Application of Bezier splines Application of Bezier splines and sensitivity analysis and sensitivity analysis

in inverse geometry in inverse geometry and boundary problemsand boundary problems

Iwona NOWAK*, Andrzej J. NOWAK*** Institute of Mathematics, ** Institute of Thermal Technology,Technical University of Silesia, Gliwice, Poland.

Inverse Problems in Engineering SymposiumInverse Problems in Engineering SymposiumTuscaloosa, June 2003Tuscaloosa, June 2003

Scope of the PresentationScope of the Presentation

• Problem formulation• Solution procedure - sensitivity analysis for geometry and boundary problems• Numerical results• Conclusions

ContinuousContinuous castingcasting

mould

water spraysolid

liquid

vx

solid

liquid

vx

x

y

F

A B

C

D

EO

Problem FormulationProblem Formulation

solid

liquid

x

y

F

A B

C

D

EO

012

xT

va

T xr 012

xT

va

T xr

FCph

EOs

DEs

CD

TT

TT

TThnT

qnT

rr

rr

r

r

,)(

,

,

,

FCph

EOs

DEs

CD

TT

TT

TThnT

qnT

rr

rr

r

r

,)(

,

,

,

IiUT ii ,,1, r IiUT ii ,,1, r

General Solution StrategyGeneral Solution Strategy

make boundary problem well-posed

solve direct problem

modify assumed datasensitivity coefficients

FDMFEMBEM

best matching

Sensitivity CoefficientSensitivity Coefficient

imeasurement

jestimated value

j

j jj

ii iY Y

Y

TT T

* *

j

j jj

ii iY Y

Y

TT T

* *

** YYZTT ** YYZTT

YT

ZYT

Zij

i

j

Main Set of EquationsMain Set of Equations

** YYZTT ** YYZTT

YWYZWZ

TUWZYWZWZ~1*1T

*1T11T

Y

min

~~ 1T

1T

YYWYY

UTWUT

Y

min

~~ 1T

1T

YYWYY

UTWUT

Y

Solution AlgorithmSolution Algorithm

direct problem formulation-assumption of vector

Y*=[y1*,..., yn*, q1*,..., qm*]„freezing”

of heat fluxes

geometry problem- iterative solution

„freezing” offront location

boundary problem- single step

convergence check

end of calculations

Sensitivity Analysis - Boundary PartSensitivity Analysis - Boundary Part

012

xZ

va

Z xr 012

xZ

va

Z xr

solid

liquid

vx

x

y

F

A B

C

D

EO

EO

FC

Z

Z

rr

rr

,0

,0 EO

FC

Z

Z

rr

rr

,0

,0

CDEjj Yxf

nZ

Yq

rr

,)()( CDE

jj Yxf

nZ

Yq

rr

,)()(

jDYD

CDExfnT

q rr ),()( CDExfnT

q rr ),()(

Sensitivity Analysis –Geometry PartSensitivity Analysis –Geometry Part

012

xZ

va

Z xr 012

xZ

va

Z xr

solid

liquid

vx

x

y

F

A B

C

D

EO

EO

DE

CD

Z

ZhnZnZ

rr

r

r

,0

,

,0

EO

DE

CD

Z

ZhnZnZ

rr

r

r

,0

,

,0

Sensitivity Analysis –Geometry PartSensitivity Analysis –Geometry Part

phTyxT ),( phTyxT ),(

jj

nj Y

yYxq

Z

sincos

jj

nj Y

yYxq

Z

sincos

jDYD

V0

V1

V2

V3

u

u u

V01

V11

V21

u

u

V02

V12

V03 = P(u)

u

Bezier Spline

Nowak I., Nowak A.J, Wrobel L.C: Identification of Phase Change Front by Bezier Splines and BEM, International Journal of Thermal Sciences, vol.41 (2002) Elsevier Science, pp.492-499

Sensor LocationSensor Locationss

geometry part- iterative solution

boundary part- single step

solid

liquid

x

y

F

A B

C

D

EO

all sensors are used

all sensors are used

Numerical ResultsNumerical Results

measurements error : 0.1%number of sensors : 35

0 0.2 0.4 0.6x

0

0.02

0.04

0.06

0.08

0.1

yExact Position

S tart Position

after Lum ping

1 loop

2 loop

3 loop

0 0.2 0.4 0.6 0.8 1x

-5.0E+006

-4.0E+006

-3.0E+006

-2.0E+006

-1.0E+006

0.0E+000

q [W

/m^2

]phase change front location

heat flux distribution

Numerical ResultsNumerical Results

measurements error : 0.1%number of sensors : 35

0

10

20

30

40

ave

rag

e e

rro

r [%

]

tem p. in sensor poin ts

tem p a long boundary heat flux

1.53 %3.06 %

36.51 %

0 20 40 60 80 100n o d e n u m b e r

1030

1040

1050

1060

1070

1080

1090

T [°

C] tem perature a long

tracked boundary

exact tem perature

Sensor LocationSensor Locationss

geometry part- iterative solution

boundary part- single step

solid

liquid

x

y

F

A B

C

D

EO

geometry sensors are used

geometry sensors are used

Numerical ResultsNumerical Results

phase change front location

0 0.2 0.4 0.6x

0

0.02

0.04

0.06

0.08

0.1

yExact Position

S tart Position

after Lum ping

1 loop

2 loop

3 loop

0 0.2 0.4 0.6 0.8 1x

-4.0E+006

-3.0E+006

-2.0E+006

-1.0E+006

0.0E+000

q [W

/m^2

]

heat flux distribution

measurements error : 0.1%number of sensors : 35sensors separated

Numerical ResultsNumerical Results

measurements error : 0.1%number of sensors : 35sensors separated

0

1

2

3

ave

rag

e e

rro

r [%

]

tem p. in sensor poin ts

tem p a long boundary heat flux

0.086 %0.344 %

2.86 %

0 20 40 60 80 100n o d e n u m b e r

1050

1060

1070

1080

1090

T [°

C] tem perature a long

tracked boundary

exact tem perature

Numerical ResultsNumerical Results

phase change front location

heat flux distribution

measurements error : 2.0 %number of sensors : 35sensors separated

0 0.2 0.4 0.6 0.8 1x

-4.0E+006

-3.0E+006

-2.0E+006

-1.0E+006

0.0E+000

q [W

/m^2

]

0 0.2 0.4 0.6x

0

0.02

0.04

0.06

0.08

0.1

yExact Position

S tart Position

after Lum ping

1 loop

2 loop

3 loop

Numerical ResultsNumerical Results

measurements error : 2.0%number of sensors : 35sensors separated

0 20 40 60 80 100n o d e n u m b e r

1068

1072

1076

1080

1084

T [°

C] tem perature a long

tracked boundary

exact tem perature

0

1

2

3

4

ave

rag

e e

rro

r [%

]

tem p. in sensor poin ts

tem p a long boundary heat flux

0.136 %

0.499 %

3.52 %

Experiment Experiment

Drezet J.-M., Rappaz M., Grun G.-U.,Gremaud M., Determination of Thermophysical Properties and Boundary Conditions of Direct Chill-Cast Aluminium Alloys Using Inverse Methods, Metallurgical and Materials Transactions A, vol.31A, June 2000, pp.1627--1634

v x

m ou ld

con tro l beads

L-rod w ith 5 the rm ocoup les

sym

me

try

axi

s

l iqu id

so lid

Numerical ResultsNumerical Results

0.07 0.08 0.09 0.1 0.11 0.12 0.13

200

300

400

500

600

Thermocouple locationbelow the surface

5 mm10 mm15 mm20 mm

Tem

pera

ture

x

Measured and predicted temperature in sensor locations

Numerical ResultsNumerical Results

Heat flux distribution along the surface

0 0.4 0.8 1.2 1.6 2

X

-5E+006

-4E+006

-3E+006

-2E+006

-1E+006

0

computationsobtained by Drezet

0 0.04 0.08 0.12

X

-5E+006

-4E+006

-3E+006

-2E+006

-1E+006

0

Hea

t Flu

x

Hea

t Flu

x

Numerical ResultsNumerical Results

Temperature distribution along the surface

0 0.4 0.8 1.2 1.6 2

X

0

200

400

600

800

measurementcalculations

0 0.04 0.08 0.12

X

0

200

400

600

800

Tem

pera

ture

Tem

pera

ture

Numerical ResultsNumerical Results

Front Location

0 0.2 0.4 0.6X

0

0.1

0.2

0.3

Y

0 0.02 0.04 0.06 0.08 0.1X

0.22

0.23

0.24

0.25

0.26

Y

ConclusionsConclusions

• BEM based algorithms for solving inverse boundary and geometry problems

• application of Bézier function and sensitivity coefficients

• stabile results does not provide accurate solution

• sensors separation permits to obtain encouraging results even with experimental measurements