xianwu ling russell keanini harish cherukuri department of mechanical engineering university of...
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Xianwu LingRussell Keanini
Harish Cherukuri
Department of Mechanical Engineering University of North Carolina at Charlotte
Presented at the 2003 IPES Symposium
A method for analyzing the stability of non-iterative inverse heat conduction algorithms
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Acknowledgements
NSF Alcoa Technical Center
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Outline
Objective
Literature Review
Inverse Problem Statement
Direct Problem
Inverse Algorithm – Sequential Function Specification Method
Derivation of Error Propagation Equation
Stability Criterion Defined
Application to 1-D Problem
Summary and Conclusions
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Objectives
Formulate a general, non-empirical approach for assessing the stability of Beck’s sequential function specification method
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Literature Review
Maciag and Al-Khatih (2000). Int. J. Num. Meths. Heat & Fluid Flow. Used integral (Green’s function) solution and backward time differencing to obtain
11 nnn AYB
Convergence, as determined by spectral radius of B, determines stability.
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Literature Review, cont’d
Liu (1996). J. Comp. Phys. Used Duhamel’s integral to obtain
1
12
1n
jjjn
n Xq
where is a response function that dependson the measured data and where the set of coefficients X are used to determine stability:
1
||k
kXE
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Inverse Heat Conduction Problem (IHCP)
1
2
Known temperatures at 1
Boundary conditions
Known Initial conditions
),,(00 zyx
Known temperature measurements
q Unknown surface heat fluxes (q) at 2
Interpolated node
Overview of IHCP
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Inverse Problem Statement
1
2
q
Known temperatures on 1
Interior temperature measurements
T],...,,[Y~ 1n
I
1n
2
1n
1
1n Y~Y~Y~ I: the total number of measurement sites (I=6).
Unknown heat fluxes to be solved
T],...,,[q 1n
J
1n
2
1n
1
1n qqq
J: the total number of nodes on (J=5). 2
Unknown heat fluxes actually solved
T]q~,...,q~,q~[q~ 1n
K
1n
2
1n
1
1n K: the total number of chosen nodes from J (K=3).
In the usual case, only some specific nodes (K<=J) are chosen from the total number (J) of nodes on 2 , while some interpolation functions (usually linear) are used for the other nodes on 2
Known initial temperatures
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Direct Heat Conduction Problem
H e a t E q u a t i o n ( n o h e a t s o u r c e s )
tck
)(
B . C . :
o n 1
qnk o n 2 I . C . : X0,X
0
I m p l i c i t c o n t i n u o u s G a l e r k i n m e t h o d , c o n d e n s e d f o r m :
1n 1n f M K)M( ntt
M : c a p a c i t y m a t r i x ; K : s t i f f n e s s m a t r i x ; f : g l o b a l f o r c e v e c t o r ;
: c o m p u t a t i o n a l t i m e s t e p t
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Inverse Algorithm
Introduction to computational time steps
Experimental time step, computational time step, and future time
t
2 4RExample: ,
t
R: the number of future temperatures used.
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Ordinary least squares error norm:
)()(
0
)()(1 ~~~~ Tmm
R
m
mmn YYs
Where
)(~mY : measurement temperatures at (m); )(~m : calculated temperatures at (m), unknown function of
1~nq ;
R : total number of future temperatures used; (m) : future time index.
Objective function
Inverse Algorithm
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0~~~ )()(
0
T (m)
mmR
m
Y X
where
1
)(
)(
q~
~~
n
m
m X
is the sensitivity coefficient matrix of dimension .
Minimization of
Inverse Algorithm
)()(
0
)()(1 ~~~~ Tmm
R
m
mmn YYs
1ns
with respect to leads to:1~ nq
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q(t)
Time0 1 2 n n+1
tq1
q2 q n+1
Introduction to function specification method
Idea: Assume a function form of the unknown, and convert IHCP into a problem in which the parameters for the function are solved for.
Piecewise constant function:
(1) q n+1 are solved for step by step;
(2) For each step from n to n+1, an unknown constant is assumed for each future temperature time; the final resultant heat flux for the step is the average of the unknown constants in the strict least squares error sense.
Inverse Algorithm
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Thus,
)(1)( c~~ mnm qDf [Linear relationship]
where
1
)(
~f~
n
j
m
PPj q
D : constant matrix determined by FE discretization;
(m)c : constant vector determined by primary conditions during condensation.
A key observation
2
qNes
i
e
i dsf e
if is a linear function of nodal heat fluxes at 2
Inverse Algorithm
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Step (1) Inversion
)()()()()( mmmnmm t fUMθUθ
where
1)()( ][ KMU mm t
Step (2) Mapping
)()()()()( ~~~ mmmnmm t fUMθUθ
where
)()(~ m
GP
m
iP UU
The local index i (spanning the I measurement sites) maps to the corresponding global node number G.
Solve for computed temperatures at the measurement sites
Inverse Algorithm
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Sensitivity coefficient matrix
)(1)( c~~ mnm qDf
)()()()()( ~~~ mmmnmm t fUMθUθ
1
)(
)(
q~
~~
n
m
m X
DUt mmm ~~~ )()()( X
tck
(m)(m) X
~)X
~(
Improvements: Time efficiency Accuracy
Approximate methods:
1
)(
1
)(
)(
q~
~
q~
~X~
n
m
n
m
m
governing sensitivity coefficient equation
fraction of two finite differences
Inverse Algorithm
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Matrix normal equation
Inverse Algorithm
)()()()()( ~~~ mmmnmm t fUMθUθ DUt mmm ~~~ )()()( X
0~~~ )()(
0
T (m)
mmR
m
Y X
R
m
mmmmnmTmnR
m
mTm Yct0
)()()()()()(1
0
)()( 0~~~
q~~~
UMUXXX
)(1)( c~~ mnm qDf
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Inverse Algorithm
(1) Given the temperatures at n, and the measured temperatures at some interior locations at some future times, the heat fluxes from n to n+1 can be solved using the matrix normal equation (together with the sensitivity coefficient matrix equation)
Inverse algorithm procedures
(2) Given the heat fluxes from n to n+1, the temperature at the end of n+1 can be updated using )()()()()( mmmnmm t fUMθUθ
(3) Go to the next time step
Characteristics
SequentialNon-iterativeFEM-basedfuture temperature regularizationexplicit calculation of sensitivity coefficient matrix
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Numerical Tests
1. Step change in heat flux:
A flat plate subjected to a constant heat flux qc at x=0 and insulated at x=L.
q/qc
0 Time, t
1qc x
L
Fictitious measurement site
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Numerical Tests
5
(a) Results from the present method
The calculated surface heat flux for const qc input for a plate. . 05.0 t
(b) Results from Beck’s function specification method
Results
smaller time step; large error suppression for large number of future temperatures;
No early time damping.
Observations
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Numerical Tests
2. Triangular heat flux:
q+
Time, t
0
Fictitious measurement site
A flat plate subjected to a triangular heat flux at x=0 and insulated at x=L.
Noise input temperatures data are simulated by (1) decimal truncating, (2) adding a random error component generated using a Gaussian probability distribution .
qc x
L
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Numerical Tests
The calculated heat flux. Decimal Truncating errors. =0.01.
The calculated heat flux. Random errors. =0.06.
Results
Observations smaller time step; less susceptible to input errors;
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Application to quenching
Drayton Quenchalyer, Inconel 600 probe, Quenchant: oil. Sampling Freq: 8 Hz, Duration: 60 S
Co8500 , Co40
Typical temperature history at the center of the probe
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Application to quenching
n
n
nnn
n
dt
YdnRkq
1
12
12
!2
1. Excellent agreement
2. Influence of small oscillations
3. Temperature comparison
Results
Burggraf’s analytical solution:
Calculated heat fluxes vs. time
Calculated temperature vs. time.
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Error Propagation Equation
)()()()()( mmmnmm t fUMθUθ
)()()()()( ~~~ mmmnmm t fUMθUθ
Globalstandard form equation
where
1)()( ][ KMU mm t
)(1)( c~~ mnm qDf
yields computed temperatures at measurement sites
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Error Propagation Equation
R
0m
(m)(m)(m)(m)n(m)T(m)1n(m) 0YcUΔtMθUXqA~~~~
)(1)( c~~ mnm qDf
]~~~
[~ )(1)( cUMθUYBDf n mnm t
Matrix normal equation
and global force vector
then yield
T1XAB~where
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Error Propagation Equation
CWYGθθ 1nn1n
MUBDUΔtUMG (m) ~~
BDUΔtW (m) ~
Substitution of )(mf into standard form eqn. then gives
where
cUBDUΔtC (m) ~~
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Error Propagation Equation
1ne
n1nne
ne
nnne
1n YδWδYWθδGδθGδθ
)G(θ)δθG(θδG ne
nne
n
nne
n δYYY
Letting nne
n δθθθ be the computed global temperature
where
and the measured temperature vector, the error
propagation equation is finally obtained:
)W(θ)δθW(θδW ne
nne
n
In linear problems 0δWδG nn
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Solution Stability to An Input Error One-dimensional axisymmetric problem
Model
Governing temperature equation (with no future temperature regularization)
11 nnn YwG
where
mN/UUv-UMG U~ mNUv/w U
and
}U, , U,U~mNm2m1{U T}1 ,0, ,0 0, {v
,
,
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Frobenius norm analysis
1. Assumption
01 Y 1,0 iY iand
2. Equations of temperature error propagations
11 nn G11 Y w
3. Temperature error propagation factors
1
1
1
Y
w 1
1
n
nn G
4. Convergence criterion
1n
,
,
5. Frobenius norm
N
i
N
iijFr
G2G
Solution Stability to An Input Error
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6. Results and discussions
1) Effect of measurement location and computational time step
Observations:
a) For the first time step, the deviation is very high for small time steps and deeply imbedded sensors;
The variation of nωwith n. 0.r
The variation of nωwith n. 0.9r
b) For small time steps, the errors are high, and suppressed slowly; for
large time steps, the errors reduced, the suppression rate extremely high;
Solution Stability to An Input Error
The variation of 1ωwith . 2/ rtγ
The variation of nωwith n. 05
c) As the sensor is far away from the surface, the initial errors increase,
yet accompanied by much higher subsequent error suppression rates.
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2) Effect of number of elements
The variation of nωwith n. 125.0t/R 0,r 2
a) Increasing the number of elements increases the error suppression rates;b) A choice of 20 element would be proper for the problem under study, as observed as J. Beck.
Spectral norm analysis
The variation of the spectral norm with , Ne=20.
11 nnn YwG
1. Governing temperature equation
2. Spectral norm
)( *
max GGG s
3. Convergence criterion
1s
G
4. Results and discussions
a) Clear indication of the allowable time steps;b) No hint of the error suppression rates.
Solution Stability to An Input Error
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Questions