Coupling Self-Adaptive Meshing-Based Regularization and Global Image Correlation for Spatially Heterogeneous Deformation Characterization

Abstract

Background

Image-based global correlation involves a class of ill-posed inverse problems associated with speckle quality and deformation gradients on specimen surfaces. However, the method used to simultaneously integrate the prior information related to images and deformations and effectively regularize these inverse problems still faces severe challenges, especially when complex heterogeneous deformation gradients exist over sample surfaces with locally degraded speckle patterns.

Objective

We propose a novel self-adaptive meshing-based regularization for global image correlation to determine spatially complex heterogeneous deformations.

Methods

A virtual truss system with a linearly elastic constitutive relationship is employed to self-adaptively implement surface meshing by numerically balancing the exerted virtual forces under the constraints of the local speckle image quality and deformation gradients. The 2-norm-based condition number of the local stiffness matrix is introduced to ensure numerical stability during meshing.

Results

The algorithms can behave as a smart regularization procedure integrating all the prior information during numerical calculations, consequently achieving an accurate, precise and robust characterization of heterogeneous deformations, as demonstrated by virtual simulations and actual experiments.

Conclusions

The regularization strategy coupled to image-based correlation is also promising for automatic quantification of complex heterogeneous deformations, particularly from images with locally degraded speckle patterns.

Appendices

Appendix A: Shape Functions of Nodes

The shape functions of nodes used in the global DIC algorithms are normally the same as those in the classical finite elements [60]. For instance, the 3-node triangular element, usually referred to as T3 element, has the following shape function

$${\varphi }_{i}=\frac{1}{2A}({a}_{i}+{b}_{i}X+{c}_{i}Y)$$

(A1)

where i=1,2,3 for the T3 element, ${a}_{1}={X}_{2}{Y}_{3}-{X}_{3}{Y}_{2}$,${a}_{2}={X}_{3}{Y}_{1}-{X}_{1}{Y}_{3}$, ${a}_{3}={X}_{1}{Y}_{2}-{X}_{2}{Y}_{1}$, ${b}_{1}={Y}_{2}{-Y}_{3}$, ${b}_{2}={Y}_{3}-{Y}_{1}$, ${b}_{3}={Y}_{1}-{Y}_{2}$, ${c}_{1}={X}_{3}{-X}_{2}$, ${c}_{2}={X}_{1}{-X}_{3}$, ${c}_{3}={X}_{2}{-X}_{1}$, $({X}_{i},{Y}_{i})$ denotes the node coordinates and A is the area of the element.

Appendix B: Discrete Form of Governing Equation in Equation (4)

To estimate the lower limit of bar length in the virtual truss structure, we first rewrote (equation (2)) as $T\left({\mathbf{p}}_{t}\right)=F\left(X,Y\right)-G(x,y)$ in an arbitrary element with

$$\left\{\begin{array}{c}F\left(X,Y\right)=\frac{f\left(X,Y\right)-{f}_{m}}{\Delta f}\\ G\left(x,y\right)=\frac{g\left(x,y\right)-{\mathrm{g}}_{m}}{\Delta \mathrm{g}}\end{array}\right.$$

(A2)

For the well-adopted first-order triangular element with three nodes, ${\mathbf{p}}_{t}=[{p}_{t1},{p}_{t2},{p}_{t3},{p}_{t4},{p}_{t5},{p}_{t6}{]}^{T}=[{u}_{t1},{v}_{t1},{u}_{t2},{v}_{t2},{u}_{t3},{v}_{t3}{]}^{T}$. We subsequently expand $G\left(x,y\right)$ as a first-order Taylor’s series approximately [31], i.e.,

$$\begin{aligned}G\left(x,y\right)= & G\left[X+u\left(X,Y,{\mathbf{p}}_{t}\right),Y+v\left(X,Y,{\mathbf{p}}_{t}\right)\right]\approx G\left(X,Y\right) \\ & +{G}_{X}u\left(X,Y,{\mathbf{p}}_{t}\right) +{G}_{Y}v(X,Y,{\mathbf{p}}_{t})\end{aligned}$$

(A3)

in which ${G}_{x}=\partial G/\partial X$ and ${G}_{Y}=\partial G/\partial Y$. Substituting the above expression into $T{(\mathbf{p}}_{t})$ yields

$$T{(\mathbf{p}}_{t})\approx F\left(X,Y\right)-\mathrm{G}\left(\mathrm{X},\mathrm{Y}\right)-{G}_{X}u(X,Y,{\mathbf{p}}_{t})-{G}_{Y}v(X,Y,{\mathbf{p}}_{t})$$

(A4)

And

$$\frac{\partial \mathrm{T}{(\mathbf{p}}_{t})}{\partial {P}_{tj}}=\left\{\begin{array}{c}{G}_{X}{\varphi }_{(j+1)/2},j=\mathrm{1,3},5\\ {G}_{Y}{\varphi }_{j/2},j=\mathrm{2,4},6\end{array}\right.$$

(A5)

In this context, we have

$$\sum_{{\Omega }_{t}}T(\mathbf{p})\frac{\partial T(\mathbf{p})}{\partial {\mathbf{p}}_{t}}=\sum_{{\Omega }_{t}}\left\{[F-G-{G}_{X}u-{G}_{Y}v]\frac{\partial T\left(\mathbf{p}\right)}{\partial {p}_{tj}}]\right\}$$

(A6)

Substituting the expression into (equation (4)) gives

$${\mathbf{C}}_{A}\mathbf{p}=\mathbf{R}$$

(A7)

where ${\mathbf{C}}_{A}$ and R are the corresponding coefficient matrix and residual vector, respectively, which have the following forms

$$C_A=\sum_{t=1}^{T_e}\mathbf H_t^T\begin{bmatrix}\sum_{\Omega_t}G_x^2\varphi_{{}_1}^2&\sum_{\Omega_t}G_xG_y\varphi_{{}_1}^2&\sum_{\Omega_t}G_x^2\varphi_1\varphi_2&\sum_{\Omega_t}G_xG_y\varphi_1\varphi_2&\sum_{\Omega_t}G_x^2\varphi_1\varphi_3&\sum_{\Omega_t}G_xG_y\varphi_1\varphi_3\\\sum_{\Omega_t}G_xG_y\varphi_{{}_1}^2&\sum_{\Omega_t}G_y^2\varphi_{{}_1}^2&\sum_{\Omega_t}G_xG_y\varphi_1\varphi_2&\sum_{\Omega_t}G_y^2\varphi_1\varphi_2&\sum_{\Omega_t}G_xG_y\varphi_1\varphi_3&\sum_{\Omega_t}G_y^2\varphi_1\varphi_3\\\sum_{\Omega_t}G_x^2\varphi_1\varphi_2&\sum_{\Omega_t}G_xG_y\varphi_1\varphi_2&\sum_{\Omega_t}G_x^2\varphi_2^2&\sum_{\Omega_t}G_xG_y\varphi_2^2&\sum_{\Omega_t}G_x^2\varphi_2\varphi_3&\sum_{\Omega_t}G_xG_y\varphi_2\varphi_3\\\sum_{\Omega_t}G_xG_y\varphi_1\varphi_2&\sum_{\Omega_t}G_y^2\varphi_1\varphi_2&\sum_{\Omega_t}G_xG_y\varphi_2^2&\sum_{\Omega_t}G_y^2\varphi_2^2&\sum_{\Omega_t}G_xG_y\varphi_2\varphi_3&\sum_{\Omega_t}G_y^2\varphi_2\varphi_3\\\sum_{\Omega_t}G_x^2\varphi_1\varphi_3&\sum_{\Omega_t}G_xG_y\varphi_1\varphi_3&\sum_{\Omega_t}G_x^2\varphi_2\varphi_3&\sum_{\Omega_t}G_xG_y\varphi_2\varphi_3&\sum_{\Omega_t}G_x^2\varphi_3^2&\sum_{\Omega_t}G_xG_y\varphi_3^2\\\sum_{\Omega_t}G_xG_y\varphi_1\varphi_3&\sum_{\Omega_t}G_y^2\varphi_1\varphi_3&\sum_{\Omega_t}G_xG_y\varphi_2\varphi_3&\sum_{\Omega_t}G_y^2\varphi_2\varphi_3&\sum_{\Omega_t}G_xG_y\varphi_3^2&\sum_{\Omega_t}G_y^2\varphi_3^2\end{bmatrix}{\mathbf H}_{\mathrm t}$$

(A8)

And

$$\mathbf{R}=\sum_{{\varvec{t}}=1}^{T_{e}}{{\mathbf{H}}_{t}}^{T}\left[\begin{array}{c}\sum_{{\Omega }_{t}}[\mathrm{F}-\mathrm{G}]{G}_{x}{\varphi }_{1}\\ \begin{array}{c}\sum_{{\Omega }_{t}}[\mathrm{F}-\mathrm{G}]{G}_{Y}{\varphi }_{1}\\ \sum_{{\Omega }_{t}}[\mathrm{F}-\mathrm{G}]{G}_{X}{\varphi }_{2}\\ \begin{array}{c}\sum_{{\Omega }_{t}}[\mathrm{F}-\mathrm{G}]{G}_{Y}{\varphi }_{2}\\ \sum_{{\Omega }_{t}}[\mathrm{F}-\mathrm{G}]{G}_{X}{\varphi }_{3}\\ \sum_{{\Omega }_{t}}[\mathrm{F}-\mathrm{G}]{G}_{Y}{\varphi }_{3}\end{array}\end{array}\end{array}\right]$$

(A9)