Abstract
Digital image correlation (DIC) is a powerful experimental technique for measuring full-field displacement and strain. The basic idea of the method is to compare images of an object decorated with a speckle pattern before and after deformation, and thereby to compute the displacement and strain fields. Local subset DIC and finite element-based global DIC are two widely used image matching methods. However there are some drawbacks to these methods. In local subset DIC, the computed displacement field may not be compatible, and the deformation gradient may be noisy, especially when the subset size is small. Global DIC incorporates displacement compatibility, but can be computationally expensive. In this paper, we propose a new method, the augmented-Lagrangian digital image correlation (ALDIC), that combines the advantages of both the local (fast) and global (compatible) methods. We demonstrate that ALDIC has higher accuracy and behaves more robustly compared to both local subset DIC and global DIC.
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References
Hild F, Roux S (2006) Digital image correlation: from displacement measurement to identification of elastic properties-a review. Strain 42:69–80
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 20:062001
Sutton MA, Orteu JJ, Schreier HW (2009) Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications. Springer, Berlin
Sutton MA, Wolters WJ, Peters WH, Ranson WF, McNeill SR (1983) Determination of displacements using an improved digital correlation method. Image Vis Comput 1:133–139
Chen DJ, Chiang FP, Tan YS, Don HS (1993) Digital speckle-displacement measurement using a complex spectrum method. Appl Opt 32:1839–1849
Dhir SK, Sikora JP (1972) An improved method for obtaining the general-displacement field from a holographic interferogram. Exp Mech 12:323–327
Kreis T (1996) Holographic interferometry: principles and methods. In: Simulation and experiment in laser metrology: proceedings of the international symposium on laser applications in precision measurements Held in Balatonfured/Hungary, June 3-6, 1996, volume 2. John Wiley & Sons, p 323
Rastogi PK (2000) Principles of holographic interferometry and speckle metrology. In: Photomechanics. Springer, pp 103–151
Dickinson AS, Taylor AC, Ozturk H, Browne M (2011) Experimental validation of a finite element model of the proximal femur using digital image correlation and a composite bone model. J Biomech Eng 133:014504
Franck C, Hong S, Maskarinec SA, Tirrell DA, Ravichandran G (2007) Three-dimensional full-field measurements of large deformations in soft materials using confocal microscopy and digital volume correlation. Exp Mech 47:427–438
Franck C, Maskarinec SA, Tirrell DA, Ravichandran G (2011) Three-dimensional traction force microscopy: a new tool for quantifying cell-matrix interactions. PloS one 6:e17833
Rehrl C, Kleber S, Antretter T, Pippan R (2011) A methodology to study crystal plasticity inside a compression test sample based on image correlation and ebsd. Mater Charact 62:793–800
Daly S (2007) Deformation and fracture of thin sheets of nitinol. Phd thesis California Institute of Technology
Bastawros AF, Bart-Smith H, Evans AG (2000) Experimental analysis of deformation mechanisms in a closed-cell aluminum alloy foam. J Mech Phys Solids 48:301–322
Jerabek M, Major Z, Lang RW (2010) Strain determination of polymeric materials using digital image correlation. Polym Test 29:407–416
Wang Y, Cuitiño A M (2002) Full-field measurements of heterogeneous deformation patterns on polymeric foams using digital image correlation. Int J Solids Struct 39:3777–3796
Zdunek J, Brynk T, Mizera J, Pakieła Z, Kurzydłowski KJ (2008) Digital image correlation investigation of portevin–le chatelier effect in an aluminium alloy. Mater Charact 59:1429–1433
Tracy J, Waas A, Daly S (2015) Experimental assessment of toughness in ceramic matrix composites using the j-integral with digital image correlation part i: methodology and validation. J Mater Sci 50:4646–4658
Kimiecik M, Jones JW, Daly S (2013) Quantitative studies of microstructural phase transformation in nickel-titanium. Mater Lett 95:25–29
Chang S, Wang CS, Xiong CY, Fang J (2005) Nanoscale in-plane displacement evaluation by afm scanning and digital image correlation processing. Nanotechnology 16:344
Erten E, Reigber A, Hellwich O, Prats P (2009) Glacier velocity monitoring by maximum likelihood texture tracking. IEEE Trans Geosci Remote Sens 47:394–405
Rubino V, Lapusta N, Rosakis A (2012) Laboratory earthquake measurements with the high-speed digital image correlation method and applications to super-shear transition. In AGU Fall Meeting Abstracts 1:06
Rubino V, Lapusta N, Rosakis AJ, Leprince S, Avouac JP (2014) Static laboratory earthquake measurements with the digital image correlation method. Exp Mech, pp 1–18
Besnard G, Leclerc H, Hild F, Roux S, Swiergiel N (2012) Analysis of image series through global digital image correlation. J. Strain Anal Eng Des 47:214–228
Blaber J, Adair B, Antoniou A (2015) Ncorr: open-source 2d digital image correlation matlab software. Exp Mech: 1–18
Jones EMC, Silberstein MN, White SR, Sottos NR (2014) In situ measurements of strains in composite battery electrodes during electrochemical cycling. Exp Mech 54:971–985
Pan B, Wang B, Lubineau G, Moussawi A (2015) Comparison of subset-based local and finite element-based global digital image correlation. Exp Mech 55:887–901
Correlated Solutions (2009) Vic-2d Reference Manual
Pan B, Asundi A, Xie HM, Gao JX (2009) Digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements. Opt Lasers Eng 47:865–874
Avril S, Feissel P, Pierron F, Villon P (2009) Comparison of two approaches for differentiating full-field data in solid mechanics. Meas Sci Technol 21:015703
Zhao JQ, Zeng P, Pan B, LP Lei HFDu, He WB, Liu Y, Xu YJ (2012) Improved hermite finite element smoothing method for full-field strain measurement over arbitrary region of interest in digital image correlation. Opt Lasers Eng 50:1662–1671
Modersitzki J (2004) Numerical methods for image registration. Oxford University Press, London
Ronovskỳ A, Vašatová A (2017) Elastic image registration based on domain decomposition with mesh adaptation. Mathematical Analysis and Numerical Mathematics 15:322–330
Bouclier R, Passieux JC (2017) A domain coupling method for finite element digital image correlation with mechanical regularization application to multiscale measurements and parallel computing. Int J Numer Methods Eng 111:123–143
Merta M, Vašatová A, Hapla V, Horák D (2014) Parallel implementation of Total-FETI DDM with application to medical image registration. In: Domain Decomposition Methods in Science and Engineering XXI. Springer, pp 917–925
Passieux JC, Perie JN, Salaun M (2015) A dual domain decomposition method for finite element digital image correlation. Int J Numer Methods Eng 102:1670–1682
Wang TY, Qian KM (2017) Parallel computing in experimental mechanics and optical measurement: a review (ii) Optics and Lasers in Engineering
Nocedal J, Wright S (2006) Numerical optimization. Springer, Berlin
Conn AR, Gould NIM, Toint PL (1991) A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J Numer Anal 28:545–572
Afonso MV, Bioucas-Dias JM, Figueiredo MAT (2011) An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans Image Process 20:681–695
Simo JC, Laursen TA (1992) An augmented Lagrangian treatment of contact problems involving friction. Comput Struct 42:97–116
Michel JC, Moulinec H, Suquet P (2000) A computational method based on augmented Lagrangians and fast Fourier Transforms for composites with high contrast. CMES-Computer Modeling in Engineering & Sciences 1:79–88
Goldstein T, O’Donoghue B, Setzer S, Baraniuk R (2014) Fast alternating direction optimization methods. SIAM J Imag Sci 7:1588–1623
Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2010) Distributed optimization and statistical learning via the alternating direction method of multipliers. Mach Learn 3:1–122
Yang JF, Zhang Y (2011) Alternating direction algorithms for l(1)-problems in compressive sensing. SIAM J Sci Comput 33:250–278
Afonso MV, Bioucas-Dias JM, Figueiredo MAT (2010) Fast image recovery using variable splitting and constrained optimization. IEEE Trans Image Process 19:2345–2356
Glowinski R, Le Tallec P (1989) Augmented Lagrangian and operator-splitting methods in nonlinear mechanics SIAM
Pan B, Xie H, Wang Z (2010) Equivalence of digital image correlation criteria for pattern matching. Applied optics 49:5501–5509
Simon B, Iain M (2004) Lucas-kanade 20 years on a unifying framework. International journal of computer vision 56:221–255
Réthoré J, Hild F, Roux S (2007) Shear-band capturing using a multiscale extended digital image correlation technique. Comput Methods Appl Mech Eng 196:5016–5030
Réthoré J, Hild F, Roux S (2008) Extended digital image correlation with crack shape optimization. International Journal for Numerical Methods in Engineerin 73:248–272
Reu PL, Toussaint E, Jones E, Bruck HA, Iadicola M, Balcaen R, Turner DZ, Siebert T, Lava P, Simonsen M (2017) Dic challenge developing images and guidelines for evaluating accuracy and resolution of 2d analyses experimental mechanics
Bornert M, Doumalin P, Dupré JC, Poilâne C, Robert L, Toussaint E, Wattrisse B (2017) Shortcut in DIC error assessment induced by image interpolation used for subpixel shifting. Opt Lasers Eng 91:124–133
Avellar L, Ravichandran G (2016) Deformation and fracture of 3d printed heterogeneous materials Society for Experimental Mechanics Annual Conference
Hackbusch W (2003) Multi-grid method and applications. Springer, Berlin
Yang J, Bhattacharya K (2018) Combining image compression with digital image correlation. Experimental Mechanics
Yang J, Bhattacharya K (2019) Fast adaptive global digital image correlation. In: Advancement of Optical Methods & Digital Image Correlation in Experimental Mechanics, volume 3. Springer, pp 69–73
Acknowledgments
We are grateful to Dr. Louisa Avellar for sharing her images of heterogeneous fracture with us. We gratefully acknowledge the support of the US Air Force Office of Scientific Research through the MURI grant ‘Managing the Mosaic of Microstructure’ (FA9550-12-1-0458).
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Appendices
Appendix A: Inverse Compositional Gauss-Newton
In this paper, we use Inverse Compositional Gauss-Newton (IC-GN) scheme to solve local subset DIC optimization. Given the current iterate of deformation map \({\mathbf y}^{k}\), we seek the updated deformation map \({\mathbf y}^{k + 1}\). It is convenient to define the inverse maps \(\boldsymbol { \phi }^{k}\) and \(\boldsymbol { \phi }^{k + 1}\), where ϕk(yk(X)) = X. We define the increment \(\boldsymbol {\psi }^{k}\) through yk+ 1 = (ψ)k ∘yk as shown in Fig. 12. We make a change of configuration and rewrite as
where \({\mathbf {z}}\) is the current iterate of deformation map \({\mathbf y}^{k}\). We obtain \(\boldsymbol { \psi }^{k}\) as the minimizer of this functional and the updated deformation map as
To minimize (27), we assume \(\boldsymbol { \psi }^{k} \approx {\mathbf z} + {\mathbf v} + {\mathbf H} ({\mathbf z} - {\mathbf z}_{0})\) for small v and \(\mathbf H\). Therefore,
Minimizing over \({\mathbf v}\) and \({\mathbf H}\), we obtain
where
and \(g_{,l} = {\partial g \over \partial _{z_{l}}}\) etc. We solve (30) for \({\mathbf v}, {\mathbf H}\) to obtain \(\boldsymbol {\psi }^{k}\). We then obtain the new (inverse) deformation \(\boldsymbol {\phi }^{k + 1}\) using Eq. 28. In practice, we don’t need to compute \({{\Omega }_{i}^{k}}\) domain at each iteration, instead we directly compute all the integrations (or discrete summations) over the final deformed configuration, which also gives us good results and saves lots of computation time.
We also use IC-GN to solve subproblem 1 or Eq. 20 in ALDIC. This reduces to Eq. 30 above with \(a_{lp}\) and \(d_{l}\) replaced with
Appendix B: The operator D
The matrix \(\mathbf {D}\) in “Augmented Lagrangian DIC (ALDIC) Method” is the discrete gradient operator. This depends on the choice of discretization. In this paper, we use first order finite difference based on an uniform square mesh. We provide explicit details for this case, but note that ALDIC is compatible with any discretization and these would lead to different matrices.
We describe it in one dimension for convenience, and the generalization to higher dimensions is obvious. We assume that the domain is discretized uniformly with the distance h between nodes \(x_{i}, x_{2} {\dots } x_{N}\). Then the (15) is explicitly written as
where \({\mathbf u}_{i}, {\mathbf F}_{i}\) are the values at node \(x_{i}\).
Appendix C: Optimality conditions
Set the first term in Eq. 18 to be
The necessary and sufficient optimality conditions for the ALDIC ADMM formulation are primal feasibility
and dual feasibility,
Since \(\mathbf {F}^{k + 1}, \mathbf {u}^{k + 1}\) minimize \(\mathcal {L}(\mathbf {F},\mathbf {u}, {\hat {\mathbf {u}}}^{k}, \mathbf {W}^{k}, \mathbf {v}^{k})\) in the ADMM Subproblem 1, we have that
Or equivalently,
This means that the quantity
can be viewed as a residual for the dual feasibility condition (41). We will refer to \(\mathbf {s}^{k + 1}\) as the dual residual at ADMM iteration \(k + 1\), and to
as the primal residual at ADMM iteration \(k + 1\). And these two residuals converge to zeros as ADMM proceeds.
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Yang, J., Bhattacharya, K. Augmented Lagrangian Digital Image Correlation. Exp Mech 59, 187–205 (2019). https://doi.org/10.1007/s11340-018-00457-0
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DOI: https://doi.org/10.1007/s11340-018-00457-0