We deal with numerical solution of a three-dimensional phase field model of solidification in single component anisotropic materials. In this contribution, we extend the model by crystal orientation transformation. A robust algorithm is then developed to simulate the growth of multiple grains with an arbitrary number of random crystallographic orientations and a fully resolved 3D dendritic geometry. In the first part, the model and the parallel implementation of the algorithms are explained. The second part is devoted to demonstrating the effect of meshrelated numerical anisotropy and the simulations of complex polycrystalline solidification on very fine meshes.
We investigate a family of phase field models for simulating dendritic growth of a pure supercooled substance. The central object of interest is the reaction term in the Allen-Cahn equation, which is responsible for the spatial distribution of latent heat release during solidification. In this context, several existing forms of the reaction term are analyzed. Inspired by the known conclusions of matched asymptotic analysis, we propose a new variant (the 'ΣP1-P' model) that is simple enough to allow mathematical and numerical analysis and robust enough to be applicable to solidification under very large supercooling. The important component of the model (the Σ limiter) can also be incorporated into the original models to extend the range of their applicability. The individual models are tested in a number of numerical simulations focusing on mesh-dependence and model parameter settings. When the phase interface thickness is kept large with respect to the microscopic capillary length to make numerical computations feasible, the parameters of the Σ limiter can be tuned to improve agreement with previous models. The results obtained using the ΣP1-P model exhibit a good quantitative agreement with experimental data from rapid solidification of nickel melts.
Phase field modeling finds utility in various areas. In optimization theory in particular, the distributed control and Neumann boundary control of phase field models have been investigated thoroughly. Dirichlet boundary control in parabolic equations is commonly addressed using the very weak formulation or an approximation by Robin boundary conditions. In this paper, the Dirichlet boundary control for a phase field model with a non-singular potential is investigated using the Dirichlet lift technique. The corresponding weak formulation is analyzed. Energy estimates and problem-specific embedding results are provided, leading to the existence and uniqueness of the solution for the state equation. These results together show that the control to state mapping is well defined and bounded. Based on the preceding findings, the optimization problem is shown to have a solution.
We propose a novel and efficient numerical approach for solving the pseudo two-dimensional multiscale model of the Li-ion cell dynamics based on first principles, describing the ion diffusion through the electrolyte and the porous electrodes, electric potential distribution, and Butler-Volmer kinetics. The numerical solution is obtained by the finite difference discretization of the diffusion equations combined with an original iterative scheme for solving the integral formulation of the laws of electrochemical interactions. We demonstrate that our implementation is fast and stable over the expected lifetime of the cell. In contrast to some simplified models, it provides physically consistent results for a wide range of applied currents including high loads. The algorithm forms a solid basis for simulations of cells and battery packs in hybrid electric vehicles, with possible straightforward extensions by aging and heat effects.
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