No abstract
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.
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.
This work concerns the development of an Algebraic Multilevel method for computing stationary vectors of Markov chains. We present an efficient Bootstrap Algebraic Multilevel method for this task. In our proposed approach, we employ a multilevel eigensolver, with interpolation built using ideas based on compatible relaxation, algebraic distances, and least squares fitting of test vectors. Our adaptive variational strategy for computation of the state vector of a given Markov chain is then a combination of this multilevel eigensolver and associated multilevel preconditioned GMRES iterations. We show that the Bootstrap AMG eigensolver by itself can efficiently compute accurate approximations to the state vector. An additional benefit of the Bootstrap approach is that it yields an accurate interpolation operator for many other eigenmodes. This in turn allows for the use of the resulting AMG hierarchy to accelerate the MLE steps using standard multigrid correction steps. Further, we mention that our method, unlike other existing multilevel methods for Markov Chains, does not employ any special processing of the coarse-level systems to ensure that stochastic properties of the fine-level system are maintained there. The proposed approach is applied to a range of test problems, involving non-symmetric stochastic M-matrices, showing promising results for all problems considered.
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.
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