Gaetano Continillo

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Gaetano Continillo

Dipartimento di Ingegneria, Università del Sannio, Benevento, Italy

Advances in the analysis of non-intrusive observations of dynamical reactive systems

ICARE/CNRS, Orléans, FRANCE 7 October 2009

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Outline

• What is POD

• POD in a vector space

• Applications to Experiments of

combustion imaging:

• Data reduction

• POD/Interpolation

• Analysis of cycle variation

• Extraction of random components

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POD

Proper Orthogonal Decomposition (POD) is a

procedure that delivers an optimal set of empirical basis functions from an ensemble of observations obtained either experimentally or from numerical

simulation, which characterize the spatio-temporal complexity of a given system.

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POD

Depending on the field of application, POD is also known as Principal Component Analysis (PCA), Karhunen–Loève (KL) decomposition, Hotelling transformation or Singular Value Decomposition (SVD).

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Experimental setup

An optically accessible engine is used to acquire high definition pixel images of luminosity

What is POD

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What is POD

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U

 

 

  

 

 

What is POD

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POD

Suppose we are given a time series, obtained from

simulation or experiment. Usually the sampled data set is a vector-valued function given as a matrix:

where N is the number of positions in the spatial domain and M is the number of samples taken in time.

1 1 2 1 1

1 2 2 2 2

1 2

( ) ( ) ( )

M M

N N M N

u x u x u x

U

u x u x u x

 

 

  

 

 



   



…

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POD

It can be shown that a suitable POD basis is obtained by solving the eigenvalue problem:

where C is the averaged autocorrelation matrix:

and angular brackets denote time-averaging operation.

C  

( , ') ( ), '( ) C x x  U x U x

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POD

The function can be expressed by a linear combination of the eigenfunctions:

where K  N is the number of POD modes used, whereas

are modal coefficients that can be determined by projection of the ensemble onto the POD modes.

t ( )

u X

1

( ) ^K ( ) ( )

t k k

k

u X a t  x







k ( ) a t

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POD

The ordering of the eigenvalues from the largest to the smallest induces an ordering in the corresponding

eigenfunctions, from the most to the least important. Hence, in order to determine the truncation degree of the POD

reduced model, we define the cumulative correlation

energy captured by the k successive modes which is given by:

1 1

k N

k i i

i i

E  

 



 

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POD

Image processing has greatly benefited of POD or PCA.

Automatic face identification is an example:

Original Image

Progressive reconstruction from 1 to 50 eigenfaces

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POD

“Eigenfaces” (functional basis for faces) are constructed from a data set of human faces:

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POD

“Eigenflames” (functional basis for flames) are constructed from a data set of ICE pictures:

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Application to ICE images: motivation

To develop new tools for analysis, reconstruction and

prediction of data from high-resolution experimental digital imaging in Internal Combustion Engines

To examine cycle-to-cycle variation of the flame by means of techniques based on POD procedure, that is:

discriminate mean, coherent and incoherent part of the flame.

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Experimental, SI

A Spark Ignition, optically accessible engine is used to acquire high definition pixel images of luminosity

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Experimental, SI

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POD, experiments on IC engines

Whenever the number of collected samples over time is smaller than the space discretization ( M << N ), it is more efficient to assume (Sirovich, 1987) that eigenfunctions are linear combinations of the snapshots:

Hence, substituting into the original POD problem we obtain:

1

( ) ^M _{i i} ( )

i

x b u x







CB  B

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POD, experiments on IC engines

where C is the space correlation matrix:

Thus the POD eigenfunctions are determined as a linear combination of the eigenvectors – obtained by solving the eigenvalue problem – with the ensemble of data, whereas the modal coefficients are calculated in the same way as in the original POD approach.

1

1 ^N ( ) ( )

ij i k j k

k

C u x u x

N _





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POD, experiments on IC engines

More precisely, the POD eigenvalue problem:

is solved and the POD eigenfunctions are determined as a combination of eigenvectors and the “snapshots”:

The coefficients of POD eigenfunctions are calculated by conducting the orthogonal projection of the data onto the set of POD basis functions.

CB  B

1

( ) ( )

M

i i i

X b u X







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POD reconstruction

2

1 _M

…

=

^a¹^j ^

+

^a²^j ^

+ …

1

( ) ( ) ( )

K j

j k k

k

u X a t  X







j ( ) u X

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POD, experiments on SI engines

The cumulative correlation energy is calculated and then the luminosity field is reconstructed by using the different number of POD eigenfunctions as follows:

where K  N is the truncation order.

1

( ) ( ) ( )

K j

j k k

k

u X a t  X







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Cycle-to-cycle analysis

Basing on the POD, a decomposition and analysis of the considered field can be conducted, by

computing some statistical properties of the coefficients.

The idea was earlier introduced by Roudnitzsky et al who applied it to PIV data of velocity components in Diesel engines obtained for cold flow, and is here attempted for cycle resolved images of luminosity in reactive flow.

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Cycle-to-cycle analysis

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Cycle-to-cycle analysis

- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0

C r a n k a n g l e [ d e g r e e s ] 0

1 0 2 0 3 0 4 0 5 0 6 0

Combustion pressure [bar]

0 1 0 2 0 3 0

Drive current [Ampere]

0 4 0 8 0 1 2 0 1 6 0

Rate Of Heat Release [kJ/kg/°]

Fuel injection pattern (dashed), and resulting pressure and heat release rate for a typical cycle.

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Typical cycle-resolved sequence

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37 complete sequences

1 sequence = 24 frames every 1.5°@1000 rpm Exposure time = 166 s= 1° @ 1000 rpm

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20 ATDC 21.5 ATDC 23 ATDC 24.5 ATDC 26 ATDC 27.5 ATDC 29 ATDC 30.5 ATDC 4 BTDC 2.5 BTDC 1 BTDC 0.5 ATDC 2 ATDC 3.5 ATDC 5 ATDC 6.5 ATDC

8 ATDC 9.5 ATDC 11 ATDC 12.5 ATDC 14 ATDC 15.5 ATDC 17 ATDC 18.5 ATDC

- 2 0 0 2 0 4 0 6 0

C r a n k a n g l e [ d e g r e e ]

0 2 0 4 0 6 0

0 1

Signal [V]

0 1

Trigger [V]

- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0

C r a n k a n g l e [ d e g r e e s ] 1 0

2 0 3 0 4 0 5 0 6 0

0 1 0 2 0 3 0 Drive current [Ampere]

0 4 0 8 0 1 2 0 1 6 0

Rate Of Heat Release []

I

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- 2 0 0 2 0 4 0 6 0

0 2 0 4 0 6 0

0 1

Signal [V]

0 1

Trigger [V]

- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0

2 0 3 0 4 0 5 0 6 0

0 4 0 8 0 1 2 0 1 6 0

II

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- 2 0 0 2 0 4 0 6 0

0 2 0 4 0 6 0

0 1

Signal [V]

0 1

Trigger [V]

- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0

2 0 3 0 4 0 5 0 6 0

0 4 0 8 0 1 2 0 1 6 0

III

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- 2 0 0 2 0 4 0 6 0

0 2 0 4 0 6 0

0 1

Signal [V]

0 1

Trigger [V]

- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0

2 0 3 0 4 0 5 0 6 0

0 4 0 8 0 1 2 0 1 6 0

IV

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- 2 0 0 2 0 4 0 6 0

0 2 0 4 0 6 0

0 1

Signal [V]

0 1

Trigger [V]

- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0

2 0 3 0 4 0 5 0 6 0

0 4 0 8 0 1 2 0 1 6 0

V

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- 2 0 0 2 0 4 0 6 0

0 2 0 4 0 6 0

0 1

Signal [V]

0 1

Trigger [V]

- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0

2 0 3 0 4 0 5 0 6 0

0 4 0 8 0 1 2 0 1 6 0

VI

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- 2 0 0 2 0 4 0 6 0

0 2 0 4 0 6 0

0 1

Signal [V]

0 1

Trigger [V]

- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0

2 0 3 0 4 0 5 0 6 0

0 4 0 8 0 1 2 0 1 6 0

VII

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I

II

III

IV

V

VI

4 BTDC 2.5 BTDC 1 BTDC 0.5 ATDC 2 ATDC 3.5 ATDC 5 ATDC 6.5 ATDC

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I

II

III

IV

V

VI

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I

II

III

IV

V

VI

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Cycle-to-cycle analysis

Cycle variations are classically related to in-cylinder engine parameters, such as pressure or global luminous intensity.

It is clear though that flame shape and position also change from cycle to cycle. Thus there is cycle variation also for space distributed parameters.

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Non-structured, 25% white

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Structured, 25% white

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Non-structured, 25% white Structured, 25% white

cycles

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Cycle-to-cycle analysis

We will see how POD can help with analysis of cycle variation of space distributed parameters.

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Cycle-to-cycle analysis

Suppose that each photograph taken at a given crank angle value can be treated as a random process over the cycles. The mean of the luminosity field can be simply computed as:

, where . Hence if we define

, then

1

1 ^N

k k

u u

N _

 

1 N

k ik i

i

u c 





1

1 ^N

i ik

k

c c

N _

 

1 N

i i i

u c 





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Cycle-to-cycle analysis

The variance of the luminosity field can be computed as:

Now substitute the expressions for and to obtain:

2 2 2

1

1 ^N

u k

k

u u

 N







uk ^u

2 2

2

1 1 1

1 ^N ^N ^N

u ik i i i

k i i

c c

 N  

  

   

     

   

  

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Cycle-to-cycle analysis

and the normalized standard deviation, also said Coefficient of Variation, finally is:

which is a function of the coefficients only.

2 2

1 1

1

N N

norm i i

i i

CV  c c

 

 



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CVglob = 0 CVPOD  0

CV^glob = 0 CV^POD  0

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Non-structured, mean Structured, mean

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Cycle-to-cycle analysis

This figure reports as a function of the crank angle. It is seen that the standard deviation shows three distinct peaks, in correspondence to the three

injection/combustion events observed.

norm

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Cycle-to-cycle analysis

The POD analysis permits to easily investigate the morphology of the fluctuation. By definition, the fluctuation at cycle k is

which is, rather obviously, given by

k k

u  u  u

 

1 N

k ik i i

i

u c c 



 





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Cycle-to-cycle analysis

It would be interesting to further discriminate coherent from incoherent fluctuations and to visualize the relevant morphologies.

The idea is that the coherent part includes all fluctuations possessing a somehow structured feature over the cycles (e.g. some luminous spot appearing in most cycles but not in all).

The incoherent part should then include all fluctuations for which no pattern can be identified over the cycles.

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Cycle-to-cycle analysis

We can assume that our fluctuation be composed of a coherent part having non-Gaussian distribution, plus an incoherent part with Gaussian distribution.

It is then possible to extract the Gaussian part by computing relevant statistical properties of each

coefficient, namely the skewness ( ) and kurtosis ( ), according to the following expressions:

ci

  ³

1 3

i i

 c



    ⁴

2 4

i i

 c



 

1 ₂

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Cycle-to-cycle analysis

-1 -0.5 0 0.5 1

-1.5 -1 -0.5 0 0.5 1 1.5 2

₁

1

Luminosity field for 17° ATDC. Top row, from left to right: Experimental field for sample No. 1;

mean field over 37 cycles; fluctuation u’ for sample No. 1. Bottom row, from left to right: scatter plot of kurtosis vs skewness of POD modes; z (non-Gaussian) part of fluctuation; w (Gaussian) part of fluctuation. Fluctuations are represented in absolute value. The w part contains all modes within the circle in the scatter plot.

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Cycle-to-cycle analysis

Luminosity field for 14° ATDC. Top row, from left to right: Experimental field for sample No. 1;

mean field over 37 cycles; fluctuation u’ for sample No. 1. Bottom row, from left to right: scatter plot of kurtosis vs skewness of POD modes; z (non-Gaussian) part of fluctuation; w (Gaussian) part of fluctuation. Fluctuations are represented in absolute value. The w part contains all modes within the circle in the scatter plot.

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Spherical bomb, ICARE-CNRS

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Spherical Bomb



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CONCLUSIONS & FUTURE WORK

A closer integration of POD decomposition with experimental

techniques will provide savings in terms of experimental efforts and more qualitative and quantitative information on flame development and morphology.

POD analysis provides information on cycle variations in terms not only of global in-cylinder quantities but also in terms of spatial

distribution of the observed variables (flame shape and location).

Higher statistical moments of the POD coefficients carry information on coherent and incoherent components that, reconstructed

separately, display the structured shapes and the underlying turbulent flow field respectively.