## Thomas Kaempfer's PhD page |

Goal : |
Development of the macrosegregation module within a metallurgy oriented software for solidification processes. |

Method : |
Solve the conservation equations for energy, mass, momentum and solute using finite elements based on an adaptive domain decomposition method with different grid-refinement levels. Apply the method to solidification problems in order to simulate macrosegregation phenomena such as freckles. |

Advisor : |
Prof. M. Rappaz |

pdf : |
the pdf-version
(4.4MB), paper version from EPFL |

A numerical method for the simulation of
buoyancy-induced
macrosegregation during solidification processes is presented.

The physical model is based on volume averaged conservation equations for energy, mass, momentum, and solute, which are coupled using linearized phase diagram data for binary alloys, a temperature-enthalpy relationship, and the lever rule as a microsegregation model.

The numerical model considers two different length-scales, on the one hand, the scale of the overall process, on the other, a small critical zone near the solidification front where solutal inhomogeneities are initiated and the fluid velocity is non-zero.

A domain decomposition method (DDM) using two adapted grids has been developed. The overall computational domain is discretized using a `coarse' finite element mesh adapted to the process scale. At each time step, the energy conservation equation is solved on this discretization and new values of temperature and solid fraction are calculated at each finite element node. Based on these values, the computational domain is divided into three subdomains: the so-called solid, mushy, and liquid regions. The mushy subdomain corresponds to the critical zone near the solidification front and is adaptively discretized with a finer finite element mesh, whereas the liquid domain uses the initial coarse grid and the solid is no longer considered. The fluid flow and solute transport equations are solved alternately in the mushy and liquid subdomains on the corresponding finite element grids. For these computations, the Galerkin least squares (GLS) method is used to stabilize the Navier-Stokes equations. The coupling of the mushy and liquid subproblems is achieved with a Dirichlet-Neumann substructuring iterative method, together with the mortar technique to deal with the non-conforming discretizations at the subdomain interfaces.

For the discretization in time, we use a finite difference method, together with a linearization procedure for the Navier-Stokes equations and split operators for the solute transport equation.

Using different test cases, with and without solidification or solute transport, the sensitivity of the numerical method with respect to the domain decomposition criterion and other numerical parameters is studied. Finally, the method is applied to several macrosegregation problems such as the Hebditch-Hunt test case, and the performances and limits of the approach are pointed out and discussed.

The physical model is based on volume averaged conservation equations for energy, mass, momentum, and solute, which are coupled using linearized phase diagram data for binary alloys, a temperature-enthalpy relationship, and the lever rule as a microsegregation model.

The numerical model considers two different length-scales, on the one hand, the scale of the overall process, on the other, a small critical zone near the solidification front where solutal inhomogeneities are initiated and the fluid velocity is non-zero.

A domain decomposition method (DDM) using two adapted grids has been developed. The overall computational domain is discretized using a `coarse' finite element mesh adapted to the process scale. At each time step, the energy conservation equation is solved on this discretization and new values of temperature and solid fraction are calculated at each finite element node. Based on these values, the computational domain is divided into three subdomains: the so-called solid, mushy, and liquid regions. The mushy subdomain corresponds to the critical zone near the solidification front and is adaptively discretized with a finer finite element mesh, whereas the liquid domain uses the initial coarse grid and the solid is no longer considered. The fluid flow and solute transport equations are solved alternately in the mushy and liquid subdomains on the corresponding finite element grids. For these computations, the Galerkin least squares (GLS) method is used to stabilize the Navier-Stokes equations. The coupling of the mushy and liquid subproblems is achieved with a Dirichlet-Neumann substructuring iterative method, together with the mortar technique to deal with the non-conforming discretizations at the subdomain interfaces.

For the discretization in time, we use a finite difference method, together with a linearization procedure for the Navier-Stokes equations and split operators for the solute transport equation.

Using different test cases, with and without solidification or solute transport, the sensitivity of the numerical method with respect to the domain decomposition criterion and other numerical parameters is studied. Finally, the method is applied to several macrosegregation problems such as the Hebditch-Hunt test case, and the performances and limits of the approach are pointed out and discussed.

Solutal variations in an alloy at the scale
of the process are known as macrosegregation and play an important
role in many solidification processes. It can be induced by deformation
of the solid skeleton in the semi-solid state, by transport of equiaxed
grains or by fluid flow in the mushy zone which redistributes
segregated
solute elements within the remaining liquid volume [1].

In order to simulate this last phenomenon in an efficient and accurate way, it is necessary to take into account two different length scales: on the one hand, the relatively small region of the mushy zone where the fluid can penetrate and transport away solute and, on the other hand, the characteristic size of the process.

In order to simulate this last phenomenon in an efficient and accurate way, it is necessary to take into account two different length scales: on the one hand, the relatively small region of the mushy zone where the fluid can penetrate and transport away solute and, on the other hand From a numerical point of view, this implies that a finer discretization is needed close to the liquidus. In the present contribution, a finite element approach is used. This has the advantage over structured finite volume or finite difference methods that complex geometries can be described with fewer nodes. However, it becomes necessary to refine the elements in the region where gradients of solute and velocity are large in order to keep the errors within reasonable bounds.

Adaptive meshing is a possibility, implying refining and coarsening algorithms. Since this is difficult to achieve in 3D, an other method has been chosen in the present study. It is based on a multidomain approach [2].

The computational domain for the fluid flow and solute transport equations is divided into two subdomains, the liquid region where a fixed coarse grid is used, and the critical zone of the mushy region where a fine, dynamic discretization is applied at each time step.

The equations to be solved with these grids are the conservation equations for heat, mass, momentum and solute, averaged over a volume element which is small with respect to the extent of the mushy zone and large with respect to the typical dendrite arm spacing [3,4].

In order to simulate this last phenomenon in an efficient and accurate way, it is necessary to take into account two different length scales: on the one hand, the relatively small region of the mushy zone where the fluid can penetrate and transport away solute and, on the other hand, the characteristic size of the process.

In order to simulate this last phenomenon in an efficient and accurate way, it is necessary to take into account two different length scales: on the one hand, the relatively small region of the mushy zone where the fluid can penetrate and transport away solute and, on the other hand From a numerical point of view, this implies that a finer discretization is needed close to the liquidus. In the present contribution, a finite element approach is used. This has the advantage over structured finite volume or finite difference methods that complex geometries can be described with fewer nodes. However, it becomes necessary to refine the elements in the region where gradients of solute and velocity are large in order to keep the errors within reasonable bounds.

Adaptive meshing is a possibility, implying refining and coarsening algorithms. Since this is difficult to achieve in 3D, an other method has been chosen in the present study. It is based on a multidomain approach [2].

The computational domain for the fluid flow and solute transport equations is divided into two subdomains, the liquid region where a fixed coarse grid is used, and the critical zone of the mushy region where a fine, dynamic discretization is applied at each time step.

The equations to be solved with these grids are the conservation equations for heat, mass, momentum and solute, averaged over a volume element which is small with respect to the extent of the mushy zone and large with respect to the typical dendrite arm spacing [3,4].

Average conservation equations for energy, mass, momentum and solute are used. The main hypotheses are

Furthermore, microsegregation is assumed
to occur with the lever rule and a constant partition coefficient
is used for the computation of the temperature, volume fraction of
solid and mass fraction of solute.

The coupled system of partial differential
equations is solved as follows. At each time step, the heat equation
is solved at the macroscopic level, thus giving the position of the
mushy zone where the refinement has to be made. The fluid flow and the
solute conservation equations are then solved on both the coarse and
fine grids:

Figure 1:Schematic view of the DDM resolution method for the fluid flow and solute transport equations. A substructuring iterative method is used.

Figure 2:Temperature field during continuous casting with superimposed fine grid around the liquidus at different stages of the computation. See also the animated gif.

Figure 3:Schematic view and snapshot of a partially solidified cavity with imposed, constant thermal gradient. Due to buoyancy forces, a velocity field is generated.

Figure 4:Velocity profiles along the horizontal midsection of the cavity and at time 200s in the overall domain (left) and the refined region (right) for the standard FE method on a 32x32 discretization compared with the results obtained by the DDM method with one and two refinements in the critical zone.

- The method has been validated with the Hebditch-Hunt macrosegregation experiment [4].
- Segregation in a solidifying channel has been computed.
- Computations showing Freckles formation have been done.

Figure 5:Freckles formation: Solid fraction (blue) and the velocity field ahead of the solidification front (left) when an instability developpes. The adaptive fine discretization which is created in the critical zone is shown on the right. See also the animated gif.

Figure 6:Freckles formation: The segregation map after the formation of a channel segregate.

By using the DDM technique in order to
refine
only the important zone close to the liquidus line, good accuracy is
achieved
while reducing the computational cost by about one order of magnitude.

Figure:Comparison of the CPU time necessary to compute 200 time-steps for a standard FE method with a 32x32 grid (1), the DDM method on the 32x32 grid with 1 refinement (2) and 2 refinements (3) close to the mushy zone, and for the standard FE method with the 128x128 discretization. The number of nodal points for the discretizations is also indicated.

**[1]** Voller, V.R., Sundarraj, S.,* Int. J. Heat
Mass Transfer*, Vol. 38, 1995, pp. 1009-18

**[2]** Quarteroni, A., Valli, A., *Domain
Decomposition
Methods for Partial Differential Equations*, Oxford Science
Publications,
1999

**[3]** Rappaz, M., Voller, V., *Metall. Trans. A*,
vol. 21 A, 1990, pp. 749-53

**[4]** Ahmad, N., Combeau, H., Desbiolles, J.-L., Jalanti,
T., Lesoult, G., Rappaz, J., Rappaz, M., Stomp, C., *Metall. Mater.
Trans. A*, vol. 29A, 1998, pp. 617-30

**[2003]** Kaempfer, Th. U. and Rappaz, M.,Modelling of
macrosegregation during solidification processes using an adaptive
domain decomposition method,
* Modelling Simul. Mater. Sci. Eng.* 11(2003), pp. 575-597

**[2002]** Kaempfer, Th. U.,* Modeling of Macrosegregation
Using
an Adaptive Domain Decomposition Method,* Thesis No 2666 (2002),
École Polytechnique Fédérale de Lausanne, 1015
Lausanne,
Switzerland

**[2000]** Kaempfer, Th. U. and Rappaz, M.,* Modeling
of Casting, Welding and Advanced Solidification Processes, *IX,
ed. P. R. Sahm, P. N. Hansen, J. G. Conley, Shaker Verlag, Aachen,
Germany, 2000, pp. 641-647

**[1999]** Kaempfer, Th. U. and Rappaz, M., *Journees
d'automne 1999 de la SF2M,* Revue de Metallurgie, Societe Francaise
de Metallurgie et de Materiaux, Les Fontenelles, Nanterre, France,
1999, p. 131