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

Lecture 10: L inear 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 Metti5 – Roscoff 1 June 13-18, 2011

Introduction  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 Metti5 – Roscoff June 13-18, 2011 2

Outline  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 Metti5 – Roscoff June 13-18, 2011 3

Semi-infinite heat conduction body The model equations (see lecture n°2) ∞ ∫ ∫ t = + − τ τ τ y ( t ) G ( x , x , t ) T ( x ) dx Z ( t ) u ( ) d 0 mo c 0 0 = + ( ) ( ) y t y t mo relax mo forced       − 2 + 2 1 ( ) ( ) x x x x     = − + − ( , , ) exp exp  c c  G x x t     c π 4 4 2  a t a t      a t   ( ) 1 = − 2 Z ( t ) exp x / 4 a t ρ π c k C t τ τ = − K exp( ) t t Metti5 – Roscoff June 13-18, 2011 4

Semi-infinite heat conduction body Figure 1 – = Z * ( t * ) Z ( t * ) / K The impulse output signal 2 x τ = c ; 4 a t = t * τ and 2 = K ρ π c x c Metti5 – Roscoff June 13-18, 2011 5

Semi-infinite heat conduction body The model equations i i ∑ ∑ = ∆ − = y ( t ) t Z ( t t ) u S u − mo i i j 1 j i j j = = j 1 j 1 ( ) ∆ − + ∆ = = =  t Z ( i j 1 ) t z ; j 1 to i , i 1 to m − + = i j 1  S i j  0 e lse   0 z 1   0 0 z z   2 1   z z z 3 2 1 =   S = 0  z z z  y S u  4 3 2 mo   0  z 1      z z z z z  − − 1 2 2 1 m m m Metti5 – Roscoff June 13-18, 2011 6

Semi-infinite heat conduction body Example of Input signal u ( t ) and output response y(t) Numerical results computed with σ = 0 . 005 K ∆ = 0 , 5 t s 1 K − − − − = = τ = λ = 6 2 1 1 x c 2 mm ; a 10 m s ; 1 s 1 W m − − ρ = − − = 3 2 1 6 1 1 0 . 564 10 K m J K c 10 J kg K Metti5 – Roscoff June 13-18, 2011 7

The IHCP in a semi-infinite body = − 1 ˆ u S y a non regularized solution Estimated heat flux – cases a, b and c - Influence of the noise level ∆ t = 0,5 s on the computed heat flux - time step Metti5 – Roscoff June 13-18, 2011 8

− = 1 ˆ u S y The IHCP in a semi-infinite body a non regularized solution ∆ t = Estimated heat flux – cases a, b and c - Influence of the noise 0,8s level on the computed heat flux - Metti5 – Roscoff June 13-18, 2011 9

The IHCP in a semi-infinite body- Influence of the time step on the stability of the solution 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 Δ t Cond(S) 46,5 292 28420 1 + Δ t t on each time step ∫ ∆ = = k Δ u 800 t d t 400 t Δ t t k The resulting increment on the output signal τ ∆ ≈ τ ∆ Δ / Δ will be “significant” y K t exp (- t ) u Δ t ∆ y > σ only if this value is greater than the level noise Metti5 – Roscoff June 13-18, 2011 10

Influence of the time step on the stability of the solution 0.8 0.5 0.4 Δ t ∆ 46,3mK 10 mK 4,7mK y Influence of the time step on the output variation - example 1 0.05 0.045 0.04 0.035 output increment 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 time step dt Metti5 – Roscoff June 13-18, 2011 11

The IHCP in a semi-infinite body − = 1 ˆ u S y a non regularized solution σ = 0 , 005 K t = Δ 0 4 s , Metti5 – Roscoff June 13-18, 2011 12

The IHCP in a semi-infinite body a regularized solution - the SVD method S = T U W V U , V are (m x m) and (n x n) orthogonal matrices ; here m = n = 51 { } = w k , k 1 ,.. m W is the matrix of the singular values The SVD regularized solution is then < = ∑ r n a = T k ˆ u V w ith a U y r k k k w = k 1 k Metti5 – Roscoff June 13-18, 2011 13

The IHCP in a semi-infinite body a regularized solution - the SVD method The truncation order r is used as the “tuning” parameter of the regularization process The expected compromise between accuracy and stability < r < will be fixed by some optimal value 0 n We have to avoid: r →  a too big error amplification , when n →  a too large bias when r 0 Metti5 – Roscoff June 13-18, 2011 14

The IHCP in a semi-infinite body a regularized solution - the SVD method σ = 0 , 005 K t = Δ 0 4 s , 2 = − ( ) ˆ f r u u r Metti5 – Roscoff June 13-18, 2011 15

The IHCP in a semi-infinite body a regularized solution - the SVD method SVD Regularized solution computed with r = 12 , 15 et 18 t = Δ 0 4 s , σ = 0 , 005 K Metti5 – Roscoff June 13-18, 2011 16

The IHCP in a sem i-infinite body a regularized solution - the SVD method     [ ] [ ] r r 0 W u = = = = r c r c     U U U ; V V V ; W a nd u c c  0    W u With c = m – r t hen the error estimate can be put in the form ( ) ( )  − 1 * T = − r r r ε c e V W U V u u c ( ) r m 1  ∑ ∑ * 2 = σ 2 + T E e e u u u 2 k w = = + 1 1 i k r k  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 of the exact heat flux signal, which have been “lost” by truncation. Metti5 – Roscoff June 13-18, 2011 17

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. Metti5 – Roscoff June 13-18, 2011 18

Heat conduction in a plane wall The model equations T d = + A T b u ( t ) d t = = 0 T ( t 0 ) = = T T 0 ∞ 0 = y ( t ) C T ( t ) Transient heat conduction in a plane wall mo Metti5 – Roscoff June 13-18, 2011 19

Heat conduction in a plane wall Solution of the direct problem [ ] = = T  T ( t ) T ( t ) T ( t ) T ( t ) avec T ( t ) T ( z , t ) 1 2 N i i e = − = Δ Δ and z ( i 1 ) z ; z − i N 1 −  2 2 0 0   1   = λ Δ Bi h z /     − 1 2 1 0 0     2 a     = = et     A b ρ 2 ( Δ )   Δ   z c z − 2 1 0         − + 0 0 2 2 ( 1 ) 0   Bi    [ ] = = ≠   C 0 0 1 0 where C 0 si i i i c C ∫ t = − τ τ τ exp y ( t ) ( A ( t ) ) b u ( ) d mo 0 Metti5 – Roscoff June 13-18, 2011 20

Heat conduction in a plane wall Discrete Solution of the direct problem { } = ∑ n = δ u ( t ) u f ( t ) ; f ( t ) j j j k ik = j 1 n ∑ = y ( t ) S u ; mo k k j j = j 1 ∫ t = − τ τ τ = k exp S C ( A ( t ) ) b f ( ) d , k 1 ,.., m k j k j 0 = y S u mo Metti5 – Roscoff June 13-18, 2011 21

Heat conduction in a plane wall Example of Numerical results = + ε y y mo = 21 nodes N = Δ t 200 s = 40 nt σ = 0 , 02 K − − − − − − = λ = ρ = = 1 1 6 3 1 2 1 e 0 , 05 m ; 0 , 3 Wm K ; c 1 . 2 10 Jm K ; h 0 Wm K Metti5 – Roscoff June 13-18, 2011 22