2,136 research outputs found
Three dimensional thermal-solute phase field simulation of binary alloy solidification
We employ adaptive mesh refinement, implicit time stepping, a nonlinear multigrid solver and parallel computation to solve a multi-scale, time dependent, three dimensional, nonlinear set of coupled partial differential equations for three scalar field variables. The mathematical model represents the non-isothermal solidification of a metal alloy into a melt substantially cooled below its freezing point at the microscale. Underlying physical molecular forces are captured at this scale by a specification of the energy field. The time rate of change of the temperature, alloy concentration and an order parameter to govern the state of the material (liquid or solid) are controlled by the diffusion parameters and variational derivatives of the energy functional. The physical problem is important to material scientists for the development of solid metal alloys and, hitherto, this fully coupled thermal problem has not been simulated in three dimensions, due to its computationally demanding nature. By bringing together state of the art numerical techniques this problem is now shown here to be tractable at appropriate resolution with relatively moderate computational resources
On the role of confinement on solidification in pure materials and binary alloys
We use a phase-field model to study the effect of confinement on dendritic
growth, in a pure material solidifying in an undercooled melt, and in the
directional solidification of a dilute binary alloy. Specifically, we observe
the effect of varying the vertical domain extent () on tip selection,
by quantifying the dendrite tip velocity and curvature as a function of
, and other process parameters. As decreases, we find that the
operating state of the dendrite tips becomes significantly affected by the
presence of finite boundaries. For particular boundary conditions, we observe a
switching of the growth state from 3-D to 2-D at very small , in both
the pure material and alloy. We demonstrate that results from the alloy model
compare favorably with those from an experimental study investigating this
effect.Comment: 13 pages, 9 figures, 3 table
Lattice Boltzmann simulations of 3D crystal growth: Numerical schemes for a phase-field model with anti-trapping current
A lattice-Boltzmann (LB) scheme, based on the Bhatnagar-Gross-Krook (BGK)
collision rules is developed for a phase-field model of alloy solidification in
order to simulate the growth of dendrites. The solidification of a binary alloy
is considered, taking into account diffusive transport of heat and solute, as
well as the anisotropy of the solid-liquid interfacial free energy. The
anisotropic terms in the phase-field evolution equation, the phenomenological
anti-trapping current (introduced in the solute evolution equation to avoid
spurious solute trapping), and the variation of the solute diffusion
coefficient between phases, make it necessary to modify the equilibrium
distribution functions of the LB scheme with respect to the one used in the
standard method for the solution of advection-diffusion equations. The effects
of grid anisotropy are removed by using the lattices D3Q15 and D3Q19 instead of
D3Q7. The method is validated by direct comparison of the simulation results
with a numerical code that uses the finite-difference method. Simulations are
also carried out for two different anisotropy functions in order to demonstrate
the capability of the method to generate various crystal shapes
A fully implicit, fully adaptive time and space discretisation method for phase-field simulation of binary alloy solidification
A fully-implicit numerical method based upon adaptively refined meshes for the
simulation of binary alloy solidification in 2D is presented. In addition we combine a
second-order fully-implicit time discretisation scheme with variable steps size control
to obtain an adaptive time and space discretisation method. The superiority of this
method, compared to widely used fully-explicit methods, with respect to CPU time
and accuracy, is shown. Due to the high non-linearity of the governing equations
a robust and fast solver for systems of nonlinear algebraic equations is needed to
solve the intermediate approximations per time step. We use a nonlinear multigrid
solver which shows almost h-independent convergence behaviour
Phase field study of the tip operating state of a freely growing dendrite against convection using a novel parallel multigrid approach
Alloy dendrite growth during solidification with coupled thermal-solute-convection fields has been studied by phase field modeling and simulation. The coupled transport equations were solved using a novel parallel-multigrid numerical approach with high computational efficiency that has enabled the investigation of dendrite growth with realistic alloy values of Lewis number ∼104 and Prandtl number ∼10−2. The detailed dendrite tip shape and character were compared with widely recognized analytical approaches to show validity, and shown to be highly dependent on undercooling, solute concentration and Lewis number. In a relatively low flow velocity regime, variations in the ratio of growth selection parameter with and without convection agreed well with theory
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