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SOLUTION OF SHAPE HOT ROLLING BY THE LOCAL RADIAL BASIS FUNCTION COLLOCATION METHOD

SOLUTION OF SHAPE HOT ROLLING BY THE LOCAL RADIAL BASIS FUNCTION COLLOCATION METHOD. Božidar Šarler, Siraj Islam, Umut Hanoglu Laboratory for Multiphase Processes University of Nova Gorica, Slovenia. SCOPE OF PRESENTATION. Introduction and Motivation Thin S trip R olling

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SOLUTION OF SHAPE HOT ROLLING BY THE LOCAL RADIAL BASIS FUNCTION COLLOCATION METHOD

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  1. SOLUTION OF SHAPE HOT ROLLING BY THE LOCAL RADIAL BASIS FUNCTION COLLOCATION METHOD Božidar Šarler, Siraj Islam, Umut Hanoglu Laboratory for Multiphase Processes University of Nova Gorica, Slovenia ICCES 2010 Las Vegas, March 28 - April 1, 2010

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

  3. Overview of Our Recent Publications on Local 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 2010 Las Vegas, March 28 - April 1, 2010 3

  4. Overview of Our Recent Publications on Local Radial Basis Function Collocation Method 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 2010 Las Vegas, March 28 - April 1, 2010 4

  5. Mainstream Research Directions CONTINUOUS CASTING HOT SHAPE ROLLING Moving of the solid-liquid interface Large deformation

  6. TWIN - ROLL CASTING PROCESS THIN STRIP CASTING convection - diffusion with phase change Šarler et al. 2007

  7. TWIN - ROLL CASTING PROCESS

  8. TWIN - ROLL CASTING PROCESS

  9. TWIN - ROLL CASTING PROCESS MACRO - MICRO APPROACH LRBFC METHOD PA METHOD

  10. INFLUENCE OF ROLLING SPEED AND CASTING TEMPERATURE

  11. INFLUENCE OF SETBACK AND STRIP THICKNESS

  12. LABORATORY FOR MULTIPHASE PROCESSES - FORESEEN RESEARCH Through process modeling optimisation process (evolution algorithm) Integrated through process model objective function - Quality - Productivity - Machine occupation process 1 process 2 proc.par. window proc.par. window product properties product properties process properties process properties + process parameters process parameters generalised cost generalised cost NN – sub model NN – sub model physical model, experience, measurements physical model, experience, measurements fizikalni model izkušnje meritve integrated neural network (NN) through process model (TPM)

  13. THIN STRIP HOT ROLLING thick plates, thin plates ICCES 2010 Las Vegas, March 28 - April 1, 2010

  14. SHAPE ROLLING rails, H beams, other complicated profiles ICCES 2010 Las Vegas, March 28 - April 1, 2010

  15. BASIC MODELLING STRATEGIES Transient Steady observe only part of the billet observe the whole billet ICCES 2010 Las Vegas, March 28 - April 1, 2010

  16. THIN STRIP ROLLING material moves slower than the roll material moves faster than the roll Homogenous compression Where the planes remain planes assumption is considered. Non - homogenous compression Might occur during high reductions with relatively small contact lenght. ICCES 2010 Las Vegas, March 28 - April 1, 2010

  17. BASIC LITERATURE REVIEW ICCES 2010 Las Vegas, March 28 - April 1, 2010 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).

  18. 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

  19. GOVERNING EQUATIONS – THERMAL MODEL Fully three dimensional steady convection - diffusion problem this shapes change Two dimensional transient diffusion problem (slice model) slice

  20. THERMO-MECHANICAL MODEL SCHEMATICS initial shape initial shape initial temperature initial velocity In rolling direction initial nodes solve temperature of the slice at the new position calculate deformation of the slice at the new position final velocity in rolling direction final shape final nodes renoding

  21. GOVERNING EQUATIONS - THERMAL MODEL Heat transfer in the direction of the billet movement is neglected

  22. GOVERNING EQUATIONS - THERMAL MODEL Governing Equation for a 2D perpendicular slice Initial Condition Boundary Conditions

  23. SOLUTION OF THE THERMAL MODEL ICCES 2010 Las Vegas, March 28 - April 1, 2010

  24. SOLUTION OF THERMAL MODEL Time discretisation of governing equation Time discretisation of boundary conditions

  25. SOLUTION OF THE THERMAL MODEL Global nodes Subdomains Subdomain nodes Relation between global index k and local indeces l and n

  26. SOLUTION OF THERMAL MODEL Collocation of temperature field on subdomain l Calculation of expansion coefficients l

  27. SOLUTION OF THE THERMAL MODEL

  28. SOLUTION OF THE THERMAL MODEL Use of indicators, completely discretised governing equation, initial and boundary conditions Left side

  29. SOLUTION OF THE THERMAL MODEL Right side

  30. SOLUTION OF THERMAL MODEL Global sparse matrix Solution mechanical model

  31. GENERATION OF NODAL POINTS INITIAL NODES DEFORMED NODES RENODED NODES

  32. GENERATION OF NODAL POINTS ICCES 2010 Las Vegas, March 28 - April 1, 2010 Nodal points are generated through the following procedures: Transfinite Interpolation Elliptic Grid Generation

  33. GENERATION OF NODAL POINTS ICCES 2010 Las Vegas, March 28 - April 1, 2010 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

  34. GENERATION OF NODAL POINTS An example of transformation from computational domain to physical domain. ICCES 2010 Las Vegas, March 28 - April 1, 2010

  35. GENERATION OF NODAL POINTS ICCES 2010 Las Vegas, March 28 - April 1, 2010 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.

  36. GENERATION OF NODAL POINTS Transfinite Interpolation Eliptic Grid Generation ICCES 2010 Las Vegas, March 28 - April 1, 2010

  37. GENERATION OF NODAL POINTS

  38. GENERATION OF NODAL POINTS Growing of a domain – Moving and inserting of nodes Moving of boundary nodes

  39. GENERATION OF NODAL POINTS Growing of a domain – Moving and inserting of nodes Moving of boundary nodes

  40. GENERATION OF NODAL POINTS Growing of a domain – Moving and inserting of nodes Moving of boundary nodes

  41. GENERATION OF NODAL POINTS Growing of a domain – Moving and inserting of nodes Moving of boundary nodes

  42. GENERATION OF NODAL POINTS Growing of a domain – Moving and inserting of nodes Moving of boundary nodes

  43. GENERATION OF NODAL POINTS Growing of a domain – Moving and inserting of nodes Inserting of nodes

  44. GENERATION OF NODAL POINTS Node generation for deformation of steel during thin strip rolling ICCES 2010 Las Vegas, March 28 - April 1, 2010

  45. CONCLUSIONS Local RBF Collocation Method is proposed to be Applied in Thermomechanical Processing (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 ICCES 2010 Las Vegas, March 28 - April 1, 2010

  46. ACKNOWLEDGEMENT Siderimpes Rolling Mill Factory, Gorizia, Italy Research Programme Modelling of Materials and Processes, Slovenian Grant Agency, Slovenia 2010 - 2013

  47. GOVERNING EQUATIONS • Mechanical Model is the friction factor ranging between 1 and 0. is the Young’s modulus and is the Poisson’s ratio. ICCES 2010 Las Vegas, March 28 - April 1, 2010

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

  49. SOLUTION OF MECHANICAL MODEL If we define strain rate in terms of velocity: , where is the matrix correlating the strain rate to the velocity. where The minimization of the total work equation can be done in terms of taking derivative with respect to the nodal velocities and Lagrange multiplier. and Newton-Raphson method: where is the acceleration coefficient usually taken between 0.1 and 1. ICCES 2010 Las Vegas, March 28 - April 1, 2010

  50. SOLUTION OF MECHANICAL MODEL Taylor series expansion Solution matrix; ICCES 2010 Las Vegas, March 28 - April 1, 2010

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