Analysis and representation of microstructure data

We focus attention on data on heterogeneous nonlinear media. Studies show how the micro-mechanical fields localize into bands, and percolate in a complex way throughout the material. Our approach for representation consists of expansion of these fields in wavelet-series, followed by truncation using an appropriate term selection. The motivation stems from the similarity between the localize nature of the developing fields, and the local support of wavelets. As a case study, we use strain data sets of polycrystalline shape-memory alloys. The implementation of our approach demonstrated how it efficiently represents experimental and simulated data sets, while reducing their size by two orders of magnitude [1].By way of example, Fig. 1 shows axial strain fields obtained during uniaxial tension of nitinol sheets (courtesy of Prof. Samantha Daly), at representative deformation states. The left and right images of each panel correspond to the original strain maps, and their truncated wavelet counterparts using 3% of the number of pixels, respectively. The reconstructions based on the compact wavelet representation have demonstrated excellent norm recovery, and reproduce the key features of the original images; namely, the location, orientation, and intensity of the strain bands. Similar results were obtained analyzing the numerical data produced by Richards et al. (Acta Mat., 2013); therein, a numerical scheme was applied to solve the evolution of strain in a polycrystal governed by a pseudo-elastic behavior, when subjected to anti-plane shear. Fig. 2 shows the local transformation at an intermediate state of deformation. The left and right columns correspond to the original field and its truncated wavelet expansion using 10% of number of original terms, respectively. The first and second rows correspond to the axial and transverse strain, respectively. The foregoing results suggest that wavelets are efficient for data storage purposes. To examine if such compact wavelet representation is sufficient for characterizing the interplay between the transformation strain, the total strain and the stress, we replace the transformation strain with its truncated wavelet-series, an order of magnitude smaller in size, in the governing equations, and calculate the resultant overall stress-strain relation (Fig. 3). The curves are shown to be in good agreement, and demonstrate the same essential trends. An examination of the terms retained during the evolution reveals how there are wavelets that are never active, i.e., they are always removed during the truncation process, while others are kept frequently. This is illustrated in Fig. 4, using an activation map, in which we denote with black marks the support of retained wavelets in wavelet space. To explore the connection between these modes and the evolution of stress, Fig. 5 shows the macroscopic stress–strain relation calculated separately across physical domain associated with active and non-active wavelets. It is observed how the overall stress at active modes is lower than at the domain of non-active ones. Also, the overall strain in non-active modes is lower than in the domain of active ones.Our second focus is on the digital image correlation technique for full-field strains measurements. The basic idea of the method is to calculate full-field strains, by tracking subsets on grids across digital images of a specimen during deformation. Together with the capability of today’s high-resolution cameras, it results in huge amount of data points to be processed, thereby increasing the computational cost. We implemented a scheme based on the Discrete Cosine Transform (DCT), to end up with a compact representation of the input images, of an order of magnitude smaller size, while accurately reproducing the strains. Fig. 6 demonstrates the strains calculated from a representative image of Daly’s experiment (left map), compared with those calculated using the DCT representation, with the sizes of 20% (central) and 5% (right) of the original image. The L2L2-norm error is less than 1%.

Fig. 1. Axial strain fields obtained during uniaxial tension of nitinol sheets, at representative deformation states. The left and right images of each panel correspond to the original strain maps, and their truncated wavelet counterparts using 3% of the number of pixels, respectively

Fig. 2. The local transformation at an intermediate state of deformation. The left and right columns correspond to the original field and its truncated wavelet expansion using 10% of number of original terms, respectively. The first and second rows correspond to the axial and transverse strain, respectively.

Fig. 3. The macroscopic stress–strain relation. The continuous and dashed curves correspond to the exact and the wavelet-approximation schemes, respectively. 

Fig. 4. Waveletactivation map. Black marks denote the active wavelets in wavelet space. White regions are associated with wavelets which are never active.  

Fig. 5. Macroscopic stress–strain relation of the spatial regions of active (dashed curves) and non-active wavelets (dash-dot curves). The stress–strain relation of the whole domain is given as a reference (continuous curve). 

Fig. 6. Strain maps calculated using an original image (left), and using its DCT representation, with the sizes of 20% (central map) and 5% (right map) of the of the original image.