Linear Solution Scheme for Microstructure Design with Process Constraints

Abstract

This paper addresses a two-step linear solution scheme to find an optimum metallic microstructure satisfying performance needs and manufacturability constraints. The microstructure is quantified using the orientation distribution function, which determines the volume densities of crystals that make up the polycrystal microstructure. The orientation distribution function of polycrystalline alloys is represented in a discrete form using finite elements, and the volume-averaged properties are computed. The first step of the solution approach identifies the orientation distribution functions that lead to the set of optimal engineering properties using linear programming. This step leads to multiple solutions, of which only a few can be manufactured using traditional processing routes such as rolling and forging. In the second step, textures from a given process are represented in a space of reduced basis coefficients called the process plane. This step involves generation of orthogonal basis functions for representing spatial variations of the orientation distribution functions during a given process using proper orthogonal decomposition. Multiple orientation distribution function solutions in step one are then projected onto these basis functions to identify which of the optimal textures are feasible through a given manufacturing process. This feasibility is determined with two approaches. The first approach finds the closest match to the orientation distribution function solutions in the material plane, whereas the second approach finds the closest match to a desired set of properties instead of the orientation distribution functions. The method is explained through an example of vibration tuning of a galfenol alloy, with the primary objective of maximizing the yield strength.

Reduced-Order Modeling Approach for Materials Design with a Sequence of Processes

“…Instead, a reduced-order representation of the texture evolution was found to be a more powerful approach to solve the process optimization problem [4]. The reducedorder maps of different deformation processes were derived to identify the optimal process to achieve the predetermined material properties [4]. The present work is the extension of our recent work [4], in which the optimal single process was identified by using these reduced-order maps.…”

“…The computational microstructural modeling has been studied extensively in the earlier works of Acar and Sundararaghavan [2][3][4][5][6], Acar et al [7,8], Acar and Sundararaghavan [9], Acar et al [10], and other works in the literature [11][12][13][14][15] by using different computational techniques. The design of microstructures has also been exercised by Acar and Sundararaghavan [2][3][4] and Acar et al [7,8] using the ODF model and a linear solution scheme to achieve optimum material properties. Multiple optimum microstructure designs were mathematically possible with the linear approach [2,3].…”

“…However, this model required a large number of spectral terms to capture sharp textural features and describe the processing paths. Instead, a reduced-order representation of the texture evolution was found to be a more powerful approach to solve the process optimization problem [4]. The reducedorder maps of different deformation processes were derived to identify the optimal process to achieve the predetermined material properties [4].…”

Stochastic Design Optimization of Microstructural Features Using Linear Programming for Robust Design

“…Introduction M ICROSTRUCTURE design has been typically addressed as a deterministic optimization problem in which features such as volume fractions of various phases are controlled to achieve the desired property. Several issues arise: First, multiple microstructures can lead to the desired property, and some microstructures are easier to manufacture than others [1,2]. Thus, the deterministic optimizer has to be capable of predicting nonunique solutions.…”

“…Thus, the materials community has recently focused on reduced-order modeling techniques. The most preferred method is proper orthogonal decomposition (POD) [1,[23][24][25][26][27]. However, POD provides a linear formulation in the reduced basis, and it is better suited for linear or weakly nonlinear problems.…”