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

Abstract

Microstructure design can have a substantial effect on the performance of critical components in numerous aerospace applications. However, the stochastic nature of metallic microstructures leads to deviations in material properties from the design point and alters the performance of these critical components. In this paper, a novel stochastic linear programming (LP) methodology is developed for microstructure design accounting for uncertainties in desired properties. The metallic microstructure is represented using a finite element discretized form of the orientation distribution function (ODF). The inverse LP problem solves the mean values and covariance matrix of the ODFs to maximize the mean values of a property, given the statistical constraints on other properties. The highlight is an analytical uncertainty quantification model via a Gaussian distribution to model propagation of microstructural uncertainties to the properties. Examples illustrate maximization of the yield strength and magnetostriction of a galfenol alloy when constrained by uncertainties in a set of stiffness constants.

Uncertainty Quantification for Ti-7Al Alloy Microstructure with an Inverse Analytical Model (AUQLin)

“…The variables of this system must be non-negative in order to satisfy the ODF non-negativity condition (A≥0), and this leads to a non-negative mean values (μA≥0) condition since the variances are already non-negative by definition. The covariance expression for the unit volume fraction constraint is enforced through the equations defined through ΣAq=0 for each row of the ODF covariance matrix, ΣboldA, as shown in the proof given in our previous work [14]. Equation (13) is solved as an LP problem.…”

“…In Equation (1), A(rm) is the ODF value at the mth integration point (that has a coordinate rm), |Jn| is the Jacobian value of the nth element, wm is the integration weight for the mth integration point, and 1(1+rm·rm)2 is the metric of the Rodrigues parameterization. The summation expression in Equation (1) can be written as a linear equation in terms of the nodal point ODF values [12,13,14,15]: qTA=1, where q is the volume normalization vector that includes the single-crystal values. Equation (2) defines a mathematical constraint that must always be satisfied when modeling with the ODF approach.…”

“…Equation (3) can also be written in the linear form in terms of the nodal points ODFs such that [12,13,14,15]: <χ>=pTA, where p is the matrix of the single-crystal values of the corresponding material property. More details about the Rodrigues parameterization and finite element discretization schemes can be found in our earlier works [12,13,14,15], and are not repeated here for brevity. In this work, the hexagonal close-packed (HCP) Ti-7Al microstructure was modeled with 50 independent ODF values, and the finite element representation of the fundamental region is shown in Figure 1.…”

“…However, given the computationally expensive nature of the multi-scale problem of interest for a potential future work, one powerful approach has been found to model the stochasticity of the microstructure by implementing an analytical solution framework. For this purpose, an analytical linear solution scheme was developed by the author and has been applied to model the propagation of the uncertainty by assuming that the microstructural variables follow a Gaussian distribution [12,13,14,15]. The analytical algorithm has also been exercised in multi-scale design problems [14,15] such that for a prescribed variation in a volume-averaged material property, the variations in the microstructure are solved and considered in optimizing the material response.…”

Design of Galfenol and Alfenol microstructures for bending mode energy harvesters

“…Similarly, in this work, we focus on controlling the microstructural texture of Alfenol and Galfenol to achieve desired material properties that maximize energy efficiency when a mechanical input is received in the bending mode. The microstructure design was previously studied by our group to optimize the vibration frequencies [40] and magnetostrictive strain [41,42] of Galfenol. However, these studies [40][41][42] addressed the optimization of different volume-averaged material properties and did not focus on the magneto-mechanical coupling.…”