Simulation of macrosegregation with mesosegregates in binary metallic casts by a meshless method

Abstract Simulation of macrosegregation with mesosegregates as a consequence of solidification of a binary Sn–10%Pb alloy in a 2-dimensional rectangular cast is tackled in the present paper. Coupled volume averaged governing equations for mass, energy, momentum and species transfer are considered by incorporating Lever solidification rule and incompressible Newtonian fluid with Darcy limit in the mushy zone. Solid phase is assumed stationary. Double diffusive effects in the melt are modeled by the thermal and solutal Boussinesq hypothesis. The physical model is solved by the meshless Local Radial Basis Function Collocation Method (LRBFCM) by using 5-noded influence domains, multiquadrics radial basis functions and explicit time stepping. Pressure–velocity coupling is based on local pressure correction. Adaptive upwinding has to be used for stabilization of the convective terms. The numerical simulations reveal instabilities during solidification process that introduce anomalies in the final segregation map that scale with the typical cast as well as sub-cast dimensions. The main advantages of choosing the represented meshless approach for solving the problem are in its simplicity and similar coding in 2D and 3D, as well as straightforward applicability in non-uniform node arrangements. The locality of the proposed numerical approach is also convenient for parallel execution. It is demonstrated that LRBFCM can be advantageously used in casting simulations where the chemical segregation exhibits industrially relevant multi-scale patterns.

[1]  W. Chen,et al.  New RBF collocation methods and kernel RBF with applications , 2001, ArXiv.

[2]  Alexandru Ioan Ciobanas,et al.  A benchmark solidification experiment on an Sn–10%wtBi alloy , 2007 .

[3]  G. Kosec,et al.  Local strong form meshless method on multiple GraphicsProcessing Units , 2013 .

[4]  Guangming Yao,et al.  Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions , 2012 .

[5]  Božidar Šarler,et al.  Modeling of macrosegregation in direct-chill casting of aluminum alloys: Estimating the influence of casting parameters , 2005 .

[6]  S. Atluri,et al.  The meshless local Petrov-Galerkin (MLPG) method , 2002 .

[7]  A. Hellawell,et al.  Channel formation in Pb-Sn, Pb-Sb, and Pb-Sn-Sb alloy ingots and comparison with the system NH4CI-H2O , 1988 .

[8]  G. Lesoult,et al.  Macrosegregation in steel strands and ingots : Characterisation, formation and consequences , 2005 .

[9]  I F Borovikov,et al.  Influence of layer-by-layer laser sintering conditions on the quality of sintered surface layer of products , 2015 .

[10]  Roman Trobec,et al.  Super linear speedup in a local parallel meshless solution of thermo-fluid problems , 2014 .

[11]  Vaughan R Voller.  On a general back-diffusion parameter , 2001 .

[12]  Gregor Kosec,et al.  Solution of Phase Change Problems by Collocation with Local Pressure Correction , 2009 .

[13]  E. J. Kansa,et al.  Multi-quadrics-a scattered data approximation scheme with applications to computational fluid dynamics-II , 1990 .

[14]  J. Ni,et al.  A volume-averaged two-phase model for transport phenomena during solidification , 1991 .

[15]  M. Cross,et al.  Modeling of casting, welding and advanced solidification processes, 7 , 1995 .

[16]  D. Drew.  Mathematical Modeling of Two-Phase Flow , 1983 .

[17]  B. Šarler.  Stefan's work on solid-liquid phase changes , 1995 .

[18]  Roman Trobec,et al.  Meshless solution of a diffusion equation with parameter optimization and error analysis , 2008 .

[19]  B. Šarler,et al.  Solution of Incompressible Turbulent Flow by a Mesh-Free Method , 2009 .

[20]  Michel Rappaz,et al.  Modeling of casting, welding and advanced solidification processes-V : proceedings of the fifth International Conference on Modeling of Casting and Welding Processes, held in Davos Switzerland, September 16-21, 1990 , 1991 .

[21]  Carlos J. S. Alves,et al.  Advances in meshfree techniques , 2007 .

[22]  Michel Quintard,et al.  Macroscopic modeling of columnar dendritic solidification , 2004 .

[23]  R. Courant,et al.  On the solution of nonlinear hyperbolic differential equations by finite differences , 1952 .

[24]  J. Hunt,et al.  Observations of ingot macrosegregation on model systems , 1974, Metallurgical and Materials Transactions B.

[25]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .

[26]  M. Flemings,et al.  Refining by partial solidification , 1974, Metallurgical and Materials Transactions B.

[27]  Satya N. Atluri,et al.  MESHLESS LOCAL PETROV-GALERKIN (MLPG) METHOD FOR CONVECTION DIFFUSION PROBLEMS , 2000 .

[28]  Joseph J Monaghan,et al.  An introduction to SPH , 1987 .

[29]  Christoph Beckermann,et al.  Equiaxed dendritic solidification with convection: Part I. Multiscale/multiphase modeling , 1996 .

[30]  Robert Vertnik,et al.  Meshfree explicit local radial basis function collocation method for diffusion problems , 2006, Comput. Math. Appl..

[31]  Roman Trobec,et al.  Computational complexity and parallelization of the meshless local Petrov-Galerkin method , 2009 .

[32]  E. Arquis,et al.  A Numerical Benchmark on the Prediction of Macrosegregation in Binary Alloys , 2011 .

[33]  Božidar Šarler,et al.  Low and Intermediate Re Solution of Lid Driven CavityProblem by Local Radial Basis Function CollocationMethod , 2013 .

[34]  Meng Yan-yu,et al.  Meshless Local Petrov-Galerkin(MLPG) Method , 2005 .

[35]  S. Atluri,et al.  The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods , 2002 .

[36]  B. Šarler,et al.  Meshless local radial basis function collocation method for convective‐diffusive solid‐liquid phase change problems , 2006 .

[37]  Vaughan R Voller,et al.  Towards a general numerical scheme for solidification systems , 1997 .

[38]  Guirong Liu,et al.  A point interpolation meshless method based on radial basis functions , 2002 .

[39]  Božidar Šarler,et al.  Simulation of Dendritic Growth with Different Orientation by Using the Point Automata Method , 2010 .

[40]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[41]  W. Kurz,et al.  Fundamentals of Solidification , 1990 .

[42]  Božidar Šarler,et al.  From Global to Local Radial Basis Function Collocation Method for Transport Phenomena , 2007 .

[43]  D. Levin,et al.  Hermite type moving-least-squares approximations , 2006, Comput. Math. Appl..

[44]  M. Založnik,et al.  An operator splitting scheme for coupling macroscopic transport and grain growth in a two-phase multiscale solidification model: Part II – Application of the model , 2010 .

[45]  C. Hong,et al.  Computer Modelling of Heat and Fluid Flow in Materials Processing , 2019 .

[46]  B. Šarler,et al.  Solution of a phase-field model for dissolution of primary particles in binary aluminum alloys by an r-adaptive mesh-free method , 2005 .

[47]  M. Cross,et al.  An explicit scheme for coupling temperature and concentration fields in solidification models , 2004 .

[48]  Martin D. Buhmann,et al.  Radial Basis Functions , 2021, Encyclopedia of Mathematical Geosciences.

[49]  B. Šarler,et al.  H-Adaptive Local Radial Basis Function Collocation Meshless Method , 2011 .

[50]  M. Flemings.  Solidification processing , 1974, Metallurgical and Materials Transactions B.

[51]  Gregor Kosec,et al.  Solution of thermo‐fluid problems by collocation with local pressure correction , 2008 .

[52]  E. Arquis,et al.  Analysis of a numerical benchmark for columnar solidification of binary alloys , 2012 .

[53]  Charles-André Gandin,et al.  Call for contributions to a numerical benchmark problem for 2D columnar solidification of binary alloys , 2009 .

[54]  Gregor Kosec,et al.  A Meshless Approach Towards Solution of Macrosegregation Phenomena , 2011 .

[55]  B. Šarler,et al.  Solution of a low Prandtl number natural convection benchmark by a local meshless method , 2013 .

[56]  Siraj-ul-Islam,et al.  Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations , 2013 .

[57]  Jim Douglas,et al.  On the numerical integration of a parabolic differential equation subject to a moving boundary condition , 1955 .

[58]  Alain J. Kassab,et al.  Localized Meshless Modeling of Natural-Convective Viscous Flows , 2008 .

[59]  Fausto Arpino,et al.  Artificial compressibility based CBS solutions for double diffusive natural convection in cavities , 2013 .

[60]  Xin Li,et al.  Trefftz Methods for Time Dependent Partial Differential Equations , 2004 .

[61]  Gregor Kosec,et al.  Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations , 2012 .

[62]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[63]  Arnaud G. Malan,et al.  An artificial compressibility CBS method for modelling heat transfer and fluid flow in heterogeneous porous materials , 2011 .