A meshless local collocation method using radial basis functions (LRBFCM)
has been developed to model the thermomechanical phenomena during the process of
DC casting. The model uses elastic-viscoplastic constitutive relations to describe the 
inhomogeneous material below the coherency isotherm. It is coupled with the results of the fluid 
flow model, which is used to determine the computational domain and to
calculate the thermal strain

Mavric, Bostjan

Sarler, Bozidar

English

A meshless local collocation method using radial basis functions (LRBFCM)

has been developed to model the thermomechanical phenomena during the process of

DC casting. The model uses elastic-viscoplastic constitutive relations to describe the 

inhomogeneous material below the coherency isotherm. It is coupled with the results of the fluid 

flow model, which is used to determine the computational domain and to

calculate the thermal strain

UPCommons

Application of local RBF collocation method to  prediction  of  mechanics-related phenomena during dc casting of aluminium alloys

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IS - Coupled Problems in Liquid-Solid SystemsApplication of local RBF collocation method to prediction of mechanics-related casting defects during DC cast-ing of aluminium alloysVII International Conference on Computational Methods for Coupled Problems in Science and EngineeringCOUPLED PROBLEMS 2017M. Papadrakakis, E. On˜ate and B. Schrefler (Eds)APPLICATION OF LOCAL RBF COLLOCATION METHODTO PREDICTION OF MECHANICS-RELATEDPHENOMENA DURING DC CASTING OF ALUMINIUMALLOYSBOSˇTJAN MAVRICˇ∗ and BOZˇIDAR SˇARLER∗,†∗Institute of Metals and TechnologyLepi pot 11, SI-1000 Ljubljana, Sloveniae-mail: bostjan.mavric@imt.si, web page: http://www.imt.si†Faculty of Mechanical Engineering, University of LjubljanaAsˇkercˇeva 6, SI-1000 Ljubljana, Sloveniae-mail: bozidar.sarler@fs.uni-lj.si, web page: http://www.fs.uni-lj.siKey words: coupled fluid flow and solid mechanics, DC casting, Radial Basis Functions,Viscoplasticity, Solidification, Local Radial Basis Function Collocation MethodAbstract. A meshless local collocation method using radial basis functions (LRBFCM)has been developed to model the thermomechanical phenomena during the process ofDC casting. The model uses elastic-viscoplastic constitutive relations to describe theinhomogeneous material below the coherency isotherm. It is coupled with the resultsof the fluid flow model, which is used to determine the computational domain and tocalculate the thermal strain.1 INTRODUCTIONDirect chill casting is a widespread technology for casting of aluminium billets andslabs, which are further used for extrusion. The achievement of high quality of the prod-uct is difficult since the thermomechanical phenomena that occur during DC casting ofaluminium billets can have a significant impact on the quality of the ingot. The maindefects caused by thermomechanics are hot tearing and cracking which can occur underspecific stress conditions [1].Because of the widespread use of the process in industry, a significant effort was ded-icated to the development of numerical models of the process with the aim of improvingthe quality of the billet at optimal productivity. This resulted in many numerical modelsof the process, which consider several aspects of the physical problem and are stated eitherwith the finite-element or the finite-volume method.1135Bosˇtjan Mavricˇ and Bozˇidar SˇarlerIn recent years, a new class of methods has emerged and has started to compete withthe classic discretization methods. Various meshless methods were successfully used tosolve a wide spectrum of scientific and engineering problems. In contrast to the estab-lished numerical methods, the meshless methods aim to reduce the complexity of thestructure associated with the node arrangement, removing the need for polygonization ofthe computational domain.The LRBFCM, which is used in the present paper, has been already successfully appliedto several coupled transport problems in solidification. The first were models of continuouscasting of steel [2] including the influence of electromagnetic braking [3, 4]. Later on, themethod was improved with a sophisticated adaptive algorithm allowing for the design ofhigh-accuracy macrosegregation models [5].Although the first implementations of the method dealt with solid mechanics problems[6], a lot of effort was invested into development of the method before the application ofthe method to the mechanical problems in metal processing. The first use was in a modelof hot-rolling process, coupling the thermal transport with the large plastic deformations[7].In this contribution we expand the mentioned efforts by developing a meshless modelof thermomechancis during DC casting of aluminium alloys. The model describes thebehavior of the non-homogeneous, viscoplastic material in steady state conditions. Themodel is one-way coupled to the results of the associated meshless heat and momentumtransfer model [13]. This model supplies the thermomechanical model with the steadystate values of liquid fraction, pressure, temperature and electromagnetic force. Thesupplied fields are used by the thermomechanical model to determine the geometry of thecomputational domain and the thermal strain, which is the driving term.2 GOVERNING EQUATIONSThe thermomechanical model is stated in small-strain approximation and considersthree contributions to the total strain: the elastic strain εe, the viscoplastic strain εp andthe thermal strain εt. The thermal strain is the driving term of the model and is givenbyεt(T ) = IT∫Tsα(T )dT = Iεt, (1)where Ts is the solidus temperature and α(T ) is the temperature dependent coefficient ofthermal expansion. The viscoplastic strain rateε˙p = −vcast∇εp + 3τ2σeε˙0(T,σ). (2)is determined by two terms. The first term is the advection term, which is the result ofthe Eulerian description of the material moving with the casting velocity vcast through2136Bosˇtjan Mavricˇ and Bozˇidar Sˇarlerthe computational domain. The second term describes the contribution of the viscoplasticdeformation, caused by the stresses in the billet. In the expression, the stress is presentin the octahedral stress tensor τ = σ − Itrσ/3 and the effective stress σe =√2τ : τ/3.The effective strain rate ε˙0(T,σ) is given byε˙0(T,σ) =A exp(− QRT)(σeσ0)n, if T < TsA′(σeσ′0)n′, otherwise.(3)Two variations of the Norton-Hoff’s law are used to describe the behavior of the materialin the mushy zone (Ts < T < Tcoh) and in the solid material (T < Ts) as proposed in[8]. By expressing the total strain with displacement vector u, the following equilibriumequation is obtained0=G∇2u+ (G+ λ)∇(∇ · u) (4)+∇λ(∇ · u) +∇G(∇u+ (∇u)T )−∇ · (2Gεp)−∇((3λ+ 2G)εt)+f g.The equation accounts for inhomogeneous material properties and coupling of deformationfield with thermal and plastic strain. The dependence of Lame parameters λ and G ontemperature is obtained from JMatPro database for the considered alloy [14]. The bodyforce f g is the gravitational body force.2.1 Boundary conditionsThe boundary conditions of the problem and the computational domain are illustratedin Figure 1. The problem is axisymmetric, so the cylindrical coordinate system with thecoordinates (r, z) is used. The computational domain is determined by the extent of thecomputational domain of the heat and mass transfer model. At the top, the computationaldomain is limited by the position of the isoline at coherency temperature Tcoh, above whichthe material cannot transfer stress and is governed by the equations of fluid dynamics.At the coherency isoline, the material has to support the metalostatic pressure of theliquid t = −pn, where p is the pressure of the liquid calculated by the heat and momentumtransfer model. The center of the billet coincides with the symmetry axis, where ur = 0and ∂uz∂r= 0. At the bottom of the computational domain we set uz = 0 and tr = 0. Theouter surface is traction free t = 0 except at the top, where the contact with the moldis possible. In that area, an iterative method is used to satisfy the boundary conditionsur ≤ 0 and tr ≤ 0.There are no specific boundary conditions for the viscoplastic strain for the most of theboundary and the rate equation is used to calculate the values on the boundary also. The3137Bosˇtjan Mavricˇ and Bozˇidar Sˇarleronly exception to this is the top boundary, where the material is solidifying. We assumethat the freshly solidified material does not accumulate any plastic strain by setting εp = 0at the top boundary.3 NUMERICAL METHOD3.1 Spatial discretizationThe spatial discretization is in this contribution achieved by the LRBFCM. Elaborationof the method for linear thermoelasticity is given in [9, 10].To formulate the method, a node arrangement is constructed with points positioned inthe domain Ω and on boundary Γ. For each point in the computational domain rl a setof nearby points, called local subdomain, is selected. The points in the subdomain areused to construct the local interpolation of the solution. The selection of the subdomaininduces mapping ls(k), which maps the local index k = 1, . . . , lN in the subdomain l tothe global index in the range 1, . . . , N , where N is the total number of points in the nodearrangement.On each subdomain an interpolation of the unknown solution y is constructed by usingmultiquadrics (MQ). A MQ centered at node l is given byΦl(r) =√2h2l|r − rl|2 + 1. (5)Here  is the shape parameter and hi is the size of the subdomain on which the interpo-lation is being constructed. It is defined ashl =√√√√ lN∑j=1|rls(j) − rl|2lN − 1 . (6)Since MQs are only conditionally positively definite, the interpolation problem formulatedsolely by MQs might be ill-conditioned. The remedy is to augment the interpolation prob-lem with m monomials pi(r), i = 1, . . . ,m, which constitute a basis for the polynomialsof certain order. The approximation of an arbitrary field y is then given byyξ(r) ≈lN∑i=1lαi,ξΦls(i)(r) +m∑i=1lα(lN+i),ξpi(r) =lN+m∑i=1lαi,ξΨi(r). (7)Here l is the index of the subdomain centered on the node closest to the evaluation pointr. The index ξ = 1, . . . , nd is running over the components of the physical field that isbeing interpolated. In this equation the basis function Ψi(r) is implicitly defined. It iseither MQ or a monomial, depending on the value of i.4138Bosˇtjan Mavricˇ and Bozˇidar SˇarlerThe values of the coefficients lαi,ξ are determined by solving an linear system Aα = γ,where the components of A arelAji,ξχ =Ψi(rls(j))δξχ if rls(j) ∈ ΩBξχ(rls(j))Ψi(rls(j)) if rls(j) ∈ Γpj(rls(i))δξχ if j ≥ lN and i ≤ lN0 otherwise(8)and the elements of the right-hand side vector γ bylγj,ξ =yξ(rls(j)) if rls(j) ∈ Ωbξ(rls(j)) if rls(j) ∈ Γ0 if j ≥ lN. (9)If a point in a subdomain lies on the boundary, linear boundary conditions of the form∑ξ Bχξ(r)yξ(r) = bχ(r) are included in the local interpolation, forcing the local interpo-lation to analytically satisfy the boundary conditions.From this construction, the result of applying a linear differential operator D to thelocal interpolation isnd∑χ=1Dξχ(rl)yχ(rl) ≈lN+m∑k=1nd∑χ=1lγk,χlN+m∑i=1nd∑ζ=1lA−1ik,ζχDξζΨi(rl). (10)By introducing discretization coefficients lwk,ξχ given bylwk,ξχ =lN+m∑i=1nd∑ζ=1lA−1ik,ζχDξζΨi(rl). (11)this givesDy(r)ξ =lN+m∑k=1nd∑χ=1lγk,χlwk,ξχ, (12)an expression similar to the finite-difference type of methods.3.2 Time discretizationThe viscoplastic strain rate equation causes the problem to be time-dependent andrequires use of a time discretization scheme. In this paper, the implicit Euler method isused. The resulting system of nonlinear equations is solved simultaneously with the stressequilibrium equations. The unknowns are ur, uz, εprr, εprz, εpzz and εpφφ.To obtain the solution, Bouaricha tensor method is used. The linear solves required bythe method are performed by FGMRES preconditioned by PARDISO solves with Jacobianthat is evaluated at the start of each time step.5139Bosˇtjan Mavricˇ and Bozˇidar SˇarlerCasting parameter ValueMelt temperature 730 ◦CCasting speed 80 mm/minDC water temperature 20 ◦CDC water flux 7 m3/hTable 1: Casting parameters used inHMTM of DC casting of Al-Cu alloys.Law parameter ValueA 9 · 105 s−1Q 145 kJ/kgn 3.8σ0 4 MPaA′ 10−6 s−1n′ 4σ′0 1 MPaTable 2: Parameters of the viscoplasticconstitutive law given by equation (3).4 RESULTSThe unknown displacement vector and viscoplastic strain are the primary results ofthe model. They are illustrated in figures 2 and 3 for the Al 4% Cu alloy. The materialproperties and process parameters are given in tables 2 and 1, respectively. The plots inthe figures demonstrate the typical results of the model.Because of the thermal contraction, the component ur is negative throughout the billet,except for a thin strip at the top of the mushy zone. The component negative values ofuz are caused by thermal contraction and the gravitational force, combined with the factthat the z coordinate of the bottom boundary is kept fixed.The components of the plastic strain have a more complicated spatial dependence.Most of the changes occur in the mushy zone or immediately below it. No changes of theplastic strain components outside this area are the result of the exponential damping ofthe viscoplastic strain rate as temperature decreases. Below this area, the r profiles ofthe stress components dependence remain constant.The impact of the mold contact can be seen in the plots of components εprr and εpzzin figures 3a and 3b, respectively. The thin strip of high plastic strain at the top of themushy zone near the outer surface of the billet is caused by the compressive forces causedby the melt pressure. They result in compressive strain in the r direction and extensionin the z direction because of the volume-preserving nature of the plastic flow.By studying the profile of εprz shown in Figure 3d, we can gain insight on the conditionsthat are experienced by the material as it travels through the computational domain. Letstrack a thin ring of material with radius r = 0.1 m. The material in the ring becomescoherent at z = −0.10m. and starts to move downwards with the casting velocity. As itsolidifies, it is positioned on the innermost part of the solid shell. The outer surface of theshell is cooled as it moves downwards, causing it to contract. This contraction puts theobserved ring under compressive stress in circumferential direction, forcing it to contract.As enough material is accumulated on the inside of the shell, the observed ring changesits role. It starts to cool faster that the newly accumulated solid on the interior, which6140Bosˇtjan Mavricˇ and Bozˇidar Sˇarlerresults in it being subjected to tensile stress conditions. This results in the increase inthe plastic strain, which can only go on while the ring is at high enough temperature inthe area above z = −0.25 m. Below this position, the strain remains constant, since thestresses there are too small to cause the plastic deformation.z0rmold contactt = 0t = −pnTcohTsur = 0∂uz∂r= 0uz = 0 tr = 0computationaldomainFigure 1: Geometry of the DC casting model and the boundary conditions.5 CONCLUSIONSThe model presented in this paper is capable of modeling the viscoplastic stress andstrain during the process of DC casting. The model couples the thermomechanical modelof a viscoplastic solid with the steady state results from the model of heat and masstransfer. The results given in terms of displacement and viscoplastic strain show thebehavior expected from the casting practice.In the future, the model will be used to analyze the effect of casting parameters onhot-tearing susceptibility and on formation of porosity.ACKNOWLEDGMENTSThe research presented in this paper was funded by the Young Researcher programby Slovenian Ministry of Higher Education, Science and Sport and by companies Impol,7141Bosˇtjan Mavricˇ and Bozˇidar Sˇarler(a) ur (b) uzFigure 2: Displacement field.Slovenska Bistrica and Sˇtore Steel.REFERENCES[1] Eskin, D.G. Physical Metallurgy of Direct Chill Casting of Aluminum Alloys. Taylor& Francis, (2008).[2] Vertnik, R. and Sˇarler, B. Simulation of continuous casting of steel by a meshlesstechnique International Journal of Cast Metals Research (2009) 22:311–313.[3] Mramor, K. and Sˇarler, B. A meshless model of electromagnetic braking for thecontinuous casting of steel Materials and Technologies (2015) 49:961–967.[4] Mramor, K., Vertnik, R. and Sˇarler, B. Low and intermediate Re solution of lid drivencavity problem by local radial basis function collocation method CMES: ComputerModeling in Engineering & Sciences (2013) 92:327–352.[5] Kosec, G., Zalozˇnik, M., Sˇarler, B. and Combeau, H. A meshless approach towardssolution of macrosegregation phenomena Computers, Materials and Continua (2011)22:169–195.[6] Lee, C. K., Liu, X. and Fan, S. C. Local multiquadric approximation for solvingboundary value problems Computational Mechanics (2003) 30:396–409.8142Bosˇtjan Mavricˇ and Bozˇidar Sˇarler(a) εprr (b) εpzz(c) εprz (d) εpφφFigure 3: Viscoplastic strain.9143Bosˇtjan Mavricˇ and Bozˇidar Sˇarler(a) σrr (b) σzz(c) σrz (d) σφφFigure 4: Stress tensor components in MPa.10144Bosˇtjan Mavricˇ and Bozˇidar Sˇarler[7] Hanoglu, U. and Sˇarler, B. Simulation of Hot Shape Rolling of Steel in ContinuousRolling Mill by Local Radial Basis Function Collocation Method CMES: ComputerModeling in Engineering & Sciences (2015) 109:447–479.[8] Nallathambi, A. K., Tyagi, M., Specht, E. and Bertram, A. Thermomechanical sim-ulation of direct chill casting Transactions of The Indian Institute of Metals (2011)64:13–19.[9] Mavricˇ, B. and Sˇarler, B. Local radial basis function collocation method for linearthermoelasticity in two dimensions International Journal of Numerical Methods forHeat & Fluid Flow (2015) 25:1488–1510.[10] Mavricˇ, B. and Sˇarler, B. Application of the RBF collocation method to transientcoupled thermoelasticity International Journal of Numerical Methods for Heat &Fluid Flow (2017) 27(5).[11] Ludwig, O., Drezet, J. M., Martin, C. L. and Suery, M. Rheological behavior of Al-Cu alloys during solidification constitutive modeling, experimental identification, andnumerical study Metallurgical and Materials Transactions A (2005) 36:1525–1535.[12] Suyitno, O., Kool, W. H. and Katgerman, L. Integrated Approach for Prediction ofHot Tearing Metallurgical and Materials Transactions A (2009) 40:2388–2400.[13] Hatic´, V., Mavricˇ, B., Kosˇnik, N., and Sˇarler, B. Simulation of direct chill castingunder the influence of a low-frequency electromagnetic field Applied MathematicalModelling (2017) (accepted).[14] Saunders, N., Guo, U. K. Z., Li, X., Miodownik, A. P. and Schille´, J.-Ph. IntegratedApproach for Prediction of Hot Tearing JOM (2003) 55:60–65.11145

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Application of local RBF collocation method to prediction of mechanics-related phenomena during dc casting of aluminium alloys

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