Lecture 10: L inear Inverse Heat Conduction Problems Two basic - - PowerPoint PPT Presentation

Lecture 10: Linear Inverse Heat Conduction Problems Two basic examples

Yvon JARNY, Denis MAILLET

LTN, CNRS & Université de Nantes- Polytech’Nantes –Nantes

LEMTA, Nancy-University & CNRS, Vandoeuvre-lès-Nancy, France

1 Metti5 – Roscoff June 13-18, 2011

Introduction

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 How to determine the time varying heat flux density entering a solid wall, from noisy data given by some temperature measurements inside (or outside) the wall, is a very standard Inverse Heat Conduction Problems (IHCP),  The choice (in practice) of a numerical method for solving such problems will depend on the “complexity” of the model equations, and the “quality” of the measurements

• Are the model equations linear or not?
• What is the dimension and/or the shape of the spatial domain?
• Which kinds of sensors? Their locations ? The output equations ?

...  In any case, some specific difficulties are “expected”, because IHCP are known to be ill-conditioned and regularized processes have to be developed for avoiding instable solutions due to noisy data, and/or biased models

Outline

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 Introduction  Inverse Heat conduction in a semi infinite body

• the linear input/output model equation
• Non regularized solutions – unstabilities
• Regularized solution – the SVD method

 Inverse Heat conduction in a plane wall

• the linear input/output model equation (single output)
• Non regularized solutions – unstabilities
• Effect of a biased model
• Effect of a multi-output sensor
• Splitting IHCP

 Conclusion

Semi-infinite heat conduction body The model equations (see lecture n°2)

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) ( ) ( d ) ( ) ( ) ( ) , , ( ) ( t y t y u t Z dx x T t x x G t y

forced mo relax mo t c mo

+ = − + =

∫ ∫

∞

τ τ τ

                + − +         − − = t a x x t a x x t a t x x G

c c c

4 ) ( exp 4 ) ( exp 2 1 ) , , (

2 2

π

( )

) exp( 4 / exp 1 ) (

2 c

t t K t a x t C k t Z τ τ π ρ − = − =

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Figure 1 – The impulse

• utput

signal

π ρ τ τ

c c

x c K t t a x 2 and * ; 4

2

= = = K t Z t Z / ) * ( ) * ( * =

Semi-infinite heat conduction body

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Semi-infinite heat conduction body The model equations

j i j j j i i j i mo

u u t t Z t y

j i 1 1 1

S ) ( ) t (

∑ ∑

= − =

= − ∆ =

( )

   = = = ∆ + − ∆ =

+ −

lse m to i i to j z t j Z t

j i

e 1 , 1 ; ) 1 i ( S

1 j i

                    =

− − 1 2 2 1 1 2 3 4 1 2 3 1 2 1

z z z z z z z z z z z z z z z

m m m

  S

u S y =

mo

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Numerical results computed with

K 005 . = σ

s t 5 , = ∆

s 1 ; s m 10 ; mm 2

1 2 6

= = =

− −

τ a xc

1 1 K

m W 1

− −

= λ

1 1 6

K kg J 10

− −

= c ρ

1 2 3

J m K 10 564 .

− −

= K

Semi-infinite heat conduction body

Example of Input signal u (t) and output response y(t)

The IHCP in a semi-infinite body a non regularized solution

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y S u

1

ˆ

−

=

Estimated heat flux – cases a, b and c - Influence of the noise level

• n the computed heat flux - time step

s 0,5 = ∆t

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0,8s t = ∆

The IHCP in a semi-infinite body a non regularized solution

y S u

1

ˆ

−

=

Estimated heat flux – cases a, b and c - Influence of the noise level on the computed heat flux -

The IHCP in a semi-infinite body- Influence of the time step on the stability of the solution

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by decreasing the time step, the sensitivity coefficients of the Toeplitz matrix S goes to zero, and the condition number grows exponentially

0.8 0.5 0.4

Cond(S) 46,5 292 28420 t Δ

• n each time step

The resulting increment on the output signal will be “significant”

• nly if this value is greater than the level noise

t t t t u

t t t

k k

Δ 400 d 800 Δ 1

Δ

= = ∆

∫

+

u ) /Δ (- exp t Δ t Δ ∆ ≈ ∆ t K y τ τ

σ > ∆y

Influence of the time step

• n the stability of the solution

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 time step dt

• utput increment

Influence of the time step on the output variation - example 1

0.8 0.5 0.4

46,3mK 10 mK 4,7mK t Δ

y ∆

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The IHCP in a semi-infinite body a non regularized solution

y S u

1

ˆ

−

=

s 4 Δ , t =

K 005 0, = σ

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The IHCP in a semi-infinite body a regularized solution - the SVD method

T

V W U S = V U, W is the matrix of the singular values

The SVD regularized solution is then

{ }

m k wk ,.. 1 , =

are (m x m) and (n x n) orthogonal matrices ; here m = n = 51

y U V u

T k k n r k k k k r

a ith w a = = ∑

< =

w ˆ

1

The IHCP in a semi-infinite body a regularized solution - the SVD method

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The truncation order r is used as the “tuning” parameter

• f the regularization process

The expected compromise between accuracy and stability will be fixed by some optimal value We have to avoid:  a too big error amplification , when  a too large bias when

→ r n r → n r < <

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The IHCP in a semi-infinite body a regularized solution - the SVD method

2

) ( u u − =

r

ˆ r f

s 4 Δ , t =

K 005 0, = σ

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The IHCP in a semi-infinite body a regularized solution - the SVD method

SVD Regularized solution computed with

18 et 15 12, r =

s 4 Δ , t =

K 005 0, = σ

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The IHCP in a sem i-infinite body a regularized solution - the SVD method

[ ] [ ]

      =       = = =

c r c r c r c r

nd V V V u u u W W W U U U a ; ;

With c = m – r then the error estimate can be put in the form

( ) ( )

*

c c T r r r u

u V ε U W V e  − =

−1

( )

2 1 1 2 2

1 E * u w

k m r k r i k u T u



∑ ∑

+ = =

+ = σ e e

 The first term is directly linked to the variance of the measurement noise,

it increases by increasing the truncation parameter r,  and the second term depends only on the c = m – r spectral components

• f the exact heat flux signal, which have been “lost” by truncation.

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The IHCP in a sem i-infinite body a regularized solution - the SVD method

Conclusion : the ill-conditioness of the inverse heat conduction problem depends both  on the mathematical model equations (singular values of S)  and on the spectral values of the input signal to be determined

The compromise in choosing the truncation parameter r takes into account these both contributions.

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Heat conduction in a plane wall The model equations

Transient heat conduction in a plane wall

= = + = ) ( ) ( d d t t u t T b T A T

) ( ) ( t t

mo

T C y =

= =

∞

T T

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                =                 + − − − − = 1 Δ 2 et ) 1 ( 2 2 1 2 1 2 1 2 2 ) (Δ

2

      z c Bi z a ρ b A

λ / z h Bi Δ =

[ ]

1 Δ ; Δ ) 1 ( z ) , z ( ) ( avec ) ( ) ( ) ( ) (

i i 2 1

− = − = = = N e z z i and t T t T t T t T t T t

i T N

 T

[ ]

c i

i i C ≠ = = si where 1   C

τ τ τ d ) ( ) ) ( ( ) ( y u t t

t mo

b A C ∫ − = exp

Heat conduction in a plane wall Solution of the direct problem

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Heat conduction in a plane wall Discrete Solution of the direct problem

{ }

ik k j n j j j

t f t f u t u δ = = ∑

=

) ( ; ) ( ) (

1

m k f t S u S t

j t k j k j n j j k k mo

k

,.., 1 , d ) ( ) ) ( ( ; ) ( y

1

= − = =

∫ ∑

=

τ τ τ b A C exp

u S y =

mo

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Heat conduction in a plane wall Example of Numerical results

1 2 1 3 6 1 1

; K Jm 10 2 . 1 ; K Wm 3 , ; m 05 ,

− − − − − −

= = = = K Wm h c e ρ λ

nodes 21 = N 40 200 Δ = = nt s t

K ,02 = σ

ε y y + =

mo

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Inverse Heat conduction in a plane wall non regularized solution

y S u

1

ˆ

−

=

Estimated heat flux – Influence of the sensor location

u ˆ

K ,02 = σ

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Inverse Heat conduction in a plane wall non regularized solution

y S u

1

ˆ

−

=

Estimated heat flux – Influence of the sensor location

s 320 Δ = t K ,02 = σ

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zc

e / 4 e / 2 3 e / 4 730 8700 1,0244 105 343* 2349 1,7 104

s 200 Δ = t

s 320 Δ = t

Condition number of the matrix S - IHCP in a plane wall

As in the previous example, by using the SVD approach,  regularized solutions could be easily computed  and the same analysis of the estimation error could be done

Inverse Heat conduction in a plane wall non regularized solution

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Inverse Heat conduction in a plane wall solution with a biased model

Output signal computed with h = 20 W m-2 K-1

(instead of h = 0)

The Influence of this parameter on the

• utput signal

becomes more and more significant when the sensor is located closer to the boundary x = e

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The numerical inversion process is performed on the original noisy output data (with h = 0 ) but with a model error ( h = 20 W m-2K-1), included in the matrix S.

Inverse Heat conduction in a plane wall solution with a biased model

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The influence of the sensor location is clearly illustrated. There is a systematic error between the solutions computed with the biased and the exact models. The mean value of this bias is evident at the end of the time interval.

) (

• )

( ˆ ) (

20

t u t u t b

h h u = =

=

) (t bu

Inverse Heat conduction in a plane wall solution with a biased model

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Inverse Heat conduction in a plane wall solution with a biased model Effect of a multi-output sensor

y S S S u

T T OLS 1

) ( ˆ

−

=

solutions

• btained with

two sensors located at z = e/4 and z = e/2 with noisy data

)

s 200 Δ = t

K ,02 = σ

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Splitting the Inverse heat conduction problems for a plane wall case

Inverse Heat conduction problems in a plane wall

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 the semi-infinite solid = simple geometry which provides a useful idealization for many practical situations.

• A numerical solution of this IHCP can be easily investigated, thanks to

the linearity of the model equation.

• The instabilities of the non regularized solutions due to: noise level,

sensor location, time step ..., can be analyzed and illustrated.

• The SVD method is a powerful approach to master the regularized

solutions.  the linear heat conduction problem in a plane wall,

• after a standard discretization of the spatial variable
• similar analysis of the instabilities of the IHCP solutions, can be

investigated

• the influence of a biased model, with/without a multi-sensor output

model is easy to illustrate

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References [1] F P Incropera, D P DeWitt, Fundamentals of Heat and Mass Transfer, (IVth edition) John Wiley and Sons, New York, 1996 [2] D Maillet, Y Jarny and D Petit, Problèmes inverses en diffusion thermique, Techniques de l’Ingénieur”, BE 8 266, Editions T.I., Paris, 2010 [3] Y Jarny and H Orlande, « Adjoint Methods », in “Thermal Measurements in Heat Transfer”, CRC Press, 2011