Modelling feasibility constraints for materials design: Application to inverse crystallographic texture problem
Graphical abstract
Introduction
Optimization of material structure and manufacturing processes for maximized performance of a material is a primary interest in the field of materials science and engineering. For numerical optimizations, a rather simplified statement of the optimization problem can be given as follows:where represents the optimal microstructure that enables a material to exhibit desirable performance, represents microstructure evolution after some processes that require an initial microstructure () and processing variables () as inputs, and represents the constraints on the feasibility of the microstructure and processing variables. To date, a tremendous amount of experimental and modelling work has been conducted in an effort to quantify the linkages among process, microstructure, and properties [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Works on microstructure quantification and structure-property correlations, as well as on process-structure linkages, have proven to be very accurate within the range of data used to fit the structure-property or process-structure models [1], [2], [3], [4], [5], [6], [7]. In addition, new opportunities for microstructure optimization and material design have opened up with the recent surge of interest in machine learning techniques [8], [9], [10], [11], [12], [13], [14], [15]. Nonetheless, there is a lack of work regarding quantification of the feasibility constraint for materials optimization.
Many engineering problems are unavoidably subject to linear or non-linear feasibility constraints. Processing variables such as imposed strain, processing temperature, and initial microstructure for a given manufacturing procedure may simply not be feasible, thereby limiting the solutions to unconstrained optimization problems. For data-driven regression models, the feasibility constraint becomes more important because the accuracy of the predictions made beyond the data domain is not guaranteed. Thus, an approach to formulate constraints suitable for optimization purposes is needed to take advantage of recent advances in process-structure-property linkages. Unfortunately, the feasibility boundary definition is an extremely complex one. Processing variables, as well as microstructural features, can be interrelated and the number of dimensions required to describe a microstructure can be immensely large [16], [17].
This work involved investigation of the utilization of a support vector machine (SVM) to model user-defined feasibility constraints for a materials-related, constrained optimization problem. The basic idea was first to model feasibility constraints in high dimensional space using SVM, then to combine an objective function with a penalty function that involves the decision function of the trained SVM. As a popular class of machine learning techniques, SVM has already found its use in limit state functions for failure domains [18], [19], feasibility prediction [20], and explicit nonlinear model predictive control [21]. Throughout this work, the use of an SVM decision function is presented to model feasibility constraints for optimizing the initial texture of body-centered cubic (BCC) polycrystalline material prior to cold-rolling.
The crystallographic texture of a BCC polycrystalline material is a key design variable in manufacturing materials with directional properties. In particular, plastic anisotropy due to crystallographic texture is central in the formability of metallic materials [22], [23], [24]. It is well-known that BCC polycrystalline materials with high formability involve a strong {1 1 1} crystal plane parallel to the normal direction (ND) of the specimen (γ-fiber orientation) and a weak {1 0 0} crystal plane parallel to the ND of the specimen (θ-fiber orientation). A common practice in processing BCC polycrystalline sheet materials with high γ-fiber texture is cold-rolling followed by recrystallization. Because it is well-established that different initial textures can result in different final microstructures and textures, the ability of a process to manufacture the most desirable final texture will strongly depend on the texture prior to the process [25], [26], [27], [28]. Therefore, it is possible that an optimal initial texture that could maximize γ-fiber and minimize θ-fiber orientations after cold-rolling exists. The optimization of the initial texture prior to cold-rolling is a typical constrained optimization problem, and can be given as follows:where an orientation distribution function (ODF) is given as with representing orientation, represents the target texture where γ-fiber orientations are maximized while θ-fiber orientations are minimized, represents the final texture after a specific process given an initial texture, and represents the constraints on what a feasible initial texture can be. In this work, we employed the processing path model proposed by Li et al. [28] as the theoretical model for and the decision boundary function of SVM as the feasibility constraint to solve the optimization problem.
Section snippets
Processing path model
Among existing theoretical models for texture evolution , the processing path model proposed by Li et al. [28] was used for this work. The model was selected because it provides an explicit form of . In a Eulerian framework for predicting texture evolution, the conservational property of ODF satisfies the continuity equation often used to describe fluid flow [29]. Because any change in an ODF due to a deformation parameter or processing variable is assumed to be
Processing path model
Only the texture evolution matrix needs to be determined for the processing path model to properly function. Because the derivative values on the left hand side of Eq. (7) cannot be obtained directly, a midpoint finite difference method is employed to numerically obtain the values
The coefficients () are calculated from a series of visco-plastic self-consistent (VPSC) model calculations with Taylor interaction. Then, the texture evolution matrix () is determined by linear least
Adding additional constraints in the processing path model
While this work mainly focused on constrained optimization of the initial texture prior to cold-rolling as a case study, the method could be extended to optimize processing parameters as well. Consider the explicit form of texture evolution in Eq. (8). If the initial texture replaces the left hand side of Eq. (8), then the initial texture can be replaced by another initial texture aswhich allows Eq. (8) to be reformulated as
The above procedure
Conclusions
The method proposed for constraint modelling provides a key option in approaches using computationally aided materials design, where numerical optimization can now take into account complex feasibility constraints in a very simple manner. As a case study, constrained optimization of the initial texture prior to cold-rolling was conducted with the aid of SVM to model user-defined feasibility constraints that reflect limitations in terms of engineering practices. The objective to maximize γ-fiber
Data availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
Author contribution
Jaimyun Jung contributed to the implementation of the computer code and supporting algorithms. Jaimyun Jung and Hyoung Seop Kim have contributed to development of the methodology and preparation of the manuscript. Jaimyun Jung, Jae Ik Yoon, Seong-Jun Park, Jun-Yun Kang, Gwang Lyeon Kim, Yi Hwa Song, Sung Taek Park, Kyeong Won Oh, and Hyoung Seop Kim have contributed to conceptualization and data preparation.
References (40)
- et al.
Reduced-order structure-property linkages for polycrystalline microstructures based on 2-point statistics
Acta Mater.
(2017) - et al.
Data science approaches for microstructure quantification and feature identification in porous membranes
J. Memb. Sci.
(2017) - et al.
Data-driven reduced order models for effective yield strength and partitioning of strain in multiphase materials
J. Comput. Phys.
(2017) - et al.
Development of high throughput assays for establishing process-structure-property linkages in multiphase polycrystalline metals: application to dual-phase steels
Acta Mater.
(2017) - et al.
Building texture evolution networks for deformation processing of polycrystalline fcc metals using spectral approaches: applications to process design for targeted performance
Int. J. Plast.
(2010) - et al.
Microstructure-based knowledge systems for capturing process-structure evolution linkages
Curr. Opin. Solid State Mater. Sci.
(2017) - et al.
Extraction of reduced-order process-structure linkages from phase-field simulations
Acta Mater.
(2017) - et al.
The onset temperature (Tg) of AsxSe1−x glasses transition prediction: a comparison of topological and regression analysis methods
Comput. Mater. Sci.
(2017) - et al.
Materials discovery and design using machine learning
J. Materiomics
(2017) - et al.
An efficient machine learning approach to establish structure-property linkages
Comput. Mater. Sci.
(2019)
Exploring the microstructure manifold: image texture representations applied to ultrahigh carbon steel microstructures
Acta Mater.
Limit state function identification using Support Vector Machines for discontinuous responses and disjoint failure domains
Probab. Eng. Mech.
Adaptive explicit decision functions for probabilistic design and optimization using support vector machines
Comput. Struct.
Influence of differential speed rolling ratio on the ridging behavior of ultra purified 17%Cr ferritic stainless steel
Mater. Charact.
Texture, microstructure and mechanical properties of aluminum modified ultra-pure 429 ferritic stainless steels
Mater. Des.
Effect of initial texture and microstructure of Mg on mechanical properties of Mg – stainless steel laminated metal composites
Mater. Charact.
Initial texture effects on the thermal stability and grain growth behavior of nanocrystalline Ni thin films
Mater. Sci. Eng. A
Effect of initial texture on texture and microstructure evolution of ME20 Mg alloy subjected to hot rolling
Mater. Sci. Eng. A
Processing path optimization to achieve desired texture in polycrystalline materials
Acta Mater.
Texture development by plastic deformation
Scr. Metall.
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