Parameter Estimation in a Mathematical Model of a Heat-Conducting Rod

8/8/2019 Parameter Estimation in a Mathematical Model of a Heat-Conducting Rod

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Parameter Estimation in a Mathematical Model of a Heat-Conducting Rod

Parameter Estimation in a Mathematical Model of aHeat-Conducting Rod

Ma. Cristina R. BargoRicardo C.H. del RosarioJose Ernie C. Lope

Department of Mathematics

University of the Philippines Diliman

MSP 2007 ConventionBohol Tropics Resort, Tagbilaran City, Bohol

May 18, 2007
http://find/http://goback/


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Objectives

Prepare the setup for the heat conduction experiment, datagathering

Create a time-dependent model for heat conduction on ametal rod

Find the solution to the model numerically

Estimate the parameters by minimizing the difference between

the actual temperature and computed temperature valuesRemark: Extension of the work presented by Ms. Margie De Pazlast year


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Objectives

Prepare the setup for the heat conduction experiment, datagathering

Create a time-dependent model for heat conduction on ametal rod

Find the solution to the model numerically

Estimate the parameters by minimizing the difference between

the actual temperature and computed temperature valuesRemark: Extension of the work presented by Ms. Margie De Pazlast year

h l d l f C d d


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Outline

1 Experimental Setup

2 Model and Parameter Estimation ProblemThe ModelSolution to the Model

Galerkin MethodParameter Estimation

3 Optimization AlgorithmsLeast Squares

Genetic Algorithm4 Results

5 Conclusions

6

Future Work

P E i i i M h i l M d l f H C d i R d


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Experimental Setup

Outline






5 Conclusions

6 Future Work

P t E ti ti i M th ti l M d l f H t C d ti R d


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Experimental Setup

Some Data

Copper Rod Specifics

Mass Radius Length

1.14 kg 0.6375 cm 0.9980 m

Aluminum Rod Specifics

Mass Radius Length

0.35 kg 0.6250 cm 1.0030 m

Thermocouple Locations (m)

x1 x2 x3 x4 x5 x6 x7

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

Parameter Estimation in a Mathematical Model of a Heat Conducting Rod


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Experimental Setup

Experimental Setup

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



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Model and Parameter Estimation Problem

Outline






5 Conclusions

6 Future Work



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The Model

Outline






5 Conclusions

6 Future Work



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The Model

Model Formulation

Assumptions:

heat is transferred on one direction, and temperature isuniform over a cross-section

constant ambient temperatureconstant heat flux at one end of the rod (due to the heatsource)heat loss along the sides of the rodheat loss at the end away from the heat source

Given parameters: radius r, length , ambient temperature ua,density

Parameters to be estimated: thermal conductivity k, heattransfer coefficient h, flux Q, specific heat capacity cp



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g


The Model

Model Formulation

Assumptions:







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g


The Model

Model Formulation

Assumptions:







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The Model

The Model

The model is given by

cpu (x, t)

t= k

2u (x, t)

x2

2h

r(u (x, t) ua)

u (x, 0) = u0 (x)u(0,t)x

= Qk

u(,t)x = hk (u (, t) ua)

(1)



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The Model

Remarks

Used Fouriers Law of Conduction and Newtons Law ofCooling to formulate the model

Solution of (1) has no analytical form use Galerkin methodto find the approximate solution (if the solution exists)

Changes in the model:

time-dependent (before, we considered the steady-state model)

heat loss at x = (before, we have u(,t)x

= 0)



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Solution to the Model

Outline






5 Conclusions

6 Future Work



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Variational Formulation

Multiply PDE in (1) by H1 (0, ), integrate over (0, ), useintegration by parts, and impose boundary conditions

The PDE becomes:

0

u (x, t)

t (x) dx =

k

cp

0

u (x, t) (x) dx

2hrcp

0

u (x, t) (x) dx hcp

u (, t) ()

+2huarcp

0

(x) dx +huacp

() +Q

cp (0)


M d l d P E i i P bl


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Variational Formulation

Multiply PDE in (1) by H1 (0, ), integrate over (0, ), useintegration by parts, and impose boundary conditions

The PDE becomes:

0

u (x, t)

t (x) dx =

k

cp

0

u (x, t) (x) dx

2hrcp

0

u (x, t) (x) dx hcp

u (, t) ()

+2huarcp

0

(x) dx +huacp

() +Q

cp (0)


M d l d P t E ti ti P bl


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Some Definitions

Let V = H1 (0, ) and H = L2 (0, ) with the following innerproducts:

, H =

0

(x) (x) dx

, V =

0

(x) (x) dx +2h

rk

0

(x) (x) dx

Define the operators : V V R and F : V R

(, ) =k

cp, V +

h

cp () ()

F () =2huarcp

0 (x) dx +

huacp

() +Q

cp (0)




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Some Definitions

Let V = H1 (0, ) and H = L2 (0, ) with the following innerproducts:

, H =

0

(x) (x) dx

, V =

0

(x) (x) dx +2h

rk

0

(x) (x) dx

Define the operators : V V R and F : V R

(, ) =k

cp, V +

h

cp () ()

F () =2huarcp

0 (x) dx +

huacp

() +Q

cp (0)

Parameter Estimation in a Mathematical Model of a Heat-Conducting RodModel and Parameter Estimation Problem


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The Weak Form

Weak form of (1)

Find u L2 (0, T; V) such thatddtu, H = (u, ) + F () , V

u (0) = u0(2)

To show well-posedness:

show existence, uniqueness and continuous dependence on thedata of the solution to (2)equivalence of the solution to (1) and (2)



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The Weak Form

Weak form of (1)

Find u L2 (0, T; V) such thatddtu, H = (u, ) + F () , V

u (0) = u0(2)

To show well-posedness:

show existence, uniqueness and continuous dependence on thedata of the solution to (2)equivalence of the solution to (1) and (2)



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Galerkin Method

Outline






5 Conclusions

6 Future Work



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Galerkin Method

Galerkin Method

Project the solution of (2) in a finite dimensional spaceVn = span {1, . . . , n}

Finite-dimensional problem: Find un L2 (0, T; Vn) such that

ddt

un, iH

= (un, i) + F (i) , i = 1, . . . , n

un (0) = PVnu0(3)

Write un (x, t) =n

j=1j (t) j (x) and substitute to (3):

M (t) = A (t) + F (t)

Initial condition: un (x, 0) =n

j=1

j (0) j (x) = PVnu0 (x)



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Galerkin Method

Galerkin Method



ddt

un, iH

= (un, i) + F (i) , i = 1, . . . , n

un (0) = PVnu0(3)

Write un (x, t) =n


M (t) = A (t) + F (t)


j=1

j (0) j (x) = PVnu0 (x)



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Galerkin Method

Galerkin Method



ddt

un, iH

= (un, i) + F (i) , i = 1, . . . , n

un (0) = PVnu0(3)

Write un (x, t) =n


M (t) = A (t) + F (t)


j=1

j (0) j (x) = PVnu0 (x)



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Galerkin Method

Galerkin Method



ddt

un, iH

= (un, i) + F (i) , i = 1, . . . , n

un (0) = PVnu0(3)

Write un (x, t) =n


M (t) = A (t) + F (t)


j=1

j (0) j (x) = PVnu0 (x)



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Galerkin Method

Galerkin Method- cont -

(t) = M1 (A (t) + F (t)) (4)

The variables in (4) are defined as follows:

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

[A]ij = k

cpj, iV

h

cpj () i ()

[M]ij =0 j (x) i (x) dx

[F (t)]i =2huarcp

0

i (x) dx +huacp

i () +Q

cpi (0)

Goal: Find (t) that satisfies (4) using an ODE solver


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l k h d


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Galerkin Method

Galerkin Method- cont -

(t) = M1 (A (t) + F (t)) (4)

The variables in (4) are defined as follows:

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

[A]ij = k

cpj, iV

h

cpj () i ()

[M]ij

= 0

j

(x) i

(x) dx

[F (t)]i =2huarcp

0

i (x) dx +huacp

i () +Q

cpi (0)

Goal: Find (t) that satisfies (4) using an ODE solver


P E i i


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Parameter Estimation

Outline






5 Conclusions

6 Future Work


P t E ti ti


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Vector of unknown parameters: q =Qk

, hk

, cpk

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

un (xi, tj; q) is the solution of the finite-dimensional problem(3) using the parameter q and evaluated at xi at time tj

Find:

minqR3

J(q) = minqR3

1

N N t

Ntj=1

Ni=1

|un (xi, tj; q) u (xi, tj)|2




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, hk

, cpk



Find:

minqR3

J(q) = minqR3

1

N N t

Ntj=1

Ni=1





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, hk

, cpk



Find:

minqR3

J(q) = minqR3

1

N N t

Ntj=1

Ni=1





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, hk

, cpk



Find:

minqR3

J(q) = minqR3

1

N N t

Ntj=1

Ni=1


Parameter Estimation in a Mathematical Model of a Heat-Conducting RodOptimization Algorithms


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Outline




3 Optimization AlgorithmsLeast SquaresGenetic Algorithm

4 Results

5 Conclusions

6 Future Work


Least Squares


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q

Outline





4 Results

5 Conclusions

6 Future Work


Least Squares


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Least Squares

Built-in optimization algorithm in Scilab (leastsq)

Solves nonlinear least squares problemsQuasi-Newton algorithm (the Jacobian of the cost functionwas not supplied)

Can be used if the errors in the measurements are randomlydistributed


Genetic Algorithm


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Outline





4 Results

5 Conclusions

6 Future Work


Genetic Algorithm


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Genetic AlgorithmOverview

Optimization algorithm inspired by the concept of evolution(survival of the fittest)

Suppose that we want to find q = (q1, q2, . . . , qn) thatminimizes the function F (q):

individual: described by qfitness of an individual: determined by F(q)

initial population: randomly selectedmain processes of GA: selection, recombination, mutation


Genetic Algorithm


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Genetic Algorithm


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Genetic Algorithm


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Genetic AlgorithmProcesses

Selection

compute the fitness of each individual

form the roulettecreate the mating pool

Recombination

randomly select two individuals from the mating pool

mating (to produce population of offsprings)probability of mating is determined by a fixed probability


Genetic Algorithm


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Selection

compute the fitness of each individual

form the roulettecreate the mating pool

Recombination

randomly select two individuals from the mating pool

mating (to produce population of offsprings)probability of mating is determined by a fixed probability


Genetic Algorithm

G l h


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Mutation

introduce small changes in the parameters of an individual(mutate)

probability of mutation is determined by a fixed probabilitythe resulting (mutated) pool will make up the initialpopulation for the next generation

Optional Process: Elitism

the fitness of the next generation doesnt necessarily improve(because of recombination and mutation)if the best parent has disappeared, introduce it again byremoving 1 arbitrary offspring


Optimization Algorithms

Genetic Algorithm

G i Al i h


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Mutation

introduce small changes in the parameters of an individual(mutate)

probability of mutation is determined by a fixed probabilitythe resulting (mutated) pool will make up the initialpopulation for the next generation

Optional Process: Elitism

the fitness of the next generation doesnt necessarily improve(because of recombination and mutation)if the best parent has disappeared, introduce it again byremoving 1 arbitrary offspring


Results


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Outline

1 Experimental Setup2 Model and Parameter Estimation Problem

The ModelSolution to the Model



4 Results

5 Conclusions

6 Future Work


Results

l


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Results

Compare the results (using Least Squares and GeneticAlgorithm) when:

end of the rod away from the heat source is insulated (BC1)there is heat loss at the end of the rod away from the heat

source (BC2)

Error Values

leastsq GA

BC1 BC2 BC1 BC2Objective Function

copper 0.1645 0.1648 0.1645 0.1648

aluminum 0.0768 0.0762 0.0758 0.0762


Results

R l


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Results

Compare the results (using Least Squares and GeneticAlgorithm) when:

end of the rod away from the heat source is insulated (BC1)there is heat loss at the end of the rod away from the heat

source (BC2)

Error Values

leastsq GA

BC1 BC2 BC1 BC2Objective Function

copper 0.1645 0.1648 0.1645 0.1648

aluminum 0.0768 0.0762 0.0758 0.0762


Results

R lt


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Results- cont -

Optimal Parameters

leastsq GA

BC1 BC2 BC1 BC2

Estimated Q/kcopper 66.1440 66.1372 66.2355 66.2739

aluminum 60.9651 60.9546 60.9729 60.9583

Estimated h/k

copper 0.0312 0.0312 0.0313 0.0313aluminum 0.0569 0.0569 0.0570 0.0569

Estimated cp/k

copper 1.1840 1.1844 1.1848 1.1864

aluminum 3.8982 3.8986 3.9000 3.8989


Results

Results


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ResultsCopper Rod: actual temperature and computed temperature

0 500 1000 1500 2000 2500 3000

300

305

310

315

320

325

Copper Rod: Data and Computed Values

time (s)

temperature(K)

data

TC1

TC2

TC3

TC4

TC5

TC6TC7

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


Results

Results


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ResultsAluminum Rod: actual temperature and computed temperature

0 1000 2000 3000 4000 5000 6000 7000 8000

306

308

310

312

314

316

318

320

322

Aluminum Rod: Data and Computed Values

time (s)

temperature(K)

data

TC1

TC2

TC3

TC4

TC5

TC6TC7

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


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Conclusions

Conclusions


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Conclusions

Obtained the data using thermocouples attached to a dataacquisition instrument.

Formulated the model for heat conduction on a metal rod,assuming that heat is lost along its entire length and at theboundary.

Obtained the solution to the model using the Galerkin method.

Obtained estimates for the parameters Q/k, h/k, cp/k usingleastsqand genetic algorithm.

Modeling the heat loss at the end of the rod away from theheat source produces the same output as the model withoutheat loss.


Future Work

Outline


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Outline

1 Experimental Setup2 Model and Parameter Estimation Problem

The ModelSolution to the ModelGalerkin MethodParameter Estimation


4 Results

5 Conclusions

6 Future Work


Future Work

Future Work


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Future Work

Create a better heat source (oven, in cooperation with NIP)

Reformulate the model to incorporate realistic assumptions

(ambient temperature, flux)Use other optimization algorithms (gradient-based, heuristic,neural networks, hierarchical Bayesian methods) for parameterestimation

Implement a faster numerical method for solving the PDE


References


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References

Instructional and Research Laboratory, Center for Research in ScientificComputation, North Carolina State University,http://www.ncsu.edu/crsc/ilfum.htm.

P. Laguitao, Estimation of Copper Rod Parameters Using Data fromHeat Conduction Experiment, Undergraduate Research Paper, College of

Science, University of the Philippines Diliman, 2001.H.T. Banks, R.C. Smith and Y. Wang, Smart Material Structures:Modeling, Estimation and Control, John Wiley & Sons, 1996.

R.R. Briones, Numerical Computations for Parameter Estimation in aSmart Beam Structure, Masters Thesis, College of Science, University of

the Philippines Diliman, 2002.R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics andPhysics, Springer-Verlag, 1997.

J. Skaar and K.M. Risvik, A Genetic Algorithm for the Inverse Problem inSynthesis of Fiber Gratings, J. Lightwave Technol., 16, 1928-1932 (1998).



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Parameter Estimation in a Mathematical Model of a Heat-Conducting Rod

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