203 research outputs found
Multiscale Problems in Solidification Processes
Our objective is to describe solidification phenomena in alloy systems. In the classical approach, balance equations in the phases are coupled to conditions on the phase boundaries which are modelled as moving hypersurfaces. The Gibbs-Thomson condition ensures that the evolution is consistent with thermodynamics. We present a derivation of that condition by defining the motion via a localized gradient flow of the entropy.
Another general framework for modelling solidification of alloys with multiple phases and components is based on the phase field approach. The phase boundary motion is then given by a system of Allen-Cahn type equations for order parameters. In the sharp interface limit, i.e., if the smallest length scale ± related to the thickness of the diffuse phase boundaries converges to zero, a model with moving boundaries is recovered. In the case of two phases
it can even be shown that the approximation of the sharp interface model by the phase field model is of second order in ±. Nowadays it is not possible to simulate the microstructure evolution in a whole workpiece. We present a two-scale model derived by homogenization methods including a mathematical justification by an estimate of the model error
Relating phase field and sharp interface approaches to structural topology optimization
A phase field approach for structural topology optimization which allows for topology
changes and multiple materials is analyzed. First order optimality conditions are
rigorously derived and it is shown via formally matched asymptotic
expansions that these conditions converge to classical first order conditions obtained in
the context of shape calculus. We also discuss how to deal with triple junctions where
e.g. two materials and the void meet. Finally, we present several
numerical results for mean compliance problems and a cost involving the least square error
to a target displacement
Stable phase field approximations of anisotropic solidification
We introduce unconditionally stable finite element approximations for a phase
field model for solidification, which take highly anisotropic surface energy and kinetic
effects into account. We hence approximate Stefan problems with anisotropic
Gibbs{Thomson law with kinetic undercooling, and quasi-static variants thereof.
The phase field model is given by
#wt + � %(') 't = r: (b(')rw) ;
c
a
� %(')w = " �
� �(r') '
An adaptive, fully implicit multigrid phase-field model for the quantitative simulation of non-isothermal binary alloy solidification
Using state-of-the-art numerical techniques, such as mesh adaptivity, implicit time-stepping and a non-linear multi-grid solver, the phase-field equations for the non-isothermal solidification of a dilute binary alloy have been solved. Using the quantitative, thin-interface formulation of the problem we have found that at high Lewis number a minimum in the dendrite tip radius is predicted with increasing undercooling, as predicted by marginal stability theory. Over the dimensionless undercooling range 0.2–0.8 the radius selection parameter, σ*, was observed to vary by over a factor of 2 and in a non-monotonic fashion, despite the anisotropy strength being constant
The viscous Cahn-Hilliard equation. I. Computations
The viscous Cahn-Hilliard equation arises as a singular limit of the phase-field model of phase transitions. It contains both the Cahn-Hilliard and Allen-Cahn equations as particular limits. The equation is in gradient form and possesses a compact global attractor A, comprising heteroclinic orbits between equilibria. Two classes of computation are described. First heteroclinic orbits on the global attractor are computed; by using the viscous Cahn-Hilliard equation to perform a homotopy, these results show that the orbits, and hence the geometry of the attractors, are remarkably insensitive to whether the Allen-Cahn or Cahn-Hilliard equation is studied. Second, initial-value computations are described; these computations emphasize three differing mechanisms by which interfaces in the equation propagate for the case of very small penalization of interfacial energy. Furthermore, convergence to an appropriate free boundary problem is demonstrated numerically
Unified derivation of phase-field models for alloy solidification from a grand-potential functional
In the literature, two quite different phase-field formulations for the
problem of alloy solidification can be found. In the first, the material in the
diffuse interfaces is assumed to be in an intermediate state between solid and
liquid, with a unique local composition. In the second, the interface is seen
as a mixture of two phases that each retain their macroscopic properties, and a
separate concentration field for each phase is introduced. It is shown here
that both types of models can be obtained by the standard variational procedure
if a grand-potential functional is used as a starting point instead of a
free-energy functional. The dynamical variable is then the chemical potential
instead of the composition. In this framework, a complete analogy with
phase-field models for the solidification of a pure substance can be
established. This analogy is then exploited to formulate quantitative
phase-field models for alloys with arbitrary phase diagrams. The precision of
the method is illustrated by numerical simulations with varying interface
thickness.Comment: 36 pages, 1 figur
Towards a 3-dimensional phase-field model of non-isothermal alloy solidification
We review the application of advanced numerical techniques such as adaptive mesh refinement, implicit time-stepping, multigrid solvers and massively parallel implementations as a route to obtaining solutions to the 3-dimensional phase-field problem for coupled heat and solute transport during non-isothermal alloy solidification. Using such techniques it is shown that such models are tractable for modest values of the Lewis number (ratio of thermal to solutal diffusivities). Solutions to the 3-dimensional problem are compared with existing solutions to the equivalent 2-dimensional problem
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