cauchy ecuacion
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Numerical Solution of Cauchy Problems for
Elliptic Equations in Rectangle-likeGeometries
Fredrik Berntsson and Lars Elden
Department of Mathematics, Linkoping University
Oktober 2005
Femlab Conference 2005
Presented at the COMSOL Multiphysics User's Conference 2005 Boston
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Overview
Introduction, Motivating Example
Illposedness, Stabilization
Numerical Test problem in FEMLAB
Transformation to rectangular geometry. Mapping of Normal derivatives.
Numerical solution of Test problem
Conclusions, Future Work
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Motivating example: Ilmenite iron melting furnace
Thermocouple
Electrode
Level K to
level Dthermocouples
highest
with level K
Under
electrodes
Between
electrodesCenter
The furnace material properties are temperature dependent.
Problem: find the inner shape of the furnace.
Nonlinear, and (rather) complex geometry
PhD thesis: I M Skaar, Monitoring the Lining of a Melting Furnace, NTNU, Trondheim, 2001
Femlab Conference 2005 2
Presented at the COMSOL Multiphysics User's Conference 2005 Boston
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Irregular geometry - non-constant coefficient
Map the region to a rectangle:
liquid steel
liquid steel
Cauchy data
Find lining: the interface between iron and furnace
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The Cauchy Problem for Laplaces equation
Classical ill-posed problem (Hadamard, 1900-1930)
uxx + uyy = 0,
u(x, 0) = 0, x ,
u
x (x, 0) =
1
n sin(nx), x .
Solution:
u(x, y) =1
n2sin(nx) sinh(ny).
Large n: arbitrarily small data, arbitrarily large solution.
The solution does not depend continuously on the data!
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Illposed Cauchy Problem on Unit Square
1
1 x
y
(a(u)ux)x+(a(u)uy)y
u=g, uy =h
ux=0 ux=0
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Fourier analysis
By separation of variables:
T(x, y) =A0y + B0 +
k=1
(Akeky + Bke
ky)cos(kx),
Fourier coefficients {Ak} and {Bk} satisfy (for k>0): 1 1k k
AkBk
= gkhk
,
where {gk} and {hk} are the Fouriercosine coefficients ofg(x) and h(x)Since eky as k the problem is severely illposed!
Femlab Conference 2005 6
Presented at the COMSOL Multiphysics User's Conference 2005 Boston
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Rewrite as an Initial value problem:u
auy
y
=
0 a1
xa
x0
u
auy
, 0y1,
u(x, 0)uy(x, 0)
=
g
h
.
The unbounded x-derivative makes the problem ill-posed!
Approximation of derivative:
x
a(u)
v
x
D(ADV)
A diagonal matrix, depends on u
D bounded differentiation matrix, spectral or otherwise
Resulting stable problem is solved using standard code (ode45) in Matlab.
Femlab Conference 2005 7
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Approximate u(x) by a least squares cubic spline s(x).
Then set u(x)s(x).
3
2
1
0
1
2
3
4
5
6
7
The basis functions B3j (x), for j=1, . . . , 5.
Choose a coarse grid = bounded differentiation operator.
Advantage: flexible at boundaries.
Femlab Conference 2005 8
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A Numerical Model Problem
The Heat Equation
(a(u)ux)x + (a(u)uy)y = 0, in
Boundary Conditions
u = g, un = h, on L1,un
= 0, on L2 and L3,
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Presented at the COMSOL Multiphysics User's Conference 2005 Boston
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Test problem generated in FEMLAB
Sides: insulated
Upper boundary: 1450o
Lower boundary: 800o
Temperature-dependent
conductivity
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Thermal conductivity
0 200 400 600 800 1000 1200 1400 16002
4
6
8
10
12
14
16
18
Temperature [oC ]
The
rmalconductivity[W/moC]
The thermal conductivity for magnesia brick, as a function of temperature.
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Constructed solution (direct problem solved using FEMLAB):
Temperature and Heat-Flux data along the lower boundary is used toidentify the upper boundary!
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Transformation of the Problem
The Conformal Mapping
(x, y) = (, ) = [x(, ), y(, )]
The problem is transformed into
(a(v)v) + (a(v)v) = 0, 0
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Transformation of the area
3 2 1 0 1 2 32
1.5
1
0.5
0
0.5
1
1.5
2
2.5
3
3.5
Conformal mapping of the auxilliary domain onto a rectangle. Actual gridis finer!
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Computed flux data on the square
2 1.5 1 0.5 0 0.5 1 1.5 23200
3400
3600
3800
4000
4200
4400
4600
Transformed using Conformal mapping (blue curve). No noise added!
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Computed heat-flux on original Domain
0 1 2 3 4 5 6500
1000
1500
2000
2500
3000
3500
The Heat-Fluxu
non L1 computed by FEMLAB
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Transformation of normal derivativesBy the Conformal mapping
u
n|L1= |(1
)
|v
(, 0)
The SchwarzChristoffel mapping function has a singularity at every
corner of the polygonal domain.
The Singularities in |(1)| almost cancel out the spikes in FEMLABsnormal derivative.
Femlab Conference 2005 17
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P t d t th COMSOL M lti h i U ' C f 2005 B t
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Numerical Solution of the Test problem
liquid steel
Cauchy data
liquid steel
Solve Cauchy problem. Find the isotherm v=1450 oC.
Conformal mapping of the identified curve.
Femlab Conference 2005 18
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P t d t th COMSOL M lti h i U ' C f 2005 B t
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Noisy flux data
2 1.5 1 0.5 0 0.5 1 1.5 23000
3500
4000
4500
Normally distributed noise added. Realistic noise level!
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Presented at the COMSOL Multiphysics User's Conference 2005 Boston
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Computed temperatures at 0.6 and 0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11140
1160
1180
1200
1220
1240
1260
1280
1300
1320
1340
Temperature[oC
]
Numerical solution at y=0.60
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11260
1280
1300
1320
1340
1360
1380
1400
1420
1440
1460
Temperature[oC
]
Numerical solution at y=0.80
Derivatives computed using 11 B-splines. Differential equation only validwhen temperature is below 1450 oC.
Femlab Conference 2005 20
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Presented at the COMSOL Multiphysics User's Conference 2005 Boston
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Noisy data, identified boundary
3 2 1 0 1 2 3
2
1
0
1
2
3
4
Derivatives computed using 11 B-splines.
Femlab Conference 2005 21
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Presented at the COMSOL Multiphysics User's Conference 2005 Boston
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Less regularization, temperature at 0.62 and 0.81
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11150
1200
1250
1300
1350
1400
Temperature[oC
]
Numerical solution at y=0.62
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11260
1280
1300
1320
1340
1360
1380
1400
1420
1440
1460
Temperature[oC
]
Numerical solution at y=0.81
Derivatives computed using 19 B-splines!
Impossible to identify boundary when the temperature is not smooth
Femlab Conference 2005 22
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Presented at the COMSOL Multiphysics User's Conference 2005 Boston
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Conclusions Efficient solution: replace a derivative by bounded approximation, solve
Cauchy problem stably using a method of lines
General geometries: transform rectangle-like region to rectangle(equivalent to creating a curvilinear mesh). Solve the problem onthe rectangle
Transformation cheap to compute for rather general geometries(approximated by polygon)
Nonlinear problems can be solved rather easily
Stability theory: A stability estimate for a Cauchy problem for an ellipticpartial differential equation, Inverse Problems, vol. 21, no 5, pp. 1643-1653, October 2005.
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Presented at the COMSOL Multiphysics User's Conference 2005 Boston
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Future Work
More detailed FEMLAB model of the Furnace. More realistic simulationof measurements.
Identify boundaries given other conditions (e.g. insulated boundary).
Applications: The Melting Furnace
Use FEMLAB to creat Test problem. Finding good test problems is very
important!
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p y
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