# On the identification of rigid body boundary conditions from full-field measurements

Antoine Vintache, Pierre Baudoin, Renaud Gras, Florent Mathieu, François Hild

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Antoine Vintache, Pierre Baudoin, Renaud Gras, Florent Mathieu, François Hild. On the identification of rigid body boundary conditions from full-field measurements. International Journal of Solids and Structures, 2026, 327, pp.113768. 10.1016/j.ijsolstr.2025.113768. hal-05393047

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# On the identification of rigid body boundary conditions from full-field measurements

Antoine Vintache<sup>a,b, ID*</sup>, Pierre Baudoin<sup>a</sup>, Renaud Gras<sup>a ID</sup>, Florent Mathieu<sup>a</sup>, François Hild<sup>b ID</sup>

## ARTICLE INFO

**Keywords:**

Boundary condition

Finite element model updating

Sensitivity analysis

Uncertainties

## ABSTRACT

This study presents a method to identify Dirichlet boundary conditions (BCs) and constitutive parameters from heterogeneous experimental data, solving a multi-objective optimization problem. The BCs are parameterized as remote rigid body motions leading to a very low number of unknowns and allowing for faster convergence. This method enables for accurate models of material and structural tests to be obtained. A comparison with classical methods of BC determination is introduced, demonstrating the advantages of the proposed approach. The identified models are evaluated by introducing a validation metric based on experimental data and their uncertainties. The approach is illustrated by using a multi-instrumented tensile test on a well-characterized material. Beyond this validation example, the introduced method paves the way for improved model updating of large-scale tests.

## 1. Introduction

During the last decades, multiple methods have been developed to calibrate constitutive parameters for numerical simulations to match experimental observations (Avril et al., 2008; Roux and Hild, 2020; Chen et al., 2024). One of the crucial steps before conducting constitutive parameter identification is to apply reliable boundary conditions (BCs) in numerical simulations (Chen et al., 2024). They are generally associated with higher sensitivities compared to constitutive parameters (Bertin et al., 2016). When full-field displacement measurements are available (e.g., using Digital Image Correlation (Sutton et al., 2009; Sutton, 2013)), one would directly apply measured displacements on the boundary of the measurement area in FE models (Sztefek and Olsson, 2008; Trellu et al., 2020). This method brings two major downsides. First, it is subjected to displacement uncertainties. As the accuracy of the boundary conditions is paramount to that of the identified parameters (Rokoš et al., 2018), measurement uncertainties are usually mitigated by a filtering step (Carpiuc-Prisacari et al., 2017) (e.g., using low-order polynomials (Hild et al., 2016; Neggers et al., 2017, 2019; Henriques et al., 2022)), leading to higher complexity and manual adjustments from users to choose the right polynomial order fit. Second, the use of intermediate boundary conditions may be necessary as the true boundary conditions are usually hidden by grips or fixtures. One may not be fulfilling Saint-Venant's principle anymore by applying boundary conditions too close to the region of interest. Last, measuring

displacements on boundary conditions spanning multiple sample faces or in large-scale tests is usually challenging.

Multiple approaches have been developed in order to overcome the previous drawbacks. In the work of Fedele (2015), a procedure to calibrate kinematic boundary conditions on the edges of the Region-of-Interest (ROI) was proposed. The minimization of the discrepancy between experimental and simulated data was based on the Finite Element Model Updating (FEMU) method (Kavanagh and Clough, 1971). As the determination of nodal displacements on the boundary would lead to large inverse problems, the author proposes to regularize the displacement of boundary conditions using polynomials, leading to lower sized problems. Bertin et al. (2016) performed a Dirichlet boundary condition correction identification. Using Digital Image Correlation (DIC) measurements, a first evaluation of the rigid body motions was carried out. Global corrections applied to the boundary conditions were identified using integrated-DIC (I-DIC (Roux and Hild, 2008; Neggers et al., 2015)). Within that framework, the identification was performed at the same time as DIC analyses, meaning that the boundary conditions were computed in order to minimize the gray level residuals between the reference image and the deformed images corrected by FE computed displacements. Rokoš et al. (2018) followed the work of Fedele (2015), including multiple novelties. The proposed method is based on I-DIC and adds the determination of an optimum boundary kinematics

<footnote><sup>a</sup> EikoSim, 92120 Montrouge, France</footnote>

<footnote><sup>b</sup> Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS – Laboratoire de Mécanique Paris-Saclay, 91190 Gif-sur-Yvette, France</footnote>

<footnote>* Corresponding author at: Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS – Laboratoire de Mécanique Paris-Saclay, 91190 Gif-sur-Yvette, France.</footnote>

<footnote>E-mail address: antoine.vintache@ens-paris-saclay.fr (A. Vintache).</footnote>
regularization. The regularization level was determined by gradually refining the boundary conditions, which involved high computational costs.

Cosseron et al. (2021) identified the pressure distribution on a tire-rim interface using full field displacement data obtained through stereocorrelation and with strain gauge data. The method was based on an extension of FEMU, called weighted Finite Element Model Updating (Gras et al., 2015), which accounts for measurement uncertainties. This method proved to be robust to identify constitutive parameters when using multiple measurement data sources (Mathieu et al., 2015; Berny, 2020) or even multiple experiments (Neggers et al., 2019). Dufour et al. (2023) identified kinematic boundary conditions applied to a composite plate under shear. Using Tikhonov regularized I-DIC, the authors were able to determine nodal kinematic boundary conditions for FE nodes that were inside the camera field of view but also for non visible nodes.

The objective of this paper is to propose a method allowing for the estimation of remote kinematic boundary conditions not limited to the boundary of the measurement area. This method is to enable for the identification of BCs for complex structural tests. In this framework, the BCs are considered as rigid body motions applied to a chosen set of FE nodes distant from the area of interest. This method has been developed for cases where Saint-Venant's principle may apply. More complex, longer and computationally intensive identification methods (Rokos et al., 2018) may be necessary for cases when the previous hypothesis does not apply. This study takes advantage of weighted FEMU, which is equivalent to I-DIC (Mathieu et al., 2015), to properly aggregate multi-modal data sources for parameter identification purposes, thus solving a multi-objective optimization problem. Moreover, the choice of weighted FEMU allows for the use of stereocorrelation data combined with 2D DIC measurements in the global cost function. In Section 2, a tensile test is introduced to validate the proposed BC identification method. In Section 3, the methods used to perform identifications are presented and explained. Section 4 showcases the simultaneous identification of kinematic BCs and constitutive parameters. The results of the introduced method are discussed and compared to standard approaches.

## 2. Studied experiment

In this first section, the test case that will be later used to validate the proposed BC identification method is introduced and described.

### 2.1. Experimental protocol

The studied specimen was a drilled 10 mm thick dogbone sample made of AA2219 aluminum alloy. The location of the holes was optimized to maximize local plasticity and sensitivity to constitutive parameters. The sample was subjected to a monotonic loading at a stroke rate of 0.1 mm min⁻¹.

Multiple instruments were monitoring the test, namely, a load cell, one camera capturing images of one side (Face A), two cameras acquiring images of the opposite side (Face B) and multiple strain gauges. In this study, only the load cell (Fig. 1(c)), mono-camera (Fig. 1(d)) and stereo-camera (Fig. 1(e)) data will be exploited. The camera setup is detailed in Fig. 1(b). The load was measured at a high frequency (10 Hz) compared to the acquisition of images (0.22 Hz). The combination of these heterogeneous data sources is discussed in Sections 3.2 and 3.3.

The test was carried out until failure, which occurred after about 8 min and 30 s (i.e., between images 106 and 107) or approximately 0.85 mm of prescribed stroke. The fracture propagated through both holes near the neck of the sample. In addition to measurements during the test, multiple images as well as force data were acquired before and after the test in static configurations in order to assess uncertainties of the various sensors.

### 2.2. DIC analyses

The DIC analyses were carried out using EikoTwin DIC (EikoSim, 2024), a global (FE-based) software. As the displacement results were measured with a unique FE mesh, to avoid the need for regularization methods in the DIC computations, a DIC specific mesh with uniform element sizes (1.5 mm) was generated. The obtained mesh had 1,875 degrees of freedom on Face A (mono-vision) and 2,418 on Face B (stereo-vision). A preliminary Point and Perspective (PnP) alignment step was performed for each camera in the DIC software using the 2D positions in the image of white dots drawn on the sample (Figs. 1(d) and 1(e)) and their respective 3D positions reported in the FE-DIC model. Brightness and contrast corrections were applied automatically by the software on the images and no regularization was used. The displacement field was then measured for each deformed image with respect to the reference images. The measured longitudinal (tension direction) displacement field for image no. 100 ($t \approx 457$ s, few images before failure) is displayed in Fig. 2 for both faces. As expected, the highest displacement level is measured at the bottom of the sample near the moving grip. The displacement measured on the top of the sample is not exactly zero for multiple reasons, namely, the upper measurement nodes are not along the top grip but in a straining area, the sample slipped in the grips, and the testing machine is not perfectly rigid.

All DIC (hardware and software) parameters are summarized in Table 1. The displacement uncertainties were evaluated via a “noise floor” estimate on a series of “static images” (International Digital Image Correlation Society et al., 2018). This method was applied to a series of 55 static images (negatively numbered in later figures) where DIC was run to compute the displacement covariance matrix (EN 4861, 2020). The DIC parameters were left unchanged. Due to the use of a servohydraulic testing machine, the DIC measured fields on the static images were corrected for rigid body motions to only consider measurement uncertainties. However, as there were only 55 measurements in time and much more degrees of freedom in space, the obtained covariance matrix was rank deficient and thus non-invertible. To allow for the inversion of the covariance matrix, only the covariances between first neighbors was considered.

## 3. Calibration methods

In this section, the methods used to match test data and FE simulations (Section 3.1) are introduced (Sections 3.2 and 3.3). A metric of model quality is also defined in Section 3.4.

### 3.1. FE simulations

This section details the Abaqus/Standard (Dassault Systèmes, 2023) FE model used to represent the test of interest. This model will be later updated to determine BCs, and eventually update constitutive parameters to better match the experimental results.

#### 3.1.1. Constitutive law

The material behavior is considered isotropic using Hooke's law for elasticity and Ludwik's law (Ludwik, 1909) to account for hardening. This model uses 5 parameters, namely, 2 elastic parameters (Young's modulus $E$, and Poisson's ratio $\nu$) and 3 plastic parameters (yield stress $\sigma_0$, hardening modulus $h$, and hardening power $m$). Hooke's law reads

$$ \sigma = \frac{E}{1 + \nu} \left( \epsilon + \frac{\nu}{1 - 2\nu} I_1(\epsilon) \mathbb{I} \right) \quad (1) $$

where $\sigma$ is the stress tensor, $\epsilon$ the strain tensor, $I_1(\epsilon)$ the first invariant of the strain tensor, and $\mathbb{I}$ the identity tensor. Ludwik's law is expressed as

$$ J_2(\sigma) = \frac{1}{3} (\sigma_0 + h \bar{\epsilon}_p^m) \quad (2) $$

where $\bar{\epsilon}_p$ denotes the equivalent plastic strain, and $J_2(\sigma)$ the second invariant of the deviatoric stress tensor.
<figure><img src="image_3.png" /><figcaption>Fig. 1. (a) Test setup before loading. (b) Top view of camera setup wrt. the sample. (c) Measured force as a function of image number, failure occurred between images 106 and 107. (d) Reference image for mono-vision camera (Face A). (e) Reference image for stereovision camera 2 (Face B).</figcaption></figure>

<table><caption>**Table 1**

DIC hardware and software parameters.</caption><tr><td>Cameras</td><td>3 Allied Vision Manta G-223</td></tr><tr><td>Definition</td><td>4112 × 3008 pixels</td></tr><tr><td>Color filter</td><td>none</td></tr><tr><td>Gray levels rendering</td><td>8bits</td></tr><tr><td>Lens</td><td>stereo: Tamron 50mm f/1.8 | mono: Edmund Optics Telecentric 0.125x</td></tr><tr><td>Field of view</td><td>stereo: 130mm × 90mm | mono: disk of diameter 80mm</td></tr><tr><td>Image Scale</td><td>stereo (avg in the center): 0.037mm/px | mono: 0.027mm/px</td></tr><tr><td>Stereo-angle</td><td>≈30°</td></tr><tr><td>Stand-off distance</td><td>stereo: ≈500mm | mono: ≈200mm</td></tr><tr><td>Image acquisition rate</td><td>0.22 Hz</td></tr><tr><td>Patterning technique</td><td>B/W paint</td></tr><tr><td>Pattern feature size</td><td>stereo: 3pixels to 20pixels | mono: 2pixels to 23pixels</td></tr><tr><td>DIC Software</td><td>EikoTwin DIC 2024.1 (EikoSim, 2024)</td></tr><tr><td>Image filtering</td><td>none</td></tr><tr><td>Element size (avg)</td><td>1.44mm or 39px (stereo) or 53px (mono)</td></tr><tr><td>Shape functions</td><td>Linear (T3 elements)</td></tr><tr><td>Interpolant</td><td>Linear</td></tr><tr><td>Displacement noise-floor</td><td>[σ(U<sub>x</sub>), σ(U<sub>y</sub>), σ(U<sub>z</sub>)] stereo: [0.6, 1, 2]μm | mono: [0.3, 0.4, NA] μm</td></tr></table>
<figure><img src="image_4.png" /><figcaption>Fig. 2. Displacement fields along the tensile direction *Y*, measured by DIC for image no. 100, for (a) Face A (mono-vision) and (b) Face B (stereo-vision).</figcaption></figure>

<table><caption>**Table 2**
Initial values of the constitutive parameters (Neggers et al., 2017).</caption><thead><tr><th colspan="2">Hooke</th><th colspan="2">Ludwik</th></tr><tr><th><i>E</i><br>GPa</th><th><i>ν</i></th><th><i>σ</i><sub>0</sub><br>MPa</th><th><i>h</i><br>MPa</th><th><i>m</i></th></tr></thead><tbody><tr><td>71.8</td><td>0.28</td><td>331</td><td>487</td><td>0.34</td></tr></tbody></table>

In Abaqus (Dassault Systèmes, 2023), the predefined Johnson-Cook law was used for plasticity, with viscous and thermal parameters set to zero (hence equivalent to Ludwik's law). As this paper focuses on boundary conditions, the initial values of constitutive parameters have been taken as the result of a previous identification study (Neggers et al., 2017) on the same alloy (Table 2). In that work, the parameters were determined via I-DIC and the measured boundary conditions were filtered using a cubic polynomial interpolation. The study of the impact of unknown constitutive parameters in the present framework will be performed in a later work.

### 3.1.2. Mesh

The simulation mesh used linear brick elements with reduced integration (C3D8R). A convergence study was carried out with the objective of minimizing the computational cost while maintaining acceptable accuracy. Three parameters (largest element size, smallest element size and number of layers) were modified and the associated simulation outputs were compared to those with the finest mesh. The selected mesh was the coarser still showing Root Mean Square displacement (resp. force) errors less than 0.1 % of the maximum absolute displacement (resp. force). It was refined around the holes and in the neck with an element size of 0.2 mm. Then, the element size was increased linearly up to the grips where it was set to 2 mm. The third

dimension is an extrusion of 4 layers of the previously described 2D mesh. The converged mesh resulted in approximately 80,000 degrees of freedom and is shown in Fig. 3(a).

Moreover, a scaling study of the FE simulations was carried out to optimize the computing resource usage. It resulted in an optimal parallelization using 4 cores, allowing the efficiency to be kept above 40% to 50% while minimizing the execution time.

### 3.2. Boundary conditions

The standard way to apply boundary conditions in the framework of global FE-based DIC is to prescribe the measured displacements on the boundary of the measurement area to the FE simulations (Sztefek and Olsson, 2008; Trellu et al., 2020), on a node-to-node basis. However, this method is sensitive to measurement uncertainties. One may filter these data to mitigate this downside or even consider rigid body motions of the measurement area border nodes. The true sample BCs being often hidden behind fixtures or grips, the intermediate BCs used in the previous methods may not be compatible with Saint-Venant's principle. It is proposed to develop a method considering rigid body boundary conditions away from the area of interest (Fig. 3(a)), and minimizing the difference of displacement fields between the experiment (i.e., DIC measurement) and FE simulations. The use of far away boundary conditions may not perfectly represent the true boundary conditions as the material in-between the grips is allowed to deform in the simulation. Moreover, this simplified model of BCs allows for very small inverse problems, leading to faster computations compared to the identification of nodal displacements. A later work may explore the determination of the optimal location and simplification (Rokoš et al., 2018) of the boundary conditions.

The simulation is parameterized with two rigid body motions applied on both ends of the sample, thus resulting in 12 parameters
<figure><img src="image_5.png" /><figcaption>Fig. 3. (a) Simulation mesh with Dirichlet nodes for the proposed methods (highlighted in red) and for the classical methods (highlighted in blue). (b) Simulation mesh and parameterization of boundary conditions.</figcaption></figure>

to identify. The 12 parameters, displayed in Fig. 3(b), are defined as 3 translations per boundary condition ($U_{T_X}$, $U_{T_Y}$, $U_{T_Z}$ for the top BC and $U_{B_X}$, $U_{B_Y}$, $U_{B_Z}$ for the bottom BC) and 3 rotations per boundary condition ($\theta_{T_X}$, $\theta_{T_Y}$, $\theta_{T_Z}$ for the top BC and $\theta_{B_X}$, $\theta_{B_Y}$, $\theta_{B_Z}$ for the bottom BC).

A weighted FEMU algorithm is used to estimate these BCs. The cost function to minimize the displacement gap on Face A (mono-vision) of the sample is defined as

$$ \chi_{U_A}^2(\{\mathbf{p}_{BC}\}) = \frac{1}{N_{dof,U_A}} \sum_t \|\{\mathbf{U}_{A,meas}(t)\} - \{\mathbf{U}_{A,sim}(t, \{\mathbf{p}_{BC}\})\}\|_{[C_{U_A}]}^2 \quad (3) $$

where {$\mathbf{p}_{BC}$} is the BC parameter set, $N_{dof,U_A}$ the number of spatio-temporal displacement Degrees Of Freedom (DOFs) for Face A (mono-vision), {$\mathbf{U}_{A,meas}$} the vector of measured nodal displacements on Face A (mono-vision), {$\mathbf{U}_{A,sim}$} the vector of computed nodal displacements, and {$[C_{U_A}]$} the covariance matrix of measured displacements on Face A (mono-vision) computed in the previous section. It is worth noting that this cost function is weighted by the measurement uncertainties through the use of the displacements covariance matrices in the norm and nondimensionnalized by the number of spatio-temporal degrees of freedom (DOFs). The cost function for Face B (stereo-vision) is defined in the same way, and designated as $\chi_{U_B}^2$.

The global cost function to minimize is written as a weighted sum of the previously defined cost functions

$$ \chi_{U}^{2}(\{\mathbf{p}_{BC}\}) = \frac{N_{dof,U_A}}{N_{dof,U_A} + N_{dof,U_B}} \chi_{U_A}^{2}(\{\mathbf{p}_{BC}\}) + \frac{N_{dof,U_B}}{N_{dof,U_A} + N_{dof,U_B}} \chi_{U_B}^{2}(\{\mathbf{p}_{BC}\}) \quad (4) $$

The minimization is performed using a Gauss-Newton scheme updating the boundary conditions parameter increment {$\delta \mathbf{p}_{BC}$}

$$ [\mathbf{H}]\{\delta \mathbf{p}_{BC}\} = \{\mathbf{b}\} \quad (5) $$

where the Hessian matrix is defined as

$$ [\mathbf{H}] = \frac{N_{dof,U_A}}{N_{dof,U_A} + N_{dof,U_B}} [\mathbf{H}_{U_A}] + \frac{N_{dof,U_B}}{N_{dof,U_A} + N_{dof,U_B}} [\mathbf{H}_{U_B}] \quad (6) $$

with {$[\mathbf{H}_{U_A}]$} (resp. {$[\mathbf{H}_{U_B}]$}) the Hessian matrix of displacements on Face A (resp. B) expressed as

$$ [\mathbf{H}_{U_A}] = \frac{1}{N_{dof,U_A}} \sum_t [\mathbf{S}_{U_A}(t)]^T [C_{U_A}]^{-1} [\mathbf{S}_{U_A}(t)] \quad (7) $$

and {$[\mathbf{S}_{U_A}]$} the sensitivity matrix constructed from sensitivity vectors such that

$$ [\mathbf{S}_{U_A}(t)] = [\{\mathbf{S}_{U_A, p_{BC_1}}(t)\} \dots \{\mathbf{S}_{U_A, p_{BC_n}}(t)\}] \quad (8) $$

The sensitivity vectors are defined, for each boundary condition parameter of interest $p_{BC_n}$, as

$$ \{\mathbf{S}_{U_A, p_{BC_n}}(t)\} = \frac{\{\mathbf{U}_{A,sim}(t, \{\mathbf{p}_{BC_{dist,n}}\})\} - \{\mathbf{U}_{A,sim}(t, \{\mathbf{p}_{BC}\})\}}{\epsilon} \quad (9) $$

where {$\{\mathbf{p}_{BC_{dist,n}}\}$} corresponds to {$\{\mathbf{p}_{BC}\}$} offset by $\epsilon$ (taken equal to 5%) on the nth parameter.

The second member reads

$$ \{\mathbf{b}\} = \frac{N_{dof,U_A}}{N_{dof,U_A} + N_{dof,U_B}} \{\mathbf{b}_{U_A}\} + \frac{N_{dof,U_B}}{N_{dof,U_A} + N_{dof,U_B}} \{\mathbf{b}_{U_B}\} \quad (10) $$

with

$$ \{\mathbf{b}_{U_A}\} = \frac{1}{N_{dof,U_A}} \sum_t [\mathbf{S}_{U_A}(t)]^T [C_{U_A}]^{-1} (\{\mathbf{U}_{A,meas}(t)\} - \{\mathbf{U}_{A,sim}(t, \{\mathbf{p}_{BC}\})\}) \quad (11) $$

and {$\{\mathbf{b}_{U_B}\}$} defined in the same way as {$\{\mathbf{b}_{U_A}\}$}.

The cost function (4) is minimized for each time step independently, and the result is used as an initialization for the next time step. As the minimization is based on computed sensitivities, the procedure can be fast for linear cases, requiring the sensitivities to be computed only once. However, for nonlinear cases, the sensitivities are updated for each time step until convergence. In the case of kinematic boundary conditions identification, the reference BC parameter set is taken to be all zeros, leading to null displacements thus avoiding the need for the reference simulation to be computed. Only one simulation per BC parameter n is needed to determine one set of sensitivity matrices. In the case of two kinematic rigid body boundary conditions, because the sensitivity matrices are updated for each time step until convergence, at least 12 simulations will be needed. However, as the constitutive model is nonlinear, multiple iterations will be required at each time step in order to converge, leading to a large number of simulations. This amount of simulations, even when using parallel computing on a cluster, leads to intensive and lengthy computations. One way to mitigate this cost is to consider a linear model (e.g., elasticity) as a first approach. This model induces constant sensitivities with respect to the BC parameters over the whole domain of solutions, allowing the minimization to converge in a single iteration for each time step,
<table><caption>**Table 3**
Studied cases.</caption><thead><tr><th rowspan="2"></th><th rowspan="2">BC determination</th><th rowspan="2">FE Mesh</th><th colspan="2">Constitutive model for identification of</th></tr><tr><th>• boundary conditions</th><th>• constitutive parameters</th></tr></thead><tbody><tr><td>C1</td><td>Measured</td><td>Converged</td><td>–</td><td>Elastoplastic</td></tr><tr><td>C2</td><td>Measured RBMs</td><td>Converged</td><td>–</td><td>Elastoplastic</td></tr><tr><td>C3</td><td>Identified RBMs</td><td>Converged</td><td>Elastic</td><td>Elastoplastic</td></tr><tr><td>C4</td><td>Identified RBMs</td><td>DIC</td><td>Elastoplastic</td><td>Elastoplastic</td></tr><tr><td>C5</td><td>Identified RBMs</td><td>Converged</td><td>Elastoplastic</td><td>Elastoplastic</td></tr></tbody></table>

and the sensitivities to be computed only once, the number of needed simulations is only 12. The linear case is not expected to perform well over the whole test, in particular in the plastic domain. Yet, it is significantly faster to compute compared to nonlinear cases.

### 3.3. Constitutive parameter identification

The constitutive parameters are identified after the boundary conditions are calibrated, this choice will be later justified (Section 4.2). However, in this case, not only the measured and simulated displacement field will be used but also the measured and simulated forces. The cost function becomes

$$ \chi^2(\{\mathbf{p}_C\}) = \frac{N_{dof,U_A}}{N_{dof,U_A} + N_{dof,U_B} + N_{dof,F}} \chi_{U_A}^2(\{\mathbf{p}_C\}) + \frac{N_{dof,U_B}}{N_{dof,U_A} + N_{dof,U_B} + N_{dof,F}} \chi_{U_B}^2(\{\mathbf{p}_C\}) + \frac{N_{dof,F}}{N_{dof,U_A} + N_{dof,U_B} + N_{dof,F}} \chi_{F}^2(\{\mathbf{p}_C\}) \quad (12) $$

with $\{\mathbf{p}_C\}$ the constitutive parameter set, $\chi_{U_A}^2$ and $\chi_{U_B}^2$ defined in Eq. (3), and $\chi_F^2$ reads

$$ \chi_F^2(\{\mathbf{p}_C\}) = \frac{1}{N_{dof,F}} \sum_t \left( \frac{F_{meas}(t) - F_{sim}(t, \{\mathbf{p}_C\})}{\sigma_F} \right)^2 \quad (13) $$

where $F_{meas}$ corresponds to the measured force, $F_{sim}$ the computed force, and $\sigma_F$ the force uncertainty.

A Gauss-Newton scheme is still used as before, and the Hessian matrix as well as the second member vector are determined in the same way, but this time using constitutive parameters rather than instantaneous boundary condition parameters. Moreover, a regularization scheme may be needed to mitigate the effect of low sensitivity of constitutive parameters (Gras et al., 2015). Tikhonov regularization consists in adding a term $\chi_{Tik}^2$ to the previously defined cost function (12) in order to ensure that the sought parameters stay close to their initialization and do not diverge to nonphysical values (Tikhonov et al., 1995)

$$ \chi_{Tik}^2 = \| (\{\mathbf{p}_{C_0}\} - \{\mathbf{p}_C\}) \ominus \{\mathbf{p}_{C_0}\} \|_2^2 \quad (14) $$

where $\{\mathbf{p}_{C_0}\}$ is the initial set of constitutive parameters, $\{\mathbf{p}_C\}$ the current set, and $\ominus$ the (term by term) Hadamard division. Since an L-2 norm was considered, such regularization is also known as ridge regression (Hoerl and Kennard, 1970b,a). The penalty term is weighted in order to control the impact of regularization, using $\lambda_{Tik}$ based on the maximum eigenvalue $\lambda_{max}$ of the global Hessian matrix

$$ \lambda_{Tik} = \lambda_{max}(|\mathbf{H}|) \times 10^{\tau} \quad (15) $$

The regularization is relaxed each time the computation converges, starting with $\tau = -1$ and lowered 1 by 1 until the computation does not converge anymore. The choice of an optimal value for $\tau$ is discussed in Section 4.6.

### 3.4. Validation metric

In order to compare different identification cases, the square roots of the previously defined costs functions (i.e., $\chi_{U_A}$ and $\chi_{U_B}$ (3), $\chi_{U}$ (4) and $\chi_F$ (13)) are computed. A global validation metric is defined as the square root of the global cost function $\chi$ (12), which will be referred to as validation residual. These metrics are expressed as Signal-to-Noise Ratios (SNRs) allowing for easy comparisons to the uncertainty levels. Moreover, they can be decomposed into spatio-temporal values for finer analyses.

## 4. Results and discussion

In this section, the previously described methods are applied to the experimental case. The first part defines multiple strategies to be analyzed and compared to assess the performance of the proposed methodology. A sensitivity analysis of the constitutive and BC parameters is conducted. Then, the BC determination and the constitutive parameter identification step are performed. The studied cases are compared along the analyses. In the final part, the results of iterating the procedure, namely, BC determination followed by constitutive parameter identification, are presented.

### 4.1. Studied cases

Multiple identification strategies are defined in Table 3. Each case varies on one parameter compared to the method proposed in this paper (i.e., C5 case). The varying parameters are the BC determination method, the FE discretization, and the constitutive model used for BC identification.

Two main groups are highlighted. First, cases C1 and C2 correspond to classical methods with measured BCs applied on the FE model. Case C1 is a direct application of BC measurements to the FE model. Case C2 utilizes heavily regularized BC measurements in the FE model as they are considered as rigid body motions (RBMs) obtained from top and bottom boundary nodal displacements. The second group, which is composed of cases C3, C4 and C5, corresponds to BCs identified from experimental data. For these cases, the boundary conditions and the constitutive parameters are identified sequentially using FEMU-U, where measured quantities are the displacements obtained from 2D DIC and stereocorrelation. In the next section, the choice of identification strategy (first boundary conditions and second constitutive parameters) will be explained based on a sensitivity analysis. Case C3 allows for fast BC identification as the computed displacements are obtained from a linear (elastic) FE model. Case C4 uses computed data from the same FE mesh as global DIC measurements, revealing the importance of a converged mesh in the determination of BCs and constitutive parameters. Case C5 allows for an accurate determination of FE parameters, as computed quantities are obtained using a converged mesh and an elastoplastic model. For all cases, the constitutive parameters are then identified using FEMU-UF, where measured quantities are the displacements, obtained from 2D DIC and stereocorrelation, and the force measured during the experiment. The corresponding computed quantities are obtained from elastoplastic FE simulations.
<figure><img src="image_6.png" /><figcaption>Fig. 4. Log-10 of absolute Hessian matrix (a) expressed as Signal-to-Noise ratios, and corresponding correlation matrix (b) using kinematic and force data.</figcaption></figure>

<figure><img src="image_7.png" /></figure>

The methods will be evaluated by comparison of the quality of the output model along the procedures using the previously defined comparison metric (Section 3.4).

## 4.2. Sensitivity analysis of the sought parameters

For cases C3 to C5, the goal is to identify both constitutive and boundary condition parameters. In order to achieve this identification with a large number of parameters, a specific strategy should be followed. First, a sensitivity analysis on a single instant of time is performed as the boundary conditions are time-dependent. The studied instant is chosen to be in the plastic regime of the test, at image 100 (or step 100 in the elastoplastic simulation). The results of this analysis are compiled in the form of the Hessian matrix (Fig. 4(a)) and the correlation matrix (Fig. 4(b)). The Hessian matrix indicates the (auto)sensitivity of each parameter on the diagonal terms as well as the coupling between parameters on the off-diagonal terms. It is observed, from the diagonal of the Hessian matrix, that the boundary conditions have generally higher sensitivity levels compared to the constitutive parameters. To have a better analysis of this matrix, its decomposition in the eigenspace is proposed in Fig. 5. The covariance matrix indicates that couplings between constitutive and BC parameters are weak or negligible. The strongest couplings occur between constitutive parameters themselves, and within BC parameters.

The eigenvalues (Fig. 5(a)) and eigenparameters (Fig. 5(b)) are sorted in descending order. Looking at the eigenparameters, the constitutive parameters are mainly present in the last ones (starting from eigenparameter no. 10). The BC parameters are mainly expressed in the first eigenparameters (up to eigenparameter no. 14). Each eigenparameter aligns with either constitutive or BC parameters, further demonstrating the minimal coupling between these two types of parameters. This eigenspace analysis confirms the previous conclusion that the boundary conditions have higher sensitivities compared to constitutive parameters, and that the two types of parameters are only mildly coupled. This comparison justifies the choice to first perform boundary condition identification, and in a second step constitutive parameter update for cases C3 to C5.

In cases C3 to C5, as the boundary conditions are described as 2 rigid body motions using 12 degrees of freedom on more than 100 time steps, the total number of parameters to identify would be over 1,200. However, each time step can be considered as independent and the minimization is computed for each one independently. This choice leads to 100+ simpler minimization problems to be solved with only 12 BC parameters to be calibrated at each step.

## 4.3. Boundary condition determination

In this section, the constitutive parameters are set to their initial values (Table 2) and will not be updated until a later analysis (Section 4.6).

For case C1, Dirichlet boundary conditions are applied to the nodes located on the highest and lowest cross-sections (defined by contiguous element faces) within the measurement areas (Fig. 3(a)). On each face, the displacements measured via DIC are interpolated onto the boundary condition (BC) nodes belonging to that face. For the BC nodes located between the two faces (i.e., within the depth of the sample), the displacements are obtained through linear interpolation between the displacements of the BC nodes belonging to both faces. One particularity to note is that the displacement along the Z-direction was only measured on Face B (stereo-vision). In that specific case, it is extruded and not interpolated in the depth of the sample. For case C2, a rigid body motion is computed from the interpolated measured displacements on each of the top and bottom lines of nodes previously specified. The obtained rigid body motions are applied to the same cross-sections of nodes. For the last three cases, the boundary conditions are defined away from the measurement area, and use 12 degrees of freedom (three translations, and three rotations for each end of the specimen). These degrees of freedom are determined by minimization of the previously defined cost function (4) for each time step. The constitutive parameters are those of Table 2. The resulting boundary conditions are shown in Fig. 6 for the bottom BC and Fig. 7 for the top BC.

For the bottom boundary conditions (Fig. 6), it is observed that the largest displacement appears for the $U_{B_y}$ component, which coincides
<figure><img src="image_8.png" /><figcaption>Fig. 5. Diagonalization of the Hessian matrix. Eigenvalues (a) (expressed as Signal-to-Noise Ratios) and corresponding eigenparameters (b) (eigenvectors).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Identified rigid body translations (a–c) and rotations (d–f) of the bottom boundary condition along X (a,d), Y (b,e) and Z (c,f) directions for cases C3, C4 and C5. Negative numbers correspond to static images acquired prior to the test.</figcaption></figure>

with the translation in the tensile direction applied to the bottom grip during the test. In cases C4 and C5, it is observed that the tensile

displacement component is quasi-linear, except when some slip in the grips occurred (confirmed by the pictures of the test).
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Identified rigid body translations (a-c) and rotations (d-f) of the top boundary condition along X (a,d), Y (b,e) and Z (c,f) directions for cases C3, C4 and C5. Negative numbers correspond to static images acquired prior to the test.</figcaption></figure>

For the top boundary conditions (Fig. 7), it is observed that the displacement are mainly close to zero except for $U_{T_X}$ and $U_{T_Y}$. The displacement $U_{T_Y}$ matches with the tensile direction and highlights the rigidity of the testing machine. The displacement $U_{T_X}$ reveals an in-plane misalignment of the sample ($U_{B_X}$ being of opposite sign).

In an ideal setup, the tensile displacement component of the bottom boundary condition ($U_{B_Y}$) should be the only one active. However, it is observed that the other components also have non-zero amplitudes, highlighting misalignments of the sample, testing machine rigidity, and even some slip in the grips. It is noted that on a majority of components, while cases C4 and C5 (differing solely on the mesh size) are close, case C3 always diverges starting from steps 40–60. This difference is due to an unsuitable constitutive law used to identify the boundary conditions, and will be later confirmed in the analysis of the quality of the models (Section 4.5). Yet, case C3 leads to very fast computations (i.e., a low number of simulations are needed) compared to the other two cases and could be used in further studies as an initialization for cases C4 and C5. Last, among the noisiest components are $U_{B_Z}$, $\theta_{B_X}$, $\theta_{B_Y}$, $U_{T_Z}$, $\theta_{T_X}$ and $\theta_{T_Y}$, all of which were mainly determined by DIC measurements along the Z-direction, namely, normal to the measurement planes. This result is justified by the fact that the Z displacement is only measured on the stereovision face (B). As it is the out-of-plane measurement, it is also the noisiest component (Table 1). As the identification cost function is being weighted by the number of spatial measurements as well as the uncertainties, the out-of-plane measured displacement has the lowest impact on the results.

### 4.4. Force uncertainty

The force uncertainty is composed of two parts. The first part is the uncertainty due to the load cell ($\sigma_{LC}$), which is evaluated to be

<table><caption>**Table 4**
Force uncertainties for each test case.</caption><thead><tr><th>Cases</th><th>σ<sub>LC</sub><br>N</th><th>σ<sub>BC</sub><br>N</th><th>σ<sub>F</sub><br>N</th></tr></thead><tbody><tr><td>C1</td><td>1.6</td><td>118</td><td>118</td></tr><tr><td>C2</td><td>1.6</td><td>117</td><td>117</td></tr><tr><td>C3</td><td>1.6</td><td>89</td><td>89</td></tr><tr><td>C4</td><td>1.6</td><td>78</td><td>78</td></tr><tr><td>C5</td><td>1.6</td><td>89</td><td>89</td></tr></tbody></table>

1.6 N using static measurements before the beginning of the test. The second part derives from a propagation of uncertainties ($\sigma_{BC}$). It is due to the fact that measured boundary conditions were considered herein. To evaluate this second contribution, the boundary conditions are also determined with the static measurements prior to the test, and the corresponding forces are evaluated through FE simulations. By summing the variances of each part, it is possible to evaluate the resulting force uncertainty $\sigma_F$ for each studied case (Table 4)

$$ \sigma_F = \sqrt{\sigma_{LC}^2 + \sigma_{BC}^2} \qquad (16) $$

It is observed that when the boundary conditions are measured or identified using experimental data, the force uncertainty is mainly driven by the second term. Starting from the classical method (C1), the propagated uncertainty is really high, and using a strong filtering such as rigid body motions (C2) does not lower it. The kineamtic measurement uncertainty is composed of multiple sources, among which two will be described. First, the camera sensor uncertainties, and second, the displacement uncertainties resulting from the small loading applied by the proportional-integral-derivative (PID) controller of the
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Global $\chi$ levels (a) and residual values (displacement (b) and force (c)) for all studied cases after boundary condition determination and before constitutive parameter updating.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Temporal $\chi$ for displacement on Face A (monovision) (a), Face B (stereovision) (b) and their weighted sum (c) for cases C1, C2, C3, C4 and C5. Negative numbers correspond to static images acquired prior to the test.</figcaption></figure>

hydraulic testing machine. The first uncertainty has been lowered by the strong filtering applied in case C2 but the second uncertainty is of dominant importance compared to the first one, thus leading to a high force uncertainty. Overall the best case is to use the proposed method that lowers the force uncertainty by 25%.

### 4.5. Model quality after determination of boundary conditions

The quality of each updated model (with the considered boundary conditions but initial constitutive parameters) is evaluated by comparing the $\chi$ levels. The global results are reported in Fig. 8 and their temporal variations are displayed in Figs. 9 and 10.

It is observed in Fig. 8(a) that the proposed C5 method has the lowest $\chi_{global}$, hence the highest quality with respect to the measured data and the measurement uncertainties. Case C4 shows higher force residuals $\chi_F$ than C5 due to its coarser mesh but has similar levels in terms of displacements. Case C3 is the worst due to the wrongly represented plasticity during boundary condition identification, which will be illustrated later on (Fig. 9). Last, cases C1 and C2 lead to good and similar results compared to case C5, but with a slightly lower quality. From the displacement residuals ($\delta_{U_A}$ and $\delta_{U_B}$, Fig. 8(b)), it is concluded that the displacements on Face A (mono-vision) are better represented than on Face B (stereo-vision). However, the displacement uncertainties (Table 1) on Face B (stereo-vision) are higher than on
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Temporal $\chi_F$ for cases C1, C2, C3, C4 and C5. Negative numbers correspond to static images acquired prior to the test.</figcaption></figure>

Face A (mono-vision) leading to better $\chi$ levels for Face B compared to Face A.

Looking at the temporal analysis of the displacement residuals (Fig. 9), the main observation is that as soon as plasticity begins (around image 40), the FE model of case C3 is not a good representation of the measured displacements as the residuals are significantly growing. The other cases are all similar regarding the temporal displacement residuals, with a global increase along the test due to accumulating model and constitutive parameter errors. The higher values for cases C1 and C2 are due to a misrepresentation of the boundary conditions that is explained in more details in Appendix A.

The temporal force residuals $\chi_F$ (Fig. 10) show that the FE models of cases C4 (coarse mesh) and C3 are unable to faithfully represent

the measured forces. It is also noted that the FE model of case C2 is not performing very well at the transition between elastic and plastic regimes (time steps 30 to 50). This bias is due to the use of rigid body boundary conditions close to the region of interest (in a highly strained area) leading to inaccurate computed forces.

### 4.6. Constitutive parameter identification

The next step is to calibrate the constitutive parameters for the five models having their boundary conditions evaluated. First, a sensitivity analysis for the constitutive parameters of each model over the whole timescale is performed and summarized by the study of the weighted FEMU-UF Hessian matrices (Fig. 11) and corresponding correlation matrices (Fig. 12).

Based on the diagonal terms of the Hessian matrices, in all studied cases, the Poisson's ratio $\nu$ is the parameter with the lowest sensitivity level. Conversely, the Young's modulus $E$ is the parameter with the highest sensitivity. Looking at the off-diagonal terms of the Hessian matrices, and at the correlation matrices, the plastic parameters (i.e., $\sigma_0$, $h$ and $m$) are strongly correlated while the elastic parameters (i.e., $E$ and $\nu$) are uncoupled from each other and from the plastic parameters. The Hessian matrices can be sorted in two groups. First, cases C1 and C2 and second, cases C3, C4 and C5 that show higher sensitivities compared to the first group. These two groups correspond to measured boundary condition cases (first group) and identified boundary conditions (second group). All the correlation matrices are similar except for case C3, which is the one with the least representative boundary conditions.

These Hessian matrices are then studied in their eigenspaces. The first quantities of interest are the eigenvalues plotted in Fig. 13. The

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Weighted FEMU-UF Hessian matrices for cases C1 (a), C2 (b), C3 (c), C4 (d) and C5 (e) expressed as Signal-to-Noise ratios.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Fig. 12. Weighted FEMU-UF correlation matrices for cases C1 (a), C2 (b), C3 (c), C4 (d) and C5 (e).</figcaption></figure>

eigenvalues represent the sensitivity of the eigenparameters (i.e., eigen-vectors) displayed in Fig. 14. A similar conclusion is drawn as with the Hessian matrices, namely, cases C3, C4 and C5 show higher sensitivities than cases C1 and C2. Moreover, as the eigenvalues are also expressed as Signal-to-Noise ratios, they can easily be compared to the measure-ment uncertainties. Only the first eigenvalue is higher than the noise level (10⁰), all subsequent eigenvalues reach low SNR levels (i.e., 10⁻⁴). The presence of eigenvalues less than the noise level indicates chal-lenges to identify the associated parameters as their sensitivities are not strong enough to differentiate them from measurement uncertainties. It is then necessary to add regularization to help the algorithm to converge to a solution of the inverse problem. In the present case, a ridge scheme is chosen, allowing the parameters to be kept close to their initialization when their sensitivities are too low.

Regarding the eigenparameters (Fig. 14), while the first two eigen-parameters are similar in all cases, two distinct sets of cases appear for the last three eigenparameters, namely, cases C1 and C2 on one side and, on the other side, cases C3, C4 and C5:

* For all cases, the first and second eigenparameters mainly char-acterize the Young's modulus *E*, and the yield stress σ₀ as well as the hardening parameters *h* and *m* to a lesser extent.

* For cases C1 and C2 (with classically measured BCs), the third eigenparameter mainly characterizes the Poisson's ratio *v*, and to a lesser extent some plastic parameters showing a stronger cou-pling compared to the first three cases. The fourth eigenparameter mainly characterizes the hardening modulus *h*, and the fifth the hardening power *m*.

* For cases C3, C4 and C5 (identified BCs), the third eigenpa-rameter mainly characterizes the hardening modulus *h*, and the

<figure><img src="image_16.png" /><figcaption>Fig. 13. Eigenvalues of weighted FEMU-UF Hessian matrices for cases C1, C2, C3, C4 and C5.</figcaption></figure>

fifth the hardening power *m*. The fourth characterizes only the Poisson's ratio *v* showing its decoupling from other parameters.

The use of remote BCs allows for the Poisson's ratio to be better de-coupled from other constitutive parameters. It is mainly characterized by the fourth eigenparameter, compared to measured BC cases where it is mainly dependent upon the third and fourth eigenparameters. Moreover, the sensitivity (represented by eigenvalues) of the fourth eigenparameter in the cases of identified BCs is higher than those of the third and fourth eigenparameters in the cases of measured BCs (Fig. 13).
<figure><img src="image_17.png" /><figcaption>Eigenparameter number</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Eigenparameter number</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Eigenparameter number</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Eigenparameter number</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Eigenparameter number</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 14. Eigenparameters of weighted FEMU-UF Hessian matrices for cases C1 (a), C2 (b), C3 (c), C4 (d) and C5 (e).</figcaption></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /><figcaption>Fig. 15. Relative change of constitutive parameters for cases C1 (a), C2 (b), C3 (c), C4 (d) and C5 (e) during the identification iterations.</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>Fig. 16. Stress-strain curves for each converged parameter set for cases C1 (a), C2 (b), C3 (c), C4 (d) and C5 (e).</figcaption></figure>

<table><caption>**Table 5**
Identified constitutive parameters for each case compared to the initialization.</caption><thead><tr><th rowspan="2">Cases</th><th rowspan="2">τ</th><th colspan="2">Hooke</th><th colspan="2">Ludwik</th></tr><tr><th>E<br>GPa</th><th>ν<br>-</th><th>σ<sub>0</sub><br>MPa</th><th>h<br>MPa</th><th>m<br>-</th></tr></thead><tbody><tr><td>Initialization set (Neggers et al., 2017)</td><td></td><td>71.8</td><td>0.28</td><td>331</td><td>487</td><td>0.34</td></tr><tr><td>C1</td><td>-3</td><td>70.1</td><td>0.27</td><td>347</td><td>457</td><td>0.34</td></tr><tr><td>C2</td><td>-3</td><td>66.2</td><td>0.26</td><td>340</td><td>479</td><td>0.32</td></tr><tr><td>C3</td><td>-2</td><td>51.3</td><td>0.26</td><td>528</td><td>645</td><td>0.27</td></tr><tr><td>C4</td><td>-3</td><td>68.5</td><td>0.30</td><td>319</td><td>468</td><td>0.35</td></tr><tr><td>C5</td><td>-4</td><td>71.8</td><td>0.33</td><td>322</td><td>474</td><td>0.31</td></tr></tbody></table>

It is then possible to proceed with the constitutive parameter identification for each case. The ridge regularization is relaxed until the convergence of the procedure is no longer reached in less than 15 iterations. The relative change of the constitutive parameters is displayed in Fig. 15. For each case, the last relaxation step leads to fluctuating changes in parameters, either through divergence or oscillations. The earlier presented correlations (Fig. 12) are also visible in the variations of the same parameters.

A first way to compare constitutive models at each relaxation step, is to run FE simulations on a single quadratic element in tension for each of them. Fig. 16 shows the stress-strain curves for each case compared to the initial parameter set. All cases look similar except C3 for which the identified constitutive parameters compensate the offsets due to wrongly determined boundary conditions leading to high stress states. For cases C2 and C4, the relaxation of the regularization has a small impact on the overall constitutive behavior where for cases C1 and C5 it does not have any significant impact. Hence, it does not allow one to choose a relaxation value.

Another way to compare constitutive models, using the results of relaxed convergence, is to plot the so-called L-curve allowing one to choose an optimum relaxation step (Hansen, 2001). In L-curves, for each converged state, the penalty cost function is compared to the weighted FEMU-UF cost function. Multiple routes are possible regarding the choice of the proper relaxation step. For example, if one has more confidence in one or the other cost function, the advised choice would be to select the relaxation step minimizing the cost function in which there is more confidence. In the present case, the choice

was made to select the relaxation step minimizing both cost-functions, hence maximizing the curvature of the L-curve (Hansen, 2001). The curvature is geometrically determined using, for example, the Menger curvature (Léger, 1999), which is defined as the inverse of the radius of the circle passing through three consecutive converged states in the L-curve plot. The L-curves as well as the optimal relaxation step are plotted in Fig. 17.

In the L-curves, the L is pretty well defined and the optimum easy to spot. For case C3 (Fig. 17(c)), only two points are drawn because starting from $τ = -3$, the computation did not converge anymore due to the wrongly determined boundary conditions. In that case, the last converged state is chosen, hence $τ = -2$.

Last, after choosing the optimum relaxation steps, the identified constitutive parameters are compared in Table 5 to the initialization set obtained from a previous work on the same alloy (Neggers et al., 2017). Case C1 is staying the closest to the initialization with maximum variations of -6% for the hardening modulus $h$. This result was expected as it was the closest method to that used by the authors of the paper providing the initialization set (Neggers et al., 2017). For case C2, the highest variations are on the elastic parameters with -7% to -8% for each of them. Saint-Venant's principle was not fulfilled because the boundary conditions were prescribed close to the zone of interest with rigid body motions, leading to untrustworthy results.

For case C3, as the results on the boundary condition are not accurate, it is, as expected, not very good in terms of constitutive parameter identification with high variations ranging from -30% to +60%. In that case, the Poisson's ratio is varying the least as it is strongly regularized with $τ = -2$. The Young's modulus is lowered and the yield stress raised in order to accommodate the elastic model on a larger part of the experiment due to the determination of the boundary conditions considering an elastic model.

For case C4, the parameters stayed closer to their initialization with a maximum variation of +11% for the Poisson's ratio $v$. It is noted that the variation of the Young's modulus is much larger compared to case C5 (-5% vs. +0%) due to the coarse mesh wrongly representing local effects. Last, for case C5 the maximum variation is +18% for the Poisson's ratio $v$, which is the least sensitive parameter. The other parameters had less than ±10% variations. This result shows that the
<figure><img src="image_30.png" /><figcaption>Fig. 17. L-curves for cases C1 (a), C2 (b), C3 (c), C4 (d) and C5 (e). The red circle depicts the optimum point based on Menger's curvature.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Fig. 18. Global $\chi$ levels (a) and residual values (displacement (b) and force (c)) for all studied cases after boundary condition determination and constitutive parameter identification.</figcaption></figure>
<table><caption>**Table 6**
Identified constitutive parameters and $\chi$ levels for case C5 compared to their initialization.</caption><thead><tr><th rowspan="2">Iteration no.</th><th rowspan="2">τ</th><th colspan="2">Hooke</th><th colspan="4">Ludwik</th><th rowspan="2">χ<sub>U</sub><sub>A</sub></th><th rowspan="2">χ<sub>U</sub><sub>B</sub></th><th rowspan="2">χ<sub>U</sub></th><th rowspan="2">χ<sub>F</sub></th><th rowspan="2">χ</th></tr><tr><th>E<br>GPa</th><th>ν<br>-</th><th>σ<sub>0</sub><br>MPa</th><th>h<br>MPa</th><th>m<br>-</th><th>SNR</th><th>SNR</th><th>SNR</th><th>SNR</th><th>SNR</th></tr></thead><tbody><tr><td colspan="12">Initialization set (Neggers et al., 2017)</td></tr><tr><td>1</td><td>-4</td><td>71.8</td><td>0.28</td><td>331</td><td>487</td><td>0.34</td><td>6</td><td>3</td><td>4</td><td>3</td><td>4</td></tr><tr><td>2</td><td>-4</td><td>71.7</td><td>0.34</td><td>321</td><td>471</td><td>0.31</td><td>5</td><td>3</td><td>4</td><td>3</td><td>4</td></tr></tbody></table>

<figure><img src="image_32.png" /><figcaption>**Fig. 19.** Temporal $\chi$ for displacement on (mono-vision) Face A (a), (stereo-vision) Face B (b) and their weighted sum (c) for cases C1, C2, C3, C4 and C5 after constitutive parameter updating.</figcaption></figure>

selected initialization set is of good quality leading to adjustments mainly on the least sensitive parameters.

## 4.7. Model quality after constitutive parameter identification

From the identified parameters, it is possible to update the models and assess their quality both globally (Fig. 18) and for each time step (Figs. 19 and 20).

For the global values, the proposed method of boundary condition determination and constitutive parameter identification (case C5) performed the best to match the FE model to the experimental data. It also performed the best for the displacements. For the force residuals $\chi_F$, the classical case C1 performed slightly better. However, the force uncertainty was not the same for each case due to the displacement uncertainties being propagated through the updated boundary conditions. Looking at the raw residuals between computed and measured forces, case C5 performs better than the other cases, with $\delta_F = 247$ N for case C5 and $\delta_F = 272$ N for case C1.

Comparing Fig. 19 to Fig. 9, and Fig. 20 to Fig. 10, an analysis of the impact of the identification of constitutive parameters is conducted. All residuals were lowered except for the force residual of case C3. In that case, due to a constitutive model error, the procedure led to a high force residual and it also impacted the displacement residuals. The temporal results highlight the offset of model C3 from the measurement data of the test. Cases C1 and C5 are similar in terms of residuals, both global and temporal, highlighting the quality of the proposed method compared to classical procedures.

## 4.8. Iterative procedure

The proposed procedure implies sequential identifications of BCs and constitutive parameters. The resulting residuals arise from two primary causes, namely, model error and weak coupling between BCs and constitutive parameters, as observed in the global sensitivity analysis (Section 4.2). While a more complex model could potentially reduce the model error and improve the fit to measured data, the global residual of 4 SNR is remarkably low (i.e. the model error is only four times

<figure><img src="image_33.png" /><figcaption>**Fig. 20.** Temporal $\chi_F$ for cases C1, C2, C3, C4 and C5 after constitutive parameter identification.</figcaption></figure>

the measurement uncertainties). Introducing additional complexity to the model at this stage would complicate the identification process and would likely offer limited benefit. In order to mitigate the weak coupling effect, it is possible to iterate over the procedure to improve the accuracy of both identifications and, in the end, the FE model. The boundary conditions of case C5 were identified a second time using the model updated in the previous section. Then, using the newly identified boundary conditions, a constitutive parameter identification step was performed. The resulting parameters as well those previously identified are gathered in Table 6, which also includes the residuals expressed as SNRs.

Because the initialization set was already of good quality, iterating the procedure had a minor impact on the identified parameters and no impact on the residuals, thus attributing the residual to model error. However, in a case where the constitutive parameters are wrongly initialized, the boundary condition identification would be impacted by this new initialization. Iterating the procedure may then improve more significantly the accuracy of the FE model.
## 5. Conclusion

In this paper, a method allowing for the identification of rigid body boundary conditions from heterogeneous test data was introduced. Its operation was demonstrated through the post-processing of a tensile test on a thick aluminum alloy sample. Boundary condition and constitutive parameter identifications were based on weighted-FEMU, thus accounting for the measurement uncertainties and data density. The boundary conditions were identified using displacement data, and the constitutive parameters were determined by adding load data. Through a sensitivity study, it was shown that the boundary condition parameters were of key importance compared to constitutive parameters, justifying their identification before the constitutive parameters. The introduced method allows the boundary conditions to be estimated outside of the measurement area, whereas classical methods (based on boundary condition measurements) are limited to the border of the measurement area.

A thorough comparison between the introduced method and classical approaches was performed showcasing on par or better results for the identification based method. The quality of the identified models was evaluated using the weighted-FEMU cost function (discrepancy between measured and computed data) and expressed as Signal-to-Noise ratio allowing for a simple comparison to the uncertainty level. The comparison was conducted up to constitutive parameter identification. As this study focused on the identification of the boundary conditions, the initial constitutive parameters were not offset and the inferred constitutive parameters with both methods were close to the initialization highlighting the stability of the methods. A later work will study the impact of offset constitutive parameters on the method.

While no regularization was used for the determination of boundary conditions, the identification of constitutive parameters used relaxed regularization in order for the minimization to converge. The choice of the relaxation step was performed by using so-called L-curves and the study of their curvature. The possibility of iterating the identification procedure was also illustrated. Although this method delivered high accuracy at high computational costs, the paper introduced the use of an elastic model to mitigate this limitation. Future work will be needed to evaluate the use of this linear model results as an initialization for the boundary condition determination. To further assess the quality of the developed procedure, a later work may add a comparison to other boundary condition identification methods and the application of the proposed method to virtual test cases where the ground truth is available.

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /><figcaption>**Fig. A.21.** Temporal $\chi_U$ levels before constitutive parameter identification evaluated on static images for all cases without (a) and with (b) rigid body motion corrections in the discrepancy computation.</figcaption></figure>

Ultimately, this method is expected to improve the identification of structural tests simulations by enabling the boundary conditions application to be outside of experimentally measured areas, thereby enhancing the accuracy of constitutive parameter identification and the FE model fidelity to real world data.

### CRediT authorship contribution statement

**Antoine Vintache:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Pierre Baudoin:** Writing – review & editing, Validation, Supervision, Methodology.

**Renaud Gras:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Methodology, Funding acquisition.

**Florent Mathieu:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Methodology, Funding acquisition.

**François Hild:** Writing – review & editing, Validation, Supervision, Methodology, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgment

This work has been financially supported by Eikosim and ANRT.

## Appendix A. Offset on measured boundary conditions

In multiple instances (Figs. 9 and 19), it is observed that measured BCs (C1 and C2) have higher $\chi_U$ values for the static images acquired prior to the test compared to identified BCs (C3, C4 and C5). For these static images used to evaluate measurement uncertainties, the expected $\chi$ values are 1 SNR. However, as shown in Fig. A.21(a), the $\chi$ levels are higher. The first explanation is that the test was performed on a hydraulic testing machine. Thus, for the static images, not only the measurement uncertainties were captured but also the effect of the proportional–integral–derivative (PID) controller. This controller leads to minute rigid body motions and loading. The rigid body motions can easily be corrected from the DIC measured fields, but not the loading that is then neglected.

In order to better approximate the measurement uncertainties, the rigid body motions were corrected in the evaluation of the covariance

<figure><img src="image_36.png" /></figure>
<figure><img src="image_37.png" /><figcaption>Fig. A.22. First normalized POD spatial modes of the residual fields between measured and computed displacements on static images for case C1 (a) and C5 (b).</figcaption></figure>

matrix. However, they were not removed in the evaluation of the discrepancy between test and simulations. This choice leads to $\chi_U$ values that are not equal to 1 SNR, and correcting for this offset allows the values to be lowered (Fig. A.21(b)).

A second explanation is found by studying the first spatial mode of the proper order decomposition (POD) of the residual fields between measured and computed displacements on the static images for cases C1 (Fig. A.22(a)) and C5 (Fig. A.22(b)). The spatial modes for both cases are different for the Z-field. It is observed that, for case C1, bending of the sample in the out of plane (Z) direction is over or under represented by the simulation. This result is explained by the choice of linearly interpolating measured displacements between the two measured faces, and even more by the choice of extruding the Z-measured displacement on Face B (stereo-vision) in the depth of the sample.

In conclusion, the elevated $\chi_U$ values for static images stem from both the hydraulic testing machine's PID controller and methodological limitations in interpolating and extruding displacement fields. The second justification is mitigated by the use of identified boundary conditions (case C5).

## Data availability

Data will be made available on request.

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