Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment

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Mathematical Modeling, Analysis and ParameterEstimation on a Heat Conduction Experiment

Ma. Cristina R. Bargo Ricardo C.H. del RosarioJose Ernie C. Lope

Institute of MathematicsUniversity of the Philippines Diliman

7th Sino-Philippine Symposium in MathematicsMeralco Development Center (MMLDC)

October 26, 2007

Outline

1 Introduction

2 Experimental Setup

3Model Formulation

4 Well-Posedness

5 Galerkin Method and Convergence

6 Parameter Estimation

7 Results

8 Conclusions and Future Work

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentIntroduction

Outline

1 Introduction

3Model Formulation

4 Well-Posedness

7 Results

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentIntroduction

Objectives

Prepare setup for heat conduction experiment, data gatheringFormulate a model for heat conduction on a metal rod (takingheat loss into account)Find the solution to the modelEstimate the parameters by minimizing the difference betweenthe actual temperature and computed temperature values

Compare the parameters and the error using differentoptimization algorithms

h l d l l d d

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentExperimental Setup

Outline

1 Introduction

3Model Formulation

4 Well-Posedness

7 Results

M h i l M d li A l i d P E i i H C d i E i

Experimental Setup

Figure: The experimental setup: metal rod, heat source, thermocouplesand data acquisition instrument.

Mathematical Modeling Analysis and Parameter Estimation on a Heat Conduction Experiment

Some Data

Copper Rod Specics

Mass Radius Length1.14 kg 0.6375 cm 0.9980 m

Aluminum Rod Specics

Mass Radius Length0.35 kg 0.6250 cm 1.0030 m

Thermocouple Locations (m)

x 1 x 2 x 3 x 4 x 5 x 6 x 7

copper 0.0500 0.1800 0.3100 0.5640 0.6940 0.8245 0.9980

aluminum 0.0420 0.1720 0.3020 0.4320 0.6930 0.8220 0.9520

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentModel Formulation

Outline

1 Introduction

3Model Formulation

4 Well-Posedness

7 Results

Preliminaries

Given parameters: radius r , length , ambient temperature ua ,density ρParameters to be estimated: ux Q, thermal conductivity k,heat transfer coefficient h, specic heat capacity c p

Assumptions:heat is transferred in one dimension, and temperature isuniform over a cross-sectionconstant ambient temperature

constant heat ux at x = 0 (due to the heat source)heat loss along the sides of the rodtwo possibilities at x = : insulated (BC1), heat loss (BC2)

Used Fourier’s Law of Conduction and Newton’s Law of Cooling

Preliminaries

g, y pModel Formulation

Preliminaries

g, y pModel Formulation

The Model

Model (BC1):

ρc p ∂u (x, t )∂t

= k ∂ 2u (x, t )∂x 2 − 2h

r(u (x, t ) − ua )

u (x, 0) = u0 (x)∂u (0, t )

∂x= −

k∂u ( , t)∂x

g y pModel Formulation

The Model

Model (BC2):

ρc p ∂u (x, t )∂t

= k ∂ 2u (x, t )∂x 2 − 2h

r(u (x, t ) − ua )

u (x, 0) = u0 (x)∂u (0, t )

= −Q

k∂u ( , t)∂x

= −hk

(u ( , t) − ua )

Model Formulation

Solution to the Model

Usual method: separation of variables Will not work for our problem because:

nonzero boundary conditionsheat loss term makes it impossible to separate the PDE intotwo ODEs

Prove that the problem is well-posed (using functional analysis)Find the approximate (nite-dimensional) solution usingGalerkin methodProve the convergence of the nite-dimensional solution to theinnite-dimensional solution

Well-Posedness

Outline

1 Introduction

3 Model Formulation

4 Well-Posedness

6 Parameter Estimation7 Results

Well-Posedness

Well-Posedness of (2) - BC2

Let V = H 1 (0, ) and H = L2 (0, ) with the following innerproducts:

ψ, φ H = 0ψ (x) φ (x) dx

ψ, φ V = 0ψ (x) φ (x) dx +

2hrk 0

ψ (x) φ (x) dx

W (0, T ) = f | f ∈L2 (0, T ; V ) , df dt ∈L2 (0, T ; V ∗) , a

Hilbert space with norm

f 2W (0 ,T ) =

0f (t) 2

V dt + T

V ∗dt

Remark: W (0, T ) → C 0 ([0, T ] ;H )

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentW ll P d

Well-Posedness

ψ, φ H = 0ψ (x) φ (x) dx

ψ, φ V = 0ψ (x) φ (x) dx +

2hrk 0

ψ (x) φ (x) dx

f 2W (0 ,T ) =

0f (t) 2

V dt + T

V ∗dt

Remark: W (0, T ) → C 0 ([0, T ] ;H )

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentW ll P d

Well-Posedness

ψ, φ H = 0ψ (x) φ (x) dx

ψ, φ V = 0ψ (x) φ (x) dx +

2hrk 0

ψ (x) φ (x) dx

f 2W (0 ,T ) =

0f (t) 2

V dt + T

V ∗dt

Remark: W (0, T ) → C 0 ([0, T ] ;H )

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentWell Posedness

Well-Posedness

Dene the operators σ : V × V → R and F : V → R

σ (ψ, φ) =k

ρc pψ, φ V +

hρc p

ψ ( ) φ ( )

F (φ) = 2hu arρc p 0

φ (x) dx + hu aρc p

φ ( ) + Qρc pφ (0)

Weak form of (2)

Find u ∈W (0, T ) such that

, φV ∗ ,V

= − σ (u, φ ) + F (φ) , ∀φ ∈V

u (0) = u0

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentWell Posedness

Well-Posedness

Dene the operators σ : V × V → R and F : V → R

σ (ψ, φ) =k

ρc pψ, φ V +

hρc p

ψ ( ) φ ( )

F (φ) = 2hu arρc p 0

φ (x) dx + hu aρc p

φ ( ) + Qρc pφ (0)

Weak form of (2)

Find u ∈W (0, T ) such that

, φV ∗ ,V

= − σ (u, φ ) + F (φ) , ∀φ ∈V

u (0) = u0

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentWell-Posedness

Well Posedness

F ∈V ∗, so F ∈L2 (0, T ; V ∗)σ is a bilinear form, which is continuous andV -ellipticThere is a unique bijective operator A : V → V ∗ such that forall ψ, φ ∈V ,

σ (ψ, φ) = A (ψ) (φ)

We can write (3) as

du (t)dt

= −A u (t) + F (in V ∗)

u (0) = u0

Well Posedness

TheoremLet F ∈L2 (0, T ; V ∗ ) and suppose that the following conditions are satised:

1 For all ψ, φ ∈V , the function t → σ (t; ψ, φ) is measurable on(0, T ) and for t ∈(0, T ),

|σ (t; ψ, φ) | ≤ c ψ V φ V .

2 There exists a λ ∈R , α > 0 such that for all ψ ∈V , t ∈(0, T ),

σ (t; ψ, ψ) + λ ψ 2H ≥ α ψ 2

Then the problem ( 4 ) admits a unique solution in W (0, T ). Furthermore,the solution depends continuously on the data, i.e. the bilinear map F, u 0 → u is continuous from L2 (0, T ; V ) × H to W (0, T ).

Use the inner products for V and H dened in the previouscase

Dene the space W (0, T )Dene the operators σ : V × V → R and F : V → R

σ (ψ, φ) =k

ρc pψ, φ V

F (φ) = 2hu a

rρc p 0φ (x) dx + Q

ρc pφ (0)

Weak form of (1)Find u ∈W (0, T ) such that

, φV ∗ ,V

= − σ (u, φ ) + F (φ) , ∀φ ∈V

u (0) = u0

¯F ∈V

, so¯F ∈L

(0, T ; V ∗

)σ is a bilinear form, which is continuous andV -ellipticWe can also write (5) in the form (4)Proof of well-posedness is similar to the previous case

Weak form of (1)Find u ∈W (0, T ) such that

, φV ∗ ,V

= − σ (u, φ ) + F (φ) , ∀φ ∈V

u (0) = u0

¯F ∈V

, so¯F ∈L

(0, T ; V ∗

)σ is a bilinear form, which is continuous andV -ellipticWe can also write (5) in the form (4)Proof of well-posedness is similar to the previous case

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence

Outline

1 Introduction

3 Model Formulation

4 Well-Posedness

Galerkin Method for BC2

Project the solution of (3) in a nite-dimensional spaceV n = span {φ1, . . . , φ n }Finite-dimensional problem for (3): Find un such that

dt, φ

−σ (un , φ

i) + F (φ

i) , i = 1 , . . . , n

un (0) = P nV u0(6)

Write un (x, t ) =n

α i (t) φi (x) and substitute to ( 6):

M α (t) = Aα (t) + F (t)

Initial condition: un (x, 0) =n

α i (0) φi (x) = P nV u0

dt, φ

i H = − σ (un , φ

i) + F (φ

i) , i = 1 , . . . , n

un (0) = P nV u0(6)

Write un (x, t ) =n

M α (t) = Aα (t) + F (t)

α i (0) φi (x) = P nV u0

dt, φ

i H = − σ (un , φ

i) + F (φ

i) , i = 1 , . . . , n

un (0) = P nV u0(6)

Write un (x, t ) =n

M α (t) = Aα (t) + F (t)

α i (0) φi (x) = P nV u0

dt, φ

i H = − σ (un , φ

i) + F (φ

i) , i = 1 , . . . , n

un (0) = P nV u0(6)

Write un (x, t ) =n

M α (t) = Aα (t) + F (t)

α i (0) φi (x) = P nV u0

α (t) = M − 1 (Aα (t) + F (t)) (7)

The variables in (7) are dened as follows:

α (t) = [α 1 (t) , α 2 (t) , . . . , α n (t)]T

[A]ij = −k

ρc pφ j , φi V −

hρc p

φ j ( ) φi ( )

[M ]ij

0φ j (x) φi (x) dx

[F (t)]i =2hu a

rρc p 0φi (x) dx +

ρc pφi ( ) +

Qρc p

φi (0)

α (t) = M − 1 (Aα (t) + F (t)) (7)

α (t) = [α 1 (t) , α 2 (t) , . . . , α n (t)]T

[A]ij = −k

ρc pφ j , φi V −

hρc p

φ j ( ) φi ( )

[M ]ij =

[F (t)]i =2hu a

ρc pφi ( ) +

Qρc p

φi (0)

Finite-dimensional problem for (5): Find un such that

dt, φi

H = − σ (un , φi ) + F (φi ) , i = 1 , . . . , n

(0) = P nV u0

Write un (x, t ) =n

i=1α i (t) φi (x) and substitute to ( 8):

M α (t) = Aα (t) + F (t)

i=1α i (0) φi (x) = P nV u0

dt, φi

H = − σ (un , φi ) + F (φi ) , i = 1 , . . . , n

(0) = P nV u0

Write un (x, t ) =n

M α (t) = Aα (t) + F (t)

i=1α i (0) φi (x) = P nV u0

dt, φi

H = − σ (un , φi ) + F (φi ) , i = 1 , . . . , n

(0) = P nV u0

Write un (x, t ) =n

M α (t) = Aα (t) + F (t)

i=1α i (0) φi (x) = P nV u0

α (t) = M − 1 Aα (t) + F (t) (9)

α (t) = [α1 (t) , α 2 (t) , . . . , α n (t)]T

A ij = −k

ρc pφ j , φi V

[M ]ij =

F (t) i =2hu a

Qρc p

φi (0)

α (t) = M − 1 Aα (t) + F (t) (9)

α (t) = [α1 (t) , α 2 (t) , . . . , α n (t)]T

A ij = −k

ρc pφ j , φi V

[M ]ij =

F (t) i =2hu a

Qρc p

φi (0)

Convergence Result

Assumptions on V n :(H 1 ) V n ⊂V ;(H 2 ) For each φ ∈V , φ − P n

V → 0 as n → ∞ ;

(H 3 ) The spaces satisfy the monotonicity condition V n ⊂V n +1 .

TheoremUnder the assumptions ( H 1)-( H 3), the sequence un converges to

u ∈C 0 ([0, T ] ;H ), where un

is the solution of ( 6 ) or ( 8 ) and u is the unique solution of ( 3 ) or ( 5 ), respectively.

Convergence Result

Assumptions on V n :(H 1 ) V n ⊂V ;(H 2 ) For each φ ∈V , φ − P n

V → 0 as n → ∞ ;

(H 3 ) The spaces satisfy the monotonicity condition V n ⊂V n +1 .

TheoremUnder the assumptions ( H 1)-( H 3), the sequence un converges to

u ∈C 0 ([0, T ] ;H ), where un

is the solution of ( 6 ) or ( 8 ) and u is the unique solution of ( 3 ) or ( 5 ), respectively.

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentParameter Estimation

Outline

1 Introduction

3 Model Formulation

4 Well-Posedness

Objective Function

Vector of unknown parameters: q = Qk , h

k , cp

T ∈Λ⊂ R 3

Set of data points: {u (x i , t j ) | i = 1 , . . . N, j = 1 , . . . , N t }(may contain errors)

u (x i , t j ; q) is the solution of (3) or (5) using the parameter qand evaluated at x i at time t j

minq∈Λ

J (q) = minq∈Λ

1N · Nt

i=1|u (x i , t j ; q) − u (x i , t j )|2 (10)

Objective Function

Vector of unknown parameters: q = Qk , h

k , cp

T ∈Λ⊂ R 3

Set of data points: {u (x i , t j ) | i = 1 , . . . N, j = 1 , . . . , N t }(may contain errors)

u (x i , t j ; q) is the solution of (3) or (5) using the parameter qand evaluated at x i at time t j

minq∈Λ

J (q) = minq∈Λ

1N · Nt

i=1|u (x i , t j ; q) − u (x i , t j )|2 (10)

Finite-Dimensional Objective Function

May involve innite-dimensional admissible parameter spacesΛ: consider nite-dimensional approximating subspacesΛm⊂Λ

un (x i , t j ; q) is the solution of the nite-dimensional problem(6) or (8) using the parameter q and evaluated at x i at time t j

minq∈Λm

J (q) = minq∈Λm

1N · Nt

i=1|un (x i , t j ; q) − u (x i , t j )|2

Finite-Dimensional Objective Function

May involve innite-dimensional admissible parameter spacesΛ: consider nite-dimensional approximating subspacesΛm⊂Λ

un (x i , t j ; q) is the solution of the nite-dimensional problem(6) or (8) using the parameter q and evaluated at x i at time t j

minq∈Λm

J (q) = minq∈Λm

1N · Nt

i=1|un (x i , t j ; q) − u (x i , t j )|2

Remarks

Let {qn,m } be the sequence of parameter estimates for (11)

(H 4) Requirements for parameter spaces:the sets Λ and Λm lie in a metric space Λ with metric d andare compact in this metricthere is a mapping im : Λ → Λm such that Λm = im (Λ)the mapping im converges to the identity on Λ

Remarks

Let {qn,m } be the sequence of parameter estimates for (11)

(H 4) Requirements for parameter spaces:the sets Λ and Λm lie in a metric space Λ with metric d andare compact in this metricthere is a mapping im : Λ → Λm such that Λm = im (Λ)the mapping im converges to the identity on Λ

Convergence Result

Theorem

To obtain convergence of at least a subsequence of {qn,m

} to asolution minimizing ( 10 ), it suffices under assumption ( H 4) to argue that for arbitrary sequences {qn,m } in Λ with qn,m → q in Λ,we have

un (x, t ; qn,m ) → u (x, t ; q) .

Summary

TheoremSuppose V n satises ( H 1 )-( H 3 ) and suppose Λ is compact with metric d. Assume σ is continuous and V -elliptic and suppose further that it is continuous with respect to the parameters in the following sense: there

exists a constant ξ > 0 such that for any ψ, φ ∈V and q, q∈Λ, we have |σ (q) (ψ, φ) − σ (q) (ψ, φ)| ≤ ξd (q, q) ψ V φ V .

Furthermore, suppose that the function F is continuous with respect to q, i.e. q → F (t; q) is continuous from Λ to L2 (0, T ; V ∗ ). Let {qm } be

arbitrary in Λ such that qm

→ q in Λ. Then for t > 0,

un (x, t ; qm ) → u (x, t ; q)

in V norm.

O i i i Al i h

Optimization Algorithms

Gradient-basedBuilt-in algorithm in Scilab (leastsq ) - solves nonlinear leastsquares problems

Quasi-Newton algorithm (the Jacobian of the cost functionwas not supplied)

Genetic AlgorithmOptimization algorithm inspired by the concept of evolution

(“survival of the ttest”)Main processes of GA: selection, recombination, mutation,elitism (optional)

O i i i Al i h

Optimization Algorithms

Gradient-basedBuilt-in algorithm in Scilab (leastsq ) - solves nonlinear leastsquares problems

Quasi-Newton algorithm (the Jacobian of the cost functionwas not supplied)

Genetic AlgorithmOptimization algorithm inspired by the concept of evolution

(“survival of the ttest”)Main processes of GA: selection, recombination, mutation,elitism (optional)

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentResults

Outline

1 Introduction

3 Model Formulation

4 Well-Posedness

Parameter Estimation7 Results

N i l R lt

Numerical Results

Optimal Parameters and Error

leastsq GA

BC1 BC2 BC1 BC2Objective Function

copper 0.1645 0.1648 0.1645 0.1648aluminum 0.0768 0.0762 0.0758 0.0762

Estimated Q/kcopper 66.1440 66.1372 66.2355 66.2739aluminum 60.9651 60.9546 60.9729 60.9583

Numerical Results

Optimal Parameters and Error

leastsq GA

BC1 BC2 BC1 BC2Estimated h/k

copper 0.0312 0.0312 0.0313 0.0313aluminum 0.0569 0.0569 0.0570 0.0569

Estimated c p /kcopper 1.1840 1.1844 1.1848 1.1864aluminum 3.8982 3.8986 3.9000 3.8989

0 500 1000 1500 2000 2500 3000300

Copper Rod: Data and Computed Values

time (s)

t e m p e r a

t u r e (

Figure: Plot of temperature vs. position (copper rod)

0 1000 2000 3000 4000 5000 6000 7000 8000306

Aluminum Rod: Data and Computed Values

time (s)

t e m p e r a

t u r e

Figure: Plot of temperature vs. position (aluminum rod)

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentConclusions and Future Work

Outline

1 Introduction

3 Model Formulation

4 Well-Posedness

7 Results

Conclusions

Obtained the data using thermocouples attached to a dataacquisition instrumentFormulated a modied model for heat conduction on a metalrod

Showed the well-posedness of the modelObtained an approximate solution using the Galerkin methodand showed its convergence to the true solutionObtained estimates for the parameters Q/k , h/k , c p/k using

leastsq and genetic algorithmModeling the heat loss at the end of the rod away from theheat source produces the same output as the model withoutheat loss.

Future Work

Improve the experimental setupReformulate the model to incorporate “realistic” assumptions(ambient temperature, ux), and show well-posedness of model

Implement optimization problem, solving for time-dependentparametersUse other optimization algorithms (gradient-based, heuristic,neural networks, hierarchical Bayesian methods) for parameterestimationImplement a faster numerical method for solving the PDEExtend the modied model to 2 or 3 dimensions

References

H.T. Banks, R.C. Smith and Y. Wang, “Smart Material Structures: Modeling, Estimation andControl,” John Wiley & Sons, 1996.

M. Braun, “Differential Equations and Their Applications, fourth ed,” Springer-Verlag, 1993.

R.R. Briones, “Numerical Computations for Parameter Estimation in a Smart Beam Structure,”Master’s Thesis, College of Science, University of the Philippines Diliman, 2002.

R. Haupt and S.E. Haupt, “Practical Genetic Algorithms, 2nd ed.,” John Wiley & Sons, Inc., 2004.P. Laguitao, “Estimation of Copper Rod Parameters Using Data from Heat ConductionExperiment,” Undergraduate Research Paper, College of Science, University of the PhilippinesDiliman, 2001.

J.L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,”Springer-Verlag, 1971

J.L. Lions and E. Magenes, “Non-Homogeneous Boundary Value Problems and Applications, vol.I,” Springer-Verlag, 1972.

D.V. Widder, “The Heat Equation,” Academic Press, Inc., 1975

”Instructional and Research Laboratory, Center for Research in Scientic Computation, NorthCarolina State University,” http://www.ncsu.edu/crsc/ilfum.htm.

The End

Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment

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