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• A rotated Gaussian weight strain filter (RGW-SF) based on the point-wise least squares (PLS) is proposed to tackle the selection of the proper smooth window size for complex strain field measurement. • The systematic and random strain errors of the RGW-SF method are derived. • three weight parameters (i.e., σ x , σ y , and θ) in the RGW function cooperate to select an optimum smooth window by minimizing the derived total strain error of RGW-SF, which is composed of the systematic and random strain errors. • The RGW-SF method produces more accurate strain fields than PLS method and the other method using changeable smooth window size. and it needs no additional effort to tune the smooth window size as required in PLS. The point-wise least squares (PLS) has been widely accepted as a simple and trustworthy strain computation method that smooths and differentiates the noisy displacement field calculated directly by digital image correlation (DIC). Smooth window size and kernel function need to be determined before using PLS. The selection of the proper smooth window size has long been a tricky issue for practitioners, which has an obvious impact on the final total strain error of complex deformation. A small smooth window may reduce the system/bias error but increase the random error, and vice versa. In this paper, a rotated Gaussian weight strain filter (RGW-SF) is proposed for heterogeneous strain field measurement. The motivation of RGW-SF is to adapt the weights of each point in tradition PLS smooth window to obtain lower total strain error, which is derived and then minimized to select an optimum strain filter with three weight parameters (i.e., σ x , σ y , and θ). The effectiveness of the proposed RGW-SF is verified using simulated sinusoidal deformation images and the Star 6 image set from DIC-challenge 2.0. Experiments show that the RGW-SF can find a trade-off between the system error and the random error. The displacement and strain field accuracy, especially for the metrological efficiency indicator, by RGW-SF is much better than that of the PLS and the self-adaptive strain algorithm (SSA). The proposed RGW-SF is strongly suggested for unknown severely heterogeneous strain measurement. [ABSTRACT FROM AUTHOR]

• The Hermite dataset is designed for producing the displacement and strain training data for deep learning based DIC, which is convenient to control the magnitude, deformation mode and spatial frequency. • Proposed DIC-Net applies deeper downsampling layers to extract more feature for improving accuracy of deformation measurement, deconvolution is replaced for avoiding Checkerboard Artifacts. • The predicted result of Star5 image set from DIC-Challenge2.0 achieves a spatial resolution of 17.25 pixels and low noise level σ u of 0.0136. Digital image correlation (DIC) is a non-contact optical method that tracks the speckle pattern on specimen surface to calculate the displacement and strain by image correlation algorithm. Although the traditional DIC method can conveniently measure surface deformation, it still has many limitations: (1) the accuracy of displacement and strain calculation needs to be improved in the case of high deformation gradient; (2) under match or over-match can hardly be avoided when the filters are used to reconstruct smooth displacement or strain field, and (3) boundary effect remains unresolved in computing the deformation near the boundary of region of interest or the discontinuous area (e.g. area near crack tip or crack face). Recently, the deep learning based DIC (Deep-DIC) has revealed its attractive ability in handling above issues in traditional DIC, and impressive results have been achieved. The mean value of the absolute error (MAE) on the test set has been optimized to 0.0361 pixels using existing Deep-DIC approaches, which are accompanied by a real-time measurement speed. The network structure and training dataset are two key factors for the deep learning method. However, the current working networks have been modified from other image tasks and cannot fully adapt to the demands of the DIC tasks, and the dataset they generated still has evident flaws, limiting the method's accuracy and generalization performance which is utilized to assess performance on samples outside the training set. In this paper, we firstly proposed a new Hermite dataset that is created by using the high-order Hermite element to take account more complex deformation, then a new network architecture designed for the DIC task has been developed to extract richer deformation features. A test set of 3216 examples containing six different modes of displacement is used to compare the performance of our network with others. The proposed DIC-Net-d achieves the lowest MAE in the test set. Meanwhile, in the Star5 image sets from DIC-Challenge, the proposed DIC-Net-d achieves a spatial resolution of 17.25 pixels and a noise level of 0.0136 which outperforms existing traditional and non-traditional methods. Finally, the strain network trained by our Hermite dataset is also successful in predicting the strain field of Star6 in the DIC challenge. The experiment results show the superiority of the proposed Hermite dataset and new network with respect to other Deep-DIC methods. [ABSTRACT FROM AUTHOR]

• A new rotated sigmoid weight (RSW) function based DIC (RSW-DIC) is proposed to tackle the problem that the results of the weight function based DIC are influenced by the initial subset size (here 'initial' means 'original', the initial subset size keepsconstant over iterations, but the equivalent subset size is varying according to the weight distribution). The RSW-DIC has much better performance than other weight function based methods for unknown heterogeneous displacement. • The spatial resoltion of the proposed method is much better than other DIC methods using the same subset and shape functionorder. • The proposed RSW-DIC has better noise resistance than other participated methods. The Measurement resolution can be reduced by using a big initial subset size without increasing spatial resolution much. Conventional digital image correlation (C-DIC) combined with a rotated Gaussian weight (RGW) function for subset has demonstrated attractive ability in resolving heterogeneous deformation parameters. To further improve the performance, the selection of an optimum weight function becomes the key issue. In this paper, a novel rotated sigmoid weight (RSW) function is proposed. RSW function aims to get a more uniform weight distribution near the subset center, and to realize the continuous change of the equivalent subset size as well. The performance of the RSW function is compared with the Gaussian weight (GW) function, RGW function, the rotated inverse distance weight (RIDW) function and the inverse distance square weight (RIDSW) function through Star 5 image set from the DIC challenge 2.0 and the simulated image set. A total of six methods with different weight functions and rotated weights are systematically compared. The experiment results clearly show that DIC combined with RSW function (i.e. RSW-DIC) has the best spatial resolution without sacrificing much measurement resolution. The spatial resolution of RSW-DIC is only about half of the other methods for both first- and second-order shape functions when a big initial subset size is adopted. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• A regularized polynomial smoothing (RPS) method based on Tikhonov regularization is proposed to improve previous local Hermite (LH) method and measure noisy high-gradient strain field (HSF). • RPS is nearly insensitive to window size and produces lower error than point-wise least-squares (PLS) method in resolving local HSF. • In match/overmatch case, Tikhonov regularization in RPS greatly decreases the degree of the morbidity caused by ill-posedness in PLS and ensure the smoothness and stability of solution. Digital image correlation (DIC) is an effective tool for studying the deformation of tested materials. However, high-gradient strain field (HSF) measurement remains a challenging issue in DIC, because multifold factors have a significant effect on the first correlation calculation step of the original displacement and the exact strain reconstruction in the second post-processing step. In this study, we focus on the second step. The formerly proposed local Hermite has been proven as an accurate and robust method, yet substantial programming work is involved in constructing a Hermite element. To solve this issue, a regularized polynomial smoothing (RPS) method is proposed, which replaces the Hermite function of the polynomial. The RPS applies the same kernel function as the point-wise least-squares (PLS) method for easy implementation. Tikhonov regularization and the generalized cross-validation (GCV) function are also used to decrease unavoidable noise and weaken the dependence on window size. The experiments demonstrate that, in the matched/overmatched case, Tikhonov regularization in RPS can greatly decrease the degree of morbidity caused by ill-posed problems in the PLS method, thereby ensuring smoothness and stability of the solution. Moreover, the influences of window size and polynomial order of both the RPS and PLS are also closely studied. In conclusion, the proposed RPS provides a practical, robust and accurate method for HSF measurement. [ABSTRACT FROM AUTHOR]

Highlights • Two types of Local Hermite (low-order method LH-1 and high-order method LH-2) smoothing method based on Hermite element and Tikhonov regularization are proposed to measure noisy high-gradient strain field (HSF). • The proposed LH-2 method is nearly insensitive to smoothing window size. • The proposed LH-2 method generally produces lower error than point-wise least-squares (PLS) method in resolving local HSF. Abstract High-gradient strain field (HSF) measurement is attracting more and more attention from engineering and scientific fields, where digital image correlation (DIC) is widely adopted to calculate the dramatically changing displacement fields on stressed samples. Accurate reconstruction of reliable strain field from noisy displacement field is one of the key issues, since strain calculation is highly sensitive to displacement noise. The formally proposed improved Hermite finite element smoothing method (IHFESM) and fast Hermite element method (FHEM) are effective global smoothing techniques based on Tikhonov regularization. However, they involve complex meshing and global regularization matrix assembly processes. In this work, we present a local version of FHEM, called local Hermite (LH) method, in which one Hermite element is taken as a smoothing window centered on one displacement data, and the whole field is handled point-wisely. Two types of LH method are obtained by employing Hermite element with two types of continuities, i.e. C1 and C2. The smoothed displacement and strain are consequently computed from the regularized local surface as done in FHEM. Two kinds of simulated noisy displacement fields and one high-gradient displacement field obtained from shape memory alloy (SMA) tensile test are used to compare the performance of the proposed LH methods with the widely used point-wise least-squares (PLS) method. Experiments show that LH method with C2 element is nearly insensitive to window size, which is a quite useful property for practical engineering test without much prior knowledge. More importantly, it generally produces lower error than PLS in resolving local HSF. Thus, LH is a robust and accurate method which is highly recommended for practical HSF measurement. [ABSTRACT FROM AUTHOR]

• The estimations of random error and under-matched error caused by two strain calculation methods (point-wise least squares (PLS) and regularized polynomial smoothing method (RPS)) are proposed, based on two assumptions on the noise error of calculated displacement. • For the typical kernel function of 3rd order polynomial, a self-adaptive algorithm minimizing the total error is proposed to choose the optimal parameters. • The self-adaptive algorithm can give the optimal parameters in restoring the displacement and strain field, and obtain a more accurate result if the provided displacement field conforms strictly to assumptions. The strain error analysis is greatly concerned recently as digital image correlation (DIC) is used to measure the heterogeneous deformation. This paper focuses on the estimation of random error and under-matched error caused by two strain calculation methods, i.e. the point-wise least squares (PLS) and the regularized polynomial smoothing method (RPS). Two assumptions are put forward on the noise error of the calculated displacement that are: a) it is pure random error without bias and b) in each strain window, it is the independent Gaussian white noise with zero-mean. Based on the assumptions, the random error of displacement and strain is estimated, and the under-matched error of displacement and strain is theoretically analyzed by the aid of Laplacian operator. These two error solutions are verified by some stimulated experiments. Then for the typical kernel function of 3rd order polynomial, a self-adaptive algorithm minimizing the total error is proposed to choose the optimal parameters, i.e. window size and parameter λ. Experiments show that when the original displacement noise conforms to the assumptions strictly, 1) the estimated random error and under-matched error agrees very well with the experimental value, 2) the self-adaptive algorithm can give the optimal parameters in restoring the displacement and strain field, and 3) the estimation of random error and under-matched error is affected by DIC noise greatly, and it is better to use low-pass Gaussian filter before utilizing self-adaptive algorithm. [ABSTRACT FROM AUTHOR]

Highlights • Three novel strategies are proposed to minimize the thread competition for the existing path-dependent parallel DIC. • The proposed method could upgrade the 2D-DIC speed up to 145,000 points per second using 48 threads. • The proposed fast DIC method can be used to monitor the structural health with ROI containing less than 5000 points in real time. Abstract With the increasing demands on the real-time measurement using digital image correlation (DIC), the fast and ultra-fast computation are attracting much more attention from academic and engineering fields in last decades. Multi-thread parallelization is a recognized effective method to raise the correlation speed of traditional single-thread DIC up to higher order of magnitude. However, for the path-dependent parallel DIC based on specific initial guess propagation method, competition between different threads greatly cuts down the overall speed, because different threads visit the same queue or status matrix at the same time. Therefore, the desired linear speed-up ratio growth could hardly be achieved especially for large thread number. In this work, three novel strategies are proposed to minimize the thread competition for the existing path-dependent parallel DIC, including i) allocating private queues for private thread, ii) restarting idle thread if others are busy, and iii) collecting more uncorrelated neighboring points for subsequent correlation. In this way, the conflict between threads is obviously decreased and experiment results reveal that the proposed method could upgrade the 2D-DIC speed up to 145,000 points per second using 48 threads, which makes the real-time measurement feasible for region of interest (ROI) with less than 5000 points. [ABSTRACT FROM AUTHOR]