Multiscale Finite-Difference-Diffusion-Monte-Carlo Method for Simulating Dendritic Solidification

We present a novel hybrid computational method to simulate accurately dendritic solidification in the low undercooling limit where the dendrite tip radius is one or more orders of magnitude smaller than the characteristic spatial scale of variation of the surrounding thermal or solutal diffusion field. The first key feature of this method is an efficient multiscale diffusion Monte-Carlo (DMC) algorithm which allows off-lattice random walkers to take longer and concomitantly rarer steps with increasing distance away from the solid-liquid interface. As a result, the computational cost of evolving the large scale diffusion field becomes insignificant when compared to that of calculating the interface evolution. The second key feature is that random walks are only permitted outside of a thin liquid layer surrounding the interface. Inside this layer and in the solid, the diffusion equation is solved using a standard finite-difference algorithm that is interfaced with the DMC algorithm using the local conservation law for the diffusing quantity. Here we combine this algorithm with a previously developed phase-field formulation of the interface dynamics and demonstrate that it can accurately simulate three-dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance.Comment: RevTeX, 16 pages, 10 eps figures, submitted to J. Comp. Phy

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

Modeling Dendritic Solidification using Lattice Boltzmann and Cellular Automaton Methods

This dissertation presents the development of numerical models based on lattice Boltzmann (LB) and cellular automaton (CA) methods for solving phase change and microstructural evolution problems. First, a new variation of the LB method is discussed for solving the heat conduction problem with phase change. In contrast to previous explicit algorithms, the latent heat source term is treated implicitly in the energy equation, avoiding iteration steps and improving the formulation stability and efficiency. The results showed that the model can deal with phase change problems more accurately and efficiently than explicit LB models. Furthermore, a new numerical technique is introduced for simulating dendrite growth in three dimensions. The LB method is used to calculate the transport phenomena and the CA is employed to capture the solid/liquid interface. It is assumed that the dendritic growth is driven by the difference between the local actual and local equilibrium composition of the liquid in the interface. The evolution of a threedimensional (3D) dendrite is discussed. In addition, the effect of undercooling and degree of anisotropy on the kinetics of dendrite growth is studied. Moreover, effect of melt convection on dendritic solidification is investigated using 3D simulations. It is shown that convection can change the kinetics of growth by affecting the solute distribution around the dendrite. The growth features of twodimensional (2D) and 3D dendrites are compared. Furthermore, the change in growth kinetics and morphology of Al-Cu dendrites is studied by altering melt undercooling, alloy composition and inlet flow velocity. The local-type nature of LB and CA methods enables efficient scaling of the model in petaflops supercomputers, allowing the simulation of large domains in 3D. The model capabilities with large scale simulations of dendritic solidification are discussed and the parallel performance of the algorithm is assessed. Excellent strong scaling up to thousands of computing cores is obtained across the nodes of a computer cluster, along with near-perfect weak scaling. Considering the advantages offered by the presented model, it can be used as a new tool for simulating 3D dendritic solidification under convection

Phase Field Model for Three-Dimensional Dendritic Growth with Fluid Flow

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 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 two-dimensional (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

Efficient simulation of neural development using shared memory parallelization

The Neural Development Simulator, NeuroDevSim, is a Python module that simulates the most important aspects of brain development: morphological growth, migration, and pruning. It uses an agent-based modeling approach inherited from the NeuroMaC software. Each cycle has agents called fronts execute model-specific code. In the case of a growing dendritic or axonal front, this will be a choice between extension, branching, or growth termination. Somatic fronts can migrate to new positions and any front can be retracted to prune parts of neurons. Collision detection prevents new or migrating fronts from overlapping with existing ones. NeuroDevSim is a multi-core program that uses an innovative shared memory approach to achieve parallel processing without messaging. We demonstrate linear strong parallel scaling up to 96 cores for large models and have run these successfully on 128 cores. Most of the shared memory parallelism is achieved without memory locking. Instead, cores have only write privileges to private sections of arrays, while being able to read the entire shared array. Memory conflicts are avoided by a coding rule that allows only active fronts to use methods that need writing access. The exception is collision detection, which is needed to avoid the growth of physically overlapping structures. For collision detection, a memory-locking mechanism was necessary to control access to grid points that register the location of nearby fronts. A custom approach using a serialized lock broker was able to manage both read and write locking. NeuroDevSim allows easy modeling of most aspects of neural development for models simulating a few complex or thousands of simple neurons or a mixture of both.journal articl

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

Solving multi-physics problems using adaptive finite elements with independently refined meshes

In this thesis, we study a numerical tool named multi-mesh method within the framework of the adaptive finite element method. The aim of this method is to minimize the size of the linear system to get the optimal performance of simulations. Multi-mesh methods are typically used in multi-physics problems, where more than one component is involved in the system. During the discretization of the weak formulation of partial differential equations, a finite-dimensional space associated with an independently refined mesh is assigned to each component respectively. The usage of independently refined meshes leads less degrees of freedom from a global point of view. To our best knowledge, the first multi-mesh method was presented at the beginning of the 21st Century. Similar techniques were announced by different mathematics researchers afterwards. But, due to some common restrictions, this method is not widely used in the field of numerical simulations. On one hand, only the case of two-mesh is taken into scientists\' consideration. But more than two components are common in multi-physics problems. Each is, in principle, allowed to be defined on an independent mesh. Besides that, the multi-mesh methods presented so far omit the possibility that coefficient function spaces live on the different meshes from the trial and test function spaces. As a ubiquitous numerical tool, the multi-mesh method should comprise the above circumstances. On the other hand, users are accustomed to improving the performance by taking the advantage of parallel resources rather than running simulations with the multi-mesh approach on one single processor, so it would be a pity if such an efficient method was only available in sequential. The multi-mesh method is actually used within local assembling process, which should not be conflict with parallelization. In this thesis, we present a general multi-mesh method without the limitation of the number of meshes used in the system, and it can be applied to parallel environments as well. Chapter 1 introduces the background knowledge of the adaptive finite element method and the pioneering work, on which this thesis is based. Then, the main idea of the multi-mesh method is formally derived and the detailed implementation is discussed in Chapter 2 and 3. In Chapter 4, applications, e.g. the multi-phase flow problem and the dendritic growth, are shown to prove that our method is superior in contrast to the standard single-mesh finite element method in terms of performance, while accuracy is not reduced

Simulations of energetic beam deposition: from picoseconds to seconds

We present a new method for simulating crystal growth by energetic beam deposition. The method combines a Kinetic Monte-Carlo simulation for the thermal surface diffusion with a small scale molecular dynamics simulation of every single deposition event. We have implemented the method using the effective medium theory as a model potential for the atomic interactions, and present simulations for Ag/Ag(111) and Pt/Pt(111) for incoming energies up to 35 eV. The method is capable of following the growth of several monolayers at realistic growth rates of 1 monolayer per second, correctly accounting for both energy-induced atomic mobility and thermal surface diffusion. We find that the energy influences island and step densities and can induce layer-by-layer growth. We find an optimal energy for layer-by-layer growth (25 eV for Ag), which correlates with where the net impact-induced downward interlayer transport is at a maximum. A high step density is needed for energy induced layer-by-layer growth, hence the effect dies away at increased temperatures, where thermal surface diffusion reduces the step density. As part of the development of the method, we present molecular dynamics simulations of single atom-surface collisions on flat parts of the surface and near straight steps, we identify microscopic mechanisms by which the energy influences the growth, and we discuss the nature of the energy-induced atomic mobility