Liu, Weitao (2005) Finite Element Modelling of Macrosegregation and Thermomechanical Phenomena in Solidification Processes. PhD thesis Mécanique Numérique, ENSMP - CEMEF Centre de Mise en Forme des Matériaux, ENSMP.

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Abstract

This work is dedicated to the modeling of macrosegregation and deformation during solidification of castings. A two-dimensional finite element model to simulate macrosegregation due to thermal-solutal convection in the case of columnar dendritic solidification is presented. A set of volume-averaged conservation equations of energy, solute, momentum and mass is solved in conjunction with the use of the lever rule as a microsegregation model. Several formulations have been implemented, permitting a resolution with either weak or strong coupling, closed or open system. In order to improve the prediction accuracy, an algorithm for dynamic remeshing is proposed. The basic idea is to generate fine elements near the liquidus isotherm. The norm of the gradient of solid fraction is used for piloting the remeshing in the mushy zone; while the objective mesh size in the liquid is considered as a function of the distance to the liquidus isotherm. The numerical approach has been validated with a benchmark test of macrosegregation in Pb-Sn alloys taken from the literature. The influences of mesh size, time step and coupling scheme have been investigated. Sufficient fine meshes, small time step and possibly coupling iterations should be applied in order to predict segregated channels. Moreover, the efficiency of mesh adaptation is demonstrated by predictions of freckles in a case of unidirectional solidification, and of ‘A-type' segregation bands in a large industrial carbon steel ingot. In the last part of this work, regarding fluid flow in the liquid induced by solidification shrinkage and thermo-convection and deformation in the solid, a thermal mechanical model has been implemented with a Eulerian-Lagrangian formulation. The alloy in the liquid state is Newtonian, and in the mushy state it is modeled by a viscoplastic continuum. Below a critical temperature the alloy is considered by a thermal elastic viscoplastic model. The thermo-mechanical simulation is used to predict the shrinkage pipe, air gap, strains and stresses.

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Table of content

Chapter 1 General introduction
1.1 Background
1.2 Solidification phenomena
1.2.1 Solidification and structure
1.2.2 Shrinkage
1.2.3 Macrosegregation
1.2.4 Liquid movement /and solid deformation
1.3 Related previous work
1.4 Objectives and outline
1.4.1 Objectives
1.4.2 Outline
Chapter 2 Bibliographic review
2.1 Macrosegregation models
2.1.1 Flemings' macrosegregation model
2.1.2 Coupling fluid flow in the mushy and bulk liquid zones
2.1.3 Modeling of solidification with mesh adaptation
2.2 Solid deformation and pipe formation
2.2.1 Fluid mechanical models
2.2.2 A thermal mechanical model
Chapter 3 Modeling of macrosegregation
3.1 Governing equations
3.1.1 Hypotheses
3.1.2 Conservation equations
3.2 Resolution strategy
3.2.1 Coupling the equations
3.2.2 The finite element solver
3.3 Resolution of the energy equation
3.3.1 Resolution with the nodal upwind method
3.3.2 Resolution with the SUPG method
3.3.3 Improvement of convergence
3.3.4 Treatment of thermal shock
3.4 Resolution of microsegregation equations
3.4.1 Binary alloys with eutectic transformation
3.4.2 Multicomponent alloys
3.5 Resolution of the solute transport equation
3.5.1 Approach 1 - resolution for the average mass concentration in liquid
3.5.2 Approach 2 - resolution for the average mass concentration
3.6 Resolution of momentum equation
3.6.1 Resolution of fluid mechanics with the nodal upwind method
3.6.2 Axisymmetric formulation
3.6.3 Resolution of momentum equation with the SUPG-PSPG formulation
3.7 Validations
3.7.1 Axisymmetric formulation in the case of Navier-Stokes flow
3.7.2 Validation of Darcy term (axisymmetric case, computed by R2SOL and PHOENICS)
3.7.3 Validations of the SUPG-PSPG formulation
3.7.4 A solidification test case
Chapter 4 Mesh adaptation
4.1 Tracking liquidus isotherm
4.1.1 Tracking procedure
4.1.2 Distance to liquidus isotherm
4.2 Isotropic remeshing
4.2.1 Definitions of isotropic mesh size
4.2.2 Domain decomposition
4.2.3 Computation of the nodal objective mesh size
4.3 Anisotropic remeshing
4.3.1 Metric tensor and anisotropic mesh
4.3.2 Determination of parameters for anisotropic remeshing
Chapter 5 Numerical results of macrosegregation
5.1 Benchmark test of Hebditch and Hunt
5.2 Results for the Sn-5%Pb alloy
5.2.1 Numerical setup
5.2.2 Study of the mesh size influence
5.2.3 Study of the time step influence
5.2.4 Study of the influence of coupling iterations within each time step
5.2.5 No-coupling resolutions
5.2.6 Comparison between P1+/P1 and SUPG-PSPG formulations
5.2.7 Confrontation with experiments
5.2.8 Discussion on results for the Sn-5%Pb alloy
5.3 Results for the Pb-48%Sn alloy
5.3.1 Numerical setup
5.3.2 Mesh size influence
5.3.3 Time step influence
5.3.4 Influence of coupling iterations within each time step
5.3.5 No-coupling resolutions
5.3.6 Confrontation with experiments
5.3.7 Concluding remarks
5.4 Modelling of freckles
5.4.1 Numerical setup
5.4.2 Results
5.4.3 Discussion
5.5 Application to a steel ingot
Chapter 6 Thermomechanical stress-strain modeling
6.1 Thermal mechanical model
6.1.1 The mechanical equilibrium
6.1.2 Constitutive equations
6.1.3 Local resolution of constitutive equations
6.2 Resolution of mechanics
6.2.1 Weak form and time discretization
6.2.2 P1+/P1 formulation
6.2.3 Implementation of axisymmetric formulation
6.2.4 ALE formulation
6.3 Validations
6.3.1 Thermoelastic test
6.3.2 Uniaxial tension test
6.4 Applications
6.4.1 Svensson solidification test
6.4.2 Solidification of industrial ingots
6.5 Conclusion
Chapter 7 Conclusion and perspectives
References
Appendix A Data for the validation of diffusion split method
Appendix B Organization for Macrosegregation computation
Appendix C Tangent modulus

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