LAUSR

M ICROSTRUCTURE design optimization is important for improving the performance of critical components in numerous aerospace applications. These critical components involve performance indices that are directly related to microstructures obtained during processing. This calls for direct control of microstructure evolution using well-designed processes. Recent developments in materials by design have allowed a more advanced systems approach that integrates the processing, structure, and property through multiscale computational material models [1]. The techniques that allow the tailoring of properties of polycrystalline alloys involve tailoring of the preferred orientation of crystals manifested as the crystallographic texture. During forming processes, mechanisms such as crystallographic slip and lattice rotation drive the formation of the texture and variability in property distributions in such materials. A useful method for designing materials is through the control of the deformation processes leading to the formation of textures that yield the desired property distributions. The microstructure modeling of the present work is based on the quantification of the microstructure using the orientation distribution function (ODF). The ODFmeasures the volumes of single crystals in different orientations. The ODF is defined based on a parameterization of the crystal lattice rotation. In this work, the ODF values are parameterized using a Rodrigues representation. The computational microstructural modeling has been studied extensively in the earlier works ofAcar and Sundararaghavan [2–6], Acar et al. [7,8], Acar and Sundararaghavan [9], Acar et al. [10], and other works in the literature [11–15] by using different computational techniques. The design of microstructures has also been exercised by Acar and Sundararaghavan [2–4] and Acar et al. [7,8] using the ODF model and a linear solution scheme to achieve optimummaterial properties. Multiple optimum microstructure designs were mathematically possible with the linear approach [2,3]. However, only a few of these optimum solutions was actually manufacturable. The texture evolution during deformation processing needed to be simulated to identify the optimum manufacturable design. The texture evolution in a deformation process was previously studied by Li et al. [16] by representing the processing paths as streamline functions in the space of the spectral coefficients. 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]. 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. It was observed that the number of manufacturable textures was limited when only one deformation process was studied and the variability in texture was higher when a sequence of deformation processes was applied. Therefore, the motivation of the present study is to find the optimal sequence of deformation processes that can produce the predetermined optimum material properties. The texture evolution is again represented by using reduced-order models in which the basis functions are derived with proper orthogonal decomposition (POD). The methodology is studied on the vibration tuning problem of a galfenol beam. The multiple optimummicrostructure solutions of the same problemwere computed in our recent works [2,3]. To determine the optimal processing sequence, the optimum solution directions are projected into the basis functions of different deformation processes. The sequence and strain rates of the deformation processes are optimized to achieve the closestmaterial properties to the previously determined values. The remainder of this Note is organized as follows. Section II summarizes the mathematical background for microstructure modeling and proper orthogonal decomposition. Section III includes the problem definition for optimal sequential process property matching, and it reports the optimization results. A summary of the Note and the potential extensions are finally discussed in Sec. IV.