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Quasi-Gauss Point Digital Image/Volume Correlation: a Simple Approach for Reducing Systematic Errors Due to Undermatched Shape Functions

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In using subset/subvolume-based digital image/volume correlation (DIC/DVC), proper shape functions must be defined to depict the underlying displacement of the target subsets/subvolumes to be tracked. Because the local deformation in a subset/subvolume cannot be known a priori, mismatched problems (i.e. undermatched or overmatched) unavoidably occur, which induce either remarkable systematic errors for undermatched cases or doubled random errors for overmatched cases in detected displacements. Also, the use of high-order shape functions (e.g. second-order shape functions) would lead to greatly increased computational complexity since more deformation parameters need to be solved. In this work, a novel and simple quasi-Gauss point DIC/DVC method that only uses first-order shape functions is proposed, which can completely avoid or significantly compromise the systematic errors due to undermatched shape functions. Specifically, based on rigorous theoretical analysis, we find that specific positions (designated as quasi-Gauss points) in a subset/subvolume deliver accurate displacement results even when undermatched issues are present. This new finding inspires us to output the displacements at these specific points rather than subset/subvolume center points. Numerical simulations with different deformation modes validate that the proposed approach can effectively reduce or even eliminate the undermatched error in these deformation modes in both DIC and DVC measurements.

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  1. 1.

    Sutton MA, Orteu J-J, Schreier HW (2009) Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications. Springer Science & Business Media

  2. 2.

    Pan B, Qian K, Xie H, Asundi A (2009) Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review. Meas Sci Technol 20(6):062001

  3. 3.

    Pan B (2018) Digital image correlation for surface deformation measurements: historical developments, recent advances and future goals. Meas Sci Technol 29(8):082001

  4. 4.

    Bay BK (2008) Methods and applications of digital volume correlation. J Strain Anal Eng Des 43(8):745–760

  5. 5.

    Fedele R, Ciani A, Fiori F (2014) X-ray microtomography under loading and 3D-volume digital image correlation. A review. Fundam Inform 135(1–2):171–197

  6. 6.

    Pan B, Xie HM, Wang ZY (2010) Equivalence of digital image correlation criteria for pattern matching. Appl Opt 49(28):5501–5509

  7. 7.

    Pan B, Li K, Tong W (2013) Fast, robust and accurate digital image correlation calculation without redundant computations. Exp Mech 53(7):1277–1289

  8. 8.

    Gao Y, Cheng T, Su Y et al (2015) High-efficiency and high-accuracy digital image correlation for three-dimensional measurement. Opt Lasers Eng 65:73–80

  9. 9.

    Schreier HW, Sutton MA (2002) Systematic errors in digital image correlation due to undermatched subset shape functions. Exp Mech 42(3):303–310

  10. 10.

    Lu H, Cary PD (2000) Deformation measurements by digital image correlation: implementation of a second-order displacement gradient. Exp Mech 40(4):393–400

  11. 11.

    Xu X, Su Y, Zhang Q (2017) Theoretical estimation of systematic errors in local deformation measurements using digital image correlation. Opt Lasers Eng 88:265–279

  12. 12.

    Wang B, Pan B (2019) Self-adaptive digital volume correlation for unknown deformation fields. Exp Mech 59(2):149–162

  13. 13.

    Yu L, Pan B (2015) The errors in digital image correlation due to overmatched shape functions. Meas Sci Technol 26(4):045202

  14. 14.

    Wang B, Pan B (2015) Random errors in digital image correlation due to matched or overmatched shape functions. Exp Mech 55:1717–1727

  15. 15.

    Davis PJ, Rabinowitz P (1984) Methods of numerical integration, 2nd edn. Academic, New York

  16. 16.

    Moan T (1974) Experiences with orthogonal polynomials and “best” numerical integration formulas on a triangle; with particular reference to finite element approximations. ZAMM Journal of applied mathematics and mechanics: Zeitschrift für angewandte Mathematik und Mechanik 54(7):501–508

  17. 17.

    Mackinnon RJ, Carey GF (1989) Superconvergent derivatives: a Taylor series analysis. Int J Numer Methods Eng 28(3):489–509

  18. 18.

    Wheeler MF, Whiteman JR (1987) Superconvergent recovery of gradients on subdomains from piecewise linear finite-element approximations. Numer Meth Part D E 3(1):65–82

  19. 19.

    Pan B, Asundi A, Xie H, Gao J (2009) Digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements. Opt Lasers Eng 47(7):865–874

  20. 20.

    Pan B, Xie HM, Xu BQ, Dai FL (2006) Performance of sub-pixel registration algorithms in digital image correlation. Meas Sci Technol 17(6):1615–1621

  21. 21.

    Wolberg G, Sueyllam HM, Ismail MA (2000) One-dimensional resampling with inverse and forward mapping functions. J Graph Tools 5(3):11–33

  22. 22.

    Beghini M, Bertini L, Fontanari V (2006) Evaluation of the stress–strain curve of metallic materials by spherical indentation. Int J Solids Struct 43(7–8):2441–2459

  23. 23.

    Sun C, Zhou Y, Chen J, Miao H (2017) Modeling and experimental identification of contact pressure and friction for the analysis of non-conforming elastic contact. Int J Mech Sci 133:449–456

  24. 24.

    Bouterf A, Adrien J, Maire E et al (2017) Identification of the crushing behavior of brittle foam: from indentation to oedometric tests. J Mech Phys Solids 98:181–200

  25. 25.

    Pan B, Wu DF, Wang ZY (2012) Internal displacement and strain measurement using digital volume correlation: a least squares framework. Meas Sci Technol 23:045002

  26. 26.

    Lava P, Cooreman S, Coppieters S et al (2009) Assessment of measuring errors in DIC using deformation fields generated by plastic FEA. Opt Lasers Eng 47:747–753

  27. 27.

    Pan B, Wang B, Wu D, Lubineau G (2014) An efficient and accurate 3D displacements tracking strategy for digital volume correlation. Opt Lasers Eng 58:126–135

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This work is supported by the National Key Research and Development Program of China (2018YFB0703500), and the National Natural Science Foundation of China (NSFC) (11872009, 11632010).

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Correspondence to B. Pan.

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Pan, B., Zou, X. Quasi-Gauss Point Digital Image/Volume Correlation: a Simple Approach for Reducing Systematic Errors Due to Undermatched Shape Functions. Exp Mech (2020). https://doi.org/10.1007/s11340-020-00588-3

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  • Digital image/volume correlation
  • Systematic error
  • Undermatched systematic error
  • Shape functions