Complete mechanical regularization applied to digital image and volume correlation

Highlights

• Each surface on a mechanical test behaves differently (Neumann, Dirichlet).

• A “complete” regularization scheme is proposed for image correlation.

• This scheme acknowledges the different surface conditions.

• The technique is applicable to 2D and 3D analyses.

Abstract

This paper presents a new regularization scheme for Digital Image Correlation (DIC) and Digital Volume Correlation (DVC) techniques based on the equilibrium gap method with reference to a linear elastic behavior. This scheme constitutes a unique framework for performing the so-called mechanical regularization for any problem dimension. “Complete regularization” refers to the fact that a specific treatment of boundaries (surfaces) is introduced here on the same footing as the bulk, independently of the complexity of their shape. The proposed treatment distinguishes the roles that different boundaries (Neumann or Dirichlet) play in mechanical tests. Numerical cases on synthetic data and a real experimental test validate the robustness and accuracy of the method. The analyzed experiment shows that only the use of (complete) regularization ensures convergence. Even in the cases where such regularization is not employed but convergence is achieved, it is at much higher cost. These results reveal the benefit of regularization on the convergence rate of DVC.

Introduction

Digital Image Correlation (DIC) and Digital Volume Correlation (DVC) are popular techniques to measure displacement fields from image pairs in respectively 2D and 3D settings [1], [2]. As a true three-dimensional technique, DVC measures the internal displacement fields from a pair of reconstructed volumes (3D images). These image registration techniques face a considerable challenge, namely, their ill-posedness [1], [3], [4]. The limited available information (i.e., intensity levels) leads to an unavoidable compromise between the measurement uncertainty and the spatial resolution [5], [6]. Hence, small scale displacement resolutions are hardly accessible.

However, different approaches have been designed to overcome this limitation. For instance, the displacement field can be assumed continuous over the entire Region Of Interest (ROI). Thus, it can be decomposed over basis functions that fulfill this constraint. A convenient choice is offered by meshes used in the Finite Element (FE) method [7], [8], [9]. It is worth mentioning that meshless techniques have also been proposed in the context of DIC [10]. Since the displacement field is determined through the solution to a problem coupling all degrees of freedom, such techniques are referred to as global DIC and DVC [9], [11]. These approaches differ from their “local” counterparts [1], [2], [12], [13], which do not assume any continuity in the sought displacement fields.

One of the advantages of this approach is that it allows for a straightforward connection between experiments and simulations [9], [14]. The same FE mesh used for measuring the displacement field can also be used in numerical analyses. For example, the measurement of “real” boundary conditions using DVC can be used for guiding micromechanical simulations without the need for additional multiscale schemes [15].

Next, given that the studied images are discretized using pixels or voxels, quadrangle and cube elements are a natural choice. Yet, those elements have been shown to lead to “soft modes” with poor conditioning, thereby leading to specific diagonal patterns either when the small element size or the poor texture challenge noise sensitivity [16].

Alternatively, current tools are capable of meshing objects with arbitrary complex boundaries using tetrahedral/triangular elements [17]. The use of such meshes for DVC opens many possibilities. For instance, the analysis would no longer be limited to ROIs with flat straight surfaces, or at the interior of the sample. Even if the sample has a complex shape, or simply because it has been slightly tilted during image acquisition, the ROI could conform to the actual boundaries.

Additionally, regularization techniques [18] are employed to further circumvent the ill-posedness of the registration [3], or to limit ill-conditioning of incremental corrections, and therefore mitigate noise sensitivity [19], [20]. Often, some a priori information on the mechanical behavior of the studied material is available. Then, it is natural to seek a displacement field that best registers the images while also being mechanically admissible. That is the goal of the so-called mechanical regularization based on the equilibrium gap method [21]. This type of regularization can be seen as a specialized filter that only acts on spurious displacements if they are inconsistent with equilibrium.

This regularization constrains the displacement field to one that locally follows a linear elastic behavior. It allows displacements to be measured on meshes with elements of size comparable to the voxel size [22]. Elastic regularization has proven useful even when the actual behavior is more complex. For instance, the study of plastic flow was reported for a controlled crack propagation in aluminum alloy sheets [23]. The proper tuning of the elastic regularization helps enforcing the isochoric constraint, and freeing the direction of easy slip.

Even though this approach is valid independently of the type of mesh, it is not capable of applying the adequate regularization to each type of surface present in the analysis. In fact, the guiding principle is only valid for the bulk and free-surfaces of the studied sample. In Ref. [20], the authors proposed an approach that mimics the bulk, as if those surfaces had an elasticity of their own in addition to the bulk (as a kind of “surface tension”). However, the link between both models (bulk and surfaces) is poor (apart from using common nodes). Moreover, the technique is only applicable to 2D cases, and only admits regions of interest with straight boundaries.

This paper presents an extension of the mechanical regularization to a low-pass filtering of surface tractions. In this context, Saint-Venant’s principle may be invoked, namely, harmonically modulated surface tractions applied to the surface of an elastic medium only affect a boundary layer whose thickness is equal to the wavelength [24]. Consequently, filtering out high frequency modes has a very limited effect, and remains in a confined region. Moreover, the resulting surface tractions over long wavelengths are to be preserved with no alteration. This is precisely what is achieved by the proposed regularization of Dirichlet surfaces. Furthermore, while the present application is 3D, the method can be easily applied to 2D configurations.

The implementation details are given in Section 2. Numerical tests are performed in Section 3, and an actual test case is presented in Section 4. These results will show the superiority of the proposed method in comparison with established DVC approaches.