Fast Hermite element method for smoothing and differentiating noisy displacement field in digital image correlation

In our previous work, an improved Hermite finite element smoothing method (IHFESM) combined with the well-known Tikhonov regularization was proposed to smooth the noisy displacement field directly calculated by digital image correlation (DIC). Even though IHFESM could reconstruct reliable strain field for arbitrary region of interest (ROI), it still suffers from three defects in practice: (i) for the large scale problems, it involves high computational burden to find the optimum regularization parameter within the generalized cross-validation (GCV) function, since the inverse and the trace of two large matrices are frequently evaluated, (ii) the search range is too wide to find the desired optimum regularization parameter quickly, and (iii) for the arbitrary ROI, the complex two dimensional meshing routines are required to locate and mesh the invalid regions into triangular elements. In current work, to overcome above deficiencies, we propose the fast Hermite element method (FHEM). The proposed FHEM avoids the redundant computation in IHFESM by decomposing the resultant two small matrices to speed up the original matrix inverse and trace computation. To narrow down the search range of regularization parameter, the magnitude of the closeness term and roughness penalty term in error function are balanced by a new parameter. The calculation of global regularization matrix is also sped up by using simple formulation without any sophisticated meshing routines. Experiments show that the FHEM is at least 50 times as fast as IHFESM with similar accuracy, and it is recommended to be adopted to smooth and differentiate the noisy displacement field for DIC in practice.