In this study, the inverse problem of reconstructing the in‐plane (2D) displacements of a monitored surface through a sequence of two‐dimensional digital images, severely ill‐posed in Hadamard's sense, is deeply investigated. A novel variational formulation is presented for the continuum 2D digital image correlation problem, and critical issues such as semi‐coercivity and solution multiplicity are discussed by functional analysis tools. In the framework of a Galerkin, finite element discretization of the displacement field, a robust implementation for 2D digital image correlation is outlined, aiming to attenuate the spurious oscillations which corrupt the deformation scenario, especially when very fine meshes are adopted. Recourse is made to a hierarchical family of grids linked by suitable restriction and prolongation operators and defined over an image pyramid. Multi‐grid cycles are performed ascending and descending along the pyramid, with only one Newton iteration per level irrespective of the tolerance satisfaction, as if the problem were linear. At each level, the conventional least‐square matching functional is herein enriched by a Tychonoff regularization provision, preserving the solution against an unstable response. The algorithm is assessed on the basis of both synthetic and truly experimental image pairs. Copyright © 2013 John Wiley & Sons, Ltd.

Ptychographic X-ray computed tomography (PXCT) is a quantitative imaging modality that non-destructively maps the 3D electron density inside an object with tens of nanometers spatial resolution. This method provides unique access to the morphology and structure of the osteocyte lacuno-canalicular network (LCN) and nanoscale density of the tissue in the vicinity of an osteocyte lacuna. Herein, we applied PXCT to characterize the lacunae and LCN in a male Wistar rat model of glucocorticoid-induced osteoporosis (GIO). The ptychographic images revealed significant (p < 0.05) differences in the number of canaliculi originating from the lacuna per ellipsoidal surface unit, Ca.Nb (p = 0.0106), and the 3D morphology of the lacuna (p = 0.0064), between GIO and SHAM groups. Moreover, the mean canalicular diameter, Ca.Dm, was slightly statistically un-significantly smaller in GIO (152 ± 6.5) nm than in SHAM group (165 ± 8) nm (p = 0.053). Our findings indicate that PXCT can non-destructively provide detailed, nanoscale information on the 3D organization of the LCN in correlative studies of pathologies, such as osteoporosis, leading to improved diagnosis and therapy.

The violent relaxation and the metastable states of the Hamiltonian Mean-Field model, a paradigmatic system of long-range interactions, is studied using a Hamiltonian formalism. Rigorous results are derived algebraically for the time evolution of selected macroscopic observables, e.g., the global magnetization.The high and low energy limits are investigated and the analytical predictions are compared with direct N -body simulations. The method we use enables us to re-interpret the out-of-equilibrium phase transition separating magnetized and (almost) unmagnetized regimes.

In this study a robust strategy for 3D-Volume Digital Image Correlation (DIC) is presented, apt to provide accurate kinematic measurements within a loaded sample on the basis of three-dimensional digital images by X-ray computed micro-tomography. In the framework of a Galerkin, finite element discretization of the displacement field, the inverse problem of estimating 3D motion inside the bulk material is solved recursively on a hierarchical family of grids, linked by suitable restriction and prolongation operators. Such structured grids are defined over an image pyramid, which is generated starting from the raw tomographic reconstructions by a reiterated application of average filters and sub-sampling operators. To achieve robust estimates of the underlying displacement fields, multi-grid cycles are performed ascending and descending along the pyramid in a selected sequence. At each scale, the least-square matching function for DIC is enriched by means of a penalty term in the spirit of Tychonoff regularization, including as a priori information the estimate achieved at the previous grid and transferred to the current scale. For each grid only one * The Authors kindly acknowledge: (i) the support provided by CARIPLO Foundation to the Project titled:"Innovative metal-ceramic joints for high and ultra-high temperature applications (UHT)" (Grant n. 2010-0432, Call for Scientific and technological research on advanced materials); (ii) the support provided by Politecnico di Milano to the project "CINEMAT" (2010,5 per mille Young researcher Call). † Address for correspondence: Piazza Leonardo da Vinci n. 32, 20133 Milan, Italy 362 R. Fedele, et al. / A Regularized, Pyramidal Multi-grid Approach to Global Digital Image CorrelationNewton iteration can be considered, implying important time savings. Results are presented concerning a laboratory X-ray micro-tomography experiment on a polymeric foam sample, subjected to loading by an apparatus ad-hoc realized.

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We investigate the finite size corrections to the equilibrium magnetization of an Ising model on a random graph with N nodes and N γ edges, with 1 < γ ≤ 2. By conveniently rescaling the coupling constant, the free energy is made extensive. As expected, the system displays a phase transition of the mean-field type for all the considered values of γ at the transition temperature of the fully connected Curie-Weiss model. Finite size corrections are investigated for different values of the parameter γ, using two different approaches: a replica-based finite N expansion, and a cavity method. Numerical simulations are compared with theoretical predictions. The cavity based analysis is shown to agree better with numerics.

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In this review paper the synergistic combination of X-ray microtomography, in situ mechanical tests on material samples and full-field kinematic measurements by 3D-Volume Digital Image Correlation is discussed. First, basic features are outlined, concerning X-ray microtomography by either laboratory sources or synchrotron radiation. The main equations for 3D-Volume Digital Image Correlation are then presented, and different provisions regularizing the ill-posed problem of motion estimation are outlined. Thereafter, a survey of the state of the art is provided, with reference to a variety of biological and engineering materials. Limitations and perspectives of the proposed methodology in diverse applications are highlighted. The rapid growth of this research topic is emphasized, due to the truly multi-disciplinary vocation, the synergy between algorithmic and technological solutions, a fusion of experiments and numerical methods.