Percolation in Two-Dimensional Grain Boundary Structures and Polycrystal Property Closures Percolation in Two-Dimensional Grain Boundary Structures and Polycrystal Property Closures
In part I, the applicability of standard lattice percolation models to a random 2-D grain structure is explored. A random network based on the triangle lattice is proposed as a more appropriate model, and results in a higher percolation threshold (0.711 compared with 0.653 for the standard hexagonal lattice). The triple junction constraint inherent in grain boundary structures is subsequently applied to the new network. This results in a lowering of the percolation threshold to 0.686, which turns out to be very close to the value obtained from the hexagonal lattice under the same constraint.
In Part II, an efficient method for finding the closure of a bi-objective optimization problem involving two material properties is formulated. The method is based upon two algorithms developed to find the Pareto front in multi-objective problems – the weighted sum, and the normal boundary intersection methods. The resultant procedure uses quadratic programming (QP) to find as many points on the closure as possible, changing to the less efficient sequential quadratic programming (SQP) only where necessary to find points on concave, or almost concave, regions.
Improvements on the method are demonstrated using extra linear constraints in the generalized weighted sum (GWS) algorithm, and multiple GWS trials at each stage.
Optimization using only the linear part of the problem is shown to give excellent results for the particular example used, and may help to achieve an approximate closure more quickly for larger problems.
An adaptation of a common Pareto front method from genetic algorithms – the maxmin algorithm – is also demonstrated. The efficiency of the method is found to be reasonable for finding closures in the test case.
Other general optimization techniques for the form of the problem in-hand are explored for completeness of the study. These include a survey of unconstrained techniques that might be useful in large-scale problems, and an in-depth application of QP for the constrained case.
School:Brigham Young University
School Location:USA - Utah
Source Type:Master's Thesis
Keywords:percolation optimization grain boundaries homogenization
Date of Publication:08/02/2005