Irene Livshits

A Variational Approach to Modeling and Simulation of Grain Growth

Most technologically useful materials arise as polycrystalline microstructures, composed of a myriad of small crystallites, called grains, separated by their interfaces, called grain boundaries. The orientations and arrangements of the grains and their network of boundaries are implicated in many properties across wide scales, for example, functional properties, like conductivity in microprocessors, and lifetime properties, like fracture toughness in structures. Simulation is becoming an important tool for understanding both materials properties and their processing requirements. Here we offer a consistent variational approach to the mesoscale simulation of these systems subject to the Mullins equation of curvature-driven growth in a two-dimensional setting. The main objective is to provide a calibration for future two-dimensional and three-dimensional efforts. We discuss several novel features of our approach, which we anticipate will render it a flexible, scalable, and robust tool to aid in microstructural prediction. Simulation results offer compelling evidence of the predictability and robustness of statistical properties of large systems, such as grain size distribution and texture, that are of immediate interest in materials science.

Mesoscale Simulation of the Evolution of the Grain Boundary Character Distribution

A mesoscale, variational simulation of grain growth in two-dimensions has been used to explore the effects of grain boundary properties on the grain boundary character distribution. Anisotropy in the grain boundary energy has a stronger influence on the grain boundary character distribution than anisotropy in the grain boundary mobility. As grain growth proceeds from an initially random distribution, the grain boundary character distribution reaches a steady state that depends on the grain boundary energy. If the energy depends only on the lattice misorientation, then the population and energy are related by the Boltzmann distribution. When the energy depends on both lattice misorientation and boundary orientation, the steady state grain boundary character distribution is more complex and depends on both the energy and changes in the gradient of the energy with respect to orientation.

Accuracy Properties of the Wave-Ray Multigrid Algorithm for Helmholtz Equations

Helmholtz equations with their highly oscillatory solutions play an important role in physics and engineering. These equations present the main computational difficulties typical to acoustic, electromagnetic, and other wave problems. They are often accompanied by radiation boundary conditions and are considered on infinite domains. Solving them numerically using standard procedures, including multigrid, is too expensive. The wave-ray multigrid algorithm efficiently solves the Helmholtz equations and naturally incorporates the radiation boundary conditions. Important accuracy properties of the wave-ray solver are discussed in this paper. Using various mode analyses, we show that, with the right choice of parameters, this algorithm can obtain an approximation to the differential solution with accuracy that equals the accuracy of the target grid discretization. Moreover, the boundary conditions can be introduced with any desired accuracy. Our theoretical conclusions are confirmed by numerical experiments.

Mesoscale Simulation of Grain Growth

Simulation is becoming an increasingly important tool, not only in materials science in a general way, but in the study of grain growth in particular. Here we exhibit a consistent variational approach to the mesoscale simulation of large systems of grain boundaries subject to Mullins Equation of curvature driven growth. Simulations must be accurate and at a scale large enough to have statistical significance. Moreover, they must be sufficiently flexible to use very general energies and mobilities. We introduce this theory and its discretization as a dissipative system in two and three dimensions. The approach has several interesting features. It consists in solving very large systems of nonlinear evolution equations with nonlinear boundary conditions at triple points or on triple lines. Critical events, the disappearance of grains and and the disappearance or exhange of edges, must be accomodated. The data structure is curves in two dimensions and surfaces in three dimensions. We discuss some consequences and challenges, including some ideas about coarse graining the simulation.

An algebraic multigrid wave–ray algorithm to solve eigenvalue problems for the helmholtz operator

Helmholtz equations with variable coefficients are known to be hard to solve both analytically and numerically. In this paper, we introduce a numerical multigrid solver for one-dimensional Helmholtz eigenvalue problems with periodic potentials and solutions. The solvers employ wave–ray methodology suggested by Brandt, Livshits for Helmholtz equations with constant coefficients. The paper concludes with numerical experiments and the discussion of future efforts for solving two-dimensional problems. Copyright

One-Dimensional Algorithm for Finding Eigenbasis of the Schrödinger Operator

Schrodinger equations are used to model numerous applications arising in quantum chemistry and physics. Most of these applications have no analytical solutions and need to be solved numerically, often an extremely challenging task. This paper offers an efficient multigrid/multiscale solver for the one-dimensional Schrodinger eigenvalue problem, a preliminary step toward developing solvers to application-rich two-dimensional problems. The solver employs a gradual multiscale eigenbasis representation that allows calculation and storage of only a small number of eigenfunction representatives on the expensive finest grids and a full eigenbasis representation and calculation on the coarsest grids. This structure not only allows calculation and storage of the entire eigenbasis in

The Surface Energy of MgO: Multiscale Reconstruction from Thermal Groove Geometry

O(N\log N)

An Approach to the Mesoscale Simulation of Grain Growth

operations, but it is also beneficial for many applications. The algorithms will be eventually adapted for solving the Schrodinger equation as it appears in the Kohn-Sham formulation for calculating electronic structure, though in this paper the nonlinearity of the problem (dependence of potentials on low eigenfunctions) is not discussed.

MULTIPLE GALERKIN ADAPTIVE ALGEBRAIC MULTIGRID ALGORITHM FOR THE HELMHOLTZ EQUATIONS

The surface energy of MgO is determined using experimental data collected from equilibrated thermal grooves circumscribing island grains. Local equilibrium assumptions at each groove require that the Herring equations be satisfied at each data site, thereby yielding a large and overdetermined system of equations involving the surface energy γ. This inverse problem is then solved using a new technique that is statistical in nature and multiscale in implementation. The resulting discrete solution represents a statistically significant representation of the surface energy of MgO as a function of surface orientation. Comparisons to results derived from a more traditional approach, along with suggested further applications, are discussed.

Remarks on the wave-ray multigrid solvers for Helmholtz equations

The simulation of curvature driven growth in grain boundary systems is becoming an important tool in understanding the behavior of microstructure evolution and there is much distinguished work in this subject. Here we address the mesoscale simulation of large systems of grain boundaries subject to the Mullins equation of curvature driven growth with the Herring force balance equation imposed at triple junctions. We discuss several novel features of our approach which we anticipate will render it a flexible, scalable, and robust tool to aid in microstructural prediction. What is the result of the simulation? We discuss what such a simulation is capable of predicting, taking as a prototype the histogram of relative area population as it changes through the simulation. We do not use this data to seek the best distribution, like Hillert, Rayleigh, or lognormal. Instead we treat the set of distributions as the solution of an inverse problem for a time varying function and determine the equation they satisfy. This results in a coarse graining of the complex simulation to simpler system governed by a Fokker-Planck Equation. Even so, fundamental questions concerning the predictability of simulations of large metastable systems arise from these considerations.