We study the effect of fluid flow on three-dimensional (3D) dendrite growth using a phase-field model on an adaptive finite-element grid. In order to simulate 3D fluid flow, we use an averaging method for the flow problem coupled to the phase-field method and the semi-implicit approximated projection method (SIAPM). We describe a parallel implementation for the algorithm, using the CHARM++ FEM framework, and demonstrate its efficiency. We introduce an improved method for extracting dendrite tip position and tip radius, facilitating accurate comparison to theory. We benchmark our results for 2D dendrite growth with solvability theory and previous results, finding them to be in good agreement. The physics of dendritic growth with fluid flow in three dimensions is very different from that in two dimensions, and we discuss the origin of this behavior.

We study dendritic microstructure evolution using an adaptive grid, finite element method applied to a phase-field model. The computational complexity of our algorithm, per unit time, scales linearly with system size, rather than the quadratic variation given by standard uniform mesh schemes. Time-dependent calculations in two dimensions are in good agreement with the predictions of solvability theory, and can be extended to three dimensions and small undercoolings. 05.70.Ln, 81.30.Fb, 64.70.Dv, 81.10.Aj Dendrites are the primary component of solidification microstructures in metals. The formation, shape, speed and size of dendritic microstructures has been a topic of intense study in the past 10-15 years. Experiments [1,2] by Glicksman and coworkers on succinonitrile (SCN) and other transparent analogues of metals have been accurate enough to provide tests of theories of dendritic growth, and have stimulated considerable theoretical progress [3][4][5]. The experiments have clearly demonstrated that naturally growing dendrites possess a unique steady state tip, characterized by its velocity, radius of curvature and shape, which leads the to time-dependent sidebranched dendrite as it propagates.The earliest theories of dendritic growth solved for the diffusion field around a self-similar body of revolution propagating at constant speed [6,7]. In these studies the diffusion field determines the product of the dendrite velocity and tip radius, but neither quantity by itself. Adding capillarity effects to the theory predicts a unique maximum growth speed, [8] but experiments showed that this point does not represent the operating state for real dendrites.The goal of contemporary research has been to predict steady state features of dendritic growth and to compute time-dependent microstructures from numerical solutions of the equations of motion. The purpose of this letter is to present a computationally efficient method for time-dependent calculations, and to verify that the steady state properties are in excellent agreement with those predicted by analysis of the steady state problem.Insight into the steady state dendrite problem was first obtained from local models [3,4,[9][10][11][12][13][14][15][16][17][18] describing the evolution of the interface, and incorporating the features of the bulk phases into the governing equation of motion for the interface. These models were the first [10] to show that a nonzero dendrite velocity is obtained only if a source of anisotropy -for example, anisotropic interface energy -is present in the description of dendritic evolution. Subsequently, it was shown that the spectrum of allowed steady state velocities is discrete, rather than continuous, and the role of anisotropy was understood theoretically, both in the local models and the full moving boundary problem [4,5,14,19]. Moreover, only the fastest of a spectrum of steady state velocities is stable, thus forming the operating state of the dendrite. It is widely believed that sidebranching is generated by thermal or other ...

We propose a computationally-efficient approach to multiscale simulation of polycrystalline materials, based on the phase field crystal (PFC) model. The order parameter describing the density profile at the nanoscale is reconstructed from its slowly-varying amplitude and phase, which satisfy rotationally-covariant equations derivable from the renormalization group. We validate the approach using the example of two-dimensional grain nucleation and growth.PACS numbers: 81.16. Rf, 05.10.Cc, 61.72.Cc, 81.15.Aa Why is it so hard to predict the properties of real materials? Unlike simple crystalline solids, real materials, produced by a wide range of processing conditions, contain defects and multiple grains that strongly impact mechanical, thermal, electrical response, and give rise to such important phenomena as plasticity, hysteresis, work hardening and glassy relaxation. Moreover, it is frequently the case that a faithful description of materials processing requires simultaneous treatment of dynamics at scales ranging from the nanoscale up to the macroscopic. For example, dendritic growth, the generic mode of solidification of most metals and alloys, involves the capillary length at the nanoscale, the emergent pattern dimensions on the scale of microns, the thermal or particle diffusion length on the scale of 10 −4 m, in addition to the grain and sample size.Despite these obstacles, progress in rational material design requires a fundamental understanding of the way in which useful properties emerge as the mesoscale is approached. Questions that must be addressed include: What is the collective behavior of assemblies of nanoscale objects? How best to achieve target mesoscale properties from nanoscale constituents? And how can the properties at nano-, meso-and intermediate scales simultaneously be captured quantitatively and predictively?A number of computational approaches to handle the range of length scales have been proposed recently [1, 2], including quasi-continuum methods [3,4,5,6], the heterogeneous multiscale method [7,8], multi-scale molecular dynamics [9,10,11,12], multigrid variants [13] and extensions of the phase field model [14]. These techniques strive to provide a unified description of the many scales being resolved, but in some cases require non-systematic ways to link the disparate scales to enable treatment of sufficiently large mesoscale systems. This can introduce spurious modes and excitations, and difficulties associated with the transition between scales [2,8]. Most of this work is limited to crystalline materials with a few isolated defects [15].In this Letter, we propose a novel theoretical approach to these difficulties, by combining the phase field crystal (PFC) formalism [16,17] with renormalization group (RG) [18,19] and related methods (see, e.g.[20]), developed for the analysis of hydrodynamic instabilities in spatially-extended dynamical systems [21,22,23,24,25,26,27,28]. We present effective equations at the mesoscale, from which the atomic density can readily be reconstructed...

We study the evolution of solidification microstructures using a phase-field model computed on an adaptive, finite element grid. We discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase-field model at low undercooling. In particular, we show that the computational complexity of solving any phase-boundary problem scales with the interface arclength when using an adapting mesh. Moreover, the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions, which would otherwise require lattices greater than 2 17 × 2 17 elements. We examine the convergence properties of our algorithm. We also present two-dimensional, time-dependent calculations of dendritic evolution, with and without surface tension anisotropy. We benchmark our results for dendritic growth with microscopic solvability theory, finding them to be in good agreement with theory for high undercoolings. At low undercooling, however, we obtain higher values of velocity than solvability theory at low undercooling, where transients dominate, in accord with a heuristic criterion which we derive.

We derive a set of rotationally covariant amplitude equations for use in multiscale simulation of the two dimensional phase field crystal (PFC) model by a variety of renormalization group (RG) methods. We show that the presence of a conservation law introduces an ambiguity in operator ordering in the RG procedure, which we show how to resolve. We compare our analysis with standard multiple scales techniques, where identical results can be obtained with greater labor, by going to sixth order in perturbation theory, and by assuming the correct scaling of space and time.