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
ICCES 2010Las Vegas, March 28 - April 1, 2010
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)
ICCES 2010Las Vegas, March 28 - April 1, 2010
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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?
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
thick plates, thin plates
SHAPE ROLLING
ICCES 2010Las Vegas, March 28 - April 1, 2010
rails, H beams, other complicated profiles
BASIC MODELLING STRATEGIES
ICCES 2010Las Vegas, March 28 - April 1, 2010
Transient Steady
observe the whole billet observe only part of the billet
THIN STRIP ROLLING
ICCES 2010Las Vegas, March 28 - April 1, 2010
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).
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
SOLUTION OF THE THERMAL MODEL
; 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
SOLUTION OF THERMAL MODEL
( , )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
SOLUTION OF THE THERMAL MODEL
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
SOLUTION OF THE THERMAL MODEL
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
SOLUTION OF THE THERMAL MODEL
Right side
SOLUTION OF THERMAL MODEL
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
GENERATION OF NODAL POINTS
Nodal points are generated through the following procedures:
Transfinite Interpolation
Elliptic Grid Generation
ICCES 2010Las Vegas, March 28 - April 1, 2010
GENERATION OF NODAL POINTS
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
( , ) [ ( , ), ( , )]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
GENERATION OF NODAL POINTS
ICCES 2010Las Vegas, March 28 - April 1, 2010
An example of transformation from computational domain to physical domain.
( )tr
( )br
( )lr ( )rr
GENERATION OF NODAL POINTS
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
GENERATION OF NODAL POINTS
ICCES 2010Las Vegas, March 28 - April 1, 2010
Transfinite Interpolation Eliptic Grid Generation
GENERATION OF NODAL POINTS
Growing of a domain – Moving and inserting of nodesGrowing of a domain – Moving and inserting of nodes
Moving of boundary nodes
0 t v
GENERATION OF NODAL POINTS
Growing of a domain – Moving and inserting of nodesGrowing of a domain – Moving and inserting of nodes
Moving of boundary nodes
0 t v
GENERATION OF NODAL POINTS
Growing of a domain – Moving and inserting of nodesGrowing of a domain – Moving and inserting of nodes
Moving of boundary nodes
0 t v
GENERATION OF NODAL POINTS
Growing of a domain – Moving and inserting of nodesGrowing of a domain – Moving and inserting of nodes
Moving of boundary nodes
0 t v
GENERATION OF NODAL POINTS
Growing of a domain – Moving and inserting of nodesGrowing of a domain – Moving and inserting of nodes
Moving of boundary nodes
0 t v
GENERATION OF NODAL POINTS
Growing of a domain – Moving and inserting of nodesGrowing of a domain – Moving and inserting of nodes
Inserting of nodes
0 t v
GENERATION OF NODAL POINTS
ICCES 2010Las Vegas, March 28 - April 1, 2010
Node generation for deformation of steel during thin strip rolling
GENERATION OF NODAL POINTS
CONCLUSIONS
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
GENERATION OF NODAL POINTS
ICCES 2010Las Vegas, March 28 - April 1, 2010
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.
ICCES 2010Las Vegas, March 28 - April 1, 2010
Node generation for deformation of steel during rolling
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
ICCES 2010Las Vegas, March 28 - April 1, 2010
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.
ICCES 2010Las Vegas, March 28 - April 1, 2010
2
3
p
where is the effective strain rate.
GOVERNING EQUATIONS
• Thermal Model
ICCES 2010Las Vegas, March 28 - April 1, 2010
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
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