 Research Article
 Open Access
Toward 4D mechanical correlation
 François Hild^{1}Email authorView ORCID ID profile,
 Amine Bouterf^{1},
 Ludovic Chamoin^{1},
 Hugo Leclerc^{1},
 Florent Mathieu^{1},
 Jan Neggers^{1},
 Florent Pled^{3},
 Zvonimir Tomičević^{1, 2} and
 Stéphane Roux^{1}
https://doi.org/10.1186/s403230160070z
© The Author(s) 2016
 Received: 23 February 2016
 Accepted: 27 April 2016
 Published: 23 May 2016
Abstract
Background:
The goal of the present study is to illustrate the full integration of sensor and imaging data into numerical procedures for the purpose of identification of constitutive laws and their validation. The feasibility of such approaches is proven in the context of in situ tests monitored by tomography. The bridging tool consists of spatiotemporal (i.e., 4D) analyses with dedicated (integrated) correlation algorithms.
Methods:
A tensile test on nodular graphite cast iron sample is performed within a lab tomograph. The reconstructed volumes are registered via integrated digital volume correlation (DVC) that incorporates a finite element modeling of the test, thereby performing a mechanical integration in 4D registration of a series of 3D images. In the present case a nonintrusive procedure is developed in which the 4D sensitivity fields are obtained with a commercial finite element code, allowing for a large versatility in meshing and incorporation of complex constitutive laws. Convergence studies can thus be performed in which the quality of the discretization is controlled both for the simulation and the registration.
Results:
Incremental DVC analyses are carried out with the scans acquired during the in situ mechanical test. For DVC, the mesh size results from a compromise between measurement uncertainties and its spatial resolution. Conversely, a numerically good mesh may reveal too fine for the considered material microstructure. With the integrated framework proposed herein, 4D registrations can be performed and missing boundary conditions of the reference state as well as mechanical parameters of an elastoplastic constitutive law are determined in fair condition both for DVC and simulation.
Keywords
 Digital volume correlation
 Identification
 Integrated approaches
 Tomography
 Verification
Background
The emergence of simulationbased engineering sciences calls, among many challenges [1], for robust validation procedures and uncertainty quantifications to achieve reliable predictions. Fullfield measurements are one way of bridging experimental and computational mechanics. Their advantage lies in the fact that the comparison is now achieved by using huge amounts of data, including 3D imaging, to probe the predictive capacity of material models [2] and numerical frameworks [3]. The aim of the paper is to show that a seamless procedure, hereafter called “integrated 4D registration,” can be formulated to analyze an insitu test performed in a lab tomograph for the purpose of identifying a nonlinear constitutive law and unknown boundary conditions.
Computed microtomography allows 3D images of materials to be obtained, which reveal the microstructure in the bulk in a nondestructive way [4–8]. Very early on, mechanical tests were performed insitu [9–12]. For instance, the damage development in particulate composites could be analyzed [11, 13, 14]. Creep has also been studied with such techniques [15–18]. One additional feature is to quantify kinematic fields via digital volume correlation (or DVC [19, 20]). This additional piece of information can be used to analyze fatigue crack propagation [21, 22] or to validate finite element simulations of cracked samples [3].
Very few studies deal with the identification of material parameters based on volumetric analyses. A first reason is related to the measurement uncertainties and biases that are usually higher than those typically encountered in 2D analyses [23, 24]. Second, the computational environment needed to solve these inverse problems is generally very intrusive (e.g., constitutive equation gap method [25–27], virtual fields method for nonlinear constitutive laws [28], equilibrium gap method [29]). Third, nonlinear constitutive equations require spatiotemporal (i.e., 4D) analyses to be considered, which are both experimentally and computationally very demanding. The aim of the present work is to show the feasibility of such a framework to study the elastoplastic behavior of spheroidal graphite (SG) cast iron.
One of the goals of the present study is to achieve full integration of sensor and measurement data in the developed numerical procedures [1]. This type of analysis corresponds to socalled integrated approaches [30, 31], which were developed up to now mostly with 2D images of sample surfaces [32–36] in which material parameters are directly measured from image registrations. A first extension to 3D images was recently proposed to analyze an indentation experiment on plasterboard monitored via tomography and DVC [37]. When material parameters are sought, the corresponding kinematic and static sensitivities [38] are needed. Finite element simulations are used in particular to generate a set of kinematic fields that are further used for DVC purposes. It was shown that nonintrusive spatiotemporal schemes can be used for 2D images [39], whereby the incorporation of time leads to 3D integration. In a similar spirit but extended to 3D (spatial) images, the following analysis corresponds to a first step toward 4D integration.
In “Methods” section, the experimental configuration is presented. DVC analyses based upon discretizations with 4noded tetrahedra (i.e., T4DVC) are used to measure kinematic fields without any mechanical regularization. From the acquired scan in the reference configuration, a mesh is adapted to the actual shape of the sample. From the kinematic measurements, the measured nodal displacements of the top and bottom faces of the region of interest (ROI) become the boundary conditions of the integrated scheme in which the material parameters are sought. A nonintrusive setting is developed. The previous framework is applied to analyze the described tensile test on spheroidal graphite cast iron to determine unknown boundary conditions, and to identify elastoplastic parameters (“Results and discussion” section). Uncertainty quantifications are performed in addition to convergence analyses in which the mesh quality is assessed as well.
Methods
In situ mechanical test
A complete scan of the sample corresponds to a \(360^{\circ }\) rotation along its vertical axis, during which 1000 radiographs are acquired with a definition of \(1280\times 1860\) pixels. Each scan lasted less than 40 min. The physical voxel size is 6.4 \(\upmu \)m, and the reconstructed volume is encoded with 8bit deep gray levels. This test was analyzed via global approaches to DVC with regular meshes made of 8noded cubes to reveal the damage micromechanism (i.e., nodule debonding) with the correlation residuals [41]. In the following, the underlying macroscopic behavior will be studied.
Digital volume correlation
DVC is used to measure 3D displacement fields in the bulk of samples [42, 43]. In its incremental version, DVC consists of registering two volumes by evaluating displacement fields that yield the best possible match. In the early developments, small zones of interest or ZOIs (i.e., small interrogation volumes [19, 20, 44, 45]) in the considered region of interest (ROI) are registered. This type of approach is nowadays referred to as local (i.e., the only information that is kept is the mean displacement assigned to each analyzed ZOI center) and incremental since only two volumes are considered. Each registration is performed over the ZOI with various kinematic hypotheses, possibly accounting for its warping. This type of analysis is not considered herein since it does not offer an interpretation of the kinematic measurement that is “congruent” with that of FE analyses.
Global and incremental approaches, which appeared more recently [46], consist of performing a registration over the whole ROI. The kinematic bases are then defined over the whole ROI (i.e., RayleighRitz formulations) or over a discretized ROI (i.e., Galerkin formulations). The fact that continuity is enforced allows the measurement uncertainty to be reduced in comparison with the same discretization in which the nodes are not shared [47]. Another important advantage of global DVC in comparison with local DVC is its direct link with numerical simulations of mechanical problems [3, 21]. When dealing with digital volumes it is natural to consider regular meshes made of 8noded cubes (i.e., C8DVC [46]) even though unstructured meshes may also be considered [48]. In the following, unstructured meshes made of 4noded tetrahedra are chosen to allow for a more faithful description of external surfaces (“Mesh of the ROI” section) in comparison with structured C8 meshes.
Equation (1) expresses the underlying assumption of gray level conservation for each considered voxel \({\mathbf {x}}\) of the ROI. This least squares problem is nonlinear. It is solved with an iterative Gauss–Newton scheme [51] in which the stationarity condition is recast in the following variational formulation [52].
Problem 1
Mesh of the ROI
To perform T4DVC analyses, the ROI of the sample needs to be meshed based on its true geometry. This type of procedure usually requires the acquired scan to be processed with mathematical morphology and ad hoc procedures [55–57]. When multiphase microstructures are studied, this operation becomes more complex to make sure that the mesh is compatible with the morphology of the phases. For instance, adaptive levelset method can be used [58]. In the present case, the simulations will not be performed at the microstructural scale but using a macroscopic description of the material behavior. Consequently, the mesh will not be adapted to the underlying microstructure. However, it needs to comply with the external surface of the sample.
A digital image correlation procedure is implemented to adjust a regular mesh made of 4noded quadrilaterals to the actual surface of the sample [59]. Different transverse sections are considered. From these registrations, a 3D discretization made of 8noded hexahedra (H8) is obtained. It is subsequently converted into a T4 discretization by subdividing each H8 element into six T4 elements of equal volume. The two meshes are exactly nodeequivalent, i.e., the number of nodes and the node positions are unchanged.
The elementary matrices \([{\mathbf {M}}^e]\) are then assembled to evaluate the global DVC matrix \([{\mathbf {M}}]\) and the corresponding covariance matrix \([{\mathbf {Cov}}_{\varvec{\upsilon }}]\). In the assembly process, the nodal displacements will now be evaluated over more than one element (i.e., all the elements sharing the considered node). Consequently, the number of voxels \(\ell _g^3\) considered for the evaluation of the nodal displacements will become larger than that of each individual element. For a regular mesh made of C8 elements, the corresponding volume is equal to \((2\ell )^3\) [47]. With the chosen procedure to transform H8 meshes into T4 meshes, inner T4 nodes are shared by 24 elements. This discussion shows that when using a T4 mesh with all its connectivities enforced, a reduction factor of the order of \(2\sqrt{6}\) is to be expected on the standard displacement uncertainty in comparison with the same discretization in which each T4 element is considered independently. An additional gain is to be expected thanks to the overall continuity requirement of the displacement field [47, 54]. Last, if the same element sizes as defined herein are considered for T4 and C8 meshes, the measurement uncertainties of the former are expected to be \(2\sqrt{3}\) times lower than the latter.
The present results show that the finer the mesh (i.e., the smaller \(\ell \)), the higher the measurement uncertainties. From a purely metrological point of view, finer meshes are to be discarded since they will lead to very high measurement uncertainties or even to singular DVC matrices that will not lead to trustworthy results, if any is obtained. However, on the mechanical side of the problem, coarse meshes may degrade the quality of the simulations.
In the context of the Finite Element Method, verification procedures have been introduced since the late 1970s [60–62]. They enable a posteriori discretization error estimates to be computed and mesh adaptivity to be driven. In this work, a verification procedure based on the concept of constitutive relation error (CRE) [61] is implemented. It leads to robust error estimates for both linear and nonlinear timedependent problems [63–66]. The CRE concept, which is particularly suited to Computational Mechanics models, is based on duality and rests on a simple idea, namely, after constructing socalled admissible fields satisfying all equations of the model except (part of) the constitutive law, the residual associated with the constitutive relationship is evaluated.

constitutive equations related to the free energy (i.e., balance equations, kinematic constraints, initial conditions, state laws) that define the concept of admissibility;

constitutive equations related to the dissipation (i.e., growth laws).
The uncertainty analysis and the verification steps are not necessarily compatible (i.e., the discretization used for DVC purposes requires not too fine meshes to be considered). Therefore a compromise needs to be found if the two approaches are performed sequentially, otherwise the two meshes will be different and the interpolation quality from one mesh to another will not be checked against experimental data. In the following, it will be shown that integrated approaches will no longer require this tradeoff to be implemented since both steps will be performed in a unique calculation in which the mesh is no longer a limitation since the number of unknowns will be drastically reduced.
4D mechanical correlation
Up to now, the implicit regularization of the correlation procedure is related to the definition of ZOIs (in a local approach) or elements (in a global approach) and the displacement interpolations therein since the inversion problem cannot be solved at the voxel scale. Incremental FEbased DVC can be seen as a strong regularization of local incremental DVC.
An additional way of regularizing correlation procedures is to consider mechanical admissibility. This is, for instance, possible by requiring that the measured displacement field minimize a weighted sum of the DVC objective functional and the equilibrium gap by assuming, say, an elastic behavior [72]. These socalled mechanicsaided (i.e., regularized) DVC procedures enable the high frequency kinematic components that are not mechanically admissible to be filtered out [47]. This penalization acts as mechanical filter in registration procedures. Because mechanical admissibility controls the high spatial frequencies, fine meshes (that cannot be considered in standard DVC calculations) become accessible [41, 72]. They also allow registrations to be performed at the voxel level [73]. In this extreme situation, implementations on Graphics Processing Units (GPUs) enable very large computations to be run [47]. One alternative route consists in considering PGDbased DVC [74].
Since measured boundary conditions drive the finite element simulations, a first T4DVC analysis is required. It uses the same MEX files as IDVC (Fig. 2b). However, there is no need for any mechanical finite element code since no mechanical regularization is performed herein via T4DVC. If the mesh used in IDVC is finer than that used in T4DVC, the measured displacement fields are interpolated by using the T4 shape functions.
Results and discussion
In situ mechanical test
Mesh of the ROI
Different meshes studied in this work and their corresponding quality with measured and linearly interpolated boundary conditions
Density  DOF #  \(\varvec{\ell }\) (voxels)  \( \varvec{\ell }_{\varvec{g}}\) (voxels)  \(\overline{{\varvec{\theta }}}_{\varvec{mea}}\) (%)  \(\overline{{\varvec{\theta }}}_{\varvec{Q3}}\) (%) 

1  792  37  89  174  35 
2  4851  18  48  134  19 
3  14,880  12  33  104  13 
4  33,579  9  25  85  11 
5  63,648  7  21  73  9 
Finer meshes will also be considered in integrated analyses (Table 1). For each mesh, two different equivalent lengths are given. The first one corresponds to the cubic root of the mean volume of the T4 elements. This quantity \(\ell \) was introduced for the evaluation of the measurement uncertainty when each T4 element is considered independently. However, to define the spatial resolution associated with displacement measurements of global DVC a second length is also evaluated. It corresponds to the cubic root of the mean number of voxels considered for the nodal displacement measurement. This length \(\ell _g\) defines the spatial resolution of the measurement technique. Table 1 shows that \(\ell _g\) is less than three times the element size \(\ell \).
Measured displacement fields
Before studying the displacement field between two scans, it is important to evaluate the level of uncertainty attached to the natural texture of the 3D images. Since the studied SG cast iron has a coarse microstructure it is necessary to make a compromise between the spatial resolution and displacement uncertainties [41]. For C8DVC it is found that the smallest element size \(\ell \) is 32 voxels to reach convergence. The standard displacement uncertainty is equal to 0.2 voxel. With regularized C8DVC [47, 72], 16voxel elements can be considered with a regularization length \(\ell _m=16\) voxels. The corresponding standard displacement uncertainty is again equal to 0.2 voxel. When the T4DVC code is run, the standard displacement uncertainty is equal to 0.07 voxel when \(\ell =37\) voxels. A gain of about a factor 3 is observed in the present case, which is close to the a priori estimate (i.e., \(2\sqrt{3}\approx 3.5\)) obtained from the uncertainty analysis (see “Mesh of the ROI” section). The root mean square (RMS) residual is equal to 6.5 gray levels, which is an estimate of acquisition and reconstruction noise (i.e., \(\sqrt{2}\gamma _I\)).
Integrated DVC
In the sequel all reported results will consider displacement fields that are mechanically admissible in a finite element sense. Consequently, the first question to address is the quality of the finite element simulations. The next issues will be related to the analysis of the experiment per se. It is worth remembering that when finite element simulations are carried out, the material model and the parameters need to be known for cast iron. Two sets of parameters (see Table 2) will be considered for Ludwik’s law (“Mesh of the ROI” section). The first one was obtained via weighted FEMUUF [41] to analyze a cyclic tensile test on the same material at the macroscale. The second one was determined from the analysis of a standard monotonic tensile test [40].
Verification
Different sets of material parameters for the studied cast iron
Set  \(\varvec{E}\) (GPa)  \(\varvec{\nu }\)  \(\varvec{\sigma }_{\varvec{y}}\) (MPa)  \(\varvec{K}\)  \(\varvec{n}\) 

1\(^{\text {a}}\)  158  0.28  210  1300  0.44 
2\(^{\text {b}}\)  156  0.33  290  1260  0.62 
Effect of the interpolation of boundary conditions on mesh quality and identification results for mesh density 1
Interpolation c  \(\overline{{\varvec{\theta }}}\) (%)  RMS error (voxels)  \(\varvec{\chi }_{\varvec{F}}\)  \(\varvec{\chi }_{\varvec{I}}\) 

Q1  42  2.55  14.7  5.02 
Q3  35  0.23  14.4  1.40 
Q4  45  0.22  14.3  1.40 
Q6  117  0.21  14.3  1.46 
Q8  117  0.20  14.2  1.44 
None  174  –  14.2  1.45 
To evaluate the effect of boundary conditions, the measured fields are fitted with low order polynomials on both surfaces where they are prescribed. Several interpolations, which are referred to as Q\(\#\), are defined by the number of monomials picked in the list \(\{1,x,y,xy,x^2,y^2,x^2y,xy^2\}\), where x and y denote the two spatial coordinates over which the interpolation is performed. Therefore, Q1 corresponds to constant interpolation, Q3 to linear interpolation, and Q6 to quadratic interpolation. For a linear interpolation, there is a clear gain in terms of discretization errors when compared to the initial results (Fig. 7). Table 3 shows that the RMS interpolation error between the measured displacements and their interpolations is very high for the uniform case (i.e., more than 36 times the standard displacement uncertainty). This is due to the fact that rigid body rotations are not accounted for. The interpolation errors for the other descriptions are close and significantly lower than in the previous case (i.e., of the order of 3 times the standard displacement uncertainty).
Effect of displacement amplitudes on static residual for a mesh density of 1 and the first set of material parameters
\(\varvec{\alpha }\)  \(\varvec{\gamma }_{{\varvec{\alpha }}}\)  \(\varvec{\chi }_{\varvec{F}}\) 

0.000  –  26 
1.222  –  20 
0.785  \(5 \,\times \,10^{3}\)  18 
Boundary conditions
Since the reference scan was acquired for a load level equal to 165 N (Fig. 1), the zero load configuration is not known. This lack of information is of no consequence when standard DVC analyses are run (i.e., C8DVC and T4DVC) or even regularized DVC. Conversely, when integrated approaches are performed in which the constitutive law is nonlinear, the zero load configuration has to be either known or estimated. In the present case, the second route has to be followed.
For a mesh density of 1 and the first set of material parameters, Table 4 shows the effect of different values of the amplitude \(\alpha \) on the static residual \(\chi _F\). There is a clear influence of \(\alpha \) on \(\chi _F\), and more importantly, the standard uncertainty \(\gamma _{\alpha }\) is very low in comparison with the reported level of \(\alpha \). The fact that \(\chi _F\) is significantly larger than unity is an indication of model errors. This first analysis allows the load residuals to be decreased from 26 (when \(\alpha =0\)) to 18 when \(\alpha \) is optimized via the load residuals, which corresponds to a 30% decrease.
Effect of mesh density
Since T4DVC results are also available for the coarsest mesh, the correlation residuals are compared to those of IDVC with different mesh densities. In the present case, the history of instantaneous dimensionless registration residuals is again analyzed (Fig. 5). Contrary to T4DVC it is observed that the registration quality degrades as the scan number increases, i.e., more yielding has occurred. The fact that this trend is identical for all five mesh densities is an indication of a material model error and not a discretization error. To obtain the results discussed in this section the computation time ranges from a few minutes for mesh density 1 to two hours for the finest mesh on a workstation (with 2.6GHz Intel Xeon E52650 processor). Let us emphasize that this computation time remains small as compared to the duration of the experiment, and even smaller if preparation time is considered.
Comparison of two sets of material parameters
Table 5 shows the load and registration residuals for the two sets of material parameters. For the load residual, the second set of material parameters is better than the first one. There is a clear sensitivity of \(\chi _F\) to the two sets. It is worth noting that the finer the meshes the sightly higher the load residuals in both cases. For the registration residuals, the first set of material parameters is better than the second one. This is also true for the global residual \(\chi \). These results indicate that even though the number of scans is very small (i.e., less than ten), the two residuals are sensitive to the material parameters.
In both cases, it is worth noting that the load, registration and total residuals are virtually unchanged for the last two densities. All of them have converged so that density 4 can be considered as the finest mesh needed for identification purposes.
Identification of material parameters
Static and registration residuals for the two sets of material parameters and different mesh densities
Set 1  \({\varvec{\chi }}_{\varvec{F}}\)  \(\varvec{\chi }_{\varvec{I}}\) 

Density 1  18.3  1.47 
Density 2  18.5  1.45 
Density 3  18.6  1.44 
Density 4  18.6  1.43 
Density 5  18.6  1.43 
Set 2  \({\varvec{\chi }}_{\varvec{F}}\)  \({\varvec{\chi }}_{\varvec{I}}\) 

Density 1  14.9  1.51 
Density 2  15.4  1.51 
Density 3  15.6  1.50 
Density 4  15.6  1.50 
Density 5  15.6  1.50 
Material parameters determined via FEMUF. Static, registration and global residuals for different mesh densities
Density  \(\varvec{\sigma }_{\varvec{y}}\) (MPa)  \(\varvec{K}\) (MPa)  \(\varvec{n}\)  \(\varvec{\chi }_{\varvec{F}}\)  \(\varvec{\chi }_{\varvec{I}}\) 

1  \(250\pm 6\)  \(1300\pm 109\)  \(0.55\pm 0.03\)  14.17  1.49 
2  \(258\pm 7\)  \(1300\pm 160\)  \(0.57\pm 0.05\)  14.20  1.48 
3  \(257\pm 2\)  \(1300\pm 40\)  \(0.57\pm 0.01\)  14.27  1.46 
4  \(256\pm 5\)  \(1300\pm 83\)  \(0.57\pm 0.03\)  14.30  1.46 
5  \(256\pm 7\)  \(1300\pm 140\)  \(0.57\pm 0.04\)  14.31  1.45 
Table 6 shows the results of the identification via FEMUF. The values of the optimal parameters are virtually identical for the four mesh densities. In the present case, the quality of the mesh has only a minor influence on the identification results. The simple geometry of the studied sample may explain such results. As expected from the sensitivity analysis, the maximum uncertainties are of the order of 10% at most. Two out of three parameters have varied significantly from the initial guess (i.e., from the first set of parameters) whereas the hardening modulus has remained virtually unchanged. The static residual has been lowered to 14.2, which is still a very high value in comparison with acquisition noise. However, it is lower than what was observed for the two sets of material parameters (Table 5).
In the following, 4D registration is coupled with load measurements to identify material parameters. If a Bayesian framework is followed, the weight associated with load data is very small and the results are identical to those in which no load data is accounted for. As a consequence, the load residual reaches very high values (i.e., more than 100), which is not acceptable. It was therefore chosen to give equal weight to all the static and kinematic data by minimizing \(\chi ^2 = 1/2(\chi ^2_I+\chi ^2_F)\) instead [39]. Table 3 shows that the residuals at convergence for mesh density 1 are close to those observed in FEMUF for the same density. The material parameters are not significantly altered in comparison to the identified values reported in Table 6.
Sensitivity to boundary conditions
The verification procedure has shown that the errors are located close to the boundaries where the measured displacements are prescribed. To evaluate the effect of the prescribed displacement interpolation on the identification results, a first analysis is performed for the coarsest mesh density. Since the boundary conditions have been interpolated, they may induce changes on the identified material parameters. Figure 12 shows the relative changes of the three identified material parameters for the different interpolations. The reference set of parameters is that when no interpolation is performed. Except for the uniform interpolation, the total residual varies less than 1%. The registration and load residuals are also similar for all analyzed cases but the uniform interpolation (Table 3). The latter is clearly not acceptable in terms of registration residuals. From this analysis, the linear interpolation (Q3) is the best compromise between mesh quality and identification results when mesh density 1 is studied.
In the following, only the linear interpolation is studied for the different mesh densities. When the load and registration residuals are computed for all these cases with the same material parameters (identified when the measured boundary conditions are considered), the registration residuals are identical and the load residuals slightly increase with the mesh density as already observed (see Fig. 11). These results indicate that the identification procedure does not need such fine meshes. One of the reasons of this result is likely to be related to the fact that the load residuals are still very high (i.e., there is a significant model error). Figure 13 summarizes the identification results for the finest mesh (i.e., leading to the smallest CRE error). The gray level residuals are very low in comparison with the reference microstructure (see Fig. 3b). This is an additional proof that the registration is successful not only globally but also locally. When the measured load history is compared to the corresponding predictions, higher differences are observed. This result is to be expected from the global residuals, which are significantly higher than the measurement uncertainties.
Conclusions
It has been shown herein that full integration of sensor and measurement data can be achieved in numerical procedures via socalled 4D mechanical correlation. In the present case, load measurements are combined with reconstructed volumes thanks to computed tomography. Local and global residuals have been constructed in which each measured or predicted result is compared to the corresponding noise level. In 4D registrations, the mechanical finite element code was utilized in a nonintrusive way to generate elastoplastic fields. This type of approach is very generic and can be deployed under very different conditions.
Within the proposed framework, uncertainty quantifications have been carried out for the measured kinematic degrees of freedom and the sought parameters. For incremental correlation procedures, it is shown that the finer the mesh, the more uncertain the measured quantities. This trend is opposite to that of verification procedures where refined meshes better capture mechanical fields. This limitation can be overcome by considering kinematic fields that are mechanically admissible, i.e., to integrated DVC (i.e., 4D mechanical correlation). Consequently, convergence studies can be conducted when dealing with experimental data.
For the analyzed tensile test on nodular graphite cast iron, unknown boundary conditions and plastic parameters have been identified. By performing sensitivity analyses, it is shown that only some of the parameters can be identified for a given uncertainty. This is due to the fact that only a very limited number of scans is available for the reported analyses. However, even under these very difficult conditions, it is possible to carry out identifications with rather low uncertainties. Further, the verification analyses also show that most of the errors are due to the fact that measured (i.e., noisy) boundary conditions are prescribed on the numerical model. Yet, the identification results are virtually insensitive to the quality of the mesh because the geometry of the tested sample is very simple. In the present analyses the boundary conditions were measured and directly applied to the finite element model to compute kinematic bases. However, the boundary conditions may themselves become part of the set of unknowns. The present framework is suitable to such extensions.
Alternative 4D routes may be considered as well. For instance, by resorting to projectionbased registration procedures [48], it is possible to reduce the number of projections needed to evaluate 3D kinematic fields. When extended to (many) more time steps, it would allow the experimentalist not to interrupt the test and acquire the radiographs on the fly. A truly 4D formulation would be required in which reconstruction and registration procedures would be coupled with mechanically admissible fields.
Another direction of progress can be envisioned thanks to the scale at which microtomography is performed, namely, the study of the mechanical behavior at the microscale. For the studied material, this would allow the behavior of the ferrite/pearlite matrix to be analyzed in conjunction with the debonding of the interface between the brittle nodules and their surrounding matrix. The same framework as that introduced herein can be used. However, the meshes will needed to be significantly finer and adapted to the microstructure [58].
Declarations
Author's contributions
AB and ZT carried out the insitu test. FP and LC performed the verification analyses. The computational framework (Correli 3.0) was implemented by HL, FM and JN. FH wrote all the IDVC routines within the Correli 3.0 framework. SR and FH introduced the theoretical background and performed the uncertainty quantifications. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by BPI France (DICCIT project) and by the French “Agence Nationale de la Recherche” through the “Investissements d’avenir” program (ANR10EQPX37 MATMECA grant). ZT thanks Campus France for supporting his stay at LMT through an Eiffel scholarship.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Oden JT, Belytschko T, Fish J, Hughes TJR, Johnson C, Keyes D, Laub A, Petzold L, Srolovitz D, Yip S. Simulationbased engineering sciences. Final report, NFS. 2006. http://www.nsf.gov/pubs/reports/sbes_final_report
 Avril S, Bonnet M, Bretelle AS, Grédiac M, Hild F, Ienny P, Latourte F, Lemosse D, Pagano S, Pagnacco E, Pierron F. Overview of identification methods of mechanical parameters based on fullfield measurements. Exp Mech. 2008;48(4):381–402.View ArticleGoogle Scholar
 Rannou J, Limodin N, Réthoré J, Gravouil A, Ludwig W, Baïetto MC, Buffière JY, Combescure A, Hild F, Roux S. Three dimensional experimental and numerical multiscale analysis of a fatigue crack. Comp Meth Appl Mech Eng. 2010;199:1307–25.MATHView ArticleGoogle Scholar
 Baruchel J, Buffière JY, Maire E, Merle P, Peix G, editors. Xray tomography in material sciences. Paris: Hermès Science; 2000.Google Scholar
 Salvo L, Cloetens P, Maire E, Zabler S, Blandin JJ, Buffiere JY, Ludwig W, Boller E, Bellet D, Josserond C. Xray microtomography an attractive characterisation technique in materials science. Nucl Instr Meth Phys Res B. 2003;200:273–86.View ArticleGoogle Scholar
 Desrues J, Viggiani G, Bésuelle P, editors. Advances in Xray tomography for geomaterials. London: Wiley / ISTE; 2006.Google Scholar
 Stock SR. Recent advances in XRay microtomography applied to materials. Int Mat Rev. 2008;53(3):129–81.View ArticleGoogle Scholar
 Maire E, Withers PJ. Quantitative Xray tomography. Int Mat Rev. 2014;59(1):1–43.View ArticleGoogle Scholar
 Desrues J, Chambon R, Mokni M, Mazerolle MF. Void ratio evolution inside shear bands in triaxial sand specimens studied by computed tomography. Géotechnique. 1996;46(3):529–46.View ArticleGoogle Scholar
 Guvenilir A, Breunig TM, Kinney JH, Stock SR. Direct observation of crack opening as a function of applied load in the interior of a notched tensile sample of AlLi 2090. Acta Mater. 1997;45(5):1977–87.View ArticleGoogle Scholar
 Buffière JY, Maire E, Cloetens P, Lormand G, Fougères R. Characterisation of internal damage in a MMCp using Xray synchrotron phase contrast microtomography. Acta Mater. 1999;47(5):1613–25.View ArticleGoogle Scholar
 Buffière JY, Maire E, Adrien J, Masse JP, Boller E. In situ experiments with X ray tomography: an attractive tool for experimental mechanics. Exp Mech. 2010;50(3):289–305.View ArticleGoogle Scholar
 Babout L, Maire E, Buffière JY, Fougères R. Characterisation by XRay computed tomography of decohesion, porosity growth and coalescence in model metal matrix composites. Acta Mater. 2001;49(11):2055–63.View ArticleGoogle Scholar
 Babout L, Maire E, Fougères R. Damage initiation in model metallic materials: Xray tomography and modelling. Acta Mater. 2004;52:2475–87.View ArticleGoogle Scholar
 Pyzalla A, Camin B, Buslaps T, Di Michiel M, Kaminski H, Kottar A, Pernack A, Reimers W. Simultaneous tomography and diffraction analysis of creep damage. Science. 2005;308(5718):92–5.View ArticleGoogle Scholar
 Isaac A, Sket F, Reimers W, Camin B, Sauthoff G, Pyzalla AR. In situ 3D quantification of the evolution of creep cavity size, shape, and spatial orientation using synchrotron Xray tomography. Mat Sci Eng A. 2008;478(1–2):108–18.View ArticleGoogle Scholar
 Huppmann M, Camin B, Pyzalla AR, Reimers W. Insitu observation of creep damage evolution in AlAl\(_2\)O\(_3\) MMCs by synchrotron Xray microtomography. Int J Mat Res. 2010;101:372–9.View ArticleGoogle Scholar
 Borbély A, Dzieciol K, Sket F, Isaac A, di Michiel M, Buslaps T, KaysserPyzalla AR. Characterization of creep and creep damage by insitu microtomography. JOM. 2011;63(7):78–84.View ArticleGoogle Scholar
 Bay BK, Smith TS, Fyhrie DP, Saad M. Digital volume correlation: threedimensional strain mapping using Xray tomography. Exp Mech. 1999;39:217–26.View ArticleGoogle Scholar
 Smith TS, Bay BK, Rashid MM. Digital volume correlation including rotational degrees of freedom during minimization. Exp Mech. 2002;42(3):272–8.View ArticleGoogle Scholar
 Limodin N, Réthoré J, Buffière JY, Gravouil A, Hild F, Roux S. Crack closure and stress intensity factor measurements in nodular graphite cast iron using 3D correlation of laboratory X ray microtomography images. Acta Mat. 2009;57(14):4090–101.View ArticleGoogle Scholar
 Limodin N, Réthoré J, Buffière JY, Hild F, Roux S, Ludwig W, Rannou J, Gravouil A. Influence of closure on the 3D propagation of fatigue cracks in a nodular cast iron investigated by Xray tomography and 3D volume correlation. Acta Mat. 2010;58(8):2957–67.View ArticleGoogle Scholar
 Limodin N, Réthoré J, Adrien J, Buffière JY, Hild F, Roux S. Analysis and artifact correction for volume correlation measurements using tomographic images from a laboratory Xray source. Exp Mech. 2011;51(6):959–70.View ArticleGoogle Scholar
 Morgeneyer TF, Helfen L, Mubarak H, Hild F. 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: an initial feasibility study. Exp Mech. 2013;53(4):543–56.View ArticleGoogle Scholar
 Allix O, Feissel P, Nguyen HM. Identification strategy in the presence of corrupted measurements. Eng Comput. 2005;22(5–6):487–504.MATHView ArticleGoogle Scholar
 Feissel P, Allix O. Modified constitutive relation error identification strategy for transient dynamics with corrupted data: the elastic case. Comput Meth Appl Mech Eng. 2007;196(13/16):1968–83.MATHView ArticleGoogle Scholar
 Latourte F, Chrysochoos A, Pagano S, Wattrisse B. Elastoplastic behavior identification for heterogeneous loadings and materials. Exp Mech. 2008;48(4):435–49.View ArticleGoogle Scholar
 Pierron F, Grédiac M. The virtual fields method. New York: Springer; 2012.MATHView ArticleGoogle Scholar
 Claire D, Hild F, Roux S. A finite element formulation to identify damage fields: the equilibrium gap method. Int J Num Meth Eng. 2004;61(2):189–208.MATHView ArticleGoogle Scholar
 Hild F, Roux S. Digital image correlation: from measurement to identification of elastic propertiesa review. Strain. 2006;42:69–80.View ArticleGoogle Scholar
 Roux S, Hild F. Stress intensity factor measurements from digital image correlation: postprocessing and integrated approaches. Int J Fract. 2006;140(1–4):141–57.MATHView ArticleGoogle Scholar
 Leclerc H, Périé JN, Roux S, Hild F. In: Gagalowicz A, Philips W, editors. Integrated digital image correlation for the identification of mechanical properties, vol. LNCS 5496. Berlin: Springer; 2009. p. 161–71.Google Scholar
 Réthoré J. A fully integrated noise robust strategy for the identification of constitutive laws from digital images. Int J Num Meth Eng. 2010;84(6):631–60.MATHView ArticleGoogle Scholar
 Réthoré J. Muhibullah, Elguedj T, Coret M, Chaudet P, Combescure A. Robust identification of elastoplastic constitutive law parameters from digital images using 3D kinematics. Int J Solids Struct. 2013;50(1):73–85.View ArticleGoogle Scholar
 Dufour JE, Hild F, Roux S. Shape, displacement and mechanical properties from isogeometric multiview stereocorrelation. J Strain Anal. 2015;50(7):470–87.View ArticleGoogle Scholar
 Neggers J, Hoefnagels JPM, Geers MGD, Hild F, Roux S. Timeresolved integrated digital image correlation. Int J Num Meth Eng. 2015;203(3):157–82.MathSciNetView ArticleGoogle Scholar
 Bouterf A, Roux S, Hild F, Adrien J, Maire E. Digital volume correlation applied to Xray tomography images from spherical indentation tests on lightweight gypsum. Strain. 2014;50(5):444–53.View ArticleGoogle Scholar
 Tarantola A. Inverse problems theory. Methods for data fitting and model parameter estimation. Southampton: Elsevier Applied Science; 1987.MATHGoogle Scholar
 Mathieu F, Leclerc H, Hild F, Roux S. Estimation of elastoplastic parameters via weighted FEMU and integratedDIC. Exp Mech. 2015;55(1):105–19.View ArticleGoogle Scholar
 Tomičević Z. Identification of the mechanical properties of nodular graphite cast iron via multiaxial tests. PhD thesis. 2015.Google Scholar
 Tomičević Z, Kodvanj J, Hild F. Characterization of the nonlinear behavior of nodular graphite cast iron via inverse identification. Analysis of uniaxial tests. Europ J Mech A/Solids. 2016;59:140–54.Google Scholar
 Sutton MA. Computer visionbased, noncontacting deformation measurements in mechanics: a generational transformation. Appl Mech Rev. 2013;65(AMR–13–1009):050802.Google Scholar
 Sutton MA, Hild F. Recent advances and perspectives in digital image correlation. Exp Mech. 2015;55(1):1–8.View ArticleGoogle Scholar
 Bornert M, Chaix JM, Doumalin P, Dupré JC, Fournel T, Jeulin D, Maire E, Moreaud M, Moulinec H. Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l’analyse des matériaux et des structures. Inst Mes Métrol. 2004;4:43–88.Google Scholar
 Verhulp E, van Rietbergen B, Huiskes R. A threedimensional digital image correlation technique for strain measurements in microstructures. J Biomech. 2004;37(9):1313–20.View ArticleGoogle Scholar
 Roux S, Hild F, Viot P, Bernard D. Three dimensional image correlation from XRay computed tomography of solid foam. Comp Part A. 2008;39(8):1253–65.View ArticleGoogle Scholar
 Leclerc H, Périé JN, Hild F, Roux S. Digital volume correlation: What are the limits to the spatial resolution? Mech Indust. 2012;13:361–71.View ArticleGoogle Scholar
 Leclerc H, Roux S, Hild F. Projection savings in CTbased digital volume correlation. Exp Mech. 2015;55(1):275–87.View ArticleGoogle Scholar
 Kak AC, Slaney M. Principles of computerized tomographic imaging, Society of Industrial and Applied Mathematics. 2001.Google Scholar
 Hild F, Fanget A, Adrien J, Maire E, Roux S. Three dimensional analysis of a tensile test on a propellant with digital volume correlation. Arch Mech. 2011;63(5–6):1–20.Google Scholar
 Lucas BD, Kanade T. An iterative image registration technique with an application to stereo vision. In: 7th international joint conference on artificial intelligence. 1981. p. 674–79.Google Scholar
 Fedele R, Galantucci L, Ciani A. Global 2D digital image correlation for motion estimation in a finite element framework: a variational formulation and a regularized, pyramidal, multigrid implementation. Int J Num Meth Eng. 2013;96(12):739–62.View ArticleGoogle Scholar
 Hild F, Roux S. In: Rastogi P, Hack E, editors. Digital image correlation. Weinheim: WileyVCH; 2012. p. 183–228.Google Scholar
 Hild F, Roux S. Comparison of local and global approaches to digital image correlation. Exp Mech. 2012;52(9):1503–19.View ArticleGoogle Scholar
 Maire E, Fazekas A, Salvo L, Dendievel R, Youssef S, Cloetens P, Letang JM. Xray tomography applied to the characterization of cellular materials related finite element modeling problems. Compos Sci Tech. 2003;63(16):2431–43.View ArticleGoogle Scholar
 Youssef S, Maire E, Gaertner R. Finite element modelling of the actual structure of cellular materials determined by xray tomography. Acta Mater. 2005;53(3):719–30.View ArticleGoogle Scholar
 Zhang Y, Bajaj C, Sohn BS. 3D finite element meshing from imaging data. Comput Meth Appl Mech Eng. 2005;194(48–49):5083–106.MATHView ArticleGoogle Scholar
 Shakoor M, Bouchard PO, Bernacki M. An adaptive levelset method with enhanced volume conservation for simulations in multiphase domains. Int J Num Meth Eng. 2016. (accepted for publication).Google Scholar
 Gras R, Leclerc H, Hild F, Roux S, Schneider J. Identification of a set of macroscopic elastic parameters in a 3d woven composite: uncertainty analysis and regularization. Int J Solids Struct. 2015;55:2–16.View ArticleGoogle Scholar
 Babus̆ka I, Rheinboldt WC. Error estimates for adaptive finite element computation. SIAM J Num Anal. 1978;15(4):736–54.Google Scholar
 Ladevèze P, Leguillon D. Error estimate procedure in the finite element method and applications. SIAM J Num Anal. 1983;20(3):485–509.MathSciNetMATHView ArticleGoogle Scholar
 Zienkiewicz OC, Zhu JZ. A simple error estimator and adaptive procedure for practical engineerng analysis. Int J Num Meth Eng. 1987;24(2):337–57.MathSciNetMATHView ArticleGoogle Scholar
 Ladevèze P, Moës N. A new a posteriori error estimation for nonlinear timedependent finite element analysis. Comp Meth Appl Mech Eng. 1998;157:45–68.MathSciNetMATHView ArticleGoogle Scholar
 Destuynder P, Métivet B. Explicit error bounds in a conforming finite element method. Math Comput. 1999;68(288):1379–96.MathSciNetMATHView ArticleGoogle Scholar
 Ladevèze P, Pelle J. Mastering calculations in linear and nonlinear mechanics. New York: Springer; 2004.MATHGoogle Scholar
 Chamoin L, Diez P. Verifying calculations, forty years on: an overview of classical verification techniques for FEM simulations. Switzerland: Springer; 2015.MATHGoogle Scholar
 Ludwik P. Elemente der Technologischen Mechanik. Leipzig: Verlag Von Julius Springer; 1909.MATHView ArticleGoogle Scholar
 Germain P, Nguyen QS, Suquet P. Continuum thermodynamics. ASME J Appl Mech. 1983;50:1010–20.View ArticleGoogle Scholar
 Lemaitre J, Chaboche JL. Mechanics of solid materials. Cambridge: Cambridge University Press; 1990.MATHView ArticleGoogle Scholar
 Prager W, Synge JL. Approximation in elasticity based on the concept of functions spaces. Q Appl Math. 1947;5:261–9.MathSciNetMATHGoogle Scholar
 Pled F, Chamoin L, Ladevèze P. On the techniques for constructing admissible stress fields in model verification: performances on engineering examples. Int J Num Meth Eng. 2011;88(5):409–41.MATHView ArticleGoogle Scholar
 TaillandierThomas T, Roux S, Morgeneyer TF, Hild F. Localized strain field measurement on laminography data with mechanical regularization. Nucl Inst Meth Phys Res B. 2014;324:70–9.View ArticleGoogle Scholar
 Leclerc H, Périé JN, Roux S, Hild F. Voxelscale digital volume correlation. Exp Mech. 2011;51(4):479–90.View ArticleGoogle Scholar
 Perini LAG, Passieux JC, Périé JN. A multigrid PGDbased algorithm for volumetric displacement fields measurements. Strain. 2014;50(4):355–67.View ArticleGoogle Scholar
 Besnard G, Guérard S, Roux S, Hild F. A spacetime approach in digital image correlation: movieDIC. Optics Lasers Eng. 2011;49:71–81.View ArticleGoogle Scholar
 Besnard G, Leclerc H, Roux S, Hild F. Analysis of image series through digital image correlation. J Strain Anal. 2012;47(4):214–28.View ArticleGoogle Scholar
 Collins JD, Hart GC, Hasselman TK, Kennedy B. Statistical identification of structures. AIAA J. 1974;12(2):185–90.MATHView ArticleGoogle Scholar
 Pagnacco E, CaroBretelle AS, Ienny P. In: Grédiac M, Hild F, editors. Parameter identification from mechanical field measurements using finite element model updating strategies. London: ISTE / Wiley; 2012. p. 247–74.Google Scholar
 Leclerc H, Neggers J, Mathieu F, Hild F, Roux S. Correli 3.0. IDDN.FR.001.520008.000.S.P.2015.000.31500, Agence pour la Protection des Programmes, Paris; 2015.Google Scholar
 Babus̆ka I, Strouboulis T. The finite element method and its reliability. Oxford: Oxford University Press; 2001.Google Scholar
 Bertin M, Hild F, Roux S, Mathieu F, Leclerc H, Aimedieu P. Integrated digital image correlation applied to elastoplastic identification in a biaxial experiment. J Strain Anal. 2016;51(2):118–31.View ArticleGoogle Scholar
 Tikhonov AN, Arsenin VY. Solutions of Illposed problems. New York: J. Wiley; 1977.MATHGoogle Scholar