A Cartesian grid-based boundary integral method for moving interface problems

This paper proposes a Cartesian grid-based boundary integral method for efficiently and stably solving two representative moving interface problems, the Hele-Shaw flow and the Stefan problem. Elliptic and parabolic partial differential equations (PDEs) are reformulated into boundary integral equations and are then solved with the matrix-free generalized minimal residual (GMRES) method. The evaluation of boundary integrals is performed by solving equivalent and simple interface problems with finite difference methods, allowing the use of fast PDE solvers, such as fast Fourier transform (FFT) and geometric multigrid methods. The interface curve is evolved utilizing the $\theta-L$ variables instead of the more commonly used $x-y$ variables. This choice simplifies the preservation of mesh quality during the interface evolution. In addition, the $\theta-L$ approach enables the design of efficient and stable time-stepping schemes to remove the stiffness that arises from the curvature term. Ample numerical examples, including simulations of complex viscous fingering and dendritic solidification problems, are presented to showcase the capability of the proposed method to handle challenging moving interface problems

Phase-field-lattice Boltzmann method for dendritic growth with melt flow and thermosolutal convection–diffusion

We propose a new phase-field model formulated within the system of lattice Boltzmann (LB) equation for simulating solidification and dendritic growth with fully coupled melt flow and thermosolutal convection–diffusion. With the evolution of the phase field and the transport phenomena all modeled and integrated within the same LB framework, this method preserves and combines the intrinsic advantages of the phase-field method (PFM) and the lattice Boltzmann method (LBM). Particularly, the present PFM/LBM model has several improved features compared to the existing phase-field models including: (1) a novel multiple-relaxation-time (MRT) LB scheme for the phase-field evolution is proposed to effectively model solidification coupled with melt flow and thermosolutal convection–diffusion with improved numerical stability and accuracy, (2) convenient diffuse interface treatments are implemented for the melt flow and thermosolutal transport which can be applied to the entire domain without tracking the interface, and (3) the evolution of the phase field, flow, concentration, and temperature fields on the level of microscopic distribution functions in the LB schemes is decoupled with a multiple-time-scaling strategy (despite their full physical coupling), thus solidification at high Lewis numbers (ratios of the liquid thermal to solutal diffusivities) can be conveniently modeled. The applicability and accuracy of the present PFM/LBM model are verified with four numerical tests including isothermal, iso-solutal and thermosolutal convection–diffusion problems, where excellent agreement in terms of phase-field and thermosolutal distributions and dendritic tip growth velocity and radius with those reported in the literature is demonstrated. The proposed PFM/LBM model can be an attractive and powerful tool for large-scale dendritic growth simulations given the high scalability of the LBM

Mechanistic selection and growth of twinned bicrystalline primary Si in near eutectic Al-Si alloys

Morphological evolution and selection of angular primary silicon is investigated in near-eutectic Al-Si alloys. Angular silicon arrays are grown directionally in a Bridgman furnace at velocities in the regime of 10-3 m/sec and with a temperature gradient of 7.5x103 K/m. Under these conditions, the primary Si phase grows as an array of twinned bicrystalline dendrites, where the twinning gives rise to a characteristic 8-pointed star-shaped primary morphology. While this primary Si remains largely faceted at the growth front, a complex structure of coherent symmetric twin boundaries enables various adjustment mechanisms which operate to optimize the characteristic spacings within the primary array. In the work presented here, this primary silicon growth morphology is examined in detail. In particular, this thesis describes the investigation of (1) morphological selection of the twinned bicrystalline primary starshape morphology; (2) primary array behavior, including the lateral propagation of the starshape grains and the associated evolution of a strong \u3c100\u3e texture; (3) the detailed structure of the 8-pointed star-shaped primary morphology, including the twin boundary configuration within the central core; (4) the mechanisms of lateral propagation and spacing adjustment during array evolution; and (5) the thermosolutal conditions (i.e. operating state) at the primary growth front, including composition and phase fraction in the vicinity of the primary tip;Experimental methods include directional solidification, high-resolution serial milling, electron-probe microanalysis, backscattered electron diffraction, and high resolution transmission electron microscopy techniques. The following key observations are made here;1. It is found that slightly hypereutectic Al-Si alloys, directionally solidified at rate below 1mum/s, exhibit rapid emergence of a strong \u3c100\u3e texture, associated with the selection and lateral propagation of the star-shaped growth mode. The dominant silicon morphology in this array is an 8-pointed star-shape dendrite with sideplates at alternating angles of 37° and 53°. Large domains of identically oriented silicon dendrites evolve, suggesting that the entire twinned domain originates from one twin-pair of silicon grains;2. Analysis of the detailed structure within the twinned core reveals that 210/310 coherent twin boundaries comprise the core structure of the 8-pointed star-shape. These crystal defects, with a high migration tendency, provide the core with an essential mechanism for the faceted interfaces to grow. Thus, the twin structure plays a crucial role in the morphological/mechanistic selection of the primary silicon morphology in these Al-Si alloys at low grow rates;3. Examination of the overall array structure and dynamics with regard to several mechanisms of branching and spacing adjustment shows that primary tip splitting/joining, tertiary branching, and related twin boundary migration in the core are the main mechanisms for array evolution. Although the Al phase has no preferred orientation with respect to the axial growth direction, its grain structure seems to be influenced by the side branching of silicon, resulting in the polycrystalline grains with no more than one grain between two silicon secondary plates;4. Primary and inter-plate spacing have been measured and the result suggests that overall growth is a diffusively coupled cellular growth. In order to model the overall low-velocity growth of Al-Si alloys, which have a kinetically limited twinned Si array, we propose a mesoscopic envelope and some relevant experiments such as corner growth experiments and directional growth at hypoeutectic compositions for the quantification of supersaturaton and tip temperature, combining with phase fraction measurements near the primary tip;Through the observations and analyses highlighted above, the growth of the faceted silicon bycrystalline structure in near eutectic Al-Si has been comprehensively understood. Phase fraction, composition, interface structure, and spacing in the interdendritic region behind the growing tip suggest that overall microstructure evolution occurred by a very tiny decoupled primary tip, followed by an Al halo-like structure surrounding the silicon star-shape. Overall microstructure evolution proceeds with the assistance of crystal defects (i.e. twinning in the silicon dendrite core) and this mechanistic self-optimization by twinning allows the strongly anisotropic faceted crystal-melt interfaces to respond to local fluctuations, selecting the distinct star-shaped silicon morphology which best satisfy long-range diffusion fields

Phase Field Models Versus Parametric Front Tracking Methods: Are They Accurate and Computationally Efficient?

We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification. Here we focus on Stefan problems, and their quasi-static variants, with applications to crystal growth. New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem