## Abstract

**Background** Uncertainty quantifications are required for any measurement result to be meaningful.

**Objective** The present work aims at deriving and comparing a priori estimates of displacement uncertainties in T3-stereocorrelation for a setup to perform high temperature tests.

**Methods** Images acquired prior to the actual experiment (i.e.,at room temperature) were registered using 3-noded triangular elements (T3-stereocorrelation) to determine displacement uncertainties for different positions of the experimental setup.

**Results** The displacement uncertainties were then compared to their a priori estimates.

**Conclusions** For the analyzed experiment, it is shown that noise floor estimates only differed by a factor 2 when compared to a posteriori measurements of standard displacement uncertainties.

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## Acknowledgements

This work was supported within PRC MECACOMP, a French research project co-funded by DGAC and SAFRAN Group, managed by SAFRAN Group and involving SAFRAN Group, ONERA and CNRS.

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The authors have no conflict of interest to declare.

## Appendices

### Appendix A: Calibration Target

To calibrate the stereovision system, a new calibration target (Fig. 17) was designed. One face of the calibration target is composed of a series of six inclined planes with a vertical offset of 3 mm, corresponding to the cameras depth of field. The shape of the opposite face is identical, with an offset of 1 mm. The object was speckled with black paint RAL 9005 by Ront. The calibration target that includes 14 corners (i.e.,fiducials) at different heights within the field of view of the cameras allows for a good initialization of the projection matrices \([{\varvec{P}}^1]\) and \([{\varvec{P}}^2]\), and the calibration of the stereovision system with a *single* pair of images (contrary to what is usually carried out with planar calibration targets [1] that require full sets of different positions). Moreover, this shape enables for the calibration within a volume corresponding to the expected deformation of CMC samples. Last, in the present configuration, the calibration target was kept in the field of view during whole experiments [39].

To perform the first calibration step, the real model of the manufactured calibration target was identified. First, a 3D surface measurement was performed using a Coordinate-Measuring Machine. For each main facet of the object (inclined planes, denoted as \(F_i\) in Fig. 17(a)), the space coordinates of 50 points, uniformly distributed over the surface, were acquired. For each of the two small surfaces (horizontal planes \(S_i\)), 10 points were acquired.

A total of 620 points was used to build the model of the calibration target. Then, an algorithm based on least squares and least distances methods was implemented to find the best surface interpolating all the points of a facet [39]. It is a two-step algorithm, where the first step was based on least squares fit between the measurements and the estimated plane, and the second step on the least distances method to update the plane normal direction. With such procedure, a more faithful parameterization of the surface is achieved in comparison to the nominal shape, *and* minimizes the distance of the measured points to the sought plane. Once the optimization of each face model was performed, all planes were combined to determine the corners of every facet. Then, the global model for the calibration target was constructed and can be expressed in different frameworks, either using an FE (T3) mesh or NURBS patches (Fig. 18).

### Appendix B: Calibration of the Two Cameras

The calibration procedure aims to minimize the cost function [12]

with respect to the components of the projection matrices \([{\varvec{P}}^{1}]\) and \([{\varvec{P}}^{2}]\), where \(\mathcal S_j\) denotes the \(n_s=6\) surfaces defining one side of the calibration target. The calibration procedure used herein was based an the FE formulation of global stereocorrelation [13]. A fine mesh of the calibration target was considered. It was composed of 558 T3 elements 1.7 mm in size (Fig. 19). The total number of integration points was equal to 528 (i.e., \(N_{eq}\approx 23\)) so that the projected surface of one integration point had the size of roughly one pixel in the images (Table 3).

The minimization procedure was initialized with a direct estimation (via SVD [12, 48]) of the projection matrices, using the 3D positions of the 14 fiducials (i.e.,corners of the calibration target) and their associated projection in each image, see Fig. 20.

The associated gray level (GL) residuals using this initialization of the projection matrices are shown in Fig. 21(a) for each camera. The initialization led to a medium level of RMS residuals (\(\approx 21.4\) gray levels, i.e.,8.4% of the dynamic range) and to errors on the description of the calibration target edges. This was expected since the initialization was obtained by using only 14 fiducials, which were manually selected on the images.

At the end of the optimization procedure, the calibration converged to a solution that considerably reduced the residuals (Fig. 21(b)) to an RMS level of 3.1 gray levels (i.e.,1.2% of the dynamic range). As opposed to Fig. 21(a), the edges were well-defined and the errors on the speckled parts were erased, hence leading to a good stereosystem calibration. Figure 22 illustrates how each node was correctly projected onto each image plane and matched the calibration target shapes on the images (especially for the edges, the corners and the inclined planes).

### Appendix C: Shape Correction for the CMC Sample

Displacements were then measured for the top surface of the CMC sample. The first step consisted in expressing its model in the calibration target frame. The correction procedure seeks to determine the (new) nodal positions gathered in the column vector \(\{{\varvec{N}}\}\) of the surface mesh by globally minimizing the cost function [13].

Since the goal was to correct the global position of the CMC surface with respect to the calibration target, the FE mesh was very simple (i.e.,made of two T3 elements covering the CMC surface). Because the discretization was coarse, a large number of integration points (i.e.,8,256 corresponding to \(N_{eq}\approx 91\)) was selected. The nodal positions were initialized using in-situ distances between the sample and the calibration target. This initial model led to an RMS residual of 12.1 gray levels, corresponding to 4.7% of the dynamic range. The maps show that the residuals were higher on the right part (free edge) of the sample with a line of evaluation points with very high residuals (Fig. 23(a)). They indicated that the real model of the CMC sample was shorter and needed to be corrected.

At convergence of the approach, the RMS residuals were reduced to 2.1% of the dynamic range (i.e.,5.3 gray levels) and the previous errors at the free edge were virtually erased (Fig. 23(b)). It was even possible to see the residuals associated with CMC 3D woven architecture of the central unspeckled part of the sample, which was a further proof of the good quality of the shape corrections.

The corrected position of the sample is shown in Fig. 24. The main contribution is on the out-of-plane component \(U_x\) with a mean translation due to a wrong estimation of the sample thickness. There is also a gradient of \(\approx ~\)180 µm between the *X*-position of nodes in the clamp (corresponding to \(Y\approx 60\) mm) and the nodes of the free edge, explained by the fact that the CMC sample was slightly inclined at the free edge. Regarding the components on the longitudinal *Y* and transverse *Z*-directions, the corrections mainly corresponded to an adjustment of the shape in terms of length and width.

Once the position of the sample was corrected, it was possible to measure its 3D surface displacements. A fine mesh was used with 498 T3 elements of mean size 1.5 mm (Table 3). To avoid convergence issues on displacement measurements, this mesh did not contain the unspeckled part and a line of nodes at the clamped edge, since they were hidden on one camera plane. Figure 25 shows the mesh with its projection onto both camera planes.

With this last step, the cameras were calibrated and the reference configuration of the CMC surface was expressed in the same frame. Displacement fields of the CMC surface and the calibration target can now be measured and are reported in a unique frame.

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Berny, M., Archer, T., Beauchêne, P. *et al.* Displacement Uncertainty Quantifications in T3-Stereocorrelation.
*Exp Mech* (2021). https://doi.org/10.1007/s11340-021-00690-0

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### Keywords

- Covariance matrix
- Displacement
- Noise floor level
- Uncertainty quantification