SpatioTemporally Adaptive Quadtree Mesh (STAQ) Digital Image Correlation for Resolving Large Deformations Around Complex Geometries and Discontinuities

Abstract

Background

Digital image correlation (DIC) is a powerful experimental tool for measuring full-field material deformations. Inherent limitations of typical DIC algorithms can cause a multitude of errors when analyzing the displacement field of samples containing complex geometries or discontinuities. Most adaptations rely on either splitting or augmenting the local DIC subsets that pass through the discontinuity path. However, these methods are challenging to generalize and automate, often requiring significant user intervention.

Objective

To address these shortcomings, we present a new, user-friendly automatic experimental approach for resolving the deformation fields around complex geometries and displacement discontinuities, which we call the spatiotemporally adaptive quadtree mesh (STAQ) DIC method.

Methods

In this method, the adaptive quadtree mesh is automatically generated from a mask file of the DIC image itself to handle the inherent complex geometry. Subsets that span either geometric or displacement discontinuities are automatically split to improve their DIC accuracy. A binary image mask is also used to inform an interpolation scheme for displacement and strain calculations. Furthermore, we also propose a data-driven reduced order modeling (ROM) approach to further reduce the computational costs by skip** unnecessary image frames thus achieving temporal adaptability for efficiently processing large image sequences.

Results

We demonstrate that our STAQ method has high accuracy in solving complex geometric and discontinuous deformation fields in an automated fashion. We find that the proposed data-driven ROM method can provide up to 60% in computational cost savings while maintaining the same level of accuracy compared to a fully processed image set.

Conclusions

STAQ DIC is a computationally efficient method for accurately solving geometrically complex and discontinuous deformation fields. Using the data-driven ROM method as part of STAQ can further reduce computational costs for processing large image sequences. An open-source Matlab implementation is freely available.

Appendices

Appendix 1

Augmented Lagrangian (AL) DIC Method

The Augmented Lagrangian (AL) DIC method [6] is used to solve the optimization problem equations (1), and (2) by adding a global kinematic constraint such that

$$\begin{aligned} \mathbf {F}_i = \nabla \hat{\mathbf {u}}(\mathbf {X}_{i0}), \quad \mathbf {u}_i = \hat{\mathbf {u}}(\mathbf {X}_{i0}), \end{aligned}$$

(12)

where $\hat{\mathbf {u}}$ is the introduced auxiliary displacement field which is always globally kinematically compatible. $\mathbf {X}_{i0}$ is the center of the $i^{\text {th}}$ subset; $\mathbf {u}_i$ and $\mathbf {F}_i$ are the local $i^{\text {th}}$ subset displacement and displacement gradient tensor. In the ALDIC method, the augmented Lagrangian correlation function $\mathcal {L}$ in equation (13) is formulated by adding the global kinematic constraint as a combination of linear Lagrange multipliers and quadratic penalties to equation (1):

$$\begin{aligned} \begin{aligned} \mathcal {L} = & \sum _{i} \int _{\Omega _i}\Big ( \left| f(\mathbf {X}) - g(\mathbf {X}+\mathbf {u}_i + \mathbf {F}_i(\mathbf {X}- \mathbf {X}_{i0}) ) \right| ^2\\&+ \varvec{\nu }_i \cdot ( (\nabla \hat{\mathbf {u}})_i - \mathbf {F}_i ) + \frac{\beta }{2} |(\nabla \hat{\mathbf {u}})_i - \mathbf {F}_i |^2 \\ & + \varvec{\lambda }_i \cdot ( \hat{\mathbf {u}}_i - \mathbf {u}_i ) + \frac{\mu }{2} | \hat{\mathbf {u}}_i - \mathbf {u}_i |^2 \Big )\,d\mathbf {X}, \end{aligned} \end{aligned}$$

(13)

where $\beta$, $\mu$ are coefficients of the quadratic penalties; $\varvec{\nu }$, $\varvec{\lambda }$ are Lagrangian multipliers.

Equation (13) is solved by the iterative alternating direction method of multipliers (ADMM) [94] that iterates between updating the local deformation ($\mathbf {u}$, $\mathbf {F}$), updating the global deformation ($\hat{\mathbf {u}}$), and updating Lagrange multipliers ($\varvec{\nu }_i$, $\varvec{\lambda }_i$). At the $(k+1)^{\text {th}}$ ADMM iteration, we solve following subproblems:

$$\begin{aligned}\lbrace \mathbf {F}^{k+1}_i \rbrace , \lbrace \mathbf {u}^{k+1}_i \rbrace = & \arg \min _{ \lbrace \mathbf {F}_i \rbrace , \lbrace \mathbf {u}_i \rbrace } \mathcal {L} ( \lbrace \mathbf {F}_i \rbrace , \lbrace \mathbf {u}_i \rbrace , \lbrace \hat{\mathbf {u}}^k_i \rbrace , \lbrace \varvec{\nu }^k_i \rbrace , \lbrace \varvec{\lambda }^k_i \rbrace ), \\ & \text {for all { i}}, \quad\text {(Subproblem 1)} \end{aligned}$$

(14)

$$\begin{aligned}\lbrace \hat{ \mathbf {u}}^{k+1} \rbrace = & \arg \min _{ \lbrace \mathbf {\hat{u}} \rbrace } \mathcal {L} ( \lbrace \mathbf {F}^{k+1} \rbrace , \lbrace \mathbf {u}^{k+1} \rbrace , \lbrace \hat{\mathbf {u}} \rbrace , \lbrace \varvec{\nu }^k \rbrace , \lbrace \varvec{\lambda }^k \rbrace ) \\ & \text {(Subproblem 2)} \end{aligned}$$

(15)

$$\begin{aligned}\left\{ \begin{aligned}\varvec{\nu }^{k+1}_i &=\varvec{\nu }^{k}_{i} + \beta ( (\nabla \hat{\mathbf {u}})^{k+1}_{i} - \mathbf {F}^{k+1}_i ), \\\varvec{\lambda }^{k+1}_i &=\varvec{\lambda }^{k}_{i} + \mu ( \hat{\mathbf {u}}^{k+1}_{i} - \mathbf {u}^{k+1}_i ). \end{aligned} \right. \ \qquad & \text {(Subproblem 3)} \end{aligned}$$

(16)

In the local step equation (14), all the subsets are solved independently and in parallel, where an automatic subset splitting technique (cf. “Automatic Subset Splitting Scheme”) can be automatically applied to subsets that span discontinuities or sample edges. The global step equation (15) leads to a simple linear problem

$$\begin{aligned} (\beta \nabla ^2 + \mu \mathbf {I}) \hat{\mathbf {u}}^{k+1} = - \nabla \cdot (\beta \mathbf {F}^{k+1} - \varvec{\nu }^{k} ) + (\mu \mathbf {u}^{k+1} - \varvec{\lambda }^k), \end{aligned}$$

(17)

where $\nabla ^2(\bullet )$ is the Laplace operator. Following Yang and Bhattacharya [18], equation (17) is solved globally using the finite element method where we apply the Gupta’s method [95] to modify quadtree transition elements with hanging nodes.

Appendix 2

A Posteriori Error Estimate and Quadtree Mesh Refinement

To estimate the error of a DIC analysis, we note that the point-wise residual of the global kinematic compatibility equation (17) is

$$\begin{aligned} \mathfrak {R}_{\text {AL}}(\hat{\mathbf {u}}) := \beta \nabla ^2 \hat{\mathbf {u}} + \tilde{\mathbf {f}}_{\text {AL}} \end{aligned}$$

(18)

where $\tilde{\mathbf {f}}_{\text {AL}}$ has an explicit formula:

$$\begin{aligned} \tilde{\mathbf {f}}_{\text {AL}} = - \nabla \cdot (\beta \mathbf {F} - \varvec{\nu }) - \mu (\hat{\mathbf {u}} - \mathbf {u}) - \varvec{\lambda } \end{aligned}$$

(19)

Assuming that $\mathcal {J}$ is a quadtree mesh of an ROI and supposing that $\tilde{\mathbf {f}}_{\text {AL}} \in L^2(\text {ROI})$, a posteriori error estimate of the adaptive quadtree mesh DIC method⁹ can be defined as [18, 96]

$$\begin{aligned} \mathcal {E}_{\text {AL}}^2 (\hat{\mathbf {u}}, T) = h_T^2 \left\| \mathbf {r}_{\text {AL}} \right\| ^2_{L^2(T)} + h_T \left\| \mathbf {j}_{\text {AL}} \right\| _{L^2(\partial T \backslash \partial \Omega )}^2 \end{aligned}$$

(20)

$$\begin{aligned} \begin{aligned} \text {where}&\ \left\{ \begin{aligned} \mathbf {r}_{\text {AL}}&= \varvec{ \mathfrak {R}}_{\text {AL}} ( \hat{\mathbf {u}} ) , \quad \text {Interior residual in any element } T \in \mathcal {J}, \\ \mathbf {j}_{\text {AL}}&= [\![ \beta \nabla \hat{\mathbf {u}} ]\!] , \quad \ \ \text {Jump residual on any element's internal side } \Gamma \in \mathcal {J}, \end{aligned} \right. \end{aligned} \end{aligned}$$

(21)

where $[\![ \beta \nabla \hat{\mathbf {u}} ]\!] = \beta \mathbf {n}^{+} \cdot \nabla \hat{\mathbf {u}}|_{T^{+}} + \beta \mathbf {n}^{-} \cdot \nabla \hat{\mathbf {u}}|_{T^{-}}$, and $\mathbf {n}^{+}$, $\mathbf {n}^{-}$ are unit normal vectors pointing towards $T^{+}$, $T^{-}$ $\in$ $\mathcal {J}$, respectively.

We mark elements with large a posteriori error estimates $\mathcal {E}_{\text {AL}}$ to satisfy Dörfler’s strategy [97]:

$$\begin{aligned} \mathcal {E}_{\mathcal {J}}(\mathcal {M}) \ge \theta \mathcal {E}_{\mathcal {J}}(\mathcal {J}), \end{aligned}$$

(22)

where $\mathcal {J}$ is the current quadtree mesh; $\mathcal {M}$ is a set formed by all the marked elements with large element estimate errors; $\mathcal {E}_{\mathcal {J}}(\bullet )$ is total error estimate of set ($\bullet$), and $\theta \in (0,1]$ is a positive parameter. Then all the marked square elements will be recursively divided into 2 $\times$ 2 children elements to refine the quadtree mesh.

In the mode II dynamic rupture experiment (cf. “Mode II Dynamic Rupture Experiment”), the first level adaptive quadtree mesh is initially generated using the binary mask file from the reference image. Then the mesh is automatically refined near the center crack interface (see Fig. 17(a–c)). After solving the DIC problem on this level, all interior errors in the elements, jump residuals on element sides and total a posteriori error estimate of Level 1 mesh are calculated and plotted in Fig. 17(d–f). Elements with large errors are marked following Dörfler’s strategy (22) where $\theta = 0.9$, and further refined. Similarly, the quadtree mesh can be further refined from the second to the third level, as shown in Fig. 17(j–o).

Fig. 17

(a) Generated binary mask file of the reference image in Fig. 14(b). (b) Generated adaptive quadtree mesh on the first level. Inset red box is zoomed in (c). (d-f) Interior errors in the elements, jump residuals on element sides, and total a posteriori error estimate of DIC results using Level 1 mesh. (g) Elements with large errors are marked following Dörfler’s strategy. (h) Marked elements are adaptively refined. Inset red box is zoomed in (i). (j-l) Interior errors in the elements, jump residuals on element sides, and total a posteriori error estimate of DIC results using Level 2 mesh. (m) Elements with large errors are marked following Dörfler’s strategy. (n) Marked elements are adaptively refined. Inset red box is zoomed in (o)