1. Konferanse i beregningsorientert mekanikk, Trondheim, 11-12 Mai, 2005
Coupled stress, thermal and fluid flow modelling in materials processing
Dag Mortensen, Magne Rudshaug and Hallvard Fjær
Institute for Energy Technology, Kjeller

2. Content of this presentation
DC casting
Heat and fluid flow modelling
Modelling of stresses and deformations
Coupling effects
Finite element package IfeFEM
Numerical solutions
Applications
DC-casting of aluminium
Welding of an aluminium windshield frame
Hydro-electromagnetic stability in electrolysis cells

3. DC-casting of aluminium
Direct Chill casting or semi-continuous casting

4. Heat and fluid flow modelling - I
Mass and momentum conservation
Buoyancy term
Solidification term
Permeability

5. Heat and fluid flow modelling - II
Turbulence modelling - Low Reynolds Number k-model
Transport equations for k and e
Turbulent production terms
Damping functions
Turbulent kinematic viscosity

6. Heat and fluid flow modelling - III
Heat flow start-up period (Eulerian-Lagrangian)
The velocity of the grid is equal to the solid phase
Grid expansion:
Eulerian
Eulerian-Lagrangian

7. Stress modelling Momentum balance omitting inertial term
Compatibility and strain decomposition
Constitutive equations
Thermal strain
Elastic strain
Plastic strain (Modelling of casting)

8. Constitutive equations for plastic strain depending on microstructure
Modelling heating cycles in 6xxx aluminium alloys (e.g. welding)
Flow stress expressed as a function of the temperature, particle strength parameters, a work hardening parameter and the plastic strain rate
Particle strengthening factor
Strain rate factor
Work hardening
Partial work hardening and recovery
Steady state creep

9. Coupling effects - I Mechanical calculations of contact/air-gap
Deformations of both sides are included
Coupling by interpolation in a general grid
Automatic calculations of water hit points
Water outlet edge
Water outlet angle
Speed of water jet
Gravity forces on water jet
Displacement of ingot surface

10. Coupling effects - II
Energy and the mechanical momentum equations are solved on the real geometry (”updated Lagrangian approach”)
The effect of the smaller narrow side is included
Real mould opening is applied

11. Finite element package – IfeFEM
In-house general purpose software package for implementation of FEM simulators
Programming language: Fortran 95
Object based style
Scalable/Dynamic allocation
Application programmer responsible for integration of the element matrices defining the problem
A number of modules supporting this task
The mesh are imported from meshing tools or created parametrically
The mesh may be changed dynamically
Elements may be appended, merged and deleted

12. IfeFEM – Adaptive mesh refinement
General algorithm based on moulds defining the refinement of the standard elements of the mesh
Continuity is maintained by introducing constraint equations for certain non-compatible inter-element nodes
Novel method based on node multiplicities has been introduced to keep track of the non-compatible nodes
Method is easily extended to adaptive coarsening of mesh

13. Numerical implementation – Fluid flow
Segregated velocity pressure method for the mass and momentum conservation equations
Equal order interpolation for pressure and velocities
Discretization by streamline upwind perturbation in the flow direction (SUPG-method)
Step1
Step2
Step3

14. Numerical implementation - Stresses
Momentum balance in solid solved by FEM using reduced integration with hourglass control
Displacements applied as primary unknowns
Thermally induced deformation entails reasonable fast converge for a simple “initial stiffness method” using extrapolated predictions of plastic strain increments
Global equation system solved by Incomplete Cholesky Conjugated Gradients method
2 levels of fill-in found to be optimal
Adaptive scaling of diagonal of preconditioning matrix to ensure convergence

15. Application: DC-casting of a sheet ingot - I
Coupled heat- and fluid-flow, stresses and deformations

16. Application: DC-casting of a sheet ingot - II
The fluid flow field

17. Application: Welding of automotive parts
A case study: Welding of a windshield frame made from extruded profiles of the aluminum alloy AA6082 T6
Only an upper corner included in the solution domain
Simplified geometry

18. Welding sequence significance
In case F: Distortions at weld interface, i.e. generation and filling of gaps, compensate for deformation of welded parts
Case A
Case F

19. Application: The Stability model for aluminium electrolysis cells
Liquid aluminium
Anode
Side ledge
Electrolyte
Bath/metal interface kA

20. The STABILITY model Linearized set of equations: Simplified model:
Capital letters – stationary state
Small letters – perturbed (1. order) state
Simplified model:
Conservation of momentum:
Conservation of mass:

21. Eigenvalue problem Looking for oscillatory solutions
FEM discretized set of equations:
Generalized eigenvalue problem with singular left hand matrix – a challenging problem.
Efficient solution with Jacobi-Davidson generalized eigenvalue solver.
Needs a good preconditioner for rapid convergence of correction equation solver.

22. Example of instable eigenmodes
Calculates the slowest eigenmodes.