Macrosegregation presents a considerable defect in the continuous casting of billets and can crit... more Macrosegregation presents a considerable defect in the continuous casting of billets and can critically affect the final properties of the product. The numerical modelling can help to predict and better understand the segregation and flow patterns inside the mould. The process is modelled with a physical model described by a set of conservation equations describing the t heat transfer, turbulence, fluid flow, solidification and segregation. A two-equation low-Re kepsilon model and Abe-Kondoh-Nagano closures are used to close governing equations in this incompressible fluid flow example. The Boussinesq approximation is applied to account for the thermo-solutal buoyancy effects, and the Darcy approximation is applied for the description of the flow through the porous mushy zone. On a microscale, a lever rule solidification model is used to couple liquid fraction, temperature and concentration. The three-dimensional model is solved with the method based on local collocation with multiquadric radial basis functions on seven-nodded subdomains. The aim of this contribution is to explore the three-dimensional macrosegregation patterns of 0.51 wt% carbon steel in the solidified shell of the steel in the mould.
A two-dimensional two-scale slice model has been developed to predict the microstructure evolutio... more A two-dimensional two-scale slice model has been developed to predict the microstructure evolution in the solidifying strand with an arbitrary cross section geometry during continuous casting of steel. The enthalpy equation is solved at the macro level by using meshless local radial basis function collocation method (LRBFCM) based on multiquadrics for spatial discretization and explicit Euler scheme for temporal discretization. The temperature and the solid fraction in computational nodes are calculated by using a continuum model formulation while the lever rule is used as the supplementary microsegregation relation. The temperature field is interpolated to the micro level by using LRBFCM. At the micro level, the normal distribution and Kurz-Giovanola-Trivedi model are proposed to determine temperature dependent nucleation rate and grain growth velocity, respectively. Meshless point-automata algorithm is applied to implement nucleation and grain growth equations. Several examples of computations of the strand with different cross-sections are shown.
WIT Transactions on the Built Environment, Sep 1, 2005
This paper uses a recently developed upgrade of the classical meshless Kansa method for solution ... more This paper uses a recently developed upgrade of the classical meshless Kansa method for solution of the transient heat transport in direct-chill casting of aluminium alloys. The problem is characterised by a moving mushy domain between the solid and the liquid phase and a moving starting bottom block that emerges from the mould during the process. The solution of the thermal field is based on the mixture continuum formulation. The movement of the bottom block is approximated by the movement of the boundary condition through the computational domain. The moving domain and boundary of interest are divided into overlapping influence areas. On each of them, the fields are represented by the multiquadrics radial basis function collocation on a related subset of nodes. Time-stepping is performed in an explicit way. The governing equation is solved in its strong form, i.e. no integrations are performed. The polygonisation is not present and the method is practically independent on the problem dimension. Realistic boundary conditions and temperature variation of material properties are included. Two-dimensional test case solution is shown at different times, verified by comparison with the finite volume method results.
WIT transactions on modelling and simulation, Jun 11, 2007
Industrial twin-roll casting of aluminium alloys requires optimisation of connections between pro... more Industrial twin-roll casting of aluminium alloys requires optimisation of connections between process parameters and product properties. To a large extent, the cast strip properties depend on the spatial and temporal characteristics of heat transfer from the cast strip, which needs estimation of the influence of process parameters on the temperature field and temperature gradients in a strip for various alloys, which are cast on an industrial machine. For this purpose, the recently developed meshless Local Radial Basis Function Collocation Method (LRBFCM) was used in the simulation of heat transfer from the strip. The solution of the nonlinear convection-diffusion equation is based on a mixture continuum formulation of the energy transport in solid-liquid phase change systems. Simulation results were compared with experimental values, which were obtained during the industrial production process. They were measured in situ by the specially designed apparatus with fast response thermocouples, which can measure the time-dependent temperature of the cast strip in several locations on the moving strip surface simultaneously. The time-dependent measurement was coupled with a digital camera recording to gather the spatial domain measurement of temperature. The differences between the numerical simulation and experimental results were smaller than 6 K. An explanation for the discrepancy is given.
There is a continuously developing need for benchmarking of solidification simulations-from the t... more There is a continuously developing need for benchmarking of solidification simulations-from the theoretical as well as from the applied points of view. The history of related benchmarking shows differences of the results between different numerical methods, and differences in comparison with the experiments when solving even quite simple solidification situations. The present benchmark test proposes a three dimensional version of the recently developed two dimensional solidification benchmark of the continuous casting with turbulent fluid flow and solidification with macrosegregation. The macroscopic transport equations for mass, momentum, energy, species, turbulent kinetic energy and dissipation rate are considered. Turbulent effects are incorporated through the solution of a low-Re turbulence model. The solidification system is treated by the mixture-continuum model, where the mushy zone is modeled as a Darcy porous media with Kozeny-Karman permeability relation and columnar solid phase moving with the casting velocity. The Fe-C binary phase diagram and the lever rule microsegregation model are used to obtain the carbon segregation in the billet. The transport equations are solved by our newly developed and efficient meshless numerical technique, based on local collocation with radial basis functions and fractional step velocitypressure coupling. The presented results represent first simulations of a three-dimensional solidification problem by a meshless method.
The current work outlines application of a framework based on artificial neural networks and an i... more The current work outlines application of a framework based on artificial neural networks and an integrated optimization module to adjustment of process parameters in steel production. The framework was originally developed for adjustment of parameters of material production processes in order to obtain the desired outcomes, and was primarily intended for use in the production of carbon nanomaterials in arc discharge reactors. Further development lead to more generalized procedures, applicable to a broad spectra of material production and processing. An example of optimizing the process parameters in continuous casting of steel on basis of expert knowledge and by the developed system is presented. Further steps are made towards modeling of the whole process chain in the steel plant, rather than just the casting process. Such models are in the development stage, and some preliminary results are shown where the model is used for performing some parametric studies.
The electric arc furnace operation at the Štore Steel company, one of the largest flat spring ste... more The electric arc furnace operation at the Štore Steel company, one of the largest flat spring steel producers in Europe, consists of charging, melting, refining the chemical composition, adjusting the temperature, and tapping. Knowledge of the consumed energy within the individual electric arc operation steps is essential. The electric energy consumption during melting and refining was analyzed including the maintenance and technological delays. In modeling the electric energy consumption, 25 parameters were considered during melting (e.g., coke, dolomite, quantity), refining and tapping (e.g., injected oxygen, carbon, and limestone quantity) that were selected from 3248 consecutively produced batches in 2018. Two approaches were employed for the data analysis: linear regression and genetic programming model. The linear regression model was used in the first randomly generated generations of each of the 100 independent developed civilizations. More accurate models were subsequently obtained during the simulated evolution. The average relative deviation of the linear regression and the genetic programming model predictions from the experimental data were 3.60% and 3.31%, respectively. Both models were subsequently validated by using data from 278 batches produced in 2019, where the maintenance and the technological delays were below 20 minutes per batch. It was possible, based on the linear regression and the genetically developed model, to calculate that the average electric energy consumption could be reduced by up to 1.04% and 1.16%, respectively, in the case of maintenance and other technological delays.
The formation of macrosegregation pattern in steel is explored for the continuous casting process... more The formation of macrosegregation pattern in steel is explored for the continuous casting process. The related solidification problem is described by incompressible turbulent fluid flow, governed by mass, momentum, energy, and species conservation equations. The solid-liquid system is designed in a mixture continuum model formulation, and the mushy region is approximated with the Darcy model. The lever rule describes the formation of microsegregation of carbon. Turbulence is modelled with a large-eddy simulation approach implemented with the Smagorinsky model and van Driest correction. The synthetic isotropic fluctuations of the inlet velocity are generated by the von Karman Pao spectrum. The solution of partial differential equations is implemented locally by using collocation with radial basis functions for spatial discretization and the explicit Euler method for time discretization. The two-dimensional large eddy simulation model solutions are compared to the previously calculated examples obtained with the low-Re k − ε turbulence model.
This paper represents an overview of the elements of the user-friendly simulation system, develop... more This paper represents an overview of the elements of the user-friendly simulation system, developed for computational analysis and optimization of the quality and productivity of the electromagnetically direct-chill cast semi-products from aluminium alloys. The system also allows the computational estimation of the design changes of the casting equipment. To achieve this goal, the electromagnetic and the thermofluid process parameters are coupled to the evolution of Lorentz force, temperature, velocity, concentration, strain and stress fields as well as microstructure evolution. This forms a multi-physics and multi-scale problem of great complexity, which has not been demonstrated before. The macroscopic fluid mechanics, solid mechanics, and electromagnetic solution framework is based on local strong-form meshless formulation, involving the radial basis functions and monomials as trial functions, and local collocation or weighted least squares approximation. It is coupled to the micro-scale by incorporating the point automata solution concept. The entire macro-micro solution concept does not require meshing and space integration. The solution procedure can be easily and efficiently automatically adapted in node redistribution and/or refinement sense, which is of utmost importance when coping with fields exhibiting sharp gradients, which occur in the phase-change problems. The simulation system is coded from scratch in modern Fortran. The elements of the experimental validation of the system and the demonstration of its use for round billet casting in IMPOL Aluminium Industry are shown.
This paper tackles an improved Localized Radial Basis Functions Collocation Method (LRBFCM) for t... more This paper tackles an improved Localized Radial Basis Functions Collocation Method (LRBFCM) for the numerical solution of hyperbolic partial differential equations (PDEs). The LRBFCM is based on multiquadric (MQ) Radial Basis Functions (RBFs) and belongs to a class of truly meshless methods which do not need any underlying mesh. This method can be implemented on a set of uniform or random nodes, without any a priori knowledge of node to node connectivity. We have chosen uniform nodal arrangement due their suitability and better accuracy. Five nodded domains of influence are used in the local support for the calculation of the spatial partial derivatives. This approach results in a small interpolation matrix for each data center and hence the time integration has comparatively low computational cost than the related global method. Different sizes of domain of influence i.e. m=5,13 are considered. Shape parameter sensitivity of MQ is handled through scaling technique. The time derivative is approximated by first order forward difference formula. An adaptive upwind technique is used for stabilization of the method. Capabilities of the LRBFCM are tested by applying it to one- and two-dimensional benchmark problems with discontinuities, shock pattern and periodic initial conditions. Performance of the LRBFCM is compared with analytical solution, other numerical methods and the results reported earlier in the literature. We have also made comparison with implicit first order time discretization and first order upwind spatial discretization (FVM1) and implicit second order time discretization and first order upwind spatial discretization (FVM2) as well. Accuracy of the method is assessed as a function of time and space. Numerical convergence is also shown for both one- and two-dimensional test problems. It has been observed that the proposed method is more efficient in terms of less memory requirement and less computational efforts due to one time inversion of 5x5 (size of local domain of influence) coefficient matrix. The results obtained through LBRFCM are stable and comparable with the existing methods for a variety of problems with practical applications.
There is a continuously developing need for benchmarking of solidification simulations-from the t... more There is a continuously developing need for benchmarking of solidification simulations-from the theoretical as well as from the applied points of view. The benchmarking is usually done in two parts. The verification part confirms the proper numerical solution (Are we solving the equations correctly?) and the validation part (Are we solving the right equations?) confirms the proper response of the simulations regarding the experimental evidence. The history of related benchmarking shows differences of the results between different numerical methods, and differences in comparison with the experiments when solving even quite simple solidification situations. The present benchmark test proposes the verification benchmark for continuous casting of steel. Since the simulations of the temperatures, velocities, pressures and concentrations in the continuous casting represent a multiscale and multiphysics problem of high complexity (far away from a closed form solution), the verification can be done only by comparing the results of different numerical methods for solving the same governing equations. This has been done in the context of continuous casting by breaking the considered two dimensional problem into several sub problems by increasing complexity of geometry (straight in vertical direction, curved), boundary conditions (linear, non-linear), material properties (only Fe and Fe-C), microscopic considerations (Lever rule and Scheil rule). The governing equations complexity first involves only convective-diffusive heat transport with a predetermined velocity field, and assumed laminar and turbulent velocity calculations afterwards, modelled by a low Reynolds number turbulence model. Further complications involve the presence of the electromagnetic forces. The paper represents guidelines for the presentation of the numerical method, discretisation and results, as well as some of the results, obtained by the commercial finite volume based code and our in-house meshless method based laboratory code.
Advances in Applied Mathematics and Mechanics, Jun 1, 2011
An application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) in solutio... more An application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) in solution of incompressible turbulent combined forced and natural convection is for the first time explored in the present paper. The turbulent flow equations are described by the low-Re number к – ε model with Launder and Sharma and Abe et al. closure coefficients. The involved temperature, velocity, pressure, turbulent kinetic energy and dissipation fields are represented on overlapping 5-noded sub-domains through the collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second order partial derivatives of the fields are calculated from the respective derivatives of the RBF’s. The involved equations are solved through the explicit time stepping. The pressure-velocity coupling is based on Chorin’s fractional step method. The adaptive upwinding technique, proposed by Lin and Atluri , is used because of the convection dominated situation. The solution procedure is represented for a 2D upward channel flow with differentially heated walls. The results have been assessed by achieving a reasonable agreement with the direct numerical simulation of Kasagi and Nishimura for Reynolds number 4494, based on the channel width, and Grash of number 9.6×105. The advantages of the represented mesh-free approach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforward applicability in non-uniform node arrangements.
This paper represents a continuation of numerical results regarding the recently proposed industr... more This paper represents a continuation of numerical results regarding the recently proposed industrial benchmark test [1], obtained by a meshless method. A part of the benchmark test, involving turbulent fluid flow with solidification in two dimensions, was elaborated in [2]. A preliminary macrosegregation upgrade was presented in [3], and in [4], a first three dimensional test was performed. Previous tests were bound to calculations in mold and sub-mold regions only. In the present paper, reference calculations in two dimensions are presented for the entire strand. The physical model is established on a set of macroscopic equations for mass, energy, momentum, species, turbulent kinetic energy, and dissipation rate. The mixture continuum model is used to treat the solidification system. The mushy zone is modeled as a Darcy porous media with Kozeny-Karman permeability relation, where the morphology of the porous media is modeled by a constant value. The incompressible turbulent flow of the molten steel is described by the Low-Reynolds-Number k-ε turbulence model, closed by the Abe-Kondoh-Nagano closure coefficients and damping functions. Lever microsegregation model is used. The numerical method is established on explicit time-stepping, collocation with scaled multiquadrics radial basis functions with adaptive selection of its shape on non-uniform five-nodded influence domains. The velocity-pressure coupling of the incompressible flow is resolved by the explicit Chorin's fractional step method. The advantages of the method are its simplicity and efficiency, since no polygonisation is involved, easy adaptation of the nodal points in areas with high gradients, almost the same formulation in two and three dimensions, high accuracy and low numerical diffusion.
A large-eddy simulation (LES) based meshless model is developed for the three-dimensional (3D) pr... more A large-eddy simulation (LES) based meshless model is developed for the three-dimensional (3D) problem of continuous casting (CC) of steel billet. The local collocation meshless method based on radial basis functions (RBF) is applied in 3D. The method applies scaled multiquadric (MQ) RBF with a shape parameter on seven nodded local sub-domains. The incompressible turbulent fluid flow is described using mass, energy, and momentum conservation equations and the LES turbulence model. The solidification system is solved with the mixture continuum model. The Boussinesq approximation for buoyancy and the Darcy approximation for porous media are used. Chorin’s fractional step method is used to couple velocity and pressure. The microscopic model is closed with the lever rule model. The LES model is compared to the two-equation Low Re k−ε turbulence Reynolds Averaged Navier–Stokes (RANS) model in terms of temperature, velocity and computational times. The LES model resolves transient charact...
A two-dimensional two-scale slice model has been developed to predict the microstructure evolutio... more A two-dimensional two-scale slice model has been developed to predict the microstructure evolution in the solidifying strand with an arbitrary cross section geometry during continuous casting of steel. The enthalpy equation is solved at the macro level by using meshless local radial basis function collocation method (LRBFCM) based on multiquadrics for spatial discretization and explicit Euler scheme for temporal discretization. The temperature and the solid fraction in computational nodes are calculated by using a continuum model formulation while the lever rule is used as the supplementary microsegregation relation. The temperature field is interpolated to the micro level by using LRBFCM. At the micro level, the normal distribution and Kurz-Giovanola-Trivedi model are proposed to determine temperature dependent nucleation rate and grain growth velocity, respectively. Meshless point-automata algorithm is applied to implement nucleation and grain growth equations. Several examples of computations of the strand with different cross-sections are shown.
The purpose of this paper is a multiphysics simulation of 3D temperature and velocity fields in c... more The purpose of this paper is a multiphysics simulation of 3D temperature and velocity fields in continuous casting of steel under the influence of electromagnetic stirring by a combined meshless-finite element method approach. The transport phenomena are calculated by a meshless local radial basis function collocation technique and the magnetic force by the finite element method solver Elmer. The electromagnetic stirring increases the mixing in the molten steel. The thermal gradient is sharper and solidification is faster along the strand. The results are similar to other publications in the field. The local radial basis function collocation method is for the first time applied to 3D continuous casting problem with mold electromagnetic stirring.
Simulation and control of macrosegregation, deformation and grain size under electromagnetic (EM)... more Simulation and control of macrosegregation, deformation and grain size under electromagnetic (EM) processing conditions is important in industrial solidification systems, since it influences the quality of the casts and consequently the whole downstream processing path. Respectively, a multiphysics and multiscale model is developed for solution of Lorentz force, temperature, velocity, concentration, deformation and grain structure of the casts. The mixture equations with lever rule, linearized phase diagram, and stationary thermoelastic solid phase are assumed, together with EM induction equation for the field imposed by the low frequency EM field or Ohm’s law and charge conservation equation for stationary EM field. Turbulent effects are incorporated through the solution of a low-Re turbulence model. The solidification system is treated by the mixture-continuum model, where the mushy zone is modeled as a Darcy porous media with Kozeny-Karman permeability relation and columnar solid...
Cmes-computer Modeling in Engineering & Sciences, 2013
The purpose of the present paper is to extend and explore the application of a novel meshless Loc... more The purpose of the present paper is to extend and explore the application of a novel meshless Local Radial Basis Function Collocation Method (LRBFCM) in solution of a steady, laminar, natural convection flow, influenced by magnetic field. The problem is defined by coupled mass, momentum, energy and induction equations that are solved in two dimensions by using local collocation with multiquadrics radial basis functions on an overlapping five nodded subdomains and explicit time-stepping. The fractional step method is used to couple the pressure and velocity fields. The considered problem is calculated in a square cavity with two insulated horizontal and two differentially heated vertical walls with magnetic field applied in the horizontal direction. Numerical predictions are calculated for different Grashof numbers, ranging from 104 to 106, and Hartman numbers, ranging from 0 to 100, at Prandtl numbers 0.71 and 0.14. The results of the method are compared to predictions, obtained by ...
Continuous casting (CC) of steel is a process that nowadays produces more than 95 % [1] of crude ... more Continuous casting (CC) of steel is a process that nowadays produces more than 95 % [1] of crude steel. To further advance this process, electromagnetic (EM) field, which affects the fluid flow as well as the temperature and segregation is added to the CC process. In general, there are two types of electromagnetic devices applicable to the CC process; one are the electromagnetic breakers (EMBR) which implement direct current, and the other are the electromagnetic stirrers (EMS) which implement the alternating current. Which of the process is used depends on what are the desired effects. Both of the processes are modelled by implementing the Lorentz force into the momentum equation and if necessary the Joule heating term into the energy equation. However, the way how these two terms are modelled depend on the type of the implemented device. In case of EMBRs the assumption of low Rem is made and consequently, the current density is calculated by solving the Poisson’s equation for elec...
Macrosegregation presents a considerable defect in the continuous casting of billets and can crit... more Macrosegregation presents a considerable defect in the continuous casting of billets and can critically affect the final properties of the product. The numerical modelling can help to predict and better understand the segregation and flow patterns inside the mould. The process is modelled with a physical model described by a set of conservation equations describing the t heat transfer, turbulence, fluid flow, solidification and segregation. A two-equation low-Re kepsilon model and Abe-Kondoh-Nagano closures are used to close governing equations in this incompressible fluid flow example. The Boussinesq approximation is applied to account for the thermo-solutal buoyancy effects, and the Darcy approximation is applied for the description of the flow through the porous mushy zone. On a microscale, a lever rule solidification model is used to couple liquid fraction, temperature and concentration. The three-dimensional model is solved with the method based on local collocation with multiquadric radial basis functions on seven-nodded subdomains. The aim of this contribution is to explore the three-dimensional macrosegregation patterns of 0.51 wt% carbon steel in the solidified shell of the steel in the mould.
A two-dimensional two-scale slice model has been developed to predict the microstructure evolutio... more A two-dimensional two-scale slice model has been developed to predict the microstructure evolution in the solidifying strand with an arbitrary cross section geometry during continuous casting of steel. The enthalpy equation is solved at the macro level by using meshless local radial basis function collocation method (LRBFCM) based on multiquadrics for spatial discretization and explicit Euler scheme for temporal discretization. The temperature and the solid fraction in computational nodes are calculated by using a continuum model formulation while the lever rule is used as the supplementary microsegregation relation. The temperature field is interpolated to the micro level by using LRBFCM. At the micro level, the normal distribution and Kurz-Giovanola-Trivedi model are proposed to determine temperature dependent nucleation rate and grain growth velocity, respectively. Meshless point-automata algorithm is applied to implement nucleation and grain growth equations. Several examples of computations of the strand with different cross-sections are shown.
WIT Transactions on the Built Environment, Sep 1, 2005
This paper uses a recently developed upgrade of the classical meshless Kansa method for solution ... more This paper uses a recently developed upgrade of the classical meshless Kansa method for solution of the transient heat transport in direct-chill casting of aluminium alloys. The problem is characterised by a moving mushy domain between the solid and the liquid phase and a moving starting bottom block that emerges from the mould during the process. The solution of the thermal field is based on the mixture continuum formulation. The movement of the bottom block is approximated by the movement of the boundary condition through the computational domain. The moving domain and boundary of interest are divided into overlapping influence areas. On each of them, the fields are represented by the multiquadrics radial basis function collocation on a related subset of nodes. Time-stepping is performed in an explicit way. The governing equation is solved in its strong form, i.e. no integrations are performed. The polygonisation is not present and the method is practically independent on the problem dimension. Realistic boundary conditions and temperature variation of material properties are included. Two-dimensional test case solution is shown at different times, verified by comparison with the finite volume method results.
WIT transactions on modelling and simulation, Jun 11, 2007
Industrial twin-roll casting of aluminium alloys requires optimisation of connections between pro... more Industrial twin-roll casting of aluminium alloys requires optimisation of connections between process parameters and product properties. To a large extent, the cast strip properties depend on the spatial and temporal characteristics of heat transfer from the cast strip, which needs estimation of the influence of process parameters on the temperature field and temperature gradients in a strip for various alloys, which are cast on an industrial machine. For this purpose, the recently developed meshless Local Radial Basis Function Collocation Method (LRBFCM) was used in the simulation of heat transfer from the strip. The solution of the nonlinear convection-diffusion equation is based on a mixture continuum formulation of the energy transport in solid-liquid phase change systems. Simulation results were compared with experimental values, which were obtained during the industrial production process. They were measured in situ by the specially designed apparatus with fast response thermocouples, which can measure the time-dependent temperature of the cast strip in several locations on the moving strip surface simultaneously. The time-dependent measurement was coupled with a digital camera recording to gather the spatial domain measurement of temperature. The differences between the numerical simulation and experimental results were smaller than 6 K. An explanation for the discrepancy is given.
There is a continuously developing need for benchmarking of solidification simulations-from the t... more There is a continuously developing need for benchmarking of solidification simulations-from the theoretical as well as from the applied points of view. The history of related benchmarking shows differences of the results between different numerical methods, and differences in comparison with the experiments when solving even quite simple solidification situations. The present benchmark test proposes a three dimensional version of the recently developed two dimensional solidification benchmark of the continuous casting with turbulent fluid flow and solidification with macrosegregation. The macroscopic transport equations for mass, momentum, energy, species, turbulent kinetic energy and dissipation rate are considered. Turbulent effects are incorporated through the solution of a low-Re turbulence model. The solidification system is treated by the mixture-continuum model, where the mushy zone is modeled as a Darcy porous media with Kozeny-Karman permeability relation and columnar solid phase moving with the casting velocity. The Fe-C binary phase diagram and the lever rule microsegregation model are used to obtain the carbon segregation in the billet. The transport equations are solved by our newly developed and efficient meshless numerical technique, based on local collocation with radial basis functions and fractional step velocitypressure coupling. The presented results represent first simulations of a three-dimensional solidification problem by a meshless method.
The current work outlines application of a framework based on artificial neural networks and an i... more The current work outlines application of a framework based on artificial neural networks and an integrated optimization module to adjustment of process parameters in steel production. The framework was originally developed for adjustment of parameters of material production processes in order to obtain the desired outcomes, and was primarily intended for use in the production of carbon nanomaterials in arc discharge reactors. Further development lead to more generalized procedures, applicable to a broad spectra of material production and processing. An example of optimizing the process parameters in continuous casting of steel on basis of expert knowledge and by the developed system is presented. Further steps are made towards modeling of the whole process chain in the steel plant, rather than just the casting process. Such models are in the development stage, and some preliminary results are shown where the model is used for performing some parametric studies.
The electric arc furnace operation at the Štore Steel company, one of the largest flat spring ste... more The electric arc furnace operation at the Štore Steel company, one of the largest flat spring steel producers in Europe, consists of charging, melting, refining the chemical composition, adjusting the temperature, and tapping. Knowledge of the consumed energy within the individual electric arc operation steps is essential. The electric energy consumption during melting and refining was analyzed including the maintenance and technological delays. In modeling the electric energy consumption, 25 parameters were considered during melting (e.g., coke, dolomite, quantity), refining and tapping (e.g., injected oxygen, carbon, and limestone quantity) that were selected from 3248 consecutively produced batches in 2018. Two approaches were employed for the data analysis: linear regression and genetic programming model. The linear regression model was used in the first randomly generated generations of each of the 100 independent developed civilizations. More accurate models were subsequently obtained during the simulated evolution. The average relative deviation of the linear regression and the genetic programming model predictions from the experimental data were 3.60% and 3.31%, respectively. Both models were subsequently validated by using data from 278 batches produced in 2019, where the maintenance and the technological delays were below 20 minutes per batch. It was possible, based on the linear regression and the genetically developed model, to calculate that the average electric energy consumption could be reduced by up to 1.04% and 1.16%, respectively, in the case of maintenance and other technological delays.
The formation of macrosegregation pattern in steel is explored for the continuous casting process... more The formation of macrosegregation pattern in steel is explored for the continuous casting process. The related solidification problem is described by incompressible turbulent fluid flow, governed by mass, momentum, energy, and species conservation equations. The solid-liquid system is designed in a mixture continuum model formulation, and the mushy region is approximated with the Darcy model. The lever rule describes the formation of microsegregation of carbon. Turbulence is modelled with a large-eddy simulation approach implemented with the Smagorinsky model and van Driest correction. The synthetic isotropic fluctuations of the inlet velocity are generated by the von Karman Pao spectrum. The solution of partial differential equations is implemented locally by using collocation with radial basis functions for spatial discretization and the explicit Euler method for time discretization. The two-dimensional large eddy simulation model solutions are compared to the previously calculated examples obtained with the low-Re k − ε turbulence model.
This paper represents an overview of the elements of the user-friendly simulation system, develop... more This paper represents an overview of the elements of the user-friendly simulation system, developed for computational analysis and optimization of the quality and productivity of the electromagnetically direct-chill cast semi-products from aluminium alloys. The system also allows the computational estimation of the design changes of the casting equipment. To achieve this goal, the electromagnetic and the thermofluid process parameters are coupled to the evolution of Lorentz force, temperature, velocity, concentration, strain and stress fields as well as microstructure evolution. This forms a multi-physics and multi-scale problem of great complexity, which has not been demonstrated before. The macroscopic fluid mechanics, solid mechanics, and electromagnetic solution framework is based on local strong-form meshless formulation, involving the radial basis functions and monomials as trial functions, and local collocation or weighted least squares approximation. It is coupled to the micro-scale by incorporating the point automata solution concept. The entire macro-micro solution concept does not require meshing and space integration. The solution procedure can be easily and efficiently automatically adapted in node redistribution and/or refinement sense, which is of utmost importance when coping with fields exhibiting sharp gradients, which occur in the phase-change problems. The simulation system is coded from scratch in modern Fortran. The elements of the experimental validation of the system and the demonstration of its use for round billet casting in IMPOL Aluminium Industry are shown.
This paper tackles an improved Localized Radial Basis Functions Collocation Method (LRBFCM) for t... more This paper tackles an improved Localized Radial Basis Functions Collocation Method (LRBFCM) for the numerical solution of hyperbolic partial differential equations (PDEs). The LRBFCM is based on multiquadric (MQ) Radial Basis Functions (RBFs) and belongs to a class of truly meshless methods which do not need any underlying mesh. This method can be implemented on a set of uniform or random nodes, without any a priori knowledge of node to node connectivity. We have chosen uniform nodal arrangement due their suitability and better accuracy. Five nodded domains of influence are used in the local support for the calculation of the spatial partial derivatives. This approach results in a small interpolation matrix for each data center and hence the time integration has comparatively low computational cost than the related global method. Different sizes of domain of influence i.e. m=5,13 are considered. Shape parameter sensitivity of MQ is handled through scaling technique. The time derivative is approximated by first order forward difference formula. An adaptive upwind technique is used for stabilization of the method. Capabilities of the LRBFCM are tested by applying it to one- and two-dimensional benchmark problems with discontinuities, shock pattern and periodic initial conditions. Performance of the LRBFCM is compared with analytical solution, other numerical methods and the results reported earlier in the literature. We have also made comparison with implicit first order time discretization and first order upwind spatial discretization (FVM1) and implicit second order time discretization and first order upwind spatial discretization (FVM2) as well. Accuracy of the method is assessed as a function of time and space. Numerical convergence is also shown for both one- and two-dimensional test problems. It has been observed that the proposed method is more efficient in terms of less memory requirement and less computational efforts due to one time inversion of 5x5 (size of local domain of influence) coefficient matrix. The results obtained through LBRFCM are stable and comparable with the existing methods for a variety of problems with practical applications.
There is a continuously developing need for benchmarking of solidification simulations-from the t... more There is a continuously developing need for benchmarking of solidification simulations-from the theoretical as well as from the applied points of view. The benchmarking is usually done in two parts. The verification part confirms the proper numerical solution (Are we solving the equations correctly?) and the validation part (Are we solving the right equations?) confirms the proper response of the simulations regarding the experimental evidence. The history of related benchmarking shows differences of the results between different numerical methods, and differences in comparison with the experiments when solving even quite simple solidification situations. The present benchmark test proposes the verification benchmark for continuous casting of steel. Since the simulations of the temperatures, velocities, pressures and concentrations in the continuous casting represent a multiscale and multiphysics problem of high complexity (far away from a closed form solution), the verification can be done only by comparing the results of different numerical methods for solving the same governing equations. This has been done in the context of continuous casting by breaking the considered two dimensional problem into several sub problems by increasing complexity of geometry (straight in vertical direction, curved), boundary conditions (linear, non-linear), material properties (only Fe and Fe-C), microscopic considerations (Lever rule and Scheil rule). The governing equations complexity first involves only convective-diffusive heat transport with a predetermined velocity field, and assumed laminar and turbulent velocity calculations afterwards, modelled by a low Reynolds number turbulence model. Further complications involve the presence of the electromagnetic forces. The paper represents guidelines for the presentation of the numerical method, discretisation and results, as well as some of the results, obtained by the commercial finite volume based code and our in-house meshless method based laboratory code.
Advances in Applied Mathematics and Mechanics, Jun 1, 2011
An application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) in solutio... more An application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) in solution of incompressible turbulent combined forced and natural convection is for the first time explored in the present paper. The turbulent flow equations are described by the low-Re number к – ε model with Launder and Sharma and Abe et al. closure coefficients. The involved temperature, velocity, pressure, turbulent kinetic energy and dissipation fields are represented on overlapping 5-noded sub-domains through the collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second order partial derivatives of the fields are calculated from the respective derivatives of the RBF’s. The involved equations are solved through the explicit time stepping. The pressure-velocity coupling is based on Chorin’s fractional step method. The adaptive upwinding technique, proposed by Lin and Atluri , is used because of the convection dominated situation. The solution procedure is represented for a 2D upward channel flow with differentially heated walls. The results have been assessed by achieving a reasonable agreement with the direct numerical simulation of Kasagi and Nishimura for Reynolds number 4494, based on the channel width, and Grash of number 9.6×105. The advantages of the represented mesh-free approach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforward applicability in non-uniform node arrangements.
This paper represents a continuation of numerical results regarding the recently proposed industr... more This paper represents a continuation of numerical results regarding the recently proposed industrial benchmark test [1], obtained by a meshless method. A part of the benchmark test, involving turbulent fluid flow with solidification in two dimensions, was elaborated in [2]. A preliminary macrosegregation upgrade was presented in [3], and in [4], a first three dimensional test was performed. Previous tests were bound to calculations in mold and sub-mold regions only. In the present paper, reference calculations in two dimensions are presented for the entire strand. The physical model is established on a set of macroscopic equations for mass, energy, momentum, species, turbulent kinetic energy, and dissipation rate. The mixture continuum model is used to treat the solidification system. The mushy zone is modeled as a Darcy porous media with Kozeny-Karman permeability relation, where the morphology of the porous media is modeled by a constant value. The incompressible turbulent flow of the molten steel is described by the Low-Reynolds-Number k-ε turbulence model, closed by the Abe-Kondoh-Nagano closure coefficients and damping functions. Lever microsegregation model is used. The numerical method is established on explicit time-stepping, collocation with scaled multiquadrics radial basis functions with adaptive selection of its shape on non-uniform five-nodded influence domains. The velocity-pressure coupling of the incompressible flow is resolved by the explicit Chorin's fractional step method. The advantages of the method are its simplicity and efficiency, since no polygonisation is involved, easy adaptation of the nodal points in areas with high gradients, almost the same formulation in two and three dimensions, high accuracy and low numerical diffusion.
A large-eddy simulation (LES) based meshless model is developed for the three-dimensional (3D) pr... more A large-eddy simulation (LES) based meshless model is developed for the three-dimensional (3D) problem of continuous casting (CC) of steel billet. The local collocation meshless method based on radial basis functions (RBF) is applied in 3D. The method applies scaled multiquadric (MQ) RBF with a shape parameter on seven nodded local sub-domains. The incompressible turbulent fluid flow is described using mass, energy, and momentum conservation equations and the LES turbulence model. The solidification system is solved with the mixture continuum model. The Boussinesq approximation for buoyancy and the Darcy approximation for porous media are used. Chorin’s fractional step method is used to couple velocity and pressure. The microscopic model is closed with the lever rule model. The LES model is compared to the two-equation Low Re k−ε turbulence Reynolds Averaged Navier–Stokes (RANS) model in terms of temperature, velocity and computational times. The LES model resolves transient charact...
A two-dimensional two-scale slice model has been developed to predict the microstructure evolutio... more A two-dimensional two-scale slice model has been developed to predict the microstructure evolution in the solidifying strand with an arbitrary cross section geometry during continuous casting of steel. The enthalpy equation is solved at the macro level by using meshless local radial basis function collocation method (LRBFCM) based on multiquadrics for spatial discretization and explicit Euler scheme for temporal discretization. The temperature and the solid fraction in computational nodes are calculated by using a continuum model formulation while the lever rule is used as the supplementary microsegregation relation. The temperature field is interpolated to the micro level by using LRBFCM. At the micro level, the normal distribution and Kurz-Giovanola-Trivedi model are proposed to determine temperature dependent nucleation rate and grain growth velocity, respectively. Meshless point-automata algorithm is applied to implement nucleation and grain growth equations. Several examples of computations of the strand with different cross-sections are shown.
The purpose of this paper is a multiphysics simulation of 3D temperature and velocity fields in c... more The purpose of this paper is a multiphysics simulation of 3D temperature and velocity fields in continuous casting of steel under the influence of electromagnetic stirring by a combined meshless-finite element method approach. The transport phenomena are calculated by a meshless local radial basis function collocation technique and the magnetic force by the finite element method solver Elmer. The electromagnetic stirring increases the mixing in the molten steel. The thermal gradient is sharper and solidification is faster along the strand. The results are similar to other publications in the field. The local radial basis function collocation method is for the first time applied to 3D continuous casting problem with mold electromagnetic stirring.
Simulation and control of macrosegregation, deformation and grain size under electromagnetic (EM)... more Simulation and control of macrosegregation, deformation and grain size under electromagnetic (EM) processing conditions is important in industrial solidification systems, since it influences the quality of the casts and consequently the whole downstream processing path. Respectively, a multiphysics and multiscale model is developed for solution of Lorentz force, temperature, velocity, concentration, deformation and grain structure of the casts. The mixture equations with lever rule, linearized phase diagram, and stationary thermoelastic solid phase are assumed, together with EM induction equation for the field imposed by the low frequency EM field or Ohm’s law and charge conservation equation for stationary EM field. Turbulent effects are incorporated through the solution of a low-Re turbulence model. The solidification system is treated by the mixture-continuum model, where the mushy zone is modeled as a Darcy porous media with Kozeny-Karman permeability relation and columnar solid...
Cmes-computer Modeling in Engineering & Sciences, 2013
The purpose of the present paper is to extend and explore the application of a novel meshless Loc... more The purpose of the present paper is to extend and explore the application of a novel meshless Local Radial Basis Function Collocation Method (LRBFCM) in solution of a steady, laminar, natural convection flow, influenced by magnetic field. The problem is defined by coupled mass, momentum, energy and induction equations that are solved in two dimensions by using local collocation with multiquadrics radial basis functions on an overlapping five nodded subdomains and explicit time-stepping. The fractional step method is used to couple the pressure and velocity fields. The considered problem is calculated in a square cavity with two insulated horizontal and two differentially heated vertical walls with magnetic field applied in the horizontal direction. Numerical predictions are calculated for different Grashof numbers, ranging from 104 to 106, and Hartman numbers, ranging from 0 to 100, at Prandtl numbers 0.71 and 0.14. The results of the method are compared to predictions, obtained by ...
Continuous casting (CC) of steel is a process that nowadays produces more than 95 % [1] of crude ... more Continuous casting (CC) of steel is a process that nowadays produces more than 95 % [1] of crude steel. To further advance this process, electromagnetic (EM) field, which affects the fluid flow as well as the temperature and segregation is added to the CC process. In general, there are two types of electromagnetic devices applicable to the CC process; one are the electromagnetic breakers (EMBR) which implement direct current, and the other are the electromagnetic stirrers (EMS) which implement the alternating current. Which of the process is used depends on what are the desired effects. Both of the processes are modelled by implementing the Lorentz force into the momentum equation and if necessary the Joule heating term into the energy equation. However, the way how these two terms are modelled depend on the type of the implemented device. In case of EMBRs the assumption of low Rem is made and consequently, the current density is calculated by solving the Poisson’s equation for elec...
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Papers by Robert Vertnik