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Augmented Lagrangian Digital Image Correlation

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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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  1. https://sem.org/dic-challenge/2d-test-image-sets.asp

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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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Correspondence to K. Bhattacharya.

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)kyk as shown in Fig. 12. We make a change of configuration and rewrite as

$$ C_{i} = {\int}_{{\Omega}_{i}^{k}} | f(\boldsymbol{ \phi}^{k}(\mathbf{z})) - g(\boldsymbol{ \psi}(\mathbf{z})) |^{2} d \mathbf{z}, $$
(27)

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

$$ {\mathbf \phi}^{k + 1} = {\mathbf \phi}^{k} \circ (\boldsymbol{ \psi}^{k})^{-1}. $$
(28)

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,

$$ C_{i} =\! {\int}_{{\Omega}_{i}^{k}} | f({\mathbf \phi}^{k}(\mathbf{z})) - g({\mathbf z}) - \nabla g({\mathbf z}) \cdot ({\mathbf v} + {\mathbf H} ({\mathbf z} - {\mathbf z}_{0})) |^{2} d{\mathbf z}. $$
(29)

Minimizing over \({\mathbf v}\) and \({\mathbf H}\), we obtain

$$ \left( \begin{array}{lll} a_{lp} & b_{lqr} \\ b_{mnp} & c_{mnqr} \end{array} \right) \left( \begin{array}{l} v_{p} \\ H_{qr} \end{array} \right) = \left( \begin{array}{l} d_{l} \\ e_{mn} \end{array} \right) $$
(30)

where

$$\begin{array}{@{}rcl@{}} a_{lp} &=& 2 {\int}_{{{\Omega}_{i}^{k}}} g_{,l} g_{,p} d{\mathbf z}, \end{array} $$
(31)
$$\begin{array}{@{}rcl@{}} b_{lqr} &=& {\int}_{{{\Omega}_{i}^{k}}} g_{,l} g_{,q} (z_{r}-z_{0r}) d {\mathbf z}, \end{array} $$
(32)
$$\begin{array}{@{}rcl@{}} c_{mnqr} &=& 2 {\int}_{{{\Omega}_{i}^{k}}} g_{,m} (z_{n}-z_{0n}) g_{,q} (z_{r}-z_{0r})d{\mathbf z}, \end{array} $$
(33)
$$\begin{array}{@{}rcl@{}} d_{l} &=& {\int}_{{{\Omega}_{i}^{k}}} (f-g) g_{,l} d{\mathbf z}, \end{array} $$
(34)
$$\begin{array}{@{}rcl@{}} e_{mn} &=& {\int}_{{{\Omega}_{i}^{k}}} (f-g) g_{,m}(z_{n}-z_{0n}) d{\mathbf z} \end{array} $$
(35)

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.

Fig. 12
figure 12

The change of variables involved in the IC-GN update

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

$$\begin{array}{@{}rcl@{}} a^{\prime}_{lp} &=& 2 {\int}_{{{\Omega}_{i}^{k}}} (g_{,l} g_{,p} + {\mu \over 2} \delta_{lp} ) d{\mathbf z}, \end{array} $$
(36)
$$\begin{array}{@{}rcl@{}} d_{l}^{\prime} &=& {\int}_{{{\Omega}_{i}^{k}}} ((f-g) g_{,l} + {\mu \over 2}(u_{l} -{v_{l}^{k}}-{\hat{u}}_{l}^{k}) )d{\mathbf z}. \end{array} $$
(37)

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

$$ \begin{array}{l} \begin{array}{llll} \underbrace{ \left\{ \begin{array}{c} \mathbf{F}_{1}\\ \mathbf{F}_{2}\\ \mathbf{F}_{3}\\ \vdots \\ \mathbf{F}_{N-1}\\ \mathbf{F}_{N} \end{array} \right\} } & = &\underbrace{\frac{1}{2h} \left[\begin{array}{lllllll} -2 & 2 & & & \\ -1 & 0 & 1 & & & \\ & -1 & 0 & 1 & & \\ & & \ddots & \ddots & \ddots & \\ & & & -1 & 0 & 1 \\ & & & & -2 & 2 \end{array}\right] } & \underbrace{ \left\{ \begin{array}{c} \mathbf{u}_{1}\\ \mathbf{u}_{2}\\ \mathbf{u}_{3}\\ \vdots \\ \mathbf{u}_{N-1}\\ \mathbf{u}_{N} \end{array} \right\} } \\ ~~~~~\{\mathbf{F} \} & &~~~~~~~~~~~~~~~~~~~~\mathbf{D} & ~~~~~\{ \mathbf{u} \} \end{array} \end{array} $$
(38)

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

$$ {\Phi}(\mathbf{F},\mathbf{u}) =\! \sum\limits_{i} {\int}_{{\Omega}_{i}} \left| f(\mathbf{X}) - g\left( \mathbf{X} + \mathbf{u}_{i} + \left( \mathbf{F}_{i}(\mathbf{X}-\mathbf{X}_{i0})\right)\right) \right|^{2} d \mathbf{X}. $$
(39)

The necessary and sufficient optimality conditions for the ALDIC ADMM formulation are primal feasibility

$$ \left[\begin{array}{l} \mathbf{D} {\hat{\mathbf{u}}}^{*} - \mathbf{F}^{*} \\ {\hat{\mathbf{u}}}^{*} - \mathbf{u}^{*} \end{array}\right] = \left[\begin{array}{l} \mathbf{0} \\ \mathbf{0} \end{array}\right] $$
(40)

and dual feasibility,

$$ \left[\begin{array}{l} \frac{\partial {\Phi}(\mathbf{F}^{*},\mathbf{u}^{*})}{\partial \mathbf{F}} \\ \frac{\partial {\Phi}(\mathbf{F}^{*},\mathbf{u}^{*})}{\partial \mathbf{u}} \end{array}\right] - \left[\begin{array}{l} \beta \mathbf{W}^{*}\\ \mu \mathbf{v}^{*} \end{array}\right] = \left[\begin{array}{l} \mathbf{0} \\ \mathbf{0} \end{array}\right]. $$
(41)

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

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{l} \mathbf{0} \\ \mathbf{0} \end{array}\right] &=& \left[\begin{array}{l} \frac{\partial {\Phi}(\mathbf{F}^{k + 1},\mathbf{u}^{k + 1})}{\partial \mathbf{F}} \\ \frac{\partial {\Phi}(\mathbf{F}^{k + 1},\mathbf{u}^{k + 1})}{\partial \mathbf{u}} \end{array}\right] - \left[\begin{array}{l} \beta \mathbf{W}^{k} \\ \mu \mathbf{v}^{k} \end{array}\right] - \left[\begin{array}{l} \beta \left( \mathbf{D} {\hat{\mathbf{u}}}^{k} - \mathbf{F}^{k + 1} \right) \\ \mu \left( {\hat{\mathbf{u}}}^{k} - \mathbf{u}^{k + 1} \right) \end{array}\right] \\ &=& \left[\begin{array}{l} \frac{\partial {\Phi}(\mathbf{F}^{k + 1},\mathbf{u}^{k + 1})}{\partial \mathbf{F}} \\ \frac{\partial {\Phi}(\mathbf{F}^{k + 1},\mathbf{u}^{k + 1})}{\partial \mathbf{u}} \end{array}\right] - \left[\begin{array}{l} \beta \mathbf{W}^{k} \\ \mu \mathbf{v}^{k} \end{array}\right] - \left[\begin{array}{l} \beta \left( \mathbf{D} {\hat{\mathbf{u}}}^{k + 1} - \mathbf{F}^{k + 1} \right) \\ \mu \left( {\hat{ \mathbf{u}}}^{k + 1} - \mathbf{u}^{k + 1} \right) \end{array}\right] - \left[\begin{array}{l} \beta \left( \mathbf{D} {\hat{\mathbf{u}}}^{k} - \mathbf{D} {\hat{\mathbf{u}}}^{k + 1} \right) \\ \mu \left( {\hat{\mathbf{u}}}^{k} - {\hat{\mathbf{u}}}^{k + 1} \right) \end{array}\right] \\ &=& \left[\begin{array}{l} \frac{\partial {\Phi}(\mathbf{F}^{k + 1},\mathbf{u}^{k + 1})}{\partial \mathbf{F}} \\ \frac{\partial {\Phi}(\mathbf{F}^{k + 1},\mathbf{u}^{k + 1})}{\partial \mathbf{u}} \end{array}\right] - \left[\begin{array}{l} \beta \mathbf{W}^{k + 1} \\ \mu \mathbf{v}^{k + 1} \end{array}\right] - \left[\begin{array}{l} \beta \left( \mathbf{D} {\hat{\mathbf{u}}}^{k} - \mathbf{D} {\hat{\mathbf{u}}}^{k + 1} \right) \\ \mu \left( {\hat{\mathbf{u}}}^{k} - {\hat{\mathbf{u}}}^{k + 1} \right) \end{array}\right]. \end{array} $$
(42)

Or equivalently,

$$ \left[\begin{array}{l} \frac{\partial {\Phi}(\mathbf{F}^{k + 1},\mathbf{u}^{k + 1})}{\partial \mathbf{F}} \\ \frac{\partial SSD(\mathbf{F}^{k + 1},\mathbf{u}^{k + 1})}{\partial \mathbf{u}} \end{array}\right] - \left[\begin{array}{l} \beta \mathbf{W}^{k + 1} \\ \mu \mathbf{v}^{k + 1} \end{array}\right] = \left[\begin{array}{l} \beta \left( \mathbf{D} {\hat{\mathbf{u}}}^{k} - \mathbf{D} {\hat{\mathbf{u}}}^{k + 1} \right) \\ \mu \left( {\hat{\mathbf{u}}}^{k} - {\hat{\mathbf{u}}}^{k + 1} \right) \end{array}\right]. $$
(43)

This means that the quantity

$$ \mathbf{s}^{k + 1} = \left[\begin{array}{l} \beta \left( \mathbf{D} {\hat{\mathbf{u}}}^{k} - \mathbf{D}{\hat{ \mathbf{u}}}^{k + 1} \right) \\ \mu \left( {\hat{\mathbf{u}}}^{k} - {\hat{\mathbf{u}}}^{k + 1} \right) \end{array}\right] $$
(44)

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

$$ \mathbf{r}^{k + 1} = \left[\begin{array}{l} \mathbf{D}{\hat{\mathbf{u}}}^{k + 1}- \mathbf{F}^{k + 1}\\ {\hat{\mathbf{u}}}^{k + 1} - \mathbf{u}^{k + 1} \end{array}\right] $$
(45)

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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