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 03.12.2018

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 11.05.2005

DC-casting of aluminium Direct Chill casting or semi-continuous casting 11.05.2005

Heat and fluid flow modelling - I Mass and momentum conservation Buoyancy term Solidification term Permeability 11.05.2005

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 11.05.2005

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 11.05.2005

Stress modelling Momentum balance omitting inertial term Compatibility and strain decomposition Constitutive equations Thermal strain Elastic strain Plastic strain (Modelling of casting) 11.05.2005

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 11.05.2005

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 11.05.2005

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 11.05.2005

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 11.05.2005

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 11.05.2005

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 11.05.2005

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 11.05.2005

Application: DC-casting of a sheet ingot - I Coupled heat- and fluid-flow, stresses and deformations 11.05.2005

Application: DC-casting of a sheet ingot - II The fluid flow field 11.05.2005

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 11.05.2005

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 11.05.2005

Application: The Stability model for aluminium electrolysis cells Liquid aluminium Anode Side ledge Electrolyte Bath/metal interface 150-300 kA 11.05.2005

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: 11.05.2005

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. 11.05.2005

Example of instable eigenmodes Calculates the 10-20 slowest eigenmodes. 11.05.2005