SpeedCHEMSpeedCHEMA Sparse Analytical Jacobian Chemistry code

for Engine Simulations with Detailed Chemistry

Federico PeriniDipartimento di Ingegneria “Enzo Ferrari”

University of Modena, Italy

University of Wisconsin-Madison Engine Research Center, October 9th 2012

1

October 9th, 2012

Outline Motivation and challenges

Outline

The Sparse Analytical Jacobian approach Chemical Kinetics for Internal Combustion Engine simulations Sparsity and structure of the problem’s Jacobian matrix Effect of reaction mechanism size on sparsity

A d f h l i f h i l ki i i SA A code for the solution of chemical kinetics using SAJ Validation at adiabatic constant-volume reactors ODE solver choice, accuracy and problems

Modelling of a Heavy-Duty Diesel Engine CFD simulation methodology and grid characteristics Impact of grid resolution, and mechanism dimensions

Conclusions

2

Motivation and challengesMotivation and challenges New combustion concepts (HCCI/PCCI, RCCI) show impressive improvements in

conversion efficiencyy ICE indicated efficiency >50% strong dependency on fuel chemistry and local mixture reactivity

i l / h l i l b ti d l l k f l ti i d lli simple/phenomenological combustion models lack of resolution in modelling: the whole range of operating conditions

of practical systems presence of exhaust gases in the mixture Sizes of biofuels mechanisms presence of exhaust gases in the mixture simultaneous operation with multiple fuels

Practical reaction mechanisms for104

r MD +C H

methyl 9 decenoate (LLNL)methyl esters (NUI)

2000 to 2008before 2000after 2008

(from Perini, Ph.D.Thesis, 2011)

Internal Combustion Enginesimulations range from 30 to 200 species 103

104

f rea

ctio

ns, n

nC7H16

methyl cyclohexane (LLNL)

MB (LLNL)

MD (LLNL)methyl 5 decenoate (LLNL)

small hydrocarbons (Konnov)MB (NUI)

after 2008

p Total CPU times typically unviable for practical

engineering design (cases should run overnight)

102

10

num

ber o

fethanol (LLNL)

methyl formate (LLNL)

dimethyl carbonate (LLNL)

DME (LLNL)

ethanol (Saxena)

biodiesel (Golovitchev)

Ethers (NUI)

2012-01-0143101 102 103 104

10

number of species, n s

nr = 5 ns

ns = 30 200

Chemical Kinetics in ICE CFD

Usually part of an operator-splitting scheme Each cell is treated as an adiabatic well-stirred

reactor◦ embarassingly parallel problem

KineticsMechanism

g y p p◦ very stiff IVP◦ only overall changes in species mass fractions and

internal energy are passed to the flow solver

Approach to the Approach to the

internal energy are passed to the flow solver

FluidFlow

solverGrid

ODE solver

ppsolution should be

tailored to the problem

ppsolution should be

tailored to the problem

4

solver

Sparse Analytical Jacobian approach Chemical Kinetics is a sparse problem => Jacobian matrix

ERC n-heptane LLNL n-heptane LLNL PRF29 x 52 160 x 1540 1034 x 4236

Mechanism n n non-zero sparsity Usually not more than 3 4 Mechanism ns nr non zero sparsity1. ERC n-heptane 29 52 412 54.2%2. LLNL n-heptane 160 1540 3570 86.2%3 LLNL PRF 1034 4236 22551 97 9%

Usually, not more than 3-4 species per reaction

Significant even at small

5

3. LLNL PRF 1034 4236 22551 97.9%4. LLNL MD 2878 8555 166703 99.4% mechanisms

Jacobian matrix sparsity effectsJ p y

100

101

Analytical JacobianFinite difference JacobianODE system fun 103

104

ns3

ns2

10-2

10-1

valu

atio

n [s

]

ODE system fun.

ERC

LLNLreducedn-hept 102

10

me,

18

IVPs

[s]

LLNLdetailed

PRF

LLNLreducedn-hept

s

ns2

ns

10-5

10-4

10-3

time

per e

v

LLNLmethyl

decanoate

ERCn-hept

LLNLdetailed

PRF100

101

tota

l CPU

tim

ERCn-hept

PRF

101 102 103 10410-6

10

number of species

PRF

101 102 10310-1

number of species

Analytical JacobianFinite difference Jacobian

Reduce scaling of the computational demand for evaluating the Jacobian matrix: ns

2 => ns

Reduce matrix storage requirements: ns2 => ns

Scaling of the costs for matrix factorization: ns3 => ns

I ( d i ) f N ’ i i h d

6

Improve (quadratic) convergence of Newton’s iterative method

Tabulation of temperature-dependentquantitiesquantities

Interpolation errors can be very low e g at degree 4 interpolation Interpolation errors can be very low e.g. at degree-4 interpolation

Fixed temperature steps make storage simpler, data contiguous, and interpolating polynomial coefficients simple to compute

2012-01-1974

CPU time reduction of more than 1 order of magnitude

Tabulation of temperature-dependentquantities II effect on total CPU timequantities II – effect on total CPU time

6%CPU time efforts, LLNL n-heptane mech.

50%44%44%

numerical factorizationODE function and Jacobiansolver-related

The relative importance of tabulation decreases at increasing mechanism dimensions

Anyway, it accounts for about -50% CPU time in comparison with the non-tabulated case

At large reaction mechanisms, the solution of the linear system accounts for almost 50% time

Results: constant-volume reactors 18 ignition cases for each mechanism

p0 = 2.0, 20.0 barT 750 1000 1500 K

100LLNL PRF mech.

COC8H18

C7H16

T0 = 750, 1000, 1500 K 0 = 0.5, 1.0, 2.0 10-5

frac

tions

[-] CO

HO2

OH

8 18

H2O

100ERC n-heptane mech.

15

10-10

mol

e f

CHEMKINSpeedCHEM

OH

CO210-5

10

ns [-

]

C7H16 CO

H2O

0 0.002 0.004 0.006 0.008 0.0110-15

Time [s]

100LLNL n-heptane mech.

10-10

mol

e fr

actio

n

OH

HO2

10-5

tions

[-]

C7H16

H2O

CO

0 0.002 0.004 0.006 0.008 0.0110-15

Time [s]

m

CHEMKINSpeedCHEMCO2

10-10

mol

e fr

act

HO2OH

Time [s]

Program CHEMKIN-II present codeODE solver VODE LSODESrelative tolerance 10-4 10-4

9 0 0.002 0.004 0.006 0.008 0.0110-15

Time [s]

CHEMKINSpeedCHEMCO2

relative tolerance 10 4 10 4

absolute tolerance 10-13 10-13

Results: constant-volume reactors II Almost linear speedup in comparison with a reference code that uses FD

103

104

102

103

SpeedCHEMCHEMKINspeedup

ns

102

10

U ti

me

[s]

101

10

dup

fact

or

p p

ERCn-heptane

LLNL

100

101C

PU

10-1

100

spee

LLNLPRF

LLNLn-heptane

101

102

103

ns

At typical mechanism dimension, the speedup is of almost one order ofmagnitude

Nota Bene: two different solvers were used!

10

Nota Bene: two different solvers were used!

Results: constant-volume reactors III10

6

Chemkin-PROCantera

104

me

[s]

Present code (LSODES)Chemkin-II

LLNLMD

102C

PU ti

m

ERCn-heptane

ns3

100

p

LLNLn-heptane

ns2

Program Cantera Chemkin-PRO CHEMKIN-II SpeedCHEM

101

102

103

10410

ns

g p

ODE solver CVODE Internal(DASPK?) VODE LSODES

relative tolerance 10-4 10-4 10-4 10-4

11

absolute tolerance 10-13 10-13 10-13 10-13

* All cases run on a 32-bit 3.0GHz Pentium IV machine with 1GB RAM

Different solvers do have differentperformanceperformance…

12

14 x 104 ERC mech, ns = 29, nr = 52

integrator steps 1 9

2ERC mech, ns = 29, nr = 52

6

8

10

12

mbe

r of

[-]

integrator stepsmatrix decompfunction evalsJacobian evals

1.6

1.7

1.8

1.9

U ti

me

[s]

0

2

4

6

num

1.3

1.4

1.5CPU

DASPK GAM LSODES MEBDF RADAU5 RODAS VODES0

DASPK GAM LSODES MEBDF RADAU5 RODAS VODES

All of the solvers have been implemented sparse solution of All of the solvers have been implemented sparse solution oflinear systems (Yale sparse matrix package, 1983)

The same truncation error handling has different meaningsg g

ATOLtyRTOLtyty nnn ˆ

… and stricter tolerances do not alwaysmean better resultsmean better results

1025

error (species)40

CPU time

casesi

n tdTT 01

1015

1020

error (species)error (temperature)

30

35CPU time

casesi

sn t njj

i iT

dYY

dTt

0

10 0

1

105

1010

erro

r val

ue

20

25

CPU

tim

e [s

]

i j j

jj

iY d

Yt10

10

10-5

100

10

15

10-15

10-10

10-5

10-10

relative tolerance, RTOL

10

-1510

-1010

-55

LLNL n-heptane mech

SpeedCHEM + LSODES solver

Reference solution has RTOL = 1d 15 Reference solution has RTOL = 1d-15

Modelling of a Heavy-Duty Diesel Engineg y y g

Caterpillar SCOTE 3401 Engine detailsEngine type direct injection diesel

Extensive experimental measurementsby Hardy, UW-ERC (SAE 2006-01-

Engine type direct-injection diesel

Number of cylinders 1

Valves per cylinder 4

Bore x Stroke [mm] 137 2 x 165 1 E i ti diti

0026)

Bore x Stroke [mm] 137.2 x 165.1

Conrod length [mm] 261.6

Compression ratio [-] 16:1

Unit displacement [L] 2.44

Engine operating conditions

Rotating speed [rev/min] 1737

Engine load [-] 57%p [ ]

Chamber Turbulence Quiescent

Piston bowl geometry Mexican hat

Intake temperature [K] 305.15

Injection rate [g/s] 1.94

Pilot injection start [°atdc] -65 -50

Pilot injection length [s] 1000 1450

Main injection start [°atdc] -5 20

Main injection length [s] 1950

EGR f ti 0 %

Two-stage combustion High sensitivity to emissions No interaction between spray jets EGR fraction 0 %

Boost pressure [kPa] 186.2 220.6

No interaction between spray jets

pilot SOI, main SOI, boost pressure

14

sweeps

CFD simulation setupp KIVA-4 code developed at LANL (Torres & Trujillo, JCP 2006)

Model improvements Model improvements Detailed chemistry capability using CHEMKIN-II (CONV) orpresent code Dynamic injection spray angle computation (Reitz and Bracco, SAE790494) Smooth grid snapping algorithm

Diesel fuel modelling tetradecane (C14H30) liquid spray properties n-heptane (nC7H16) ignition chemistry

NOx formation mechanism from GRI-mech 5 additional species, 12 reactions

Grid Coarse Refined

Cells at BDC 16950 42480

15

Average resolution 2.2 mm 1.1 mm

Azimuth resolution 4.0 deg 3.3 deg

Results: Caterpillar SCOTE 3401p

140

160ERC n-heptane mech., coarse mesh

ar) 420

480

/CA

)

experimentCHEMKIN-II

t d25

30ERC n-heptane

350

400LLNL n-heptane (4CPU)

chemkinpresent code

60

80

100

120

er p

ress

ure

(b

180

240

300

360

eat R

elea

se (J

/present code

15

20

time

[h]

200

250

300

-75.4%-48.8%

0

20

40

60

In-c

ylin

d

0

60

120

180

Rat

e of

He

nj 5

10CPU

t

100

150

160ERC n-heptane mech., refined mesh

480

i t

-80 -60 -40 -20 0 20 40 60 80

min

CA degrees ATDC coarse refined0

5

coarse refined

0

50

80

100

120

140

ress

ure

(bar

)

240

300

360

420

Rel

ease

(J/C

A)experiment

CHEMKINpresent code

Excellent agreement of the new solver coupling vs. Chemkin-II

20

40

60

80

In-c

ylin

der p

r

60

120

180

240

Rat

e of

Hea

t R

Average CPU time reduction (includedflow-field part): 75 4% (LLNL) 48 8% (ERC)

16

0 0

-80 -60 -40 -20 0 20 40 60 80

min

j

CA degrees ATDC

75.4% (LLNL) 48.8% (ERC)

Caterpillar SCOTE 3401: coarse vs. refined

refined grid’s peak in AHRR due to ignition ofalmost-quiescent mixture in the squish region Injection axis cutplane, -16°ATDC

volume averaging and reduced spray penetration in the coarse grid reduces the AHRR peak and delays ignition

160LLNL n-heptane mech., present code

480

100

120

140

sure

(bar

)

300

360

420

ase

(J/C

A)experiment

coarse meshrefined mesh

40

60

80

100

linde

r pre

ss

120

180

240

300

f Hea

t Rel

ea0

20

40

In-c

yl

0

60

120

Rat

e o

nj

In both grids the fuel vapor jet isdeviated to the cylinder head

17

-80 -60 -40 -20 0 20 40 60 80

mi

CA degrees ATDC

Caterpillar SCOTE 3401: NOx emissionsp x pilot pulse SOI main pulse SOI35

40

Experimentcoarse, ERC, present coderefined ERC present code

Parameter sweeps

boost pressure

20

25

30

Ox [g

/kg f]

refined, ERC, present coderefined, ERC, CHEMKINcoarse, LLNL, present coderefined, LLNL, present codecoarse, LLNL, CHEMKIN

ERC mechanism+ refined grid case( ) outperforms the others

5

10

15NO

( ) p◦ peak in AHRR after pilot injection leads to mismatched

NOx trend

◦ Coarse grid has greater temperature

40

-68 -66 -64 -62 -60 -58 -56 -54 -52 -50 -480

pilot pulse SOI [deg. ATDC]

45Experiment

◦ Coarse grid has greater temperature distribution mixing ◦ NOx are underestimated

Very good agreement

25

30

35

g f]

Experimentcoarse, ERC, present coderefined, ERC, present codecoarse, ERC, CHEMKINrefined, ERC, CHEMKINcoarse, LLNL, present coderefined LLNL present code 25

30

35

40

g f]Experimentrefined, ERC, present codecoarse, ERC, present codecoarse, LLNL, present coderefined, LLNL, present codecoarse, LLNL, CHEMKIN

y g gbetween chemistrysolvers

10

15

20

NO

x [g/k

g refined, LLNL, present code

10

15

20

25

NO

x [g/k

18-10 -5 0 5 10 15 20 250

5

main pulse SOI [deg. ATDC]

185 190 195 200 205 210 215 220 2250

5

boost pressure [kPa]

Conclusions

A new code for the integration of sparse reaction kinetics of i t h b d l dgaseous mixtures has been developed

The code has been coupled with KIVA-4, to model a heavy-duty Diesel engine operated in a two-stage combustion modeDiesel engine operated in a two stage combustion mode

Comparison with a reference academic chemistry code showed Comparison with a reference academic chemistry code showed excellent agreement

Speedups of the order of 2 times => 30+ times were achieved at a range of mechanism dimensions vs. FD code

CPU times for refined grid + detailed mech. were < 25h on 4CPUTh l i i bl i i d il d i h i i The solver is suitable to incorporate semi-detailed reaction mechanisms in practical engine simulations

19

There’s plenty of room at the bottomp y

Find out optimal ODE integration method at different reactivity (and stiffness) conditions

Investigate the accuracy of sparse semi-implicit integrators

Find suitable Jacobian preconditioners

Explore ODE solvers accuracy using quadruple-precision arithmetics

Investigate the role of transport in ICE simulations with detailed Investigate the role of transport in ICE simulations with detailed chemistry integration

20

Thanks for your attention!Thanks for your attention!Questions?Q

Acknowledgements Prof. Giuseppe Cantore and prof. Emanuele Galligani Prof. Rolf Reitz, Dr. Youngchul Ra, Dr. A. Krishnasamy, ERC staff & students VM Motori (Cento Italy) for funding the present work with a research grant

21

VM Motori (Cento, Italy) for funding the present work with a research grant

Backup

22

Jacobian matrix sparsity assumptionJ p y p

Three-body and more complex pressure-dependent reactionsinvolve the whole mixture concentrationinvolve the whole mixture concentration

sn

itot

YC

In constant-pressure environments, Ctot is constant In constant volume environments C has non negative derivative

i i

tot W1

In constant-volume environments, Ctot has non-negative derivative with each species

s

n

i

itot njWW

YYY

C s

,...,11

Dense lines in the Jacobian matrix

ji ijj WWYY 1

23

Jacobian matrix sparsity assumptionJ p y p

Simplifying assumption: 0 tot

YC jY

Affected is the Jacobian only, not the problem formulation

24

Jacobian matrix sparsity assumptionJ p y p Comparison between complete and sparser formulation

25 2012-01-0143

Chemical Kinetics IVPs

26

Interpolation of temperature-dependent quantitiesp p p q

27

Interpolation of temperature-dependent quantitiesp p p q

28

Interpolation of temperature-dependent quantitiesp p p q

n = 1 n = 2

n = 4

nn

n

i

iin xaxaxaaxaxp

2210

0

29

Species thermodynamic properties databases35 150

4h5

25

30100

c4h5c5h3oc5h4oc5h4ohc5h5o

15

20

Cp/

R [-

]

c4h5c5h3o5h4

50

U/R

T [-]

c5h5ooc6h4o

0 500 1000 1500 2000 2500 30005

10

c5h4oc5h4ohc5h5ooc6h4o

0 500 1000 1500 2000 2500 3000-50

0

0 500 1000 1500 2000 2500 3000

temperature [K]0 500 1000 1500 2000 2500 3000

temperature [K]

100

110

c4h5c5h3o

h4

150

c4h5c5h3o

h4

70

80

90

R [-

]

c5h4oc5h4ohc5h5ooc6h4o

50

100

RT

[-]

c5h4oc5h4ohc5h5ooc6h4o

50

60

S/R

0

H/R

300 500 1000 1500 2000 2500 300030

40

temperature [K]

0 500 1000 1500 2000 2500 3000

-50

temperature [K]

Effect of RTOL and ATOL on local errorconstraintconstraint

ATOLtyRTOLtyty ˆ

100Impact of ODE solver's weighting factor on desired accuracy

ATOLtyRTOLtyty nnn

10-10

10

RTOL |y|RTOL |y| + ATOL

RTOL effect

10-20

ATOL effect

-40

10-30

ATOL effect

10-50

10-40

10-30

10-20

10-10

100

10-50

1040

solid lines: RTOL = 1e-4 ATOL = 1e-15dashed lines: RTOL = 1e-8 ATOL = 1e-30

2012-01-197431

10 10 10 10 10 10y