- Email: [email protected]
Solidification modeling in continuous casting by finite point method
Journal of Materials Processing Technology 192–193 (2007) 511–517
Solidification modeling in continuous casting by finite point method Lei Zhang a,∗ , Yi-Ming Rong a , Hou-Fa Shen b , Tian-You Huang b a
Department of Manufacturing Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, United States b Department of Mechanical Engineering, Tsinghua University, Beijing 100084, PR China
Abstract In this paper, a meshless method, finite point method, is studied and applied to model metal solidification processes in continuous casting. An additional term is added to stabilize the computation with Neumann boundary. The enthalpy method is used to calculate the latent heat and the corresponding iterative solution is given. An iteration scheme for nonlinear material calculation is also constructed. The model is verified by the classical Stefan problem and a 2D FEM solidification example. And then it is applied to the simulation of the solid shell growth in the continuous casting of a large square bland in mold. The result is coincided with the measurement. © 2007 Elsevier B.V. All rights reserved. Keywords: Meshless method; Finite point method; Solidification; Stefan problem; Continuous casting
1. Introduction Continuous casting is globally one of the most noted processes for the production of metals. In 2003, the whole steel production of China was 2.2 billion tonnes, and 95.3% of them were made from continuous casting [1]. A typical continuous casting process is shown in Fig. 1 where the molten metal is poured from the ladle into the tundish and then runs through a submerged entry nozzle into a mold cavity. The mold is watercooled so that enough heat is extracted to solidify a shell of sufficient thickness. The shell is withdrawn from the bottom of the mold at a “casting speed” that matches the inflow of the liquid metal. The process is ideally operated at a steady state. Below the mold, water is sprayed to further extract heat from the surface of strand, which eventually becomes fully solid. Finally, the solidified strand is straightened, cut, and then discharged. Cracks due to thin thickness of the solid shell at mold exit is the main defect that will not only decrease the production efficiency and but also cause fatal accident. The crack is closely related to the solidification process in mold [3]. Continuous casting is operated at high temperature, which is usually above the melting point of steel. Therefore the measurement of the temperature variations is almost impossible. Meanwhile, physical simulation of continuous casting is very expensive too. So numerical simulation tools, primarily finite element method
∗
Corresponding author. Tel.: +1 508 831 5825; fax: +1 508 831 6412. E-mail address: [email protected] (L. Zhang).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.04.092
(FEM) based, are normally used to mathematically simulate the process [3]. However, elements are the building blocks of FEM, and mesh distortion will fail the computation when FEM treats the discontinuities, i.e. the deformation of mesh which does not coincide with the original mesh lines. The discontinuities are usually observed in the large-scale deformation problem, and the prediction of crack growth with arbitrary and complex paths. The solution is to remesh in each step of the evolution. But this may leads to degradation of accuracy and complexity in the computer program, and the burden associated with the tedious adaptive and interactive re-meshing. Compared with FEM, meshless method dose not use elements. In meshless method, the approximation is constructed entirely in terms of nodes. In result, the discontinuities can be treated by free adding or deleting nodes whenever and wherever needed to simulate the new deformation. Therefore, it is possible to solve large classes of problems, such as large-scale deformation and crack growth problems, which are very awkward with mesh-based methods [4,5]. The meshless method, also called meshfree or element-free method was developed about 20 years ago. It starts with the establishment of the smooth particle (SPH) method [6], which is used for modeling astrophysical phenomena without boundaries. Later, Belytschko et al. [7] developed the element-free Galerkin method (EFG), and the successful application of EFG also triggered the great research effort devoted to the meshless method [8]. So far more than ten meshless methods or schemes have been developed, and a few books on them are available (see [9,10,5,8]).
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L. Zhang et al. / Journal of Materials Processing Technology 192–193 (2007) 511–517
Fig. 2. Schematic representation of FPM.
The boundary value problem can be expressed as
Fig. 1. Schematic representation of the continuous casting process [2].
Among the meshless methods, finite point method (FPM) is one of the easiest to implement [11,12] and one can expresses the really meaning of meshless. In fact, even be called meshless, but some methods still need “background cell”, which is required to evaluate the integration in the Galerkin weak-form. Background cell is regular mesh, which will cover the whole solution area. EFG belongs to this catalog. Compared with these methods, FPM is a pure meshless method, because it dose not need background cell integration. FPM only needs nodes, so it can totally escape from remeshing by free adding or deleting nodes whenever or wherever needed to treat the discontinuities. Therefore, it makes FPM efficient, and advantageous in solving large-scale deformation and crack growth problems. Good understanding of the solidification is the fundamental of solving crack problem in continuous casting. Therefore, in this paper, FPM is applied to continuous casting where a solidification model is constructed based on FPM. The model is verified first and then employed to simulate the solid shell growth of continuous casting large square bland in mold. The predicted results are coincided with the measurements. This indicates that meshless method is a potential numerical analysis tool and valuable for the analysis of the continuous casting process.
Lu(x) − q = 0,
x ∈ Ω,
u(x) − ud = 0,
x ∈ Γd
Lt u(x) − t = 0,
x ∈ Γt , (1)
where unknown function u(x) here represent scalar function such as temperature in solving heat transfer problem, or vector function such as displacement, stress and strain in elastic problem, L and Lt represent linear differential operators, u¯ d is the known boundary condition on Γ d , the subscripts d and t represent the Dirichlet and Neumann boundary conditions, q, t, and x are variables. In FPM, The solution area is represented by a series of nodes or collocation points which are scattered both in the area and on its boundary, as shown in Fig. 2. And at point i, the unknown function u(x) is approximated by uh (x): u(x) ≈ uh (x) =
n
ψI (x) · uI
(2)
I
where ψI is the shape function obtained using the moving leastsquare (MLS) method [13]. Eq. (1) is implemented on all the points with substituting Eq. (2) into Eq. (1), then Eq. (1) becomes: Luh (xi ) − qi = 0, uh (xi ) − u¯ d = 0,
xi ∈ Ω,
Lt uh (xi ) − ti = 0,
xi ∈ Γd
xi ∈ Γt (3)
Eq. (3) can be easily implemented in programming. And because there is no integration needed, this method is more efficient compared with the ones that need “background cell” [4,5]. 3. Model descriptions
2. Finite point method
3.1. Equilibrium equation and boundary conditions
Finite point method was proposed by Onate et al. [11,12]. For a typical boundary value problem, the weighted residual method and the idea of point collocation are used with the unknown function replaced by the moving least-square (MLS) approximation. Then the FPM collocation form of the differential equations can be obtained.
The energy equilibrium equation and boundary conditions of solidification processes are [3]. Equilibrium equation: ρcp
∂T − k∇ 2 T − qv = 0 ∂t
(4a)
L. Zhang et al. / Journal of Materials Processing Technology 192–193 (2007) 511–517
Heat flux boundary condition: −k
∂T + qn = 0 ∂n
(4b)
Initial condition: T |t=0 = Tin
3.2. Neumann boundary stabilization scheme For solving convection and diffusion problems by FPM, Neumann boundary needs to be stabilized first. For non-selfadjoint problems such as convection-diffusion problems, the domination of convection item will lead to unstable problems. So a special treatment is needed to stabilize the solution by applying the standard conservation laws expressing balance of momentum and mass over a control domain. The related works can be seen in Onate’s papers [11,12], and a stabilization item is added to Neumann boundary: ∂T h + qn − r = 0 ∂n 2
(5)
where r = k2 T + qv and h is the characteristic length [11,12]. 3.3. Discretization in space By using MLS [13], the approximation function of the unknown temperature T of node i at time t can be expressed as n Ti |t ≈ Tih = ψI · TI |t t
(6)
I
n I
3.5. Latent heat treatment In Eq. (4a), if setting qv equal to zero, and importing enthalpy [16], the energy equilibrium becomes k∇ 2 T = ρ
∂h ∂T ∂T ∂t
∇ 2 ψI T h . t
n ∂ T˜ t ∂ψI = · TIh t ∂n ∂n
(7a)
(7b)
I
3.4. Discretization in time General two point difference format of T with respect to time t [14] is substituted into the energy equilibrium Eq. (4a), then it becomes: T˜ i t+t − T˜ i t ρcp (8) − θ · ri |t+t − (1 − θ) · ri |t = 0 t
(10)
where h is enthalpy. Replacing ∂h/∂T with its difference form and ∂T/∂t with its implicit form of discretization in time, Eq. (10) becomes:
∼
˜ Ti t+t − Ti |t h T˜ i t+Δt − h T˜ i t · − ri |t+t ρ T t =0 (11) where ri |t+t = k∇ 2 T˜ i t+t . Eq. (11) cannot be solved directly, so the corresponding iterative solution is given as m+1
T˜ i t+t − T˜ i t h T˜ i i − h ( Ti |t ) ρ · − ri |m+1 t+t = 0 T t (12) m+1 m+1 where ri |t+t = k∇ 2 T˜ i t+t , and m is the number of iteration. 3.6. Nonlinear material treatment If the material properties are functions of temperature, the energy equilibrium equation for a solidification process can be expressed as k(T )∇ 2 T = ρ(T )
Then the derivatives of T with respect to space become the derivatives of the shape function ψ with respect to space: ∇ 2 t T˜ |t =
where θ is constant between 0 and 1, representing different discretization form [15]: ⎧ implicit ⎪ ⎨ 0, (9) θ = 0.5, Crank–Nicolson ⎪ ⎩ 1, explicit
(4c)
where T is the temperature (K), t the time (s), cp the specific heat (J/kg K), ρ the density (kg/m3 ), k the thermal conductivity (W/m K), qv the energy source term (W/m3 ), n the normal to the boundary, qn the known heat flux at the boundaries (W/m2 ) and Tin is the initial temperature (K).
−k
513
∂h ∂T ∂T ∂t
(13)
Same as Eq. (11), replacing ∂h/∂T and ∂T/∂t, and setting k and ρ the function of the present temperature at time t + t, then
T˜ i t+t − T˜ i t h T˜ i t+t − h T˜ i t
· ρ Ti |t+t · T t | − ri t+t = 0 (14) where ri |t+t = k T˜ i t+t · ∇ 2 T˜ i t+t .Eq. (14) cannot be solved directly either. The corresponding iterative solution is
m ˜i ˜i ˜i h T˜ i m − h T − T T t t t+t t+t m ρ T˜ i t+t · · T t − ri |m+1 t+t = 0 where ri |m+1 t+t
m+1 m = k T˜ i t+t · ∇ 2 T˜ i t+t .
(15)
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L. Zhang et al. / Journal of Materials Processing Technology 192–193 (2007) 511–517
Fig. 5. The isotherms at 150,000 s (unit: ◦ C).
Fig. 3. 1D Stefan solidification problem.
4. Verifications 4.1. 1D case Since the theoretical solution of 1D solidification problem can be derived from the analytical model, the FPM model is first validated with a typical 1D solidification problem. Fig. 3 shows a 1D Stefan problem. The problem is that the liquid metal in the one-dimension unlimited-length mold is cooled by water on the left side. The liquid metal solidifies as time goes by in x direction and the solid front (i.e., interface between liquid and solid) is the function of time R(t), which can be exactly predicted. This problem can be described by the following equations [17]: cp ρ
∂T ∂2 T =k 2, ∂t ∂x
∂T = 0, ∂x t = 0,
T = T0 ,
t > 0, x = ∞, 0
t > 0,
x = 0,
T = Tl , (16)
The exact solution for the location of the solid front is first given by Stefan [18]: kt R(t) = 2γ (17a) cp ρ where γ is obtained from the root of the transcendental equation: √ πγ exp(γ 2 ) erf(γ) = Ste (17b)
Fig. 6. 2D solidification problem [19].
and cp (Tl − T0 ) (17c) L where L is the latent heat of solidification and T0 and Tl are the boundary temperatures, T0 < Tl = melting point of the steel. The thermophysical properties used in this problem are that specific heat cp is 1079 J/kg K, density ρ is 2700 kg/m3 , thermal conductivity k is 94 W/m K, Tl is 933 K, T0 is 893 K, and L is 398 kJ/kg. The FPM calculated results and the exact solid front locations from Eq. (17a) are plotted in Fig. 4 for comparison. It shows that the FPM predicted results matched very well with the precise solution. The isotherms at t = 150,000 s when the entire domain becomes solid is shown in Fig. 5. Ste =
4.2. 2D case An FEM solidification example is employed here to verify the FPM model in 2D case. Fig. 6 is the quarter model of the Table 1 Material properties [18]
Fig. 4. FPM predicted solid front locations with the precise solution.
Parameters
Value
Conductivity of solid, ks (W/m K) Conductivity of liquid, kl (W/m K) Specific heat of solid, cp,s (J/kg K) Specific heat of liquid, cp,l (J/kg K) Density, ρ (kg/m3 ) Latent heat, L (kJ/kg) Melting point temperature, Tl (K) Initial temperature, Tin (K)
237.65 94.14 903.74 1079.47 2700 397.48 933 973
L. Zhang et al. / Journal of Materials Processing Technology 192–193 (2007) 511–517
515
Fig. 7. Solid front locations calculated with FPM in comparison to FEM results. (a) Geometry model; (b) photograph of mold.
Table 2 Production parameters Size of the mold (mm) Bevel size (mm) Meniscus (mm) Length of mold (mm) Casting speed (m/min) Carbon percentage of steel (wt.%) Pouring temperature (◦ C)
280 × 380 12 × 45◦ 80 850 0.7 0.7 1505
Fig. 8. Continuous casting of large bland. (a) Geometry model; (b) discretization model (quarter).
problem. Liquid metal in a square mold is cooled by the constant flux (equal to 2000 MW/m2 ) at boundary and solidified. The computing parameters are shown in Table 1. The solid front location at the diagonal and the y symmetrical axial of the mold against time is calculated by the FPM model. The results are displayed in Fig. 7 with the comparison to the FEM predicted results.
Fig. 9. Continuous casting large bland mold.
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L. Zhang et al. / Journal of Materials Processing Technology 192–193 (2007) 511–517
In Fig. 7, the FPM results represented by lines and the FEM results represented by symbols are well matched, that implies the FPM model works in 2D case. 5. Application Solid shell growth during a continuous casting process with large square bland in mold is analyzed with the FPM model in this research. The large square bland in continuous casting is shown in Fig. 8. And Fig. 9 is the shape and size of the mold and its quarter node discretization model for calculation. Based on the assumptions on the working conditions, the heat transfer along the casting direction is very small and ignored. And the heat flux on the surface of the solid shell will react on the change of casting speed and is known. The solidification process in the continuous casting in model can be mathematically described by Eqs. (4a)–(4c), and then it can be simulated by the FPM model. The corresponding production parameters are listed in Table 2. In Fig. 10a, the chart shows the predicted thickness, which is of the solidified shell at the narrow face center, against the distance from meniscus along the casting direction. And below the chart it shows the photo of the shell slice, which is chip off at the narrow face center along the casting direction. The
Fig. 11. Temperature contour at mold exit (unit: ◦ C).
corresponding thickness predictions and photo of the slice of the shell at the wide face center are shown in Fig. 10b. In continuous casting, the liquid metal is poured into the water-cooling mold (see Fig. 8), and solidifies due to contact with the mold. The solidification thickness increases from the surface to the center of the liquid metal and forms a solid shell with different thickness along the casting direction. Fig. 10a shows the thickness of the shell grows along y axis, and Fig. 10b shows the growth along x axis. It observed that the solid shell evenly grows in x and y direction. And at the mold exit, the thickness of the solidified shell at narrow face is 19.5 mm, and the thickness of the shell at wide face center is 19.1 mm. These predicted results are coincided with the measurements. The uniform temperature distribution of solid shell can also be seen in Fig. 11, which is the temperature contour at mold exit calculated by FPM. 6. Summary The meshless method, FPM, is studied for the first time in the simulation of solidification in continuous casting. The FPM solidification simulation model was developed with • Onate stabilization scheme for Neumann boundary; • nonlinear material treatment; • solidification enthalpy treatment.
Fig. 10. (a) Growth of the solid shell at narrow face center against the distance from meniscus. (b) Growth of the solid shell at wide face center against the distance from meniscus.
The model is validated with comparison to analytical solution in 1D and 2D cases. Then the solid shell growth of a continuous casting large square bland in mold is simulated by using the model. The calculation results are coincided with the measurement. Observations show that meshless method is a potential numerical analysis tool and would be valuable for the analysis of the continuous casting process with solving crack and large-scale deformation problems in the future.
L. Zhang et al. / Journal of Materials Processing Technology 192–193 (2007) 511–517
References [1] Chinese Steel Industry Statistics, Chinese Steel Industry Association, Beijing, 2003. [2] B.G. Thomas, Continuous Casting: Modeling’ The Encyclopedia of Advanced Materials, 2, Pergamon Elsevier Science Ltd., Oxford, UK, 1999. [3] Y. Gan, S. Chou, Z. Xiao, Numerical Simulation of Continuous Casting Processes, Metallurgy Industry Press, Beining, 2001. [4] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Eng. 139 (1–4) (1996) 3–47. [5] G.R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, Boca Raton, USA, 2002. [6] L.B. Lucy, A numerical approach to the testing of the fission hypothesis, Astron. J. 8 (12) (1977) 1013–1024. [7] T. Belytschko, Y.Y. Lu, L. Gu, Element Free Galerkin Methods, Int. J. Numer. Meth. Eng. 37 (1994) 91–100. [8] X. Zhang, Y. Liu, Meshless Methods, Tsinghua Press, Beijing, 2004. [9] S.N. Atluri, S. Shen, The Meshless Local Petrov–Galerkin (MLPG) Method, Tech Science Press, Stuttgart, 2002. [10] M. Griebel, M.A. Schweitzer, Meshfree Methods for Partial Differential Equations, 26, Springer-Verlag, Berlin, 2002.
517
[11] E. Onate, S. Idelsohn, O.C. Zienkiewics, R.L. Taylor, A finite point method in computational mechanics. Applications to convective transport and fluid flow, Int. J. Numer. Methods Eng. 39 (1996) 3839–3866. [12] E. Onate, S. Idelsohn, O.C. Zienkiewics, R.L. Taylor, S. Sacco, A stabilized finite point method for analysis of fluid mechanics problems, Comput. Methods Appl. Mech. Eng. 139 (1996) 315–346. [13] P. Lancaster, K. Salkauskas, Surfaces generated by moving least-squares methods, Math. Comput. 37 (155) (1981) 141–158. [14] X.Q. Kong, Application of FEM in Heat Transfer Process, Science Press, Beijing, 1998. [15] J. Crank, P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, in: Proceedings of the Cambridge Philosophical Society, vol. 43, 1947, pp. 50–64. [16] N.R. Eyres, D.R. Hartree, The calculation of variable heat flow in solid, Phil. Trans. R. Soc. A 240 (1946) 1–57. [17] J. Chen, Handbook of Continuous Casting of Steel, Metallurgy Industry Press, Beijing, 1995. [18] M.N. Ozisik, Heat Conduction, Wiley, New York, 1980. [19] C.Y. Li, S.V. Garimella, J.E. Simpson, Fixed-grid front-tracking algorithm for solidification problems, part I: Method and validation, Numer. Heat Transfer Part B 43 (2) (2003) 117–141.
Solidification modeling in continuous casting by finite point method Lei Zhang a,∗ , Yi-Ming Rong a , Hou-Fa Shen b , Tian-You Huang b a
Department of Manufacturing Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, United States b Department of Mechanical Engineering, Tsinghua University, Beijing 100084, PR China
Abstract In this paper, a meshless method, finite point method, is studied and applied to model metal solidification processes in continuous casting. An additional term is added to stabilize the computation with Neumann boundary. The enthalpy method is used to calculate the latent heat and the corresponding iterative solution is given. An iteration scheme for nonlinear material calculation is also constructed. The model is verified by the classical Stefan problem and a 2D FEM solidification example. And then it is applied to the simulation of the solid shell growth in the continuous casting of a large square bland in mold. The result is coincided with the measurement. © 2007 Elsevier B.V. All rights reserved. Keywords: Meshless method; Finite point method; Solidification; Stefan problem; Continuous casting
1. Introduction Continuous casting is globally one of the most noted processes for the production of metals. In 2003, the whole steel production of China was 2.2 billion tonnes, and 95.3% of them were made from continuous casting [1]. A typical continuous casting process is shown in Fig. 1 where the molten metal is poured from the ladle into the tundish and then runs through a submerged entry nozzle into a mold cavity. The mold is watercooled so that enough heat is extracted to solidify a shell of sufficient thickness. The shell is withdrawn from the bottom of the mold at a “casting speed” that matches the inflow of the liquid metal. The process is ideally operated at a steady state. Below the mold, water is sprayed to further extract heat from the surface of strand, which eventually becomes fully solid. Finally, the solidified strand is straightened, cut, and then discharged. Cracks due to thin thickness of the solid shell at mold exit is the main defect that will not only decrease the production efficiency and but also cause fatal accident. The crack is closely related to the solidification process in mold [3]. Continuous casting is operated at high temperature, which is usually above the melting point of steel. Therefore the measurement of the temperature variations is almost impossible. Meanwhile, physical simulation of continuous casting is very expensive too. So numerical simulation tools, primarily finite element method
∗
Corresponding author. Tel.: +1 508 831 5825; fax: +1 508 831 6412. E-mail address: [email protected] (L. Zhang).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.04.092
(FEM) based, are normally used to mathematically simulate the process [3]. However, elements are the building blocks of FEM, and mesh distortion will fail the computation when FEM treats the discontinuities, i.e. the deformation of mesh which does not coincide with the original mesh lines. The discontinuities are usually observed in the large-scale deformation problem, and the prediction of crack growth with arbitrary and complex paths. The solution is to remesh in each step of the evolution. But this may leads to degradation of accuracy and complexity in the computer program, and the burden associated with the tedious adaptive and interactive re-meshing. Compared with FEM, meshless method dose not use elements. In meshless method, the approximation is constructed entirely in terms of nodes. In result, the discontinuities can be treated by free adding or deleting nodes whenever and wherever needed to simulate the new deformation. Therefore, it is possible to solve large classes of problems, such as large-scale deformation and crack growth problems, which are very awkward with mesh-based methods [4,5]. The meshless method, also called meshfree or element-free method was developed about 20 years ago. It starts with the establishment of the smooth particle (SPH) method [6], which is used for modeling astrophysical phenomena without boundaries. Later, Belytschko et al. [7] developed the element-free Galerkin method (EFG), and the successful application of EFG also triggered the great research effort devoted to the meshless method [8]. So far more than ten meshless methods or schemes have been developed, and a few books on them are available (see [9,10,5,8]).
512
L. Zhang et al. / Journal of Materials Processing Technology 192–193 (2007) 511–517
Fig. 2. Schematic representation of FPM.
The boundary value problem can be expressed as
Fig. 1. Schematic representation of the continuous casting process [2].
Among the meshless methods, finite point method (FPM) is one of the easiest to implement [11,12] and one can expresses the really meaning of meshless. In fact, even be called meshless, but some methods still need “background cell”, which is required to evaluate the integration in the Galerkin weak-form. Background cell is regular mesh, which will cover the whole solution area. EFG belongs to this catalog. Compared with these methods, FPM is a pure meshless method, because it dose not need background cell integration. FPM only needs nodes, so it can totally escape from remeshing by free adding or deleting nodes whenever or wherever needed to treat the discontinuities. Therefore, it makes FPM efficient, and advantageous in solving large-scale deformation and crack growth problems. Good understanding of the solidification is the fundamental of solving crack problem in continuous casting. Therefore, in this paper, FPM is applied to continuous casting where a solidification model is constructed based on FPM. The model is verified first and then employed to simulate the solid shell growth of continuous casting large square bland in mold. The predicted results are coincided with the measurements. This indicates that meshless method is a potential numerical analysis tool and valuable for the analysis of the continuous casting process.
Lu(x) − q = 0,
x ∈ Ω,
u(x) − ud = 0,
x ∈ Γd
Lt u(x) − t = 0,
x ∈ Γt , (1)
where unknown function u(x) here represent scalar function such as temperature in solving heat transfer problem, or vector function such as displacement, stress and strain in elastic problem, L and Lt represent linear differential operators, u¯ d is the known boundary condition on Γ d , the subscripts d and t represent the Dirichlet and Neumann boundary conditions, q, t, and x are variables. In FPM, The solution area is represented by a series of nodes or collocation points which are scattered both in the area and on its boundary, as shown in Fig. 2. And at point i, the unknown function u(x) is approximated by uh (x): u(x) ≈ uh (x) =
n
ψI (x) · uI
(2)
I
where ψI is the shape function obtained using the moving leastsquare (MLS) method [13]. Eq. (1) is implemented on all the points with substituting Eq. (2) into Eq. (1), then Eq. (1) becomes: Luh (xi ) − qi = 0, uh (xi ) − u¯ d = 0,
xi ∈ Ω,
Lt uh (xi ) − ti = 0,
xi ∈ Γd
xi ∈ Γt (3)
Eq. (3) can be easily implemented in programming. And because there is no integration needed, this method is more efficient compared with the ones that need “background cell” [4,5]. 3. Model descriptions
2. Finite point method
3.1. Equilibrium equation and boundary conditions
Finite point method was proposed by Onate et al. [11,12]. For a typical boundary value problem, the weighted residual method and the idea of point collocation are used with the unknown function replaced by the moving least-square (MLS) approximation. Then the FPM collocation form of the differential equations can be obtained.
The energy equilibrium equation and boundary conditions of solidification processes are [3]. Equilibrium equation: ρcp
∂T − k∇ 2 T − qv = 0 ∂t
(4a)
L. Zhang et al. / Journal of Materials Processing Technology 192–193 (2007) 511–517
Heat flux boundary condition: −k
∂T + qn = 0 ∂n
(4b)
Initial condition: T |t=0 = Tin
3.2. Neumann boundary stabilization scheme For solving convection and diffusion problems by FPM, Neumann boundary needs to be stabilized first. For non-selfadjoint problems such as convection-diffusion problems, the domination of convection item will lead to unstable problems. So a special treatment is needed to stabilize the solution by applying the standard conservation laws expressing balance of momentum and mass over a control domain. The related works can be seen in Onate’s papers [11,12], and a stabilization item is added to Neumann boundary: ∂T h + qn − r = 0 ∂n 2
(5)
where r = k2 T + qv and h is the characteristic length [11,12]. 3.3. Discretization in space By using MLS [13], the approximation function of the unknown temperature T of node i at time t can be expressed as n Ti |t ≈ Tih = ψI · TI |t t
(6)
I
n I
3.5. Latent heat treatment In Eq. (4a), if setting qv equal to zero, and importing enthalpy [16], the energy equilibrium becomes k∇ 2 T = ρ
∂h ∂T ∂T ∂t
∇ 2 ψI T h . t
n ∂ T˜ t ∂ψI = · TIh t ∂n ∂n
(7a)
(7b)
I
3.4. Discretization in time General two point difference format of T with respect to time t [14] is substituted into the energy equilibrium Eq. (4a), then it becomes: T˜ i t+t − T˜ i t ρcp (8) − θ · ri |t+t − (1 − θ) · ri |t = 0 t
(10)
where h is enthalpy. Replacing ∂h/∂T with its difference form and ∂T/∂t with its implicit form of discretization in time, Eq. (10) becomes:
∼
˜ Ti t+t − Ti |t h T˜ i t+Δt − h T˜ i t · − ri |t+t ρ T t =0 (11) where ri |t+t = k∇ 2 T˜ i t+t . Eq. (11) cannot be solved directly, so the corresponding iterative solution is given as m+1
T˜ i t+t − T˜ i t h T˜ i i − h ( Ti |t ) ρ · − ri |m+1 t+t = 0 T t (12) m+1 m+1 where ri |t+t = k∇ 2 T˜ i t+t , and m is the number of iteration. 3.6. Nonlinear material treatment If the material properties are functions of temperature, the energy equilibrium equation for a solidification process can be expressed as k(T )∇ 2 T = ρ(T )
Then the derivatives of T with respect to space become the derivatives of the shape function ψ with respect to space: ∇ 2 t T˜ |t =
where θ is constant between 0 and 1, representing different discretization form [15]: ⎧ implicit ⎪ ⎨ 0, (9) θ = 0.5, Crank–Nicolson ⎪ ⎩ 1, explicit
(4c)
where T is the temperature (K), t the time (s), cp the specific heat (J/kg K), ρ the density (kg/m3 ), k the thermal conductivity (W/m K), qv the energy source term (W/m3 ), n the normal to the boundary, qn the known heat flux at the boundaries (W/m2 ) and Tin is the initial temperature (K).
−k
513
∂h ∂T ∂T ∂t
(13)
Same as Eq. (11), replacing ∂h/∂T and ∂T/∂t, and setting k and ρ the function of the present temperature at time t + t, then
T˜ i t+t − T˜ i t h T˜ i t+t − h T˜ i t
· ρ Ti |t+t · T t | − ri t+t = 0 (14) where ri |t+t = k T˜ i t+t · ∇ 2 T˜ i t+t .Eq. (14) cannot be solved directly either. The corresponding iterative solution is
m ˜i ˜i ˜i h T˜ i m − h T − T T t t t+t t+t m ρ T˜ i t+t · · T t − ri |m+1 t+t = 0 where ri |m+1 t+t
m+1 m = k T˜ i t+t · ∇ 2 T˜ i t+t .
(15)
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Fig. 5. The isotherms at 150,000 s (unit: ◦ C).
Fig. 3. 1D Stefan solidification problem.
4. Verifications 4.1. 1D case Since the theoretical solution of 1D solidification problem can be derived from the analytical model, the FPM model is first validated with a typical 1D solidification problem. Fig. 3 shows a 1D Stefan problem. The problem is that the liquid metal in the one-dimension unlimited-length mold is cooled by water on the left side. The liquid metal solidifies as time goes by in x direction and the solid front (i.e., interface between liquid and solid) is the function of time R(t), which can be exactly predicted. This problem can be described by the following equations [17]: cp ρ
∂T ∂2 T =k 2, ∂t ∂x
∂T = 0, ∂x t = 0,
T = T0 ,
t > 0, x = ∞, 0
t > 0,
x = 0,
T = Tl , (16)
The exact solution for the location of the solid front is first given by Stefan [18]: kt R(t) = 2γ (17a) cp ρ where γ is obtained from the root of the transcendental equation: √ πγ exp(γ 2 ) erf(γ) = Ste (17b)
Fig. 6. 2D solidification problem [19].
and cp (Tl − T0 ) (17c) L where L is the latent heat of solidification and T0 and Tl are the boundary temperatures, T0 < Tl = melting point of the steel. The thermophysical properties used in this problem are that specific heat cp is 1079 J/kg K, density ρ is 2700 kg/m3 , thermal conductivity k is 94 W/m K, Tl is 933 K, T0 is 893 K, and L is 398 kJ/kg. The FPM calculated results and the exact solid front locations from Eq. (17a) are plotted in Fig. 4 for comparison. It shows that the FPM predicted results matched very well with the precise solution. The isotherms at t = 150,000 s when the entire domain becomes solid is shown in Fig. 5. Ste =
4.2. 2D case An FEM solidification example is employed here to verify the FPM model in 2D case. Fig. 6 is the quarter model of the Table 1 Material properties [18]
Fig. 4. FPM predicted solid front locations with the precise solution.
Parameters
Value
Conductivity of solid, ks (W/m K) Conductivity of liquid, kl (W/m K) Specific heat of solid, cp,s (J/kg K) Specific heat of liquid, cp,l (J/kg K) Density, ρ (kg/m3 ) Latent heat, L (kJ/kg) Melting point temperature, Tl (K) Initial temperature, Tin (K)
237.65 94.14 903.74 1079.47 2700 397.48 933 973
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Fig. 7. Solid front locations calculated with FPM in comparison to FEM results. (a) Geometry model; (b) photograph of mold.
Table 2 Production parameters Size of the mold (mm) Bevel size (mm) Meniscus (mm) Length of mold (mm) Casting speed (m/min) Carbon percentage of steel (wt.%) Pouring temperature (◦ C)
280 × 380 12 × 45◦ 80 850 0.7 0.7 1505
Fig. 8. Continuous casting of large bland. (a) Geometry model; (b) discretization model (quarter).
problem. Liquid metal in a square mold is cooled by the constant flux (equal to 2000 MW/m2 ) at boundary and solidified. The computing parameters are shown in Table 1. The solid front location at the diagonal and the y symmetrical axial of the mold against time is calculated by the FPM model. The results are displayed in Fig. 7 with the comparison to the FEM predicted results.
Fig. 9. Continuous casting large bland mold.
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In Fig. 7, the FPM results represented by lines and the FEM results represented by symbols are well matched, that implies the FPM model works in 2D case. 5. Application Solid shell growth during a continuous casting process with large square bland in mold is analyzed with the FPM model in this research. The large square bland in continuous casting is shown in Fig. 8. And Fig. 9 is the shape and size of the mold and its quarter node discretization model for calculation. Based on the assumptions on the working conditions, the heat transfer along the casting direction is very small and ignored. And the heat flux on the surface of the solid shell will react on the change of casting speed and is known. The solidification process in the continuous casting in model can be mathematically described by Eqs. (4a)–(4c), and then it can be simulated by the FPM model. The corresponding production parameters are listed in Table 2. In Fig. 10a, the chart shows the predicted thickness, which is of the solidified shell at the narrow face center, against the distance from meniscus along the casting direction. And below the chart it shows the photo of the shell slice, which is chip off at the narrow face center along the casting direction. The
Fig. 11. Temperature contour at mold exit (unit: ◦ C).
corresponding thickness predictions and photo of the slice of the shell at the wide face center are shown in Fig. 10b. In continuous casting, the liquid metal is poured into the water-cooling mold (see Fig. 8), and solidifies due to contact with the mold. The solidification thickness increases from the surface to the center of the liquid metal and forms a solid shell with different thickness along the casting direction. Fig. 10a shows the thickness of the shell grows along y axis, and Fig. 10b shows the growth along x axis. It observed that the solid shell evenly grows in x and y direction. And at the mold exit, the thickness of the solidified shell at narrow face is 19.5 mm, and the thickness of the shell at wide face center is 19.1 mm. These predicted results are coincided with the measurements. The uniform temperature distribution of solid shell can also be seen in Fig. 11, which is the temperature contour at mold exit calculated by FPM. 6. Summary The meshless method, FPM, is studied for the first time in the simulation of solidification in continuous casting. The FPM solidification simulation model was developed with • Onate stabilization scheme for Neumann boundary; • nonlinear material treatment; • solidification enthalpy treatment.
Fig. 10. (a) Growth of the solid shell at narrow face center against the distance from meniscus. (b) Growth of the solid shell at wide face center against the distance from meniscus.
The model is validated with comparison to analytical solution in 1D and 2D cases. Then the solid shell growth of a continuous casting large square bland in mold is simulated by using the model. The calculation results are coincided with the measurement. Observations show that meshless method is a potential numerical analysis tool and would be valuable for the analysis of the continuous casting process with solving crack and large-scale deformation problems in the future.
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References [1] Chinese Steel Industry Statistics, Chinese Steel Industry Association, Beijing, 2003. [2] B.G. Thomas, Continuous Casting: Modeling’ The Encyclopedia of Advanced Materials, 2, Pergamon Elsevier Science Ltd., Oxford, UK, 1999. [3] Y. Gan, S. Chou, Z. Xiao, Numerical Simulation of Continuous Casting Processes, Metallurgy Industry Press, Beining, 2001. [4] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Eng. 139 (1–4) (1996) 3–47. [5] G.R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, Boca Raton, USA, 2002. [6] L.B. Lucy, A numerical approach to the testing of the fission hypothesis, Astron. J. 8 (12) (1977) 1013–1024. [7] T. Belytschko, Y.Y. Lu, L. Gu, Element Free Galerkin Methods, Int. J. Numer. Meth. Eng. 37 (1994) 91–100. [8] X. Zhang, Y. Liu, Meshless Methods, Tsinghua Press, Beijing, 2004. [9] S.N. Atluri, S. Shen, The Meshless Local Petrov–Galerkin (MLPG) Method, Tech Science Press, Stuttgart, 2002. [10] M. Griebel, M.A. Schweitzer, Meshfree Methods for Partial Differential Equations, 26, Springer-Verlag, Berlin, 2002.
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[11] E. Onate, S. Idelsohn, O.C. Zienkiewics, R.L. Taylor, A finite point method in computational mechanics. Applications to convective transport and fluid flow, Int. J. Numer. Methods Eng. 39 (1996) 3839–3866. [12] E. Onate, S. Idelsohn, O.C. Zienkiewics, R.L. Taylor, S. Sacco, A stabilized finite point method for analysis of fluid mechanics problems, Comput. Methods Appl. Mech. Eng. 139 (1996) 315–346. [13] P. Lancaster, K. Salkauskas, Surfaces generated by moving least-squares methods, Math. Comput. 37 (155) (1981) 141–158. [14] X.Q. Kong, Application of FEM in Heat Transfer Process, Science Press, Beijing, 1998. [15] J. Crank, P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, in: Proceedings of the Cambridge Philosophical Society, vol. 43, 1947, pp. 50–64. [16] N.R. Eyres, D.R. Hartree, The calculation of variable heat flow in solid, Phil. Trans. R. Soc. A 240 (1946) 1–57. [17] J. Chen, Handbook of Continuous Casting of Steel, Metallurgy Industry Press, Beijing, 1995. [18] M.N. Ozisik, Heat Conduction, Wiley, New York, 1980. [19] C.Y. Li, S.V. Garimella, J.E. Simpson, Fixed-grid front-tracking algorithm for solidification problems, part I: Method and validation, Numer. Heat Transfer Part B 43 (2) (2003) 117–141.