A Multigrid PGD‐based Algorithm for Volumetric Displacement Fields Measurements

Abstract

The use of a finite elements-based Digital Volume Correlation (FE-DVC) leads to lower measurement uncertainties in comparison to subset-based approaches. However, the associated computing time may become prohibitive when dealing with highresolution measurements. To overcome this limitation, a Proper Generalised Decomposition solver was recently applied to 2D digital image correlation. In this paper, this method is extended to measure volumetric displacements from 3D digital images. In addition, a multigrid Proper Generalised Decomposition algorithm is developed, which allows to use different discretisations in each term of the decomposition. Associated to a coarse graining of the digital images, this allows to avoid local minima, especially in presence of large displacements. Synthetic and practical cases are analysed with the present approach, and measurement uncertainties are compared with standard FE-DVC. Results show that such an approach reduces the computational cost (when compared to FE-DVC) whilst maintaining lower measurement uncertainties than standard subset-based DVC.

A Space-Time PGD-DIC Algorithm:

“…The interest of extending such methods to DIC has already been illustrated in previous works [27,17], where the dimensions of space were separated in order to speedup computation time. In this work, the idea is to separate space and time in the evolution problem of DIC (1).…”

“…Without loss of generality, we will simply consider a rank one approximation in the remainder of the paper, since, following [27,17], a rank m approximation is obtained in a greedy manner from successive best rank-one approximations. In addition, in the particular case of linear vibrations, a rank one approximation may be sufficient.…”

“…(17) As for classic DIC, a Gauss-Newton algorithm is used for the minimization (17). Since all the functions of time are known, the time integral of (17) can be computed, and we are left with a modified static DIC problem.…”

“…Note that the algorithm proposed in the following is based on developments regarding existing PGD methods in structural [25][19] and experimental mechanics [27] [17]. In order to do this, two mappings are defined:…”

“…Then the displacement is calculated as a sum of products of separated space and time functions over the full time interval. This can be seen as an extension of the Proper Generalized Decomposition (PGD) method previously proposed in DIC [27] and Digital Volume Correlation (DVC) [17] where the dimensions of space are separated in the same way. It is worth noting that the proposed PGD-DIC method is not limited to global DIC and could also be applied to sub-set based approaches.…”