ICCES 2010 Las Vegas, March 28 - April 1, 2010 SOLUTION OF SHAPE HOT ROLLING BY THE LOCAL RADIAL BASIS FUNCTION COLLOCATION METHOD Božidar Šarler, Siraj

ICCES 2010Las Vegas, March 28 - April 1, 2010

SOLUTION OF SHAPE HOT ROLLING BY THE LOCAL RADIAL BASIS FUNCTION

COLLOCATION METHOD

Božidar Šarler, Siraj Islam, Umut Hanoglu

Laboratory for Multiphase ProcessesUniversity of Nova Gorica, Slovenia

SCOPE OF PRESENTATION

• Introduction and Motivation

• Thin Strip Rolling

• Shape Rolling

• Modeling Assumptions

• Structure of Thermal and Mechanical Models

• Solution of Thermal Model

• Generation of Nodal Points

• Ongoing Research

• Conclusions


Kick-off of a 4 year project: Modelling of shape hot rolling of steel

Overview of Our Recent Publications onLocal Radial Basis Function Collocation Method

B.Šarler and R.Vertnik, Computers & Mathematics with Applications (2006) (Diffusion)R.Vertnik and B.Šarler, Int.J.Numer.Methods Heat & Fluid Flow (2006) (Convection - Diffusion)R.Vertnik, M.Založnik and B.Šarler, Eng.Anal.Bound.Elem. (2006) (Continuous Casting of Aluminium - Growing Comp. Domain)I. Kovačević and B. Šarler, Materials Science and Engineering A (2006) (R-adaptive Phase Field Modeling of Microstructure Evolution) J.Perko and B.Šarler, Computer Modeling in Engineering and Sciences(2007) (Irregular Node Arrangements)G.Kosec and B.Šarler, Computer Modeling in Engineering and Sciences(2008) (Navier Stokes - Local Pressure Correction)G.Kosec and B.Šarler, Int.J.Numer.Methods Heat & Fluid Flow(2008) (Porous Media Flow - Local Pressure Correction)R.Vertnik and B.Šarler, Cast Metals Research(2008) (Continuous Casting of Steel – Conduction-Convection)


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R.Vertnik, B. Šarler, Computer Modeling in Engineering and Sciences (2009) (k-epsilon turbulence)G. Kosec, B.Šarler, Computer Modeling in Engineering and Sciences (2009) (Melting - Local pressure correction)G. Kosec, B.Šarler, International Journal of Cast Metals Research (2009) (Melting of anisotropic metals - Local pressure correction)G. Kosec, B.Šarler, Materials Science Forum (2010) (Freezing with natural convection - Local pressure correction)A. Lorbiecka, B.Šarler, Materials Science Forum (2010) (Grain Growth Modelling with Point Automata Method)

Extension to solid mechanics?


Overview of Our Recent Publications onLocal Radial Basis Function Collocation Method

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CONTINUOUS CASTING HOT SHAPE ROLLING

Mainstream Research Directions

Moving of the solid-liquid interface Large deformation

TWIN - ROLL CASTING PROCESSTHIN STRIP CASTING

Šarler et al. 2007

convection - diffusion with phase change

TWIN - ROLL CASTING PROCESS

TWIN - ROLL CASTING PROCESS

TWIN - ROLL CASTING PROCESSMACRO - MICRO APPROACH

LRBFC METHOD

PA METHOD

INFLUENCE OF ROLLING SPEED AND CASTING TEMPERATURE

INFLUENCE OF SETBACK AND STRIP THICKNESS

process

1

processparameters

proc.par.window

productproperties

processproperties

generalisedcost

process 2

+

NN – sub model

physical model, experience, measurements

fizikalni modelizkušnjemeritve

optimisation process(evolution algorithm)

integrated neural network (NN) through process model (TPM)

proc.par.window

processparameters

productproperties

processproperties

generalisedcost

LABORATORY FOR MULTIPHASE PROCESSES - FORESEEN RESEARCHThrough process modeling

physical model, experience, measurements

Integratedthroughprocessmodel

objectivefunction

- Quality- Productivity

- Machine occupation

NN – sub model

THIN STRIP HOT ROLLING


thick plates, thin plates

SHAPE ROLLING


rails, H beams, other complicated profiles

BASIC MODELLING STRATEGIES


Transient Steady

observe the whole billet observe only part of the billet

THIN STRIP ROLLING


Homogenous compression

Where the planes remain planes assumption is considered.

Non - homogenous compression

Might occur during high reductions with relatively small contact lenght.

material moves faster than the roll

material moves slower than the roll

BASIC LITERATURE REVIEW

Basic contemporary literature

J.G. Lenard, Primer on Flat Rolling, Elsevier, Amsterdam, 2007.

M. Piertrzyk, L. Cser, Mathematical and Physical Simulation of the Properties of Hot Rolled Products, John G. Lenard, Elsevier, Amsterdam, 1999.

V.B. Ginzburg, High Quality Steel Rolling, Marcel Dekker, New York, 1993.

W.L. Roberts, Cold Rolling of Steel, Marcel Dekker, New York, 1993.

First models with FEM - Marcal and King (1967) and Lee and Kobayashi (1970).

(Accurate results gained for small plastic strains).


MESHLESS METHODS - LITERATURE REVIEW

• On the utilization of the reproducing kernel particle method for the numerical simulation of plane strain rollingInternational Journal of Machine Tools and Manufacture, Volume 43, Issue 1, January 2003, Pages 89-102X. Shangwu, W. K. Liu, J. Cao, J. M. C. Rodrigues, P. A. F. Martins

• Splitting Rolling Simulated by Reproducing Kernel Particle MethodJournal of Iron and Steel Research, International, Volume 14, Issue 3, May 2007, Pages 43-47Qing-ling Cui, Xiang-hua Liu, Guo-dong Wang

• Simulation of plane strain rolling through a combined element free Galerkin–boundary element approachJournal of Materials Processing Technology, Volume 159, Issue 2, 30 January 2005, Pages 214-223Xiong Shangwu, J. M. C. Rodrigues, P. A. F. Martins

• Application of the element free Galerkin method to the simulation of plane strain rollingEuropean Journal of Mechanics - A/Solids, Volume 23, Issue 1, January-February 2004, Pages 77-93Shangwu Xiong, J. M. C. Rodrigues, P. A. F. Martins

• Parallel point interpolation method for three-dimensional metal forming simulationsEngineering Analysis with Boundary Elements, Volume 31, Issue 4, April 2007, Pages 326-342Wang Hu, Li Guang Yao, Zhong Zhi Hua

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4B-4789SGT-C&_user=10&_coverDate=01%2F31%2F2003&_alid=1275226516&_rdoc=2&_fmt=high&_orig=search&_cdi=5754&_sort=r&_st=4&_docanchor=&_ct=61&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=aa6a441778aa2e412043ec05f2cca302



http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B82XP-4NT7XBJ-9&_user=10&_coverDate=05%2F31%2F2007&_alid=1275226516&_rdoc=3&_fmt=high&_orig=search&_cdi=33036&_sort=r&_st=4&_docanchor=&_ct=61&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=db003cd169798f68d184c413e0ca28c8

GOVERNING EQUATIONS – THERMAL MODEL

pc T k T Q v

Fully three dimensional steady convection - diffusion problem

p

Tc k T Q

t

Two dimensional transient diffusion problem (slice model)

this shapes change

slice

THERMO-MECHANICAL MODEL SCHEMATICS

initial temperature

calculate deformation of the slice

at the new position

solve temperature of the slice

at the new position

initial shape

initial velocityIn rolling direction

final velocityin rolling direction

0v

v

initial shape

final shape

initial nodes

final nodes renoding

GOVERNING EQUATIONS - THERMAL MODEL

Heat transfer in the direction of the billet movement is neglected

0

0( ) , ' 't

t

z t v z t dt z

00

0

( )z z

t z tv

0 0( ) , , ( , )t z f z t v z t

0 0 0( )z t z v t t

GOVERNING EQUATIONS - THERMAL MODEL

2, , , , ;pc T t k T t k T t S tt

p p p p p

Governing Equation for a 2D perpendicular slice

Initial Condition

Boundary Conditions

0 0 0, , ;T t T t p p p

;D DT T p p p

,, , ;N NT t

T t T t

p

p n p pn

,, , , ;R RT t

T t h T t T t

pp n p p p

n

SOLUTION OF THE THERMAL MODEL


N

D N RN N N N N N N

N

0;

1;

pp

p 0;

1;

DD

D

pp

p 0;

1;

NN

N

pp

p 0;

1;

RR

R

pp

p

SOLUTION OF THERMAL MODEL

0 02 20 0 0 0 0 01p pc c

T k T k T S T k T k T St t

Time discretisation of governing equation

Time discretisation of boundary conditions

0, 1 ;D D DT t T T p p

0

,, 1 ;N N NT t

T t T T

p

p n pn

0 0

,, 1 ;R R RT t

T t h T T h T T

pp n p

n

1;fully implicit

0; fully explicit


; 1, 2,...,k k Np

Global nodes

Subdomain nodes

; 1, 2,...,l l N

Subdomains

; 1, 2,...,l n ln Np

( , )k k l n

Relation between global index k and local indeces l and n

l

kp l np


( , )1

;l N

k l n l n ln

T

p p p

Collocation of temperature field on subdomain l

( , ) ( , ) ( , )1

; ; 1, 2,...,l N

k l m l k l n k l m l n l ln

T m N

p p p

Calculation of expansion coefficients l

( , )1

; , 1, 2,...,l N

k l m l mn l n ln

T m n N

( , ) ( , )l mn l k l n k l m p

1( , )

1

l N

l n l nm k l mm

T

1( , ) ( , )

1 1

l lN N

l k l n l nm k l mn m

T T

p p

1( , )

1 1

l lN N

l l ln l nm k l mn m

T T


1 1( , ) ( , )

1 1 1 1

2 1( , )

1 1

0

0

1( , )

1

1

l l l l

l l

pl l

l

N N N N

l l ln l nm k l m l ln l nm k l mn m n m

N N

l l l ln l nm k l mn m

l l

D Dl l l l

N Nl l ln l l nm k l m l l ln l l nm

cT

t

k T

k T

S

T T

T

n n

10 ( , )

1 1 1 1

1 1( , ) 0 ( , )

1 1 1 1

1

l l l l

l l l l

N N N N

k l mn m n m

N N N NR R

l l ln l l nm k l m l l ln l l nm k l mn m n m

T

T T

n n


Left side

Use of indicators, completely discretised governing equation, initial and boundary conditions

0 00

1 10 ( , ) 0 ( , )

1 1 1 1

2 10 ( , )

1 1

0

0 0

0 0

1

1

1

1

1

l l l l

l l

pl l

l

N N N N

l l ln l nm k l m l ln l nm k l mn m n m

N N

l l l ln l nm k l mn m

l l

D D Dl l l

N N Nl l l

l

cT

t

k T

k T

S

T T

T T

1( , ) 0 0

1 1

1 ; , 1,2,...,l lN N

R R Rl ln l nm k l m l l l

n m

T T T T k l N


Right side


1

A ; 1,2,...,N

li i li

T b l N

Global sparse matrix

0,i iT T t t p

Solution

mechanical model 0t 0t t

INITIAL NODES

DEFORMED NODES

RENODED NODES

GENERATION OF NODAL POINTS


Nodal points are generated through the following procedures:

Transfinite Interpolation

Elliptic Grid Generation



TRANSFINITE INTERPOLATION

Through this technique we can generate initial grid which is confirming to the geometry we encounter in different stages of plate and shape rolling.We suppose that there exists a transformation

which maps the unit square, in the computational domain onto the interior of the region ABCD in the physical domain such that the edges

map to the boundaries AB, CD and the edges are mapped to the boundaries AC, BD.

The transformation is defined as

Where represents the values at the bottom, top, left and right edges respectively


( , ) [ ( , ), ( , )]tx y r

0 1, 0 1

0, 1 0,1

0 0 1l r b t b t b t, r ( ) = (1- )r ( ) + r ( ) + (1- )r ( ) + r ( ) - (1 - )(1 - )r ( ) - (1 - ) r ( ) - (1 - ) r ( ) - r

, , ,b t l rr r r r



An example of transformation from computational domain to physical domain.

( )tr

( )br

( )lr ( )rr


ELLIPTIC GRID GENERATION

The mapping procedure defined above form the physical domain to the computational domain is described by are continuously differentiable maps of all order.

The grid generated through transfinite interpolation can be made more conformal to the geometry by using the following elliptic grid generators

where is the Jacobean of the transformation.ICCES 2010

Las Vegas, March 28 - April 1, 2010

2 2 2

22 12 112 2

2 2 2

22 12 112 2

2 0

2 0

x x xg g g

y y yg g g

2 2

22 122 2

2 2

11 2

1 1, ,

1

x x x x y yg g

J J

x xg

J

J

( , ), ( , )x y x y



Transfinite Interpolation Eliptic Grid Generation


Growing of a domain – Moving and inserting of nodesGrowing of a domain – Moving and inserting of nodes

Moving of boundary nodes

0 t v




0 t v




0 t v




0 t v




0 t v



Inserting of nodes

0 t v



Node generation for deformation of steel during thin strip rolling


CONCLUSIONS


Local RBF Collocation Method is proposed to be Applied in ThermomechanicalProcessing (Hot Rolling)

Basic physical concept of hot shape rolling has been developed

The solution procedure for the thermal field has been defined in detail

The manipulations of the nodes have been defined in detail

Ongoining research

Numerical implementation of the thermal and mechanical models

ACKNOWLEDGEMENT

Siderimpes Rolling Mill Factory, Gorizia, Italy

Research Programme Modelling of Materials and Processes, Slovenian Grant Agency, Slovenia

2010 - 2013

GOVERNING EQUATIONS

• Mechanical Model


321 1 4exp mm

i i ik m Tm

2

3

0

1

1 1 4 0 0 4exp exp

mm

V

W k m T m m dVt

32

2

1 1 4 00 4

expexp( )

3

mm

mS

k m T mmW m dS

t

v

yx

V

vvW dV

x y

i i V

V S V

J W W W dV dS dV τv

is the friction factor ranging between 1 and 0. is the Young’s modulus and is the Poisson’s ratio.

E v

3p

S

mW dS v

m

0i i

GOVERNING EQUATIONS

• Boundary conditions for the mechanical model


I. Constrained boundary conditions when the material expansion is fixed with the geometry;

When and satisfies the boundary shape equation,

II. Unconstrained boundary conditions when the material is completely free to expand

0

xi

xx

e 0

yi

yy

e

ix iy

0x y

x z yV V V

x z y

SOLUTION OF MECHANICAL MODEL


1i i v v v

Newton-Raphson method:

where is the acceleration coefficient usually taken between 0.1 and 1.

2

2 3

11

1 1 4 0 0 4

1exp exp

i

mm m

i i i i i

V

W k m T m m dVt

Bv B v Bv B v B v

2 3

2

1 1 4 00 4

expexp( )

3i

m mii i i im

S

k m T mmW m dS

t

Bv

B v Bv B v B v v

6(1 ) V

E tW CBvdV

v

1 1 0

TC where

i i B v

The minimization of the total work equation can be done in terms of taking derivative with respect to the nodal velocities and Lagrange multiplier.

0J

v0

J

and

If we define strain rate in terms of velocity: , where is the matrix correlating the strain rate to the velocity.

ii

i

v

x iBv B

SOLUTION OF MECHANICAL MODEL


0 0

2

0T

v v v v

J J J

v

v v v v

0 0

2

0v v v v

J J J

v

v

2

L

- 0

T

W W J

L L

vv v v v

v

2

V

JL C dV

B

v

Solution matrix;

Taylor series expansion



Initial node distribution Node distribution after the deformation

Rearrangement of node distribution after the deformation

INITIAL SHAPE DEFORMED SHAPE

RE-NODED SHAPE

ONGOING RESEARCH

Simulation procedure of the velocity field and temperature field, internal heat generation and strain field of the flat and shape rolling with using the thermal and mechanical model stated above.


Node generation for deformation of steel during rolling


The necessary input parameters are:

- Roll radius in mm.

- Roll’s revolution speed in rpm.

- Entry velocity of the steel in m/s.

- Entry thickness of the billet in mm.

- Total reduction in %.

- Material constants for defining stresses.

THIN STRIP ROLLING

R

r

xentryv

entryh

r

1 1 2 3 4, , , ,k m m m m

THIN STRIP ROLLING


2 22 2exith h R R x

2 2

2 2

cos

h x x

x RR x

r

2 22 2

entry

R R xr

h

0 bitex x

xentry entry x x xexit exitv h v h v h

Instantaneous reduction is:

Due to conservation of mass;

Definition of thickness:

THIN STRIP ROLLING


2

2

200%entry

bite

h rx R R

2 2

02

- 0

0

entry bite x

entry entryx bite

exit

exit x

y

v x x x

v hv x x

h R R x

v x x

v

2 2 2 2

2 0

2

0 - 0

0

bite x

entry entrybite

exit

x

x x x

v hxx x

R x h R R x

x x

Q

0

0 - 0

bite x

i i bite

x

x x x

x x

x x

THIN STRIP ROLLING

• Boundary conditions of the thermal model for hot rolling of steel


0, 0dT dT

dx dy xx x

4 4q k T h T T T T

, ,2 2

entry exitbite x

h hx x x y y

2q k T h T T

2 20 , 2bite exitx x y h R R x

4 4q k T h T T T T

0 ,2exit

x

hx x y

I. °C and when

II. when

III.

when IV.

when

0dT

q kdn

xx x n

0dT

q kdn

0y

V. when

( is the normal direction).

when

.

VI.

1150T

is the thermal conductivity in . is the Stefan Boltzmann constant which is . is the heat transfer coefficient in . is the emissivity.

kh/W mK

2/W m K 8 2 45.6697 10 /W m K

SHAPE ROLLING


i i V

V S V

J W W W dV dS dV τv

321 1 4( , , ) exp mm

i i iT k m Tm

The power equation to be optimized is:

, ,T

x y xy , ,T

x y xy

Integral equations define first the plastic deformation in terms of work done per unit volume and time, second frictional work done per unit surface area and time, third is the penalty part due to volume consistency. Is the Lagrange multiplier.

Components of stress tensor

Components of strain rate tensor

Vector of boundary traction

,T

x y τ

Vector of velocities

,T

x yv vv

MODELLING ASSUMPTIONS

• Planes Remain Planes

• Rigid Plastic Deformation

• THERMAL MODEL– 2D heat conduction– Temperature field will be

calculated

• MECHANICAL MODEL– Huber - Mises criterion which lets us to assume the effective stress is

equal to the yield stress.

– Levy-Mises criterion

– Velocity and strain fields will be calculated– Stress field will be calculated.


2

3

p

where is the effective strain rate.

GOVERNING EQUATIONS

• Thermal Model


p p

Tc c T k T Q

t

v

2 2

2 2p x y

T T k T k T T Tc v v k Q

x y x x y y x y

0

T

t

when

321 1 4exp( ) mm

i i i ik m T m

32 11 1 4exp( ) mm

i i iQ k m T m

i iQ

represents the internal heat generation rate due to plastic workQ

Documents

ICCES 2010 Las Vegas, March 28 - April 1, 2010 SOLUTION OF SHAPE HOT ROLLING BY THE LOCAL RADIAL BASIS FUNCTION COLLOCATION METHOD Božidar Šarler, Siraj