Application of Bezier splines and sensitivity analysis in inverse geometry and boundary problems...

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


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


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

]


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



0

1

2

3

ave

rag

e e

rro

r [%

]



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 [°


tracked boundary

exact tem perature


phase change front location


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



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

1068

1072

1076

1080

1084

T [°


tracked boundary

exact tem perature

0

1

2

3

4

ave

rag

e e

rro

r [%

]



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


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


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


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


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

Application of Bezier splines and sensitivity analysis in inverse geometry and boundary problems...

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