. We developed several ISAT-VLE solvers for both fully conservative (FC) and double flux (DF) schemes. New methods are proposed to delete redundant records in the ISAT-VLE table and the ISAT-VLE method performance is further improved. To improve the convergence of the VLE solvers, a modified initial guess for equilibrium constant is also introduced. Simulations of high-pressure transcritical two-phase temporal mixing layers and shock-droplet interaction were conducted using the ISAT-VLE CFD solvers. The simulation results show that the new method obtains a speed-up factor approximately from 10 to 60 and the ISAT errors can be controlled within 1%. The shock-droplet interaction results show that the DF scheme can achieve a higher speed-up factor than the FC scheme. The two sets of simulations exhibit the phase separation at high-pressure conditions. It was found that even at supercritical pressures with respect to each component, the droplet surface could still be in a subcritical two-phase state, because the mixture critical pressure is often significantly higher than each component and hence triggers phase separation. In addition, a shock wave could partially or completely convert the droplet surface from a subcritical two-phase state to a single-phase state by raising temperature and pressure.
Learning stiff chemical kinetics using extended deep neural operators
2024, Computer Methods in Applied Mechanics and Engineering
We utilize neural operators to learn the solution propagator for challenging systems of differential equations that are representative of stiff chemical kinetics. Specifically, we apply the deep operator network (DeepONet) along with its extensions, such as the autoencoder-based DeepONet and the newly proposed Partition-of-Unity (PoU-) DeepONet to study a range of examples, including the ROBERS problem with three species, the POLLU problem with 25 species, pure kinetics of a skeletal model for ${CO/H}_{2}$ burning of syngas, which contains 11 species and 21 reactions and finally, a temporally developing planar ${CO/H}_{2}$ jet flame (turbulent flame) using the same syngas mechanism. We have demonstrated the advantages of the proposed approach through these numerical examples. Specifically, to train the DeepONet for the syngas model, we solve the skeletal kinetic model for different initial conditions. In the first case, we parameterize the initial conditions based on equivalence ratios and initial temperature values. In the second case, we perform a direct numerical simulation of a two-dimensional temporally developing ${CO/H}_{2}$ jet flame with integrated chemistry modeling as a source of training data for the solution propagator. Then, we initialize the kinetic model by the thermochemical states visited by a subset of grid points at different time snapshots. Stiff chemical kinetics are computationally expensive to solve, thus, this work aims to develop a neural operator-based surrogate model to efficiently solve stiff chemical kinetics. The operator, once trained offline, can accurately integrate the thermochemical state for arbitrarily large time advancements, leading to significant computational gains compared to stiff integration schemes.
Automated adaptive chemistry for Large Eddy Simulations of turbulent reacting flows
2024, Combustion and Flame
Large Eddy Simulations (LES) of turbulent reacting flows carried out with detailed kinetic mechanisms have a key role for the discovery of the physical and chemical processes occurring in combustion systems, and are essential for the development of efficient, stable, and non-pollutant technologies. Nevertheless, these simulations require a large amount of computational resources, making their utilization for large-scale systems, such as industrial burners and gas turbines, impractical. In this work, we combine state-of-the-art machine learning algorithms and model reduction methods to deliver a fully automated strategy for performing LES with adaptive chemistry. This strategy is based on the Sample-Partitioning Adaptive Chemistry (SPARC) algorithmic procedure, which consists of four steps: the generation of a training dataset, its partitioning in clusters, the generation of a set of reduced chemical mechanisms specifically tailored to each cluster and, lastly, the numerical simulation of the case of interest with adaptive chemistry enabled by an on-the-fly classification of every grid point.
The SPARC approach has already been demonstrated to substantially reduce the computational effort of reactive flows simulations. However a non-negligible level of user interventions is needed, upon which the method’s success critically depend. Therefore, with the goal of boosting the performance of this workflow and minimise the user-specified degrees of freedom, we plug in and exploit the Local Principal Component Analysis augmented with an automated Bayesian-optimised search for optimal clustering solutions, and the Computational Singular Perturbation method with an additional layer of automation based on the Tangential Stretching Rate for minimally-sized reduced mechanisms. We employ a cheap and easy-to-generate 1-dimensional-flames training database and we demonstrate the efficiency, accuracy and robustness of this strategy with an application to LES of the Adelaide Jet in Hot Coflow (AJHC) burner, a turbulent reacting flow exhibiting intense turbulence-chemistry interactions.
Analysis of flame instability in low-emission tower-type gas turbine combustor with multi-stage methane injection
2023, Fuel
A low-emission tower-type coaxial-staged combustor with a pilot stage, first main stage and second main stage is simulated in this study. The flow field, flame structure, vortex-mixing interaction, and flame instabilities are predicted with large eddy simulation and partially stirred reactor model. Heat radiation is not considered in the simulations. The results show that the maximum of the averaged length and width of the inner recirculation zone are approximately 249.8 and 71.3 mm, respectively. The distributions of the fuel from various stages in the combustor are analyzed based on the respective mixture fractions. The pressure and heat release rate (HRR) in the combustor fluctuate periodically with a frequency of about 373 Hz. Both HRR and pressure fluctuations increase and then decrease with time, and there is a phase difference of 46.1° between them. Due to the combustion instabilities, the flame structures change periodically, including attached flame and lifted V-shaped flame, in one HRR oscillation cycle. Moreover, the swirling flame length also varies periodically with time. As the pilot stage and first main stages are turned-off, the frequency of HRR fluctuations and pressure fluctuations, as well as the flame root dynamics, are significantly affected. It is also found that the effects of the fuel ratio of the first and second main stages have a significant influence on the flame root dynamics, e.g., different degrees of flashback observed in different cases.
Use of dynamic adaptive chemistry and dynamic cell clustering in computational fluid dynamics to accelerate calculation of combustion simulation of diesel engine
2023, Fuel
Based on the theory of dynamic adaptive chemistry and dynamic cell clustering, a 3D model for fuel combustion in the cylinder is constructed to study the fast calculation algorithm to accelerate the engine combustion simulation and analyze the impacts of the basic structure and the main parameters of dynamic adaptive chemistry and dynamic cell clustering on diesel engine simulation. It is found that the error threshold value of the dynamic adaptive chemistry algorithm should be set appropriately to ensure the accuracy of the simulation, and the dynamic adaptive chemistry algorithm should employ different initial search species and threshold values to meet different research objectives. Regarding the dynamic cell clustering algorithm, the data accuracy in the overall spatial range can be guaranteed, but the local location information will be missing, which will have a great impact on the prediction of pollutants. Therefore, suitable cell clustering conditions should be selected to meet the requirements of different targets.
Zone-adaptive modeling of turbulent flames with multiple chemical mechanisms
2023, Proceedings of the Combustion Institute
A novel zone-adaptive modeling method (AdaCM) with multiple chemical mechanisms is proposed for turbulent flames to achieve high-fidelity, yet computationally efficient, predictions. Specifically, a computational economical species transport model, e.g., the well-mixed model, via finite volume algorithm is employed with a simple mechanism as the base model for the whole computational domain, while the advanced transported probability density function (TPDF) method via Lagrangian particle tracking is employed with a detailed mechanism only for spatial regions with intense turbulence-chemistry interaction (TCI), denoted as the “PDF regions”. The PDF regions are dynamically identified based on local flow and flame characteristics and may evolve with time. A two-way particle/finite volume submodel coupling is formulated to ensure the composition consistency between submodels in the PDF regions and impose the correct interface conditions for composition and mass flow rate on the boundary of the PDF regions. With regard to transformation between different species representations in the mechanisms, a species reconstruction/reduction approach based on constrained chemical equilibrium is proposed to ensure element conservation and an adequate specification of unrepresented species at the model interface. The proposed adaptive modeling method has been applied to the well-known Sandia Flame D, in which the well-mixed combustion model with a 6-species, two-step global mechanism is employed as a base model and the high-fidelity TPDF with a 25-species skeletal mechanism is employed for regions with intense TCI. Results demonstrate the consistency in PDF regions between submodels with two distinct mechanisms. The predictions from the adaptive modeling are almost identical to those of TPDF and agree well with experimental measurements, illustrating the preservation of prediction accuracy in the adaptive method. In addition, the total number of computational particles in AdaCM for Flame D is only 18% of that for the stand-alone TPDF, and the recorded computational speedup is about 2.8.

View all citing articles on Scopus

View full text

An improved algorithm for in situ adaptive tabulation

Abstract

Introduction

Section snippets

Overview of the ISAT algorithm

Retrieve search

EOA growing

Adding

Performance of ISAT

Conclusions

Acknowledgments

Prog. Energy Combust. Sci.

Combust. Flame

Proc. Combust. Inst.

Proc. Combust. Inst.

Comp. Methods Appl. Mech. Eng.

Comput. Chem. Eng.

Chem. Eng. Sci.

Comput. Chem. Eng.

Proc. Combust. Inst.

Proc. Combust. Inst.

Combust. Flame

Combust. Flame

Comput. Chem.

Int. J. Plasticity

Combust. Flame

Combust. Flame

Combust. Flame

Combust. Flame

Proc. Combust. Inst.

Comput. Chem. Eng.

Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation

Combust. Theory Modell.