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Transcript of Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
8/8/2019 Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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
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Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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
8/8/2019 Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentIntroduction
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
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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
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Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentExperimental Setup
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
M h i l M d li A l i d P E i i H C d i E i
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Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentExperimental Setup
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
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Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentExperimental Setup
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 Experiment
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Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentModel Formulation
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 Experiment
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Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentModel Formulation
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
Mathematical Modeling Analysis and Parameter Estimation on a Heat Conduction Experiment
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Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentModel Formulation
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
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentModel Formulation
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
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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g, y pModel Formulation
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
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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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= −
Q
k∂u ( , t)∂x
= 0
(1)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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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 )
∂x
= −Q
k∂u ( , t)∂x
= −hk
(u ( , t) − ua )
(2)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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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
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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Well-Posedness
Outline
1 Introduction
2 Experimental Setup
3 Model Formulation
4 Well-Posedness
5 Galerkin Method and Convergence
6 Parameter Estimation7 Results
8 Conclusions and Future Work
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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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 ) =
T
0f (t) 2
V dt + T
0
df dt
2
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
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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 ) =
T
0f (t) 2
V dt + T
0
df dt
2
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
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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 ) =
T
0f (t) 2
V dt + T
0
df dt
2
V ∗dt
Remark: W (0, T ) → C 0 ([0, T ] ;H )
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentWell Posedness
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Well-Posedness
Well-Posedness of (2) - BC2
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
dudt
, φV ∗ ,V
= − σ (u, φ ) + F (φ) , ∀φ ∈V
u (0) = u0
(3)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentWell Posedness
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Well-Posedness
Well-Posedness of (2) - BC2
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
dudt
, φV ∗ ,V
= − σ (u, φ ) + F (φ) , ∀φ ∈V
u (0) = u0
(3)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentWell-Posedness
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Well Posedness
Well-Posedness of (2) - BC2
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
(4)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentWell-Posedness
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Well Posedness
Well-Posedness of (2) - BC2
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
V .
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 ).
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentWell-Posedness
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Well-Posedness of (1) - BC1
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)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentWell-Posedness
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Well-Posedness of (1) - BC1
Weak form of (1)Find u ∈W (0, T ) such that
dudt
, φV ∗ ,V
= − σ (u, φ ) + F (φ) , ∀φ ∈V
u (0) = u0
(5)
¯F ∈V
∗
, so¯F ∈L
2
(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 ExperimentWell-Posedness
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Well-Posedness of (1) - BC1
Weak form of (1)Find u ∈W (0, T ) such that
dudt
, φV ∗ ,V
= − σ (u, φ ) + F (φ) , ∀φ ∈V
u (0) = u0
(5)
¯F ∈V
∗
, so¯F ∈L
2
(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
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Outline
1 Introduction
2 Experimental Setup
3 Model Formulation
4 Well-Posedness
5 Galerkin Method and Convergence
6 Parameter Estimation7 Results
8 Conclusions and Future Work
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence
8/8/2019 Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
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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
dun
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
i=1
α i (t) φi (x) and substitute to ( 6):
M α (t) = Aα (t) + F (t)
Initial condition: un (x, 0) =n
i=1
α i (0) φi (x) = P nV u0
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence
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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
dun
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
i=1
α i (t) φi (x) and substitute to ( 6):
M α (t) = Aα (t) + F (t)
Initial condition: un (x, 0) =n
i=1
α i (0) φi (x) = P nV u0
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence
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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
dun
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
i=1
α i (t) φi (x) and substitute to ( 6):
M α (t) = Aα (t) + F (t)
Initial condition: un (x, 0) =n
i=1
α i (0) φi (x) = P nV u0
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence
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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
dun
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
i=1
α i (t) φi (x) and substitute to ( 6):
M α (t) = Aα (t) + F (t)
Initial condition: un (x, 0) =n
i=1
α i (0) φi (x) = P nV u0
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence
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Galerkin Method for BC2
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 +
hu a
ρc pφi ( ) +
Qρc p
φi (0)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence
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Galerkin Method for BC2
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 +
hu a
ρc pφi ( ) +
Qρc p
φi (0)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence
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Galerkin Method for BC1
Finite-dimensional problem for (5): Find un such that
dun
dt, φi
H = − σ (un , φi ) + F (φi ) , i = 1 , . . . , n
un
(0) = P nV u0
(8)
Write un (x, t ) =n
i=1α i (t) φi (x) and substitute to ( 8):
M α (t) = Aα (t) + F (t)
Initial condition: un (x, 0) =n
i=1α i (0) φi (x) = P nV u0
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence
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Galerkin Method for BC1
Finite-dimensional problem for (5): Find un such that
dun
dt, φi
H = − σ (un , φi ) + F (φi ) , i = 1 , . . . , n
un
(0) = P nV u0
(8)
Write un (x, t ) =n
i=1α i (t) φi (x) and substitute to ( 8):
M α (t) = Aα (t) + F (t)
Initial condition: un (x, 0) =n
i=1α i (0) φi (x) = P nV u0
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence
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Galerkin Method for BC1
Finite-dimensional problem for (5): Find un such that
dun
dt, φi
H = − σ (un , φi ) + F (φi ) , i = 1 , . . . , n
un
(0) = P nV u0
(8)
Write un (x, t ) =n
i=1α i (t) φi (x) and substitute to ( 8):
M α (t) = Aα (t) + F (t)
Initial condition: un (x, 0) =n
i=1α i (0) φi (x) = P nV u0
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Galerkin Method for BC1
α (t) = M − 1 Aα (t) + F (t) (9)
The variables in (9) are dened as follows:
α (t) = [α1 (t) , α 2 (t) , . . . , α n (t)]T
A ij = −k
ρc pφ j , φi V
[M ]ij =
0φ j (x) φi (x) dx
F (t) i =2hu a
rρc p 0φi (x) dx +
Qρc p
φi (0)
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Galerkin Method for BC1
α (t) = M − 1 Aα (t) + F (t) (9)
The variables in (9) are dened as follows:
α (t) = [α1 (t) , α 2 (t) , . . . , α n (t)]T
A ij = −k
ρc pφ j , φi V
[M ]ij =
0φ j (x) φi (x) dx
F (t) i =2hu a
rρc p 0φi (x) dx +
Qρc p
φi (0)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentGalerkin Method and Convergence
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Convergence Result
Assumptions on V n :(H 1 ) V n ⊂V ;(H 2 ) For each φ ∈V , φ − P n
V φ
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.
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Convergence Result
Assumptions on V n :(H 1 ) V n ⊂V ;(H 2 ) For each φ ∈V , φ − P n
V φ
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.
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Outline
1 Introduction
2 Experimental Setup
3 Model Formulation
4 Well-Posedness
5 Galerkin Method and Convergence
6 Parameter Estimation7 Results
8 Conclusions and Future Work
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Objective Function
Vector of unknown parameters: q = Qk , h
k , cp
k
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
Find:
minq∈Λ
J (q) = minq∈Λ
1N · Nt
Nt
j =1
N
i=1|u (x i , t j ; q) − u (x i , t j )|2 (10)
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Objective Function
Vector of unknown parameters: q = Qk , h
k , cp
k
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
Find:
minq∈Λ
J (q) = minq∈Λ
1N · Nt
Nt
j =1
N
i=1|u (x i , t j ; q) − u (x i , t j )|2 (10)
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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
Find:
minq∈Λm
J (q) = minq∈Λm
1N · Nt
Nt
j =1
N
i=1|un (x i , t j ; q) − u (x i , t j )|2
(11)
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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
Find:
minq∈Λm
J (q) = minq∈Λm
1N · Nt
Nt
j =1
N
i=1|un (x i , t j ; q) − u (x i , t j )|2
(11)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentParameter Estimation
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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
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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
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentParameter Estimation
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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) .
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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.
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentParameter Estimation
O i i i Al i h
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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)
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O i i i Al i h
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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
O tli
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Outline
1 Introduction
2 Experimental Setup
3 Model Formulation
4 Well-Posedness
5 Galerkin Method and Convergence
6
Parameter Estimation7 Results
8 Conclusions and Future Work
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentResults
N i l R lt
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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
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentResults
Numerical Results
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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
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentResults
Plots
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Plots
0 500 1000 1500 2000 2500 3000300
305
310
315
320
325
Copper Rod: Data and Computed Values
time (s)
t e m p e r a
t u r e (
K )
data
TC1
TC2
TC3
TC4
TC5
TC6
TC7
Figure: Plot of temperature vs. position (copper rod)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentResults
Plots
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Plots
0 1000 2000 3000 4000 5000 6000 7000 8000306
308
310
312
314
316
318
320
322
Aluminum Rod: Data and Computed Values
time (s)
t e m p e r a
t u r e
( K )
data
TC1
TC2
TC3
TC4
TC5
TC6
TC7
Figure: Plot of temperature vs. position (aluminum rod)
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentConclusions and Future Work
Outline
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Outline
1 Introduction
2 Experimental Setup
3 Model 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 ExperimentConclusions and Future Work
Conclusions
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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.
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction ExperimentConclusions and Future Work
Future Work
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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
Mathematical Modeling, Analysis and Parameter Estimation on a Heat Conduction Experiment
References
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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.
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The End