Motion estimation by X-ray tomography: a variational formulation for 3D-Volume DIC and a finite element implementation

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. As an alternative to conventional Rayleigh- Ritz approach, a novel variational formulation is presented for the continuum DIC estimation. 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, with only one Newton iteration per level irrespectively of the tolerance satisfaction, as if the problem were linear. A Tychonoff regularization provision is implemented, which preserves the estimates against spurious oscillations. Results are presented concerning a laboratory Xray micro-tomography experiment on a polymeric foam sample, subjected to uniaxial loading by an apparatus ad-hoc realized.