# Reliability of the Data-Driven Identification algorithm with respect to incomplete input data

Marie Dalémat, Michel Coret, Adrien Leygue, Erwan Verron

► To cite this version:

Marie Dalémat, Michel Coret, Adrien Leygue, Erwan Verron. Reliability of the Data-Driven Identification algorithm with respect to incomplete input data. 11th European Conference on Constitutive Models for Rubber (ECCMR 2019), Jun 2019, Nantes, France. pp.311-316, 10.1201/9780429324710-55. [https://doi.org/10.5555/hal-05273631](https://doi.org/10.5555/hal-05273631)

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# Reliability of the Data-Driven Identification algorithm with respect to incomplete input data

Marie DALEMAT, Michel CORET, Adrien LEYGUE & Erwan VERRON

Institut de Recherche en Génie Civil et Mécanique (GeM) UMR CNRS 6183

École Centrale de Nantes, France

`{First_name.Last_name}@ec-nantes.fr`

**ABSTRACT:** This work focuses on the influence of the input parameters of the Data-Driven Identification algorithm developed by Leygue et al. (2018) on the results obtained. This algorithm allows to measure stresses, from displacement fields and forces applied to a structure; the particularity is the absence of underlying constitutive equation. In the case of real experiments, the data are incomplete; but it is proven here that with appropriate data handling, stress fields can be identified in a robust manner. The incompleteness of input data is twofold: some missing displacement values (close to the edges or in a noise-affected area) and also a partial force information. The study proves that recovering those missing data has to be done smartly so that no assumptions except the balance equation is made.

# 1 INTRODUCTION

Constitutive equations are essential in Mechanics of Materials to get robust simulations. In practice, it is first necessary to choose or create a constitutive model that describes well the material response. Then, the identification of the model parameters has to be done ideally on several cases, with multiple deformation states (uniaxial tension, pure shear, biaxial...), which makes difficult the experimental execution. With the evolution of full-field measurements techniques such as Digital Image Correlations (Sutton et al. 2009) and Data Sciences, identification methods are constantly being improved. For example, Avril et al. (2008) gave an overview of identification techniques. Paralelly, Furukawa and Yagawa (1998) trained a neural for identification.

Here, a new path is chosen: to work without underlying constitutive equation. Indeed, it is possible to use the previous technics (full-field methods and Data Sciences) to create rich databases, that can be used for identification and simulation. This removes the difficulties mentionned above (smart choice and experimental execution for proper identification).

A specific algorithm is considered in this paper called Data Driven Identification (DDI) and developed by Leygue et al. (2018). It allows to identify heterogeneous stress fields from measured displacement fields and external forces. It has been validated on synthetic data in (Leygue et al. 2018). Its applica-

tion to real data has been assessed by Dalemat et al. (2019), giving an innovative tool to measure stress fields without constitutive equation, from DIC gathered displacement fields and net forces measured by a load cell.

The difficulty in applying the DDI algorithm to real data lies mainly in the lack of data in some areas, which is inherent to a real problem. Indeed, unlike a synthetic problem where everything is perfectly known, neither all forces nor all displacements can be measured; these difficulties have been overcome by making choices (both on the parameters of the algorithm and on the experimental data). This work proposes a parametric study related to these choices in order to demonstrate the robustness and reliability of the DDI applied to real data.

First, a brief recall of the algorithm is done to highlight its optimal parameters and input data. Then, it is shown that experimental conditions lead to an incompleteness of the input data (kinematics fields and force information). Finally, we focus on the influence of (i) intrinsic parameters (to the algorithm) and (ii) incomplete inputs on the DDI results. For this study, synthetic data are to be used as the correct answer is well known. A degradation of the synthetic data is therefore carried out to compare them with real, i. e. incomplete, experimental data.
## 2 ALGORITHM AND STUDY CASE

The algorithm is recalled in its application to synthetic data. These perfectly known inputs are then deteriorated to look like real data.

### 2.1 Data Driven Identification

The Data Driven Identification (DDI) (Leygue et al. 2018) corresponds to the inverse method of Data Driven Computational Mechanics (DDCM) initiated by Kirchdoerfer and Ortiz (2016).

This method identifies the complete response of a structure without using constitutive equation, from a large database, in the framework of small elastic (non-linear) transformations. The extension to large strain can be naturally done in an Eulerian configuration, as explained in Dalemat et al. (2019).

**Perfect input data.** We consider a 2D meshed geometry, loaded in several increments denoted $\cdot^X$. For this geometry, the following data are available:

(I-1) the nodal displacements $\mathbf{u}_j^X, \cdot_j$ being the node number. The information used from the displacements is a strain measurement. Here, we adopt the Henky's true strain $\ln \mathbf{v}$, that is the natural extension for an Eulerian formulation, defined from $\mathbf{b}$, the left Cauchy-Green tensor, by:

$$ \ln \mathbf{v} = \frac{1}{2} \ln \mathbf{b}; \qquad (1) $$

(I-2) the matrix $\mathbf{B}_{ej}^X$ which encodes geometry and connectivity, $\cdot_e$ being the Gauss integration point number. In particular, the mechanical balance can be calculated on each node by:

$$ \sum_e w_e^X \mathbf{B}_{ej}^X \cdot \sigma_e^X = \mathbf{f}_j^X \quad \forall X, j, \qquad (2) $$

where $w_e$ is the integration weight;

(I-3) the nodal forces $\mathbf{f}_j^X$;

(I-4) the prescribed displacements (in particular the indices of the concerned nodes).

**Data to find.** To solve the mechanical problem, it is necessary to have:

(O-1) the mechanical fields $\sigma_e^X$ that satisfy the mechanical balance in each element $e$ according to the Eq. 2. The stress $\sigma_e^X$ (calculated) and the strain $\ln \mathbf{v}_e^X$ (measured) forms a mechanical state, because they are mechanically admissible (balanced and compatible);

(O-2) the material states $(\ln \mathbf{v}_i^*, \sigma_i^*)$, which are in the number $N^*$, a chosen number. These material states can be interpreted as a sampling of the material response. Their distance from mechanical states is defined by an energetic norm $|| \cdot ||_C^2$ given by Eq. 3 where $C$ is a fourth order tensor also chosen.

**Practically.** The algorithm is based on solving this problem: finding the material states that are as close as possible to the mechanical states (the latter being half known with the strain fields, and half constrained by the equilibrium equation applied to the stress field), according to the energetic norm $|| \cdot ||_C^2$ defined by:

$$ ||(\ln \mathbf{v}, \sigma)||_C^2 = \frac{1}{2}(\ln \mathbf{v} : \mathbb{C} : \ln \mathbf{v} + \sigma : \mathbb{C}^{-1} : \sigma). \quad (3) $$

The solution to the problem is defined as follows:

$$ \text{solution} = \arg \min_{\sigma_e^X, \ln \mathbf{v}_e^*, \sigma_e^*} \mathcal{E}(\sigma_e^X, \ln \mathbf{v}_e^*, \sigma_e^*), \quad (4) $$

with,

$$ \mathcal{E}(\sigma_e^X, \ln \mathbf{v}_{eX}^*, \sigma_{eX}^*) = \sum_{X} \sum_{e} w_e^X ||(\ln \mathbf{v}_e^X - \ln \mathbf{v}_{eX}^*, \sigma_e^X - \sigma_{eX}^*)||_C^2, \quad (5) $$

and under the constraints:

$$ \left\{ \begin{array}{l} \text{- of respecting the mechanical balance, given in Eq. 2,} \\ \text{- that the material state}(\ln \mathbf{v}_{eX}^*, \sigma_{eX}^*) \\ \text{associated to the element } e^X \text{ belongs to} \\ \text{the database } (\ln \mathbf{v}_i^*, \sigma_i^*)_{i=1}^{N^*}. \end{array} \right. \qquad (6) $$

Therefore, the problem outputs are:

* the mechanical states;

* the database of material states;

* the mapping between the mechanical states associated with each material state.

In the initial work (Leygue et al. 2018), the validity of the method is demonstrated with complete synthetic data (I-1) to (I-4). The two intrinsic parameters to the method are: the number of material states $N^*$ and $C$ defining the energetic norm.
## 2.2 Building of a study case

**Theoritical problem.** The above method is applied to a perforated hyperelastic membrane subjected to uniaxial tension. The mechanical problem and the different notations are provided in Figure 1. The deterioration of the synthetic data is made regarding the usual experimental conditions: we cannot measure the nodal forces but a net force; displacements are sometimes missing in areas called clusters or close to the edges, mainly due to the tool used (the DIC software).

**Deteriation of input data to obtain comparability to real ones.** With real data, it is necessary to rewrite the mechanical problem by handling properly the areas where data are missing (Dalemat et al. 2019). Several possibilities are proposed to deal with the missing data and their impact on the results are being discussed:

* In the area near the grip, where the force is measured:

(t-1) In the synthetic case, we know each nodal force $f_j^X$ at the top boundary $\Gamma_T$;

(t-2) In the real case, we only know the sum of the forces $f_j^X$ on the top boundary $\Gamma_T$ in the direction of solicitations $\mathbf{n}_{sol}$:

$$ \sum_{j \in \Gamma_T} f_j^X \cdot \mathbf{n}_{sol} = F_{cell}^X. \quad (7) $$

It is thus possible to define an equilibrium condition on the boundary:

$$ \sum_{j \in \Gamma_T} \sum_{e} w_e^X \mathbf{B}_{ej}^X \cdot \sigma_e^X \cdot \mathbf{n}_{sol} = F_{cell}^X \quad \forall X. \quad (8) $$

* For clusters of missing displacement, information cannot be accessed within these areas; only the boundary of the clusters $\Gamma_C$ is considered:

(c-1) A simple and naive solution is to not consider the equilibrium equation for these nodes. This leads the algorithm to find the stress values at element $e$ with the neighbours nodes $j$ (in the bulk, for which everything is known);

(c-2) A mechanically optimal solution is to consider that the boundary of clusters is the boundary of a mechanically balanced subset. Indeed, the material of the cluster is in balance, which is equivalent to imposing a global-balance condition on $\Gamma_C$. This is also equivalent to consider a zero net force on this boundary:

$$ \sum_{j \in \Gamma_C} \sum_{e} w_e^X \mathbf{B}_{ej}^X \cdot \sigma_e^X = 0. \quad (9) $$

* For the edges close to the holes, the imperfect selection of the zone of interest in DIC gives missing displacement values in the vicinity of the edges of the hole. Therefore, the real boundary $\Gamma_H$ is not usable and $\tilde{\Gamma}_H$ is being considered instead. On this boundary, several assumptions can be made:

(h-1) The free edge assumption can be adopted if we consider that $\tilde{\Gamma}_H$ is really close to $\Gamma_H$ so the edge is free. This is a strong assumption.

(h-2) A weaker assumption is to not consider the free edge assumption. Instead, a similar to (c-1) consideration can be done: a zero net force on this boundary is verified has the missing matter should be mechanically balanced:

$$ \sum_{j \in \tilde{\Gamma}_H} \sum_{e} w_e^X \mathbf{B}_{ej}^X \cdot \sigma_e^X = 0. \quad (10) $$

<figure><img src="image_2.png" /><figcaption>Figure 1: Theoretical problem treated with 3 particular boundaries (top boundary with grip, cluster of missing data and imperfectly defined edges close to hole).</figcaption></figure>

## 2.3 Input data to study

Finally, the input data of the algorithm are:

* The algorithm parameters (intrinsic to the resolution method): $N^*$ and $C$;

* The measured data, especially displacements and forces, which are incomplete.

First, the effect of intrinsic parameters on a case where the input data are complete are being analyzed. Secondly, the influence of the measured data in cases (t-1) and (t-2) related to the (top) grips, in cases (c-1) and (c-2) related to the clusters of missing data and finally in cases (h-1) and (h-2) related to the edges close to the holes are being compared.
## 3 PARAMETRIC STUDY

The initial and distorted meshes of the problem are presented in Figure 2. First, the stresses in each element

<figure><img src="image_3.png" /><figcaption>Figure 2: Case of study: a perforated hyperelastic membrane under tensile traction (initial on right and after 200% of total strain on left). On the top nodes information on force is available while bottom ones are clamped.</figcaption></figure>

element $\sigma_{FE}$ and the resulting forces are calculated by the conventional Finite Element method. The DDI method is then applied with the parameters and input data as described above in Section 2.3, resulting in identified stresses $\sigma_{DDI}$. The error between the stresses identified by the DDI and the reference ones calculated by Finite Element is then calculated for each loading increment $\cdot^X$ (here, $N_X = 21$) and each element $\cdot_e$ (here, $N_e = 6108$):

$$ \text{err} = \frac{\sum_{X,e} \|\sigma_{\text{FE},e^X} - \sigma_{\text{DDI},e^X}\|_2}{\sum_{X,e} \|\sigma_{\text{FE},e^X}\|_2}. \quad (11) $$

### 3.1 Influence of intrinsic parameters

<figure><img src="image_4.png" /><figcaption>Figure 3: Influence of the intrinsic parameters $N^*$ (related to $r^*$) and $C$ in a case without missing data. The error is compared to $r^*$ (left), and the minimum error is compared to $C$ (right).</figcaption></figure>

The number of material states $N^*$ is the parameter that allows to sample more or less finely the behaviour of the material from the mechanical states. It is to be compared to the total number of degrees of freedom

of the problem: $N_e \times N_X = 128 268$. We can define the ratio $r^* = (N_e \times N_X)/N^*$ and we make it vary between 2 and $10^4$. The distance to mechanical states $(\epsilon_e^X, \sigma_e^X)$ is defined by the energetic norm $||\cdot||_2^2_C$ of the Eq. 3, where $C$ is a fourth order tensor. The simplest form of this tensor is to choose a spherical tensor defined by its amplitude $C$, which can be written, adopting the Voigt notation:

$$ [C] = C \cdot I. \quad (12) $$

This form aims to favour the convergence of all components at the same time. The tensor amplitude is defined relatively to the tangent elasticity modulus of the behaviour model used in Finite Element simulations: $C_0 = 2.5 \cdot 10^6$ Pa. Thus, we choose values of $C$ ranging from $C_0$ to $10^6 C_0$.

The Figure 3 shows the results in error, after convergence.

The left subfigure 3 shows the convergence error as a function of the $r^*$ ratio for different values of $C$. For each $C$ value, the value of the minimum error (with respect to $r^*$) is reported. Thus, the minimum error (for the optimal $r^*$) in relation to $C/C_0$ is shown on the right subfigure. The error is minimal for a $r^* \approx 20$ (10 to 50 depending on the $C$ value). A ratio that is too high (too few material states) implies a sub-sampling of the response and therefore a significant error. Conversely, a too low ratio (too many material states) produces an effect comparable to the overfitting in regression: the behaviour is no longer averaged enough, which also leads to a significant error. It is therefore necessary to choose a value between these two extremes; these are results similar to those obtained in (Leygue et al. 2018).

In addition, $C$ contributes significantly to the convergence: the more it is increased (to a certain extent), the lower the error is. This is simply explained by the fact that the distance defined by the norm influences the mapping between material states and mechanical states. By choosing a high value of $C$, the mapping based on strain values is privileged, which is interesting since they are measured (and so reliable), unlike the stresses that evolve during the convergence of the algorithm. Finally, we note that the choice of $N^*$ is more consequential than the choice of $C$: without missing data, a bad choice of $N^*$ will never be compensated by a good choice of $C$.

### 3.2 Influence of the incompleteness of input database

**Force input analysis.** First, we consider the forces: either with all the nodal forces (t-1), or averaged by their net value on the boundary (t-2): the influence of $N^*$ on the error is tested in the Figure 4 (we choose to represent the results for a large value of $C$).

The error curves are essentially identical: a ratio of material states to the number of degrees of freedom of 20 to 50 is preferable. These results show that the
<figure><img src="image_5.png" /><figcaption>Figure 4: Influence of the force inputs on the error as a function of $r^*$ for (t-1) the given nodal forces and (t-2) the given net force.</figcaption></figure>

DDI algorithm is only slightly influenced by the way this input conditions is imposed on the top boundary. In DDI, the stresses are provided by the equation of equilibrium and are defined to a multiplicative constant, found by the force information. In a simple case (symmetrical and uniaxial), the average information $F_{cell}^X$ is sufficient to find the stress.

<figure><img src="image_6.png" /><figcaption>Figure 5: Influence of input data with a cluster on the error as a function of $r^*$ and $C$ for cases (top) naive and (bottom) mechanically optimal</figcaption></figure>

**Handling a cluster of missing data.** Then, we observe the influence of the input data related to a cluster: by (c-1) handling it naively and (c-2) in a mechanically optimal way. The influence of $N^*$ and $C$ is shown in Figure 5.

In the case of a naively handled cluster (c-1), it is more difficult to get a small error. Too many or too few material states lead to more important errors. Here, the choice of $C$ is essential: the larger it

is (within a certain limit), the closer we get to a mapping based on strains (known). By simply adding the zero net force condition (c-2) on the Eq. 9 boundary, a similar to previously robustness is recovered; in this case the choice of $C$ is much less important than that of $N^*$. In other words, in the naive case, the elements with a node on the boundary $\Gamma_c$ (orange on the Figure 9 (a)) have no information to mechanically balance them. This is corrected in the mechanically optimal case.

<figure><img src="image_7.png" /><figcaption>Figure 6: Influence of input data with hole edge definition on the error as a function of $r^*$ for a given $C$, for the cases of an imperfectly defined edge close to the hole (with (h-1) the assumption of free edge, (h-2) the global balance condition) and of a perfectly defined edge.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: (top) For the perfect case and the 2 imperfect cases handled differently: plots of the values of the Von Mises stress along the horizontal line going through the hole (red dots in the bottom subfigure)</figcaption></figure>

**Imperfect selection and assumptions close to the hole.** Finally, we observe the influence of inputs in the case of an imperfect selection close to the edges, on the boundary $\tilde{\Gamma}_H$. The case of (h-1) the free assumption on this boundary and the case of (h-2) zero net
<figure><img src="image_9.png" /><figcaption>Figure 8: Comparison between real and identified stress; for a geometry with well handled clusters and edges close to the holes; and the information of average force</figcaption></figure>

force over the boundary $\tilde{\Gamma}_H$ are compared to the perfect case with no missing data close to the edge. Errors are plotted for one value of $C$ with respect to the ratio $r^*$ in Figure 6. It can be noticed that considering the free edge assumption leads to a high error. The values of Von Mises stress along a horizontal line going through the hole is plotted for the three cases in Figure 7. It is noticed that in the case of the imperfect free edge assumption, the algorithm is actually able to find the correct value, but at the wrong position. All the stress values in this case are overestimated around the hole, which explains the large error. Therefore, the cleaner manner to handle a imperfect edge is to assume something which is weaker: only a zero net force condition. This gives less information but does not fake the algorithm with a different mechanical problem to solve. The optimal ratio $r^*$ is again around 20-100.

### 3.3 Summary

In conclusion, we present a result of the identified constraints (for the 3 components) with the corrections proposed on the Figure 8. The incomplete inputs have to be completed smartly with only the mechanical balance equation: (t-2),(c-2),(h-2).

## 4 CONCLUSION

In this work, the input parameters of the DDI algorithm are examined. A study of its intrinsic parameters confirms the previous work. In particular, the consideration of incomplete data (frequent for experimental data) is analysed through three aspects of the problem: the resulting forces, the clusters of missing data (areas of a few pixels/elements) and the imperfect edge definitions close to the holes. This shows that the robustness of the method is ensured when incomplete data are managed under a mechanic's perspective. Indeed, the replacement of nodal forces by their net value (measured) does not pose any additional problem: it is sufficient to choose the parameters of the algorithm carefully. For clusters of missing data, well-chosen parameters can reduce the error, but the most robust solution is to add a zero net force

(or a global equilibrium condition) over the boundary of the missing data. The imperfect edges selection leads to an imperfectly positioned boundary. It is important to not assume that the edges are free on this boundary and to deal with it as for a cluster: a zero net force expresses that the missing matter is balanced. Not all cases of deteriorated data have been treated here: noise on displacement measurement, uncertainties on thickness (common in soft materials)... Despite this, it is believed that the study validates the application of the DDI to real data, incomplete but well handled. For the application to real data, a large value of $C$ and a ratio $r^* = 100$ are chosen so that all errors can be smoothened while having a good confidence on the results.

**Acknowledgements.** This work was performed by using HPC resources of Centrale Nantes Supercomputing Center on the cluster Liger, granted and identified D1705030 by the High Performance Computing Institute (ICI).

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