Lecure 3: Models and measurements for thermal syste ms Types of - - PowerPoint PPT Presentation

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Lecure 3: Models and measurements for thermal systems Types of inverse problems

Denis Maillet, Jean-Luc Battaglia, Daniel Petit

LEMTA Nancy - I2M, Dpt. TREFLE Bordeaux - Institut P’ Poitiers

• 1. Objectives, models & direct problems, internal/external representations
• 2. Parameterizing a function & parcimony principle
• 3. State-space representation, model terminology & structure, measurements
• 4. Different types of inverse problems, measurements & noise, bias
• 5. Physical model reduction

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

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Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

• 1. Objectives of a model (in heat transfer)

First objective: Simulation of physical reality = Direct Problem

) (t u

y

: heat source/flux, variable external temperature : measured temperature at given time t and given location

mo

y

: modelled temperature at given time t and given location

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Example: model of a semi infinite medium (1 input - 1 output)

( ) ( ) ( )

2 2 s s

1 exp exp 4 4 2

s

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

( )

( )

2 s

1 exp / 4 Z t x a t b t π = −

Output equation: Solution of direct problem:

( ) ( )

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

t s mo

− + =

∫ ∫

∞

) ( ) ( ) (

forced relax

t y t y t y

mo mo mo

+ =

c c / ρ ρ k b k a = =

Perfect temperature sensor: y = yexact

k −

External representation:

Green’s function convolution product

) , ( ) ( t x T t y

s mo

=

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( ) ( ) ( ) ( ) ( ) ( )

forced

with exp d

mo

y p Z p u p f p f t pt t

∞

= = −

∫

( )

( )

2 s

1 exp / 4 Z t x a t b t π = −

Thermal impedance: Parameter list β = (a, b, xs)

          =

s

x b a β

Parameter « vector » a in W m-2

• b in J m-2 K-1 s-1/2 - xs in m

Perfect temperature sensor: y = y *

p = Laplace parameter (s-1)

( )

a / p x

• p

b p Z

s

exp 1 ) ( = )

s

x

zero initial temperature field

β

( )

2 1 2 2 2 / s

x b a + + = ⇒ β β β β

M of the white box type (internal representation) : internal parameters of physical nature

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Semi-infinite medium model: general case

List of data of Direct Problem:

{ }

) ( , ) ( ,

0 .

T . u x β =

structural parameter vector input (stimulus) initial state functions

          =

s

x b a β

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• 2. Problem of function parameterizing & Parcimony principle

) ( ) (

1

t f u t u

j j j

∑

∞ =

=

{ }

• 2

1

with f f

function

• n [ 0 tsup ]

basis of infinite number of functions: [ 0 tsup ] → R

( ) ( ) ( ) [ ]

T

t f t f t

• 2

1

= f

column vectors with an infinite number of components

[ ]

T

u u

• 2

1

= u

) ( ) ( t t u

T f

u = ⇒

) ( ), ( ) ( ) (

1

t f t u u t f u t u

j j j j j

= ⇒ = ∑

∞ =

projection of u (t) onto fj (t) Good approximation: high n ⇒ large number of parameters

) ( ) ( ) (

1 param

t u t f u t u

j n j j

≠ = ∑

=

truncation

            = ⇒

n

u u u t u

• 2

1

) ( u by replaced

in practice:

The model-builder has to choose 2 things: 1) functions fj 2) their number n

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Parameterization: 2 possible choices of local function basis

heterogeneous material

hat functions

0 if t

door functions

) (t u

linear excitation

) ( ) (

1 param

t f u t u

j n j j

∑

=

=

non uniform initial temperature

) (T u

non linear excitation

(P) T

temperature dependent thermophysical properties

) (T β

(P) β

: local discretized value → interpolated uparam (t)

• interesting for :

: averaged value over an interval

j

u

• interesting for :

j

u

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Remarks

• non-local bases available: eigenfunctions f j of the heat equation

(method of separation of variables)

• rthogonal bases (with a unit function norm Nj) interesting:
• non constant time step possible
• Extended parameter vector x

gathering the Direct Problem data:

Parameterization (continued)

= ) (

param t

u

Fourier series, for example

list

{ }

          = → =

0 (P)

, ) ( , T u β x β T . u x

parameterized functions

k j j k t t j

N t t f t f

sup inf

δ =

∫

d ) ( ) (

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Parcimony principle: limitation of the number of parameters (or of degrees of freedom) to be sought

ideal parameterization greediness: too many parameters sought / quality of measurements

• r available a priori information
• bjectives:

= parameters

realistic parameterization constraint

inverter : beginner experienced

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( )

conditions initial and interface Boundary, div

vol

+ ∂ ∂ = + t T c q T ρ grad k

State of the system = continuous temperature field:

) ( ) (P

P

t T t , T =

W/m3

distributed parameter system

Discretized state becoming = vector :

[ ]

) ( ) ( ) ( ) (

2 1

t T t T t T t

N

• =

T

in a N dimension space (number of nodes)

0)

( with ) , ( d d T T U T E T = = = t t , t t lumped parameter system:

[ ]

T p t

u t u t u t ) ( ) ( ), ( ) (

2 1

• =

U

p excitated nodes

qvol (P, t)

Linear heat source (excitation):

[ ]

T p t

T u t T u t T u t )) ( ( )) ( ( )), ( ( ) (

2 1

• =

U

p excitated nodes

qvol (T ( P, t))

Non-linear heat source:

• 3. State-space representation, model terminology & structure, measurements

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State-space representation (continued)

matrices constant : and with ) , ( B A U B T A U T E + = , t

Case of a linear heat source with temperature independent thermophysical properties and coefficients (selection of

• bserved nodes)

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Output equation: detailed

                                                =                 = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 ) ( ) ( ) ( ) ( ) ( ) (

20 19 18 17 5 4 3 2 1 5 4 3 2 1

t T t T t T t T t T t T t T t T t T t y t y t y t y t y t

mo mo mo mo mo mo

• y

q = 5 observed temperatures (output) N = 20 nodes

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τ τ τ d ) ( ) ) ( ( ) ( ( ) ( U B A T A T

∫

− + − =

t t

t ) t t t exp exp

Explicit solution for temperature field: and for model output: Remarks:

• advection case possible (dispersion in porous medium, one-temperature model):
• coupled modes transfer : radiation in semitransparent absorbing medium

(Heat equation + radiative transfer equation) ⇒ composite state X :

τ τ τ d ) ( ) ) ( ( ) ( ( ) ( U B A C T A C y

∫

− + − =

t t mo

t ) t t t exp exp

Relaxation of initial state forced (convolution) response

Linear state equation :

( )

conditions initial and interface Boundary, div

vol

+ ∂ ∂ = + − t T c q T c T

f

ρ ρ λ grad grad . v U B T A T + = t d d

      = ) ( ) ( ) ( t t t I T X

Discretized intensity field: wavelength, direction, position Discretized temperature field: position

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U B A C y U B A T

1 1 − −

− = ⇒ − =

mo

• steady state case (linear) :

= + = U B T A T t d d

Model terminology and structure

) ( x , t ymo η =

) ( x , t ymo η =

• single output:
• multiple output:

list extended parameter vector scalar scalar

) ( x , t

mo

η y =

) ( x η y , t

mo = list extended parameter vector column vector column vector

          = T u β x

{ }

=

0 (P)

, ) ( , β T . u x

Before parameterization After parameterization

• data

η (t, .) or η η η η (t, .) : scalar or vector function x or x : corresponding data (list/vector)

= structure of the model

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Structure of a parameterized model in heat transfer

Direct Problem Data Model output

      =

pos struct

β β β

location of detectors dimensions of system parametrized shape thermophysical properties contact resistances heat transfer coefficients emissivities …

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Comparison between measurements and state model

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Direct and inverse problems

MODEL MODEL known x part of x ? known y Direct problem : Inverse problem : ?

mo

y data

Extended parameter vector: Objective of inverse problem: finding a part xr of x, using additional information (output y or something else)

      =

c r

x x x

sought (researched) parameters complementary part: known

            = ) ( T U β β x t

pos struct

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• 4. Different types of inverse problems in heat transfer,

measurements & noise, bias

• inverse measurement problems → additional information stems from output signal y of sensors
• control problems

→ additional information = desired (target) values of state T or output y → sought quantity = excitation U, initial state T0 or velocity/flowrate in β β β β

• system identification problems = model construction
• model reduction

→ additional linformation = output of detailed model → sought quantity = structure + parameter vector of a reduced model 1) mathematical reduction : 2) physical reduction : In both cases: explicit for physical reduction

) ( ) (

red red det det

x x ; t ; t η η ≈ ) ; (

det det

x t η

[ ] [ ]

T red T t red red red de det det det

and : with T U β x T U β x = =

red de det red det red det red

) (P, ) (P, ) (P, ) (P, T T U U = ⇒ = = ⇒ =

t

t T t T t u t u

( )

( )

det red det red det red det red

) (P, ) (P, ) (P, ) (P, T T U U

T U

f t T t T f t u t u = ⇒ ≈ = ⇒ ≈

( )

det red

β β

β

f =

GREY BOX type WHITE BOX type

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• experimental model identification (belongs to inverse measurements problem class)

→ additional information = y, U and T0 are measured, or supposed to be known → sought quantity = parameter vector for a model of given structure 3 types of identified models:

• white box type, based on first principles: physical meaning for
• black box type: general structure, no physical meaning for

(neural networks)

• grey box type : in between, physical structure, no physical meaning for
• optimal design problems

→ additional information: quality criterion to satisfy → sought quantity = parameter vector with constaints, for a model of given structure

β

β β β β

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Inverse measurement problems in heat transfer

Extended parameter vector:

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

c r pos struct

t x x T U β β x ) (

sought (researched) parameters complementary part: known

Measurements y (t) available on interval [ , ]

final

t t

a) Inverse problems of structural parameters estimation :

• example 1: thermophyscal property « measurement »:
• example 2: calibration of a sensor/acquisition chain:

b) Inverse input problems :

• example: “inverse heat conduction” = wall heat flux “measurement”

c) Inverse initial state problems : d) Inverse shape reconstruction problems e) Inverse problems of optimal design/control (of a characterzation experiment, for example)

r r

β x ≡

(P) T xr ≡

) (P t , u xr ≡

!

• r
• r

h ... a c k xr + = ρ

) (

calib

β , T Vmo

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Measurement and noise

measured temperature signal (sensor output) « true » temperature (unknown) Temperature measurement noise (unknown)

• only discrete values available

) ( i

i

t ε ε = ⇒

• (implicit) assumptions:

→ unbiased model: → unbiased noised:

) ( ) (

exact mo exact

, t y t y x =

) ( =

i

E ε

) ( ) ( ) ( t t y t y

exact

ε + =

) ( i

i

t y y =

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• 5. Physical model reduction on an example
• homogeneous rectangular slab, thickness e, lengths
• thermal diffusivity and conductivity a and k, volumetric heat

/ c k a ρ =

• 4 lateral sides insulated, h heat exchange coefficient over rear face
• uniform initial temperature T0
• two dimensional heat flux absorption over front face
• q temperature sensors inside the slab

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* * * * * * * * , diff

( , ) . ( , , , / , , ( , , ) / , , , ) T

mo i i i i x y

y t x T T x y z t R u x y t T H η τ

∞

= = ∆ ∆ +

• T

T T∞ ∆ = −

*

/

i i

x x e =

*

/

i i

z z e =

*

/

i i

y y e =

*

/

x x

e =

• *

/

y y e

=

• 2 /

diff

e a τ = k e R / = k e h H / =

*

( ) / T T T T

∞

= − ∆

* * * * * * * * , diff

( , ) . ( , , , / , , ( , , ) / , , , ) T

mo i i i i x y

y t x T T x y z t R u x y t T H η τ

∞

= = ∆ ∆ +

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Remarks on physical reduction (for later inversion)

• the simpler the model, the higher the possible bias (for direct simulation)

but

• detailed model may be biased too /experiment
• decrease in number of parameters
• inversion more robust/noise amplification (inversion)
• parameters keep explicit physical meaning (white box): can be exported !
• first step for later finer inversion (non linear estimation)

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Thank you for your attention !