Radial basis point interpolation for strain field calculation in digital image correlation

Jiayi Du; Jian Zhao; Jian Zhao; Jian Zhao; Jiahui Liu; Jiahui Liu; Dong Zhao; Dong Zhao; Dong Zhao

doi:10.1364/AO.520232

Applied Optics
Vol. 63,
Issue 14,
pp. 3929-3943
(2024)
•https://doi.org/10.1364/AO.520232

Radial basis point interpolation for strain field calculation in digital image correlation

Jiayi Du, Jian Zhao, Jiahui Liu, and Dong Zhao

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Jiayi Du, Jian Zhao, Jiahui Liu, and Dong Zhao, "Radial basis point interpolation for strain field calculation in digital image correlation," Appl. Opt. 63, 3929-3943 (2024)

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Abstract

In order to extract smooth and accurate strain fields from the noisy displacement fields obtained by digital image correlation (DIC), a point interpolation meshless (PIM) method with a radial basis function (RBF) is introduced for full-field strain calculation, which overcomes the problems of slow calculation speed and unstable matrix inverse calculation of the element-free Galerkin method (EFG). The radial basis point interpolation method (RPIM) with three different radial basis functions and the moving least squares (MLS) and pointwise least squares (PLS) methods are compared by analyzing and validating the strain fields with high-strain gradients in simulation experiments. The results indicate that the RPIM is nearly 80% more computationally efficient than the MLS method when a larger support domain is used, and the efficiency of the RPIM is nearly 26% higher than that of the MLS method when a smaller support domain is used; the strain calculation accuracy is slightly lower than that of the MLS method by 0.3–0.5%, but the stability of the calculation is significantly improved. In contrast with the PLS method, which is easily affected by the noise and the size of the strain calculation window, the RPIM is insensitive to the displacement noise and the size of the support domain and can obtain a similar calculation accuracy. The RPIM with multiquadric (MQ) radial basis functions performs well in balancing the computational accuracy and efficiency and is insensitive to shape parameters. The application cases show that the method can effectively compute the strain field at the crack tip, validating its applicability to the study of the plastic region at the crack tip. In conclusion, the proposed RPIM-based method provides an accurate, practical, and robust approach for full-field strain measurements.

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Supplementary Material (1)

Name	Description
Code 1	Matlab code for for Radial basis point interpolation for strain field calculation in digital image correlation

Data availability

The code for the RPIM algorithm in this paper is available in Ref. [39].

39. J. Du, J. Zhao, J. Liu, et al., “RPIM code,” figshare, 2024, https://doi.org/10.6084/m9.figshare.25101479.

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Radial Basis Function	Expression^a	Shape Parameters
Multiquadric (MQ)	$R_{i} (r_{i}) = (r_{i}^{2} + a_{c}^{2} d_{c}^{2})^{q}$	$a_{c} \geq 0, q = 1.03$
Gaussian (EXP)	$R_{i} (r_{i}) = \exp (- a_{c} {(r_{i} / d_{c})}^{2})$	$a_{c} = 0.31$
Thin-plate spline (TPS)	$R_{i} (r_{i}) = r_{i}^{η} \ln (r_{i})$	$η = 2.01$

	Support Domain Size (pixel)
Method	3	5	7	9	11	13	15
MQ	$1.18 \times 10^{7}$	$7.45 \times 10^{7}$	$6.54 \times 10^{8}$	$4.49 \times 10^{9}$	$2.25 \times 10^{10}$	$8.74 \times 10^{10}$	$2.80 \times 10^{11}$
EXP	$4.83 \times 10^{5}$	$1.73 \times 10^{7}$	$1.27 \times 10^{8}$	$4.51 \times 10^{8}$	$1.10 \times 10^{9}$	$2.15 \times 10^{9}$	$3.70 \times 10^{9}$
TPS	$4.09 \times 10^{6}$	$6.65 \times 10^{7}$	$8.40 \times 10^{8}$	$6.32 \times 10^{9}$	$3.24 \times 10^{10}$	$1.27 \times 10^{11}$	$4.11 \times 10^{11}$
MLS-1 order	$8.32 \times 10^{8}$	$4.59 \times 10^{8}$	$1.98 \times 10^{8}$	$1.25 \times 10^{8}$	$8.01 \times 10^{8}$	$5.98 \times 10^{8}$	$3.89 \times 10^{8}$
MLS-2 order	$6.23 \times 10^{17}$	$7.57 \times 10^{16}$	$1.83 \times 10^{16}$	$7.21 \times 10^{15}$	$3.18 \times 10^{15}$	$1.55 \times 10^{15}$	$8.95 \times 10^{14}$
MLS-3 order	$2.23 \times 10^{23}$	$1.96 \times 10^{16}$	$5.10 \times 10^{16}$	$9.28 \times 10^{16}$	$1.87 \times 10^{17}$	$2.25 \times 10^{17}$	$1.81 \times 10^{17}$

Radial Basis Function	Expression^a	Shape Parameters
Multiquadric (MQ)	$R_{i} (r_{i}) = (r_{i}^{2} + a_{c}^{2} d_{c}^{2})^{q}$	$a_{c} \geq 0, q = 1.03$
Gaussian (EXP)	$R_{i} (r_{i}) = \exp (- a_{c} {(r_{i} / d_{c})}^{2})$	$a_{c} = 0.31$
Thin-plate spline (TPS)	$R_{i} (r_{i}) = r_{i}^{η} \ln (r_{i})$	$η = 2.01$

	Support Domain Size (pixel)
Method	3	5	7	9	11	13	15
MQ	$1.18 \times 10^{7}$	$7.45 \times 10^{7}$	$6.54 \times 10^{8}$	$4.49 \times 10^{9}$	$2.25 \times 10^{10}$	$8.74 \times 10^{10}$	$2.80 \times 10^{11}$
EXP	$4.83 \times 10^{5}$	$1.73 \times 10^{7}$	$1.27 \times 10^{8}$	$4.51 \times 10^{8}$	$1.10 \times 10^{9}$	$2.15 \times 10^{9}$	$3.70 \times 10^{9}$
TPS	$4.09 \times 10^{6}$	$6.65 \times 10^{7}$	$8.40 \times 10^{8}$	$6.32 \times 10^{9}$	$3.24 \times 10^{10}$	$1.27 \times 10^{11}$	$4.11 \times 10^{11}$
MLS-1 order	$8.32 \times 10^{8}$	$4.59 \times 10^{8}$	$1.98 \times 10^{8}$	$1.25 \times 10^{8}$	$8.01 \times 10^{8}$	$5.98 \times 10^{8}$	$3.89 \times 10^{8}$
MLS-2 order	$6.23 \times 10^{17}$	$7.57 \times 10^{16}$	$1.83 \times 10^{16}$	$7.21 \times 10^{15}$	$3.18 \times 10^{15}$	$1.55 \times 10^{15}$	$8.95 \times 10^{14}$
MLS-3 order	$2.23 \times 10^{23}$	$1.96 \times 10^{16}$	$5.10 \times 10^{16}$	$9.28 \times 10^{16}$	$1.87 \times 10^{17}$	$2.25 \times 10^{17}$	$1.81 \times 10^{17}$

Abstract

Supplementary Material (1)

Data availability

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Equations (8)

Applied Optics