The thermomechanical phenomena that occur during DC casting of aluminium billets can have a signi... more The thermomechanical phenomena that occur during DC casting of aluminium billets can have a significant impact on the quality of the ingot. Under specific stress conditions hot tearing and cracking of ingot can occur. A large deformation of the billet can lead to its instability and to the change in heat transfer efficiency on the boundary, which can lead to unwanted remelting and melt bleed-outs. Modeling of thermomechanical phenomena during this process is not trivial due to involved strong thermomechanic coupling. In addition to elastic deformation the strain field also has contributions from viscoplastic creep, plastic deformation and thermal expansion. All these phenomena occur in nonhomogeneous material with strong temperature dependence of material properties. Many models describing the DC casting process already exist [1] and can provide accurate results. Existing models mainly use the standard finite element method (FEM), which may prove inefficient in some circumstances. T...
• Adaptive finite volume method for simulation of microstructure evolution. • Adaptive algorithm ... more • Adaptive finite volume method for simulation of microstructure evolution. • Adaptive algorithm based on quadtree structure. • Novel calculation of inerphase normal and curvature. • Parametric study of mesh anisotropy. • Efficient and accurate calculation of dendritic and eutectic growth in 2D.
This paper explores, for the first time, the application of the novel mesh-free local radial basi... more This paper explores, for the first time, the application of the novel mesh-free local radial basis function collocation method (LRBFCM) to the solution of a multi-physics problem in three dimensions. A related benchmark problem is solved by considering the natural convection of an incompressible Newtonian fluid in a differentially heated cubic cavity with and without the application of a magnetic field. The research is limited to typical magnetic fields used in the magnetohydrodynamic processing of liquid metals. For this purpose the assumption of small magnetic Reynolds numbers Re m ≪ 1 is made. Spatial discretization is performed by local non-uniform collocation with scaled multiquadrics radial basis functions (RBFs) with the shape parameter set to a constant value and the explicit Euler formula used to perform the time stepping. The involved temperature, velocity and pressure fields are represented on overlapping seven-nodded sub-domains. The pressure-velocity coupling is resolved by the fractional step method. The originality of the contribution represents LRBFCM solution of the classic threedimensional steady natural convection benchmark for Rayleigh numbers from 10 5 to 10 7 and Prandtl number 0.71, and its extension to Prandtl number 0.1, and Hartman numbers 0, 10, 50 and 100. The accuracy of the LRBFCM is found to be comparable with the published benchmark results obtained using established numerical methods.
The purpose of this work is to determine, based on the computational model, whether a mixture of ... more The purpose of this work is to determine, based on the computational model, whether a mixture of a binary liquid is capable of producing longer, thinner and faster gas-focused micro-jets, compared to the mono-constituent liquids of its components. Mixtures of water with two different alcohols, water + ethanol and water + 2-propanol, are considered. The numerical study of pre-mixed liquids is performed in the double flow focusing nozzle geometry used in sample delivery in serial femtosecond crystallography experiments. The study reveals that an optimal mixture for maximizing the jet length exists both in a water + ethanol and in a water + 2-propanol system. Additionally, the use of 2-propanol instead of ethanol results in a 34% jet length increase, while the jet diameters and velocities are similar for both mixtures. Pure ethanol and pure 2-propanol are the optimum liquids to achieve the smallest diameter and the fastest jets. However, the overall aim is to find a mixture with the lo...
Formation of liquid sheets has been demonstrated as a critical capability needed in many differen... more Formation of liquid sheets has been demonstrated as a critical capability needed in many different research fields. Many different types of liquid sheets have been produced experimentally, its thickness ranging from few tens of nanometres to few micrometres. Due to the small size of such systems, where physical parameters such as thickness, velocity and temperature are difficult to measure, a need for numerical simulation of liquid sheets arises. In this paper we demonstrate such capability with sheets that can be used in experiments with synchrotrons, X-ray free electron lasers or lab sources. A modified gas dynamic virtual nozzle (GDVN) design is used in order to generate micrometre thin sheets. The system is characterised by a strongly coupled problem between the focusing gas flow and the liquid sheet flow. Investigation of varying physical properties of liquid is performed in order to demonstrate the effects on the sheet production. It was found that the primary sheet thickness is not sensitive to the variation of liquid viscosity and density. On the other hand, the variation of surface tension greatly affects the thickness and the width of a primary sheet, such as expected in flows where surface tension is the dominating force. Findings demonstrate that by lowering the surface tension of a liquid, i.e. changing liquid from water to alcohol for example, would produce thinner and wider sheets. Simulations were produced with OpenFOAM, relying on finite volume based multiphase solver "compressibleInterFoam", capable of simulating free surfaces. Mixture formulation of a multiphase system consists of an incompressible liquid phase along with a compressible ideal gaseous phase. Such model was also used in axisymmetric GDVN micro-jet simulations preformed in our previous work. Due to the need for 3D simulations and huge computational resources needed, an adaptive approach was chosen. This made the simulations of liquid sheets of thicknesses down to 500 nm possible.
The work presents verification of a numerical model for micro-jet focusing, where a coupled liqui... more The work presents verification of a numerical model for micro-jet focusing, where a coupled liquid and gas flow occurs in a gas dynamic virtual nozzle (GDVN). Nozzlesof this type are usedinserial femtosecond crystallographyexperimentsto deliver samplesintoX-ray beam. Thefollowing performance criteria are desirable: the jet to be longer than 100 μm to avoid nozzle shadowing, the diameter as small as possibleto minimize the background signal,and the jet velocityas high as possible to avoid sample'sdouble X-ray exposure.Previouscomprehensive numerical investigation has been extended to includenumerical analysis of the tip jet velocities. These simulations were then comparedwith the experimental data. The coupled numerical model of a 3D printed GDVN considers a laminar two-phase, Newtonian, compressible flow, which is solved based on the finite volume method discretization and interface tracking with volume of fluid (VOF). The numerical solution is calculated with OpenFOAM based com...
Serial femtosecond crystallography requires reliable and efficient delivery of fresh crystals acr... more Serial femtosecond crystallography requires reliable and efficient delivery of fresh crystals across the beam of an X-ray free-electron laser over the course of an experiment. We introduce a double-flow focusing nozzle to meet this challenge, with significantly reduced sample consumption, while improving jet stability over previous generations of nozzles. We demonstrate its use to determine the first room-temperature structure of RNA polymerase II at high resolution, revealing new structural details. Moreover, the double flow-focusing nozzles were successfully tested with three other protein samples and the first room temperature structure of an extradiol ring-cleaving dioxygenase was solved by utilizing the improved operation and characteristics of these devices.
In this paper, a heat transfer problem of continuous casting is solved by two BEM approaches, i.e... more In this paper, a heat transfer problem of continuous casting is solved by two BEM approaches, i.e. front tracking and fixed grid with dual reciprocity. Both techniques are compared and critically evaluated by solving two numerical examples consisting of determining the temperature field and location of phase change front. Good accuracy has been observed.
Résumé/Abstract A heat transfer problem of continuous casting is solved by two BEM approaches, ie... more Résumé/Abstract A heat transfer problem of continuous casting is solved by two BEM approaches, ie front tracking and fixed grid with dual reciprocity. Both techniques are compared and critically evaluated by solving two numerical examples consisting of determining the temperature field and location of phase change front
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.
A numerical model is developed for thesimulationofsolidificationgrain structurefor- mation (equia... more A numerical model is developed for thesimulationofsolidificationgrain structurefor- mation (equiaxed to columnar and columnar to equiaxed transitions) during the continuous cast- ing process of steel billets. The cellular au- tomata microstructure model is combined with the macroscopic heat transfer model. The cellular automata method is based on the Nastac's defini- tion of neighborhood, Gaussian nucleation rule, and KGT growth model. The heat transfer model is solved by themeshless techniqueby using local collocation with radial basis functions. The mi- croscopic model parameters have been adjusted with respect to the experimental data for steel 51CrMoV4. Simulations have been carried out for nominal casting conditions, reduced casting temperature, and reduced casting speed. Proper response of the multiscale model with respect to the observed grain structures has been proved. Keyword: continuous casting of steel, solidifi- cation, multiscale modeling, equiaxed to colum- nar tr...
This paper explores the application of the mesh-free Local Radial Basis Function Collocation Meth... more This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [1] in solution of coupled heat transfer and fluid flow problems associated with solidification of a binary alloy. The involved temperature, velocity, species and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF’s. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [2,3]. The solution procedure is demonstrated in simulation of solidification of 8 wt % Pb-Sn alloy and 5 wt % Sn-Pb alloy in rectangular cavity at conditions, defined in [4]. The numerical examples include studies with different uniform discretization and its...
Cmes-computer Modeling in Engineering & Sciences, Feb 1, 2008
ABSTRACT This paper explores the application of the mesh-free Local Radial Basis Function Colloca... more ABSTRACT This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) in solution of coupled heat transfer and fluid flow problems in Darcy porous media. The involved temperature, velocity and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF's. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation. This formulation does not require solution of pressure Poisson or pressure correction Poisson equations and thus much simplifies the Kassab and Divo formulation [Divo and Kassab (2007)]. The solution procedure is represented for a steady natural convection problem in a rectangular cavity, filled with Darcy porous media. The numerical examples include studies with different uniform discretization for differentially heated boundaries at filtration Rayleigh numbers Ra F =25, 50, 10 2, 10 3, 10 4, and aspect ratios A = 1/2, 1, 2. The solution is assessed by comparison with reference results of the fine mesh finite volume method (FVM) in terms of mid-plane velocities, mid-plane and insulated surface temperatures, mid-point streamfunction and Nusselt number. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements.
This paper represents the elements and the use of the simulation system, developed for the Štore-... more This paper represents the elements and the use of the simulation system, developed for the Štore-Steel billet caster. The simulation system is used in the context of the state-of-the-art automation and information of the twenty years old three-strand Concast billet continuous caster for dimensions 140, 180, and 220 [mm] with the capacity of 150.000 [tons/year]. The simulation system is used in the off and in the on-line mode. The off-line mode is used in order to set the proper process parameters and for the redesign of the primary and secondary cooling systems. The numerical solution of the respective models for mass, energy, momentum ans species transfer on the macroscopic and on the microscopic scales are based on the recently developed meshless technology. The options of simulation system, which enable the technologist automatic setup of process parameters, are described. The regulation coefficients were calculated based on this simulation system and installed into the casting m...
This paper explores an application of a novel mesh-free Local Radial Basis Function Collocation M... more This paper explores an application of a novel mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [Šarler and Vertnik (2006)] in solution of coupled heat transfer and fluid flow problems with solid-liquid phase change. The melting/freezing of a pure substance is solved in primitive variables on a fixed grid with convection suppression, proportional to the amount of the solid fraction. The involved temperature, velocity and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF's. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [Kosec and Šarler (2008a)]. The solution procedure is assessed on the classical rectangular 2D cavity melting benchmark test [Gobin and Le Quéré (2000)] which encompasses a low Prandtl 0.02 and Stefan number 0.01 situation (metal) with Rayleigh numbers 2.51e4 and 2.5e5, and a high Prandtl 50 and Stefan number 0.1 situation (paraffin wax) with Rayleigh numbers 10e7 and 10e8. The results of the mesh free simulation of the related four cases have been compared with the results of a spectra of different numerical methods [Bertrand, Binet, Combeau, Couturier, Delannoy, Gobin, Lacroix, Quéré, Médale, Mencinger, Sadat and Vieira (1998)] in terms of liquid-solid interphase position at a fixed time, and time evolution of the average hot side Nusselt number and average cavity liquid fraction. The results show good agreement with other approaches in terms of the dynamics of the interphase boundary and complicated flow structure, despite the simplest LRBFCM implementation. The advantages of the method are simplicity, accuracy, and straightforward applicability in non-uniform node arrangements.
International Journal of Numerical Methods for Heat & Fluid Flow, 2008
PurposeThe purpose of this paper is to explore the application of the mesh‐free local radial basi... more PurposeThe purpose of this paper is to explore the application of the mesh‐free local radial basis function collocation method (RBFCM) in solution of coupled heat transfer and fluid‐flow problems.Design/methodology/approachThe involved temperature, velocity and pressure fields are represented on overlapping five nodded sub‐domains through collocation by using multiquadrics radial basis functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time stepping.FindingsThe performance of the method is assessed on the classical two dimensional de Vahl Davis steady natural convection benchmark for Rayleigh numbers from 103 to 108 and Prandtl number 0.71. The results show good agreement with other methods at a given range.Originality/valueThe pressure‐velocity coupling is calculated iteratively, with pressure correction, predicted from the local mass conti...
International Journal of Numerical Methods for Heat & Fluid Flow, 2013
PurposeThe purpose of this paper is to present the solution of a highly nonlinear fluid dynamics ... more PurposeThe purpose of this paper is to present the solution of a highly nonlinear fluid dynamics in a low Prandtl number regime, typical for metal‐like materials, as defined in the call for contributions to a numerical benchmark problem for 2D columnar solidification of binary alloys. The solution of such a numerical situation represents the first step towards understanding the instabilities in a more complex case of macrosegregation.Design/methodology/approachThe involved temperature, velocity and pressure fields are represented through the local approximation functions which are used to evaluate the partial differential operators. The temporal discretization is performed through explicit time stepping.FindingsThe performance of the method is assessed on the natural convection in a closed rectangular cavity filled with a low Prandtl fluid. Two cases are considered, one with steady state and another with oscillatory solution. It is shown that the proposed solution procedure, despite...
Natural convection is a phenomenon where fluid motion is generated by density changes due to the ... more Natural convection is a phenomenon where fluid motion is generated by density changes due to the temperature or concentration variations in a gravity field. The computational modelling of systems with natural convection (Bejan, 2004) has become a highly popular research subject due to its pronounced influence in better understanding of nature as well as in the development of the advanced technologies. Melting of the polar ice caps, the global oceans dynamics, various weather systems, water transport, soil erosion and denudation, magma transport and manufacturing of nano-materials, improving casting processes, energetic studies, exploitation of natural resources, welding, casting and advanced solidifications are two typical contemporary example groups where natural convection plays an important role. This chapter deals with the numerical approach towards solution of this type of problems by a meshless technique. The main part of the solution procedure is focused on the general transport equation treatment and the pressure velocity coupling strategy. The transport phenomena are solved by a local meshless method and explicit time stepping. The local variant of Radial Basis Function Collocation Method (LRBFCM) has been previously developed for diffusion problems (Šarler and Vertnik, 2006), convection-diffusion solid-liquid phase change problems (Vertnik and Šarler, 2006) and subsequently successfully applied in industrial process of direct chill casting (Vertnik, et al., 2006). The fluid flow, which is generally a global problem, is treated by the proposed local iterative method. Instead of solving the pressure Poisson equation or/and pressure correction Poisson equation (Divo and Kassab, 2007) a more simplified local pressure-velocity coupling (LPVC) (Kosec and Šarler, 2008a) algorithm is proposed where the pressure-correction is predicted from the local mass continuity violation similar to the SOLA algorithm (Hong, 2004). The presented solution procedure represents a variant of already developed global approach (Šarler, et al., 2004a, Šarler, 2005). In this chapter, such a local solution procedure is tested with the standard free fluid flow benchmark test (de Vahl Davis natural convection test (de Vahl Davis, 1983)). The test is especially convenient for benchmarking purposes as there are several numerical solutions published in the literature (
The thermomechanical phenomena that occur during DC casting of aluminium billets can have a signi... more The thermomechanical phenomena that occur during DC casting of aluminium billets can have a significant impact on the quality of the ingot. Under specific stress conditions hot tearing and cracking of ingot can occur. A large deformation of the billet can lead to its instability and to the change in heat transfer efficiency on the boundary, which can lead to unwanted remelting and melt bleed-outs. Modeling of thermomechanical phenomena during this process is not trivial due to involved strong thermomechanic coupling. In addition to elastic deformation the strain field also has contributions from viscoplastic creep, plastic deformation and thermal expansion. All these phenomena occur in nonhomogeneous material with strong temperature dependence of material properties. Many models describing the DC casting process already exist [1] and can provide accurate results. Existing models mainly use the standard finite element method (FEM), which may prove inefficient in some circumstances. T...
• Adaptive finite volume method for simulation of microstructure evolution. • Adaptive algorithm ... more • Adaptive finite volume method for simulation of microstructure evolution. • Adaptive algorithm based on quadtree structure. • Novel calculation of inerphase normal and curvature. • Parametric study of mesh anisotropy. • Efficient and accurate calculation of dendritic and eutectic growth in 2D.
This paper explores, for the first time, the application of the novel mesh-free local radial basi... more This paper explores, for the first time, the application of the novel mesh-free local radial basis function collocation method (LRBFCM) to the solution of a multi-physics problem in three dimensions. A related benchmark problem is solved by considering the natural convection of an incompressible Newtonian fluid in a differentially heated cubic cavity with and without the application of a magnetic field. The research is limited to typical magnetic fields used in the magnetohydrodynamic processing of liquid metals. For this purpose the assumption of small magnetic Reynolds numbers Re m ≪ 1 is made. Spatial discretization is performed by local non-uniform collocation with scaled multiquadrics radial basis functions (RBFs) with the shape parameter set to a constant value and the explicit Euler formula used to perform the time stepping. The involved temperature, velocity and pressure fields are represented on overlapping seven-nodded sub-domains. The pressure-velocity coupling is resolved by the fractional step method. The originality of the contribution represents LRBFCM solution of the classic threedimensional steady natural convection benchmark for Rayleigh numbers from 10 5 to 10 7 and Prandtl number 0.71, and its extension to Prandtl number 0.1, and Hartman numbers 0, 10, 50 and 100. The accuracy of the LRBFCM is found to be comparable with the published benchmark results obtained using established numerical methods.
The purpose of this work is to determine, based on the computational model, whether a mixture of ... more The purpose of this work is to determine, based on the computational model, whether a mixture of a binary liquid is capable of producing longer, thinner and faster gas-focused micro-jets, compared to the mono-constituent liquids of its components. Mixtures of water with two different alcohols, water + ethanol and water + 2-propanol, are considered. The numerical study of pre-mixed liquids is performed in the double flow focusing nozzle geometry used in sample delivery in serial femtosecond crystallography experiments. The study reveals that an optimal mixture for maximizing the jet length exists both in a water + ethanol and in a water + 2-propanol system. Additionally, the use of 2-propanol instead of ethanol results in a 34% jet length increase, while the jet diameters and velocities are similar for both mixtures. Pure ethanol and pure 2-propanol are the optimum liquids to achieve the smallest diameter and the fastest jets. However, the overall aim is to find a mixture with the lo...
Formation of liquid sheets has been demonstrated as a critical capability needed in many differen... more Formation of liquid sheets has been demonstrated as a critical capability needed in many different research fields. Many different types of liquid sheets have been produced experimentally, its thickness ranging from few tens of nanometres to few micrometres. Due to the small size of such systems, where physical parameters such as thickness, velocity and temperature are difficult to measure, a need for numerical simulation of liquid sheets arises. In this paper we demonstrate such capability with sheets that can be used in experiments with synchrotrons, X-ray free electron lasers or lab sources. A modified gas dynamic virtual nozzle (GDVN) design is used in order to generate micrometre thin sheets. The system is characterised by a strongly coupled problem between the focusing gas flow and the liquid sheet flow. Investigation of varying physical properties of liquid is performed in order to demonstrate the effects on the sheet production. It was found that the primary sheet thickness is not sensitive to the variation of liquid viscosity and density. On the other hand, the variation of surface tension greatly affects the thickness and the width of a primary sheet, such as expected in flows where surface tension is the dominating force. Findings demonstrate that by lowering the surface tension of a liquid, i.e. changing liquid from water to alcohol for example, would produce thinner and wider sheets. Simulations were produced with OpenFOAM, relying on finite volume based multiphase solver "compressibleInterFoam", capable of simulating free surfaces. Mixture formulation of a multiphase system consists of an incompressible liquid phase along with a compressible ideal gaseous phase. Such model was also used in axisymmetric GDVN micro-jet simulations preformed in our previous work. Due to the need for 3D simulations and huge computational resources needed, an adaptive approach was chosen. This made the simulations of liquid sheets of thicknesses down to 500 nm possible.
The work presents verification of a numerical model for micro-jet focusing, where a coupled liqui... more The work presents verification of a numerical model for micro-jet focusing, where a coupled liquid and gas flow occurs in a gas dynamic virtual nozzle (GDVN). Nozzlesof this type are usedinserial femtosecond crystallographyexperimentsto deliver samplesintoX-ray beam. Thefollowing performance criteria are desirable: the jet to be longer than 100 μm to avoid nozzle shadowing, the diameter as small as possibleto minimize the background signal,and the jet velocityas high as possible to avoid sample'sdouble X-ray exposure.Previouscomprehensive numerical investigation has been extended to includenumerical analysis of the tip jet velocities. These simulations were then comparedwith the experimental data. The coupled numerical model of a 3D printed GDVN considers a laminar two-phase, Newtonian, compressible flow, which is solved based on the finite volume method discretization and interface tracking with volume of fluid (VOF). The numerical solution is calculated with OpenFOAM based com...
Serial femtosecond crystallography requires reliable and efficient delivery of fresh crystals acr... more Serial femtosecond crystallography requires reliable and efficient delivery of fresh crystals across the beam of an X-ray free-electron laser over the course of an experiment. We introduce a double-flow focusing nozzle to meet this challenge, with significantly reduced sample consumption, while improving jet stability over previous generations of nozzles. We demonstrate its use to determine the first room-temperature structure of RNA polymerase II at high resolution, revealing new structural details. Moreover, the double flow-focusing nozzles were successfully tested with three other protein samples and the first room temperature structure of an extradiol ring-cleaving dioxygenase was solved by utilizing the improved operation and characteristics of these devices.
In this paper, a heat transfer problem of continuous casting is solved by two BEM approaches, i.e... more In this paper, a heat transfer problem of continuous casting is solved by two BEM approaches, i.e. front tracking and fixed grid with dual reciprocity. Both techniques are compared and critically evaluated by solving two numerical examples consisting of determining the temperature field and location of phase change front. Good accuracy has been observed.
Résumé/Abstract A heat transfer problem of continuous casting is solved by two BEM approaches, ie... more Résumé/Abstract A heat transfer problem of continuous casting is solved by two BEM approaches, ie front tracking and fixed grid with dual reciprocity. Both techniques are compared and critically evaluated by solving two numerical examples consisting of determining the temperature field and location of phase change front
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.
A numerical model is developed for thesimulationofsolidificationgrain structurefor- mation (equia... more A numerical model is developed for thesimulationofsolidificationgrain structurefor- mation (equiaxed to columnar and columnar to equiaxed transitions) during the continuous cast- ing process of steel billets. The cellular au- tomata microstructure model is combined with the macroscopic heat transfer model. The cellular automata method is based on the Nastac's defini- tion of neighborhood, Gaussian nucleation rule, and KGT growth model. The heat transfer model is solved by themeshless techniqueby using local collocation with radial basis functions. The mi- croscopic model parameters have been adjusted with respect to the experimental data for steel 51CrMoV4. Simulations have been carried out for nominal casting conditions, reduced casting temperature, and reduced casting speed. Proper response of the multiscale model with respect to the observed grain structures has been proved. Keyword: continuous casting of steel, solidifi- cation, multiscale modeling, equiaxed to colum- nar tr...
This paper explores the application of the mesh-free Local Radial Basis Function Collocation Meth... more This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [1] in solution of coupled heat transfer and fluid flow problems associated with solidification of a binary alloy. The involved temperature, velocity, species and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF’s. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [2,3]. The solution procedure is demonstrated in simulation of solidification of 8 wt % Pb-Sn alloy and 5 wt % Sn-Pb alloy in rectangular cavity at conditions, defined in [4]. The numerical examples include studies with different uniform discretization and its...
Cmes-computer Modeling in Engineering & Sciences, Feb 1, 2008
ABSTRACT This paper explores the application of the mesh-free Local Radial Basis Function Colloca... more ABSTRACT This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) in solution of coupled heat transfer and fluid flow problems in Darcy porous media. The involved temperature, velocity and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF's. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation. This formulation does not require solution of pressure Poisson or pressure correction Poisson equations and thus much simplifies the Kassab and Divo formulation [Divo and Kassab (2007)]. The solution procedure is represented for a steady natural convection problem in a rectangular cavity, filled with Darcy porous media. The numerical examples include studies with different uniform discretization for differentially heated boundaries at filtration Rayleigh numbers Ra F =25, 50, 10 2, 10 3, 10 4, and aspect ratios A = 1/2, 1, 2. The solution is assessed by comparison with reference results of the fine mesh finite volume method (FVM) in terms of mid-plane velocities, mid-plane and insulated surface temperatures, mid-point streamfunction and Nusselt number. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements.
This paper represents the elements and the use of the simulation system, developed for the Štore-... more This paper represents the elements and the use of the simulation system, developed for the Štore-Steel billet caster. The simulation system is used in the context of the state-of-the-art automation and information of the twenty years old three-strand Concast billet continuous caster for dimensions 140, 180, and 220 [mm] with the capacity of 150.000 [tons/year]. The simulation system is used in the off and in the on-line mode. The off-line mode is used in order to set the proper process parameters and for the redesign of the primary and secondary cooling systems. The numerical solution of the respective models for mass, energy, momentum ans species transfer on the macroscopic and on the microscopic scales are based on the recently developed meshless technology. The options of simulation system, which enable the technologist automatic setup of process parameters, are described. The regulation coefficients were calculated based on this simulation system and installed into the casting m...
This paper explores an application of a novel mesh-free Local Radial Basis Function Collocation M... more This paper explores an application of a novel mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [Šarler and Vertnik (2006)] in solution of coupled heat transfer and fluid flow problems with solid-liquid phase change. The melting/freezing of a pure substance is solved in primitive variables on a fixed grid with convection suppression, proportional to the amount of the solid fraction. The involved temperature, velocity and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF's. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [Kosec and Šarler (2008a)]. The solution procedure is assessed on the classical rectangular 2D cavity melting benchmark test [Gobin and Le Quéré (2000)] which encompasses a low Prandtl 0.02 and Stefan number 0.01 situation (metal) with Rayleigh numbers 2.51e4 and 2.5e5, and a high Prandtl 50 and Stefan number 0.1 situation (paraffin wax) with Rayleigh numbers 10e7 and 10e8. The results of the mesh free simulation of the related four cases have been compared with the results of a spectra of different numerical methods [Bertrand, Binet, Combeau, Couturier, Delannoy, Gobin, Lacroix, Quéré, Médale, Mencinger, Sadat and Vieira (1998)] in terms of liquid-solid interphase position at a fixed time, and time evolution of the average hot side Nusselt number and average cavity liquid fraction. The results show good agreement with other approaches in terms of the dynamics of the interphase boundary and complicated flow structure, despite the simplest LRBFCM implementation. The advantages of the method are simplicity, accuracy, and straightforward applicability in non-uniform node arrangements.
International Journal of Numerical Methods for Heat & Fluid Flow, 2008
PurposeThe purpose of this paper is to explore the application of the mesh‐free local radial basi... more PurposeThe purpose of this paper is to explore the application of the mesh‐free local radial basis function collocation method (RBFCM) in solution of coupled heat transfer and fluid‐flow problems.Design/methodology/approachThe involved temperature, velocity and pressure fields are represented on overlapping five nodded sub‐domains through collocation by using multiquadrics radial basis functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time stepping.FindingsThe performance of the method is assessed on the classical two dimensional de Vahl Davis steady natural convection benchmark for Rayleigh numbers from 103 to 108 and Prandtl number 0.71. The results show good agreement with other methods at a given range.Originality/valueThe pressure‐velocity coupling is calculated iteratively, with pressure correction, predicted from the local mass conti...
International Journal of Numerical Methods for Heat & Fluid Flow, 2013
PurposeThe purpose of this paper is to present the solution of a highly nonlinear fluid dynamics ... more PurposeThe purpose of this paper is to present the solution of a highly nonlinear fluid dynamics in a low Prandtl number regime, typical for metal‐like materials, as defined in the call for contributions to a numerical benchmark problem for 2D columnar solidification of binary alloys. The solution of such a numerical situation represents the first step towards understanding the instabilities in a more complex case of macrosegregation.Design/methodology/approachThe involved temperature, velocity and pressure fields are represented through the local approximation functions which are used to evaluate the partial differential operators. The temporal discretization is performed through explicit time stepping.FindingsThe performance of the method is assessed on the natural convection in a closed rectangular cavity filled with a low Prandtl fluid. Two cases are considered, one with steady state and another with oscillatory solution. It is shown that the proposed solution procedure, despite...
Natural convection is a phenomenon where fluid motion is generated by density changes due to the ... more Natural convection is a phenomenon where fluid motion is generated by density changes due to the temperature or concentration variations in a gravity field. The computational modelling of systems with natural convection (Bejan, 2004) has become a highly popular research subject due to its pronounced influence in better understanding of nature as well as in the development of the advanced technologies. Melting of the polar ice caps, the global oceans dynamics, various weather systems, water transport, soil erosion and denudation, magma transport and manufacturing of nano-materials, improving casting processes, energetic studies, exploitation of natural resources, welding, casting and advanced solidifications are two typical contemporary example groups where natural convection plays an important role. This chapter deals with the numerical approach towards solution of this type of problems by a meshless technique. The main part of the solution procedure is focused on the general transport equation treatment and the pressure velocity coupling strategy. The transport phenomena are solved by a local meshless method and explicit time stepping. The local variant of Radial Basis Function Collocation Method (LRBFCM) has been previously developed for diffusion problems (Šarler and Vertnik, 2006), convection-diffusion solid-liquid phase change problems (Vertnik and Šarler, 2006) and subsequently successfully applied in industrial process of direct chill casting (Vertnik, et al., 2006). The fluid flow, which is generally a global problem, is treated by the proposed local iterative method. Instead of solving the pressure Poisson equation or/and pressure correction Poisson equation (Divo and Kassab, 2007) a more simplified local pressure-velocity coupling (LPVC) (Kosec and Šarler, 2008a) algorithm is proposed where the pressure-correction is predicted from the local mass continuity violation similar to the SOLA algorithm (Hong, 2004). The presented solution procedure represents a variant of already developed global approach (Šarler, et al., 2004a, Šarler, 2005). In this chapter, such a local solution procedure is tested with the standard free fluid flow benchmark test (de Vahl Davis natural convection test (de Vahl Davis, 1983)). The test is especially convenient for benchmarking purposes as there are several numerical solutions published in the literature (
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Papers by Bozidar Sarler