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

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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!

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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.

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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.

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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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    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.

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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.

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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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    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.

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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.

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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

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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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    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