Numerical Solution of Cauchy Problems for Elliptic ... - COMSOL.com

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Numerical Solution of Cauchy Problems for
Elliptic Equations in “Rectangle-like”
Geometries
Fredrik Berntsson and Lars Eldén
Department of Mathematics, Linköping University
Oktober 2005
– Femlab Conference 2005 –

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Overview
• Introduction, Motivating Example
• Ill–posedness, Stabilization
• Numerical Test problem in FEMLAB
• Transformation to rectangular geometry. Mapping of Normal derivatives.
• Numerical solution of Test problem
• Conclusions, Future Work
– Femlab Conference 2005 – 1

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Motivating example: Ilmenite iron melting furnace
Thermocouple
Electrode
Level K to
level D
thermocouples
with level K
highest
Center
Under
electrodes
Between
electrodes
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
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
– Femlab Conference 2005 – 3

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
The Cauchy Problem for Laplace’s equation
Classical ill-posed problem (Hadamard, 1900-1930)
u xx + u yy = 0,
u(x,0) = 0, −∞ ≤ x ≤ ∞,
∂u
∂x (x, 0) = 1 sin(nx), −∞ ≤ x ≤ ∞.
n
Solution:
u(x,y) = 1 sin(nx) sinh(ny).
n2 Large n: arbitrarily small data, arbitrarily large solution.
The solution does not depend continuously on the data!
– Femlab Conference 2005 – 4

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Ill–posed Cauchy Problem on Unit Square
y
1
(a(u)u x ) x +(a(u)u y ) y
u x =0 u x =0
u=g, u y =h
1 x
– Femlab Conference 2005 – 5

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
By separation of variables:
Fourier analysis
T(x,y)=A 0 y + B 0 +
∞∑
(A k e kπy + B k e −kπy )cos(kπx),
k=1
Fourier coefficients {A k } and {B k } satisfy (for k>0):
( ) ( ) ( )
1 1 Ak ĝk
= ,
kπ −kπ B k ĥ k
where {ĝ k } and {ĥk} are the Fourier–cosine coefficients of g(x) and h(x)
Since e kπy → ∞ as k → ∞ the problem is severely ill–posed!
– Femlab Conference 2005 – 6

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Rewrite as an Initial value problem:
( ) (
u 0 a
=
−1
au y
y
− ∂
∂x a ∂
∂x
0
) (
u
au y
)
, 0≤y ≤1,
(
u(x, 0)
u y (x, 0)
) ( )
g
= .
h
The unbounded x-derivative makes the problem ill-posed!
Approximation of derivative:
∂
∂x
(
a(u) ∂v )
∂x
≈ ¯D(A ¯DV )
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

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Approximate u(x) by a least squares cubic spline s(x).
Then set u ′ (x)≈s ′ (x).
α −3
α −2
α −1
α 0
α 1
α 2
α 3
α 4
α 5
α 6
α 7
The basis functions Bj 3 (x), for j =−1,...,5.
Choose a coarse grid =⇒ bounded differentiation operator.
Advantage: flexible at boundaries.
– Femlab Conference 2005 – 8

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
A Numerical Model Problem
The Heat Equation
(a(u)u x ) x + (a(u)u y ) y =0, in Ω
Boundary Conditions
∂u
u = g,
∂n = h, on L 1,
∂u
∂n = 0, on L 2 and L 3 ,
– Femlab Conference 2005 – 9

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Test problem generated in FEMLAB
Sides: insulated
Upper boundary: 1450 o
Lower boundary: 800 o
Temperature-dependent
conductivity
– Femlab Conference 2005 – 10

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
18
Thermal conductivity
16
Thermal conductivity [ W/m o C ]
14
12
10
8
6
4
2
0 200 400 600 800 1000 1200 1400 1600
Temperature [ o C ]
The thermal conductivity for magnesia brick, as a function of temperature.
– Femlab Conference 2005 – 11

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Constructed solution (direct problem solved using FEMLAB):
Temperature and Heat-Flux data along the lower boundary is used to
identify the upper boundary!
– Femlab Conference 2005 – 12

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Transformation of the Problem
The Conformal Mapping
(x, y) = φ(ξ,η) = [x(ξ, η), y(ξ, η)]
The problem is transformed into
⎧
⎪⎨
⎪⎩
(a(v)v ξ ) ξ + (a(v)v η ) η = 0, 0

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Transformation of the area
3.5
3
2.5
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−3 −2 −1 0 1 2 3
Conformal mapping of the auxilliary domain onto a rectangle. Actual grid
is finer!
– Femlab Conference 2005 – 14

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Computed flux data on the square
4600
4400
4200
4000
3800
3600
3400
3200
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
Transformed using Conformal mapping (blue curve). No noise added!
– Femlab Conference 2005 – 15

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Computed heat-flux on original Domain
3500
3000
2500
2000
1500
1000
500
0 1 2 3 4 5 6
The Heat-Flux ∂u
∂n on L 1 computed by FEMLAB
– Femlab Conference 2005 – 16

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Transformation of normal derivatives
By the Conformal mapping
∂u
∂n | L1=|(φ −1 ) ′ | ∂v (ξ, 0)
∂η
The Schwarz–Christoffel mapping function φ has a singularity at every
corner of the polygonal domain.
The Singularities in |(φ −1 ) ′ | almost cancel out the spikes in FEMLABs
normal derivative.
– Femlab Conference 2005 – 17

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Numerical Solution of the Test problem
liquid steel
liquid steel
Cauchy data
• Solve Cauchy problem. Find the isotherm v=1450 o C.
• Conformal mapping of the identified curve.
– Femlab Conference 2005 – 18

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
4500
Noisy flux data
4000
3500
3000
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
Normally distributed noise added. Realistic noise level!
– Femlab Conference 2005 – 19

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Computed temperatures at 0.6 and 0.8
1340
Numerical solution at y=0.60
1460
Numerical solution at y=0.80
1320
1440
1300
1420
1280
1400
Temperature [ o C ]
1260
1240
1220
Temperature [ o C ]
1380
1360
1340
1200
1320
1180
1300
1160
1280
1140
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1260
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Derivatives computed using 11 B-splines. Differential equation only valid
when temperature is below 1450 o C.
– Femlab Conference 2005 – 20

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Noisy data, identified boundary
4
3
2
1
0
−1
−2
−3 −2 −1 0 1 2 3
Derivatives computed using 11 B-splines.
– Femlab Conference 2005 – 21

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Less regularization, temperature at 0.62 and 0.81
1400
Numerical solution at y=0.62
1460
Numerical solution at y=0.81
1440
1350
1420
1400
Temperature [ o C ]
1300
1250
Temperature [ o C ]
1380
1360
1340
1320
1200
1300
1280
1150
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1260
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Derivatives computed using 19 B-splines!
Impossible to identify boundary when the temperature is not smooth
– Femlab Conference 2005 – 22

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
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 on
the 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 elliptic
partial differential equation, Inverse Problems, vol. 21, no 5, pp. 1643-
1653, October 2005.
– Femlab Conference 2005 – 23

Presented at the COMSOL Multiphysics User's Conference 2005 Boston
Future Work
• More detailed FEMLAB model of the Furnace. More realistic simulation
of 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!
– Femlab Conference 2005 – 24