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

Acknowledgements NSF Alcoa Technical Center

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

Objectives

Formulate a general, non-empirical approach for assessing the stability of Beck’s sequential function specification method

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.

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

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

Inverse Problem Statement 1

2

q

Known temperatures on 1

Interior temperature measurementsT],...,,[Y~ 1n

I1n

21n

11n Y~Y~Y~

I: the total number of measurement sites (I=6).

Unknown heat fluxes to be solved

T],...,,[q 1nJ

1n2

1n1

1n qqq

J: the total number of nodes on (J=5). 2

Unknown heat fluxes actually solvedT]q~,...,q~,q~[q~ 1n

K1n

21n

11n

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

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

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.

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

0~~~ )()(

0

T (m)

mmR

mY X

where

1

)()(

q~

~~

n

mm

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

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

Thus,

)(1)( c~~ mnm qDf [Linear relationship]

where

1

)(

~f~

n

j

mP

Pj qD : constant matrix determined by FE discretization;

(m)c : constant vector determined by primary conditions during condensation.

A key observation

2

qNes

ie

i dsf eif is a linear function of nodal heat fluxes at 2

Inverse Algorithm

Step (1) Inversion

)()()()()( mmmnmm t fUMθUθ

where

1)()( ][ KMU mm t

Step (2) Mapping

)()()()()( ~~~ mmmnmm t fUMθUθ where

)()(~ m

GPm

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

Sensitivity coefficient matrix

)(1)( c~~ mnm qDf

)()()()()( ~~~ mmmnmm t fUMθUθ

1

)()(

q~

~~

n

mm

X

DUt mmm ~~~ )()()( X

tck

(m)(m) X~)X~(

Improvements: Time efficiency Accuracy

Approximate methods:

1

)(

1

)()(

q~

~

q~

~X~

n

m

n

mm

governing sensitivity coefficient equation

fraction of two finite differences

Inverse Algorithm

Matrix normal equation

Inverse Algorithm

)()()()()( ~~~ mmmnmm t fUMθUθ DUt mmm ~~~ )()()( X

0~~~ )()(

0

T (m)

mmR

mY X

R

m

mmmmnmTmnR

m

mTm Yct0

)()()()()()(1

0

)()( 0~~~q~~~ UMUXXX

)(1)( c~~ mnm qDf

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

Numerical Tests1. 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

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

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

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;

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

Application to quenching

n

n

nnn

n

dtYdnRkq

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.

Error Propagation Equation

)()()()()( mmmnmm t fUMθUθ

)()()()()( ~~~ mmmnmm t fUMθUθ

Globalstandard form equation

where1)()( ][ KMU mm t

)(1)( c~~ mnm qDf

yields computed temperatures at measurement sites

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

Error Propagation Equation

CWYGθθ 1nn1n

MUBDUΔtUMG (m) ~~

BDUΔtW (m) ~

Substitution of )(mf into standard form eqn. then gives

where

cUBDUΔtC (m) ~~

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

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

,

,

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

11

Y

w 1

1

n

nn G

4. Convergence criterion

1n

,

,

5. Frobenius norm

N

i

N

iijFr

G 2G

Solution Stability to An Input Error

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.

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

G4. 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

Questions