<table><caption>**Table 7.** Averaged calibrated mean, standard-deviations and correlations $\Pi_{2,mean}$ and averaged error $\mathcal{E}_{\text{mean}}(\hat{\Pi}_2)$ over 20 repetitions of the calibration process with samples of 50 individuals each. The coefficients of variation COV in % are indicated between brackets.</caption><thead><tr><th></th><th>$S_{11}^0$ [$\times 10^{-6}\text{MPa}^{-1}$]</th><th>$\nu_{12}$ [$\times 10^{-2}$]</th><th>$S_1^C$ [$\times 10^{-6}\text{MPa}^{-1}$]</th></tr></thead><tbody><tr><td>Exact Means</td><td>85.2</td><td>33.1</td><td>155.2</td></tr><tr><td>Averaged Calibrated Means</td><td>85.4(0.424)</td><td>33.1(0.339)</td><td>155.4(1.89)</td></tr><tr><td>Averaged Errors On Calibrated Means (%)</td><td>0.376(76.5)</td><td>0.273(74.6)</td><td>1.52(74.0)</td></tr><tr><td>Exact Standard-Deviations</td><td>1.31</td><td>0.660</td><td>17.3</td></tr><tr><td>Averaged Calibrated Standard Deviations</td><td>1.01(28.6)</td><td>0.669(10.5)</td><td>17.9(7.93)</td></tr><tr><td>Averaged Errors On Calibrated Standard-Deviations (%)</td><td>26.3(67.5)</td><td>9.41(54.8)</td><td>6.68(84.9)</td></tr><tr><td>Correlations</td><td>$\rho(S_{11}^0, \nu_{12})$</td><td>$\rho(S_1^C, \nu_{12})$</td><td>$\rho(S_{11}^0, S_1^C)$</td></tr><tr><td>Exact Correlations</td><td>-0.741</td><td>0.729</td><td>-0.414</td></tr><tr><td>Averaged Calibrated Correlations</td><td>-0.726(18.6)</td><td>0.713(6.40)</td><td>-0.354(56.0)</td></tr><tr><td>Averaged Errors On Calibrated Correlations (%)</td><td>14.7(74.6)</td><td>4.91(90.4)</td><td>41.1(69.0)</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** KL-divergences of the calibrated and exact joint distribution $S_{11}^0$ and $\nu_{12}$ for the 20 repetitions in the two stages. Remember that the KL-divergence expresses a notion of distance between two distributions and the lower is its value, the more accurate the calibration of $f_{\Theta|\Pi}$ is.</figcaption></figure>

can be understood as it is rather weak (−0.41) which adds to the limited number of individuals. Otherwise, the calibration of the population parameters that involve $S_1^C$ can be considered as satisfactory, provided that this parameter is the most dispersed around its mean value (its exact coefficient of variation is about 10% which makes it more subjected to estimation errors). It is now possible to focus on the joint distribution of $S_{11}^0$ and $\nu_{12}$. From Table 7, it seems that the estimation on this joint distribution is downgraded between the two steps. Indeed, except for the mean parameters, the two standard-deviations appear to be flawed from their initial estimation (Tab. 4), in particular for the standard-deviation of $S_{11}^0$ (the averaged error increases by 30%). This downgrade can be noticed both

the averaged values or averaged errors on the population parameters. The correlation between $S_{11}^0$ and $\nu_{12}$ does not seem to be much affected and even exhibits a small improvement. Rather than focusing independently on each parameter, a more suited way to quantify this alleged degradation is the Kullback-Leibler divergence between the exact joint distribution of $S_{11}^0, \nu_{12}$ ($f_{\text{exact}}(S_{11}^0, \nu_{12})$) and the calibrated distribution ($f_{\text{calibrated}}(S_{11}^0, \nu_{12})$) in the two stages as depicted Figure 7 for the 20 repetitions.

In Figure 7, the deterioration of the joint distribution of $S_{11}^0$ and $\nu_{12}$ suggested by Table 7 seems to be confirmed. For instance, the median of the KL-divergences increases (multiplied by 3), and the ranges of variation of the KL-divergences are also increased by two (without
<table><caption>**Table 8.** Relative errors on the estimation of the individual parameters $e(\hat{\theta}_1, \dots, \hat{\theta}_n)$ averaged over 20 repetitions of the calibration process with the spread $\zeta_{\text{exact}}$. The corresponding coefficients of variation in % are indicated between brackets.</caption><thead><tr><th></th><th>$S_{11}^0$ [$\times 10^{-6} \text{MPa}^{-1}$]</th><th>$\nu_{12}$ [$\times 10^{-2}$]</th><th>$S_1^C$ [$\times 10^{-6} \text{MPa}^{-1}$]</th></tr></thead><tbody><tr><td>Relative error (%)</td><td>1.50(13.0)</td><td>2.22(9.83)</td><td>13.1(10.1)</td></tr><tr><td>Spread $\zeta_{\text{exact}}$ (%)</td><td>1.53</td><td>2.00</td><td>10.0</td></tr></tbody></table>

<table><caption>**Table 9.** Error in model space $d(\hat{\theta}_1, \dots, \hat{\theta}_n)$ for each strain component.</caption><thead><tr><th>Strain Component</th><th>$\varepsilon_{11}$</th><th>$\varepsilon_{22}$</th></tr></thead><tbody><tr><td>Order of magnitude of the mean strain value</td><td>$10^{-3}$</td><td>$10^{-3}$</td></tr><tr><td>Distance in model space</td><td>$6.36 \times 10^{-8}$</td><td>$9.00 \times 10^{-9}$</td></tr></tbody></table>

consideration of the outliers). This seems mainly due to poorer estimation in the compression phase of the standard-deviation of $S_{11}^0$. Consequently, a deterioration of the estimation of joint distribution of $S_{11}^0$ and $\nu_{12}$ can be observed in the second phase. Nevertheless, this deterioration remains limited for both the relative errors and the KL-divergences. In particular, even if they are larger in the second phase than in the first phase, the KL-divergences have the same orders of magnitude. Indeed, except for one repetition, the KL-divergences range from 0 to 0.7 in the first step and from 0 to 1 in the second step, which can be considered as a clear but small downgrade. This does not necessarily invalidate the sequential method, that would require a comparison with the direct estimation method proposed in Section 4.5.4 but rather hints that the calibration is harder to perform using the compression experiments instead of tension experiments. One possible explanation could be a poorer calibration of the individual linear compliances $S_{11}^0$ (to be investigated later). To finish the analysis, it remains to check the calibration of the individuals, either in the parameters and model space. The suited indicators are displayed in Tables 8 and 9 that also contain, as earlier, the measure of the spread of each model parameter labeled $\zeta_{\text{exact}}$ (instead of the empirical COV), that is to say $\zeta_{\text{exact}} := \frac{\sqrt{\text{diag}(\Sigma_{\text{exact}})}}{\mu_{\text{exact}}}$ with $\mu_{\text{exact}}$ and $\Sigma_{\text{exact}}$ the exact mean vector and covariance matrix.

From Tables 8 and 9, the same remarks made on Tables 5 and 6 arise. Indeed, in average, for all repetitions, all experiments seem properly calibrated either in model or parameters space. However, with respect to their exact spread, the nonlinear compliance can be considered as better calibrated than the others. The only difference is that the elastic linear parameters $S_{11}^0, \nu_{12}$ are flawed with higher errors, partly explaining the poorer calibration of their joint distribution highlighted above. This could come from the non-linearity of the strain-stress curve more pronounced in compression than in tension, making the sum of the residuals of the nonlinear part greater than those from the linear part, here reduced to few points, but this remains to be confirmed by a detailed study.

The results from this section and the previous one show that by conducting the calibration starting with the tension experiments and then continuing with the

compression experiments, it is possible to estimate accurately the full joint distribution of the longitudinal nonlinear elasticity provided the limitations of the model and of population parameters values. Still, the sequential calibration has been performed in an arbitrarily chosen order and there is no particular reason to favor this one to others. Thus, it is necessary to check to what the extent the choice of the order and more generally the calibration strategy influences the calibration results. In the next section, the sequential strategy is achieved in the reverse order to check in particular whether this choice downgrades the estimation of the model parameters distribution, before being compared to the method that implements design matrices.

### 4.5.4 Comparison of the different calibration strategies

In this section, the objective is to present the results provided by the other solutions that allow to estimate the model parameters distribution. Remember that among these solutions can be found either the implementation of the sequential strategy in the reverse order: compression first, then traction, and with the use of design matrices. Results from the sequential calibration in the reverse order are presented, before a comparison with the calibration with design matrices. Note that all the optimization settings for the sequential scheme are kept the same as in Section 4.5.2. The main issue here is to know whether one of the orders provides better results (in terms of errors). To investigate this point, the result of main interest is directly studied, i.e., how does evolve the calibration of the joint distribution of $S_{11}^0, \nu_{12}$? In particular, is the estimation downgraded or improved between the two stages? To answer this question, the KL-divergence between the exact joint distribution of $S_{11}^0, \nu_{12}$ ($f_{\text{exact}}(S_{11}^0, \nu_{12})$) and the calibrated distribution ($f_{\text{calibrated}}(S_{11}^0, \nu_{12})$) in the two stages can be estimated.

Here, contrary to the previous case, there is a clear improvement of the calibration results between the two phases. Indeed, the median of the KL-divergences is almost divided by 7. This is interesting because, here, the tension phase corrected the errors made in the compression phase that appears harder to conduct as shown by all the indicators studied so far. In fact, the observed decrease
<table><caption>**Table 8.** Relative errors on the estimation of the individual parameters $e(\hat{\theta}_1, \dots, \hat{\theta}_n)$ averaged over 20 repetitions of the calibration process with the spread $\zeta_{\text{exact}}$. The corresponding coefficients of variation in % are indicated between brackets.</caption><thead><tr><th></th><th>$S_{11}^0$ [$\times 10^{-6}\text{MPa}^{-1}$]</th><th>$\nu_{12}$ [$\times 10^{-2}$]</th><th>$S_1^C$ [$\times 10^{-6}\text{MPa}^{-1}$]</th></tr></thead><tbody><tr><td>Relative error (%)</td><td>1.50(13.0)</td><td>2.22(9.83)</td><td>13.1(10.1)</td></tr><tr><td>Spread $\zeta_{\text{exact}}$ (%)</td><td>1.53</td><td>2.00</td><td>10.0</td></tr></tbody></table>

<table><caption>**Table 9.** Error in model space $d(\hat{\theta}_1, \dots, \hat{\theta}_n)$ for each strain component.</caption><thead><tr><th>Strain Component</th><th>$\varepsilon_{11}$</th><th>$\varepsilon_{22}$</th></tr></thead><tbody><tr><td>Order of magnitude of the mean strain value</td><td>$10^{-3}$</td><td>$10^{-3}$</td></tr><tr><td>Distance in model space</td><td>$6.36 \times 10^{-8}$</td><td>$9.00 \times 10^{-9}$</td></tr></tbody></table>

consideration of the outliers). This seems mainly due to poorer estimation in the compression phase of the standard-deviation of $S_{11}^0$. Consequently, a deterioration of the estimation of joint distribution of $S_{11}^0$ and $\nu_{12}$ can be observed in the second phase. Nevertheless, this deterioration remains limited for both the relative errors and the KL-divergences. In particular, even if they are larger in the second phase than in the first phase, the KL-divergences have the same orders of magnitude. Indeed, except for one repetition, the KL-divergences range from 0 to 0.7 in the first step and from 0 to 1 in the second step, which can be considered as a clear but small downgrade. This does not necessarily invalidate the sequential method, that would require a comparison with the direct estimation method proposed in Section 4.5.4 but rather hints that the calibration is harder to perform using the compression experiments instead of tension experiments. One possible explanation could be a poorer calibration of the individual linear compliances $S_{11}^0$ (to be investigated later). To finish the analysis, it remains to check the calibration of the individuals, either in the parameters and model space. The suited indicators are displayed in Tables 8 and 9 that also contain, as earlier, the measure of the spread of each model parameter labeled $\zeta_{\text{exact}}$ (instead of the empirical COV), that is to say $\zeta_{\text{exact}} := \frac{\sqrt{\text{diag}(\Sigma_{\text{exact}})}}{\mu_{\text{exact}}}$ with $\mu_{\text{exact}}$ and $\Sigma_{\text{exact}}$ the exact mean vector and covariance matrix.

From Tables 8 and 9, the same remarks made on Tables 5 and 6 arise. Indeed, in average, for all repetitions, all experiments seem properly calibrated either in model or parameters space. However, with respect to their exact spread, the nonlinear compliance can be considered as better calibrated than the others. The only difference is that the elastic linear parameters $S_{11}^0, \nu_{12}$ are flawed with higher errors, partly explaining the poorer calibration of their joint distribution highlighted above. This could come from the non-linearity of the strain-stress curve more pronounced in compression than in tension, making the sum of the residuals of the nonlinear part greater than those from the linear part, here reduced to few points, but this remains to be confirmed by a detailed study.

The results from this section and the previous one show that by conducting the calibration starting with the tension experiments and then continuing with the

compression experiments, it is possible to estimate accurately the full joint distribution of the longitudinal nonlinear elasticity provided the limitations of the model and of population parameters values. Still, the sequential calibration has been performed in an arbitrarily chosen order and there is no particular reason to favor this one to others. Thus, it is necessary to check to what the extent the choice of the order and more generally the calibration strategy influences the calibration results. In the next section, the sequential strategy is achieved in the reverse order to check in particular whether this choice downgrades the estimation of the model parameters distribution, before being compared to the method that implements design matrices.

### 4.5.4 Comparison of the different calibration strategies

In this section, the objective is to present the results provided by the other solutions that allow to estimate the model parameters distribution. Remember that among these solutions can be found either the implementation of the sequential strategy in the reverse order: compression first, then traction, and with the use of design matrices. Results from the sequential calibration in the reverse order are presented, before a comparison with the calibration with design matrices. Note that all the optimization settings for the sequential scheme are kept the same as in Section 4.5.2. The main issue here is to know whether one of the orders provides better results (in terms of errors). To investigate this point, the result of main interest is directly studied, i.e., how does evolve the calibration of the joint distribution of $S_{11}^0, \nu_{12}$? In particular, is the estimation downgraded or improved between the two stages? To answer this question, the KL-divergence between the exact joint distribution of $S_{11}^0, \nu_{12}$ ($f_{\text{exact}}(S_{11}^0, \nu_{12})$) and the calibrated distribution ($f_{\text{calibrated}}(S_{11}^0, \nu_{12})$) in the two stages can be estimated.

Here, contrary to the previous case, there is a clear improvement of the calibration results between the two phases. Indeed, the median of the KL-divergences is almost divided by 7. This is interesting because, here, the tension phase corrected the errors made in the compression phase that appears harder to conduct as shown by all the indicators studied so far. In fact, the observed decrease
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** KL-divergences of the calibrated and exact joint distribution $S_{11}^0$ and $\nu_{12}$ for the 20 repetitions in the two stages. Remember that the KL-divergence expresses a notion of distance between two distributions and the lower the more accurate the calibration of $f_{\Theta|\Pi}$ is.</figcaption></figure>

mainly comes from the fact that, with the compression experiments, the estimation of the population parameters that concern the joint distribution of $S_{11}^0$ and $\nu_{12}$ shows higher levels of error compared to the estimate of the population parameters from the tension experiments, as it arises from Tables 4 and 7. Thus, choosing this order of calibration implies an improvement when the other was synonymous of a deterioration. It is also worth noticing that, with the compression experiments, the joint distribution of the $S_{11}^0$ and $\nu_{12}$ is estimated with increased accuracy when its calibration is carried out in the second phase rather than in the first phase, as show the KL-divergences in Figures 7 and 8. Indeed, though their medians are comparable in both situations, the range of variation of the KL-divergences (without consideration of the outliers) in the second phase of the calibration process is about half of their range of variation in the first phase of the calibration process when starting with the compression experiments. This illustrates the interest to use a sequential scheme. Indeed, the MLE calibrated in the first phase (conducted with the tension experiments and described in Sect. 4.5.2) is used to define a trust region, which reduces the size of the search space by focusing the optimization within a region that concentrates proposals of population parameters that seems to be close to the value of MLE of the second phase. This eases the optimization and seems to limit the deterioration of the population parameters estimates, contrary to the situation in which the population parameters are learnt from scratch (that is to say when the compression experiments are used first). Still, this assumption would demand to be consolidated with uncertainty quantification techniques and confirmed by appropriate convergence studies. It can be interesting to know whether this is a global improvement (i.e., the estimation of all parameters was improved)

or if it is concentrated on one specific parameter. To study this particular point, the averaged error on the population parameters of the joint distribution of $S_{11}^0$ and $\nu_{12}$ whose values are reported in Tables 10 and 11 can be investigated.

From these tables, it follows that the estimation of all the population parameters is slightly improved from the first to the second phase except $\sigma_{S_{11}^0}$, whose averaged estimation error is reduced by about 30%. This mainly comes from the fact that the linear compliances are identified with better accuracy with the tension experiments than with the compression experiments, as show Tables 5 and 8. Still, it can be noticed that the calibration of the standard-deviation of $\nu_{12}$ and of the correlation is not improved as for the other parameters given that the coefficient of variation of their corresponding averaged error is increased, indicating that this improvement is not uniform among the repetitions. However, globally, with this particular model, provided the modeling assumptions and the values chosen for population parameters, it seems better to identify first the joint distribution of the population parameters in compression, before a refinement with the tension experiments. Of course, this conclusion does not apply in all cases and is only specific to our model of interest, parameters value and modeling choices.

So far, the results provided by a sequential strategy have only been discussed. However, in this particular case, it is possible to use classical mixed-effects to estimate, with the help of design matrices, the 13 parameters of the joint distribution of $S_1^T, S_1^C, S_{11}^0$ and $\nu_{12}$ (recall that $\text{Cov}(S_1^T, S_1^C)$ cannot be estimated with the actual data). Furthermore, with the two different kind of experiments considered here, the tension and compression experiments, a reasonable assumption is to consider different residual models for each kind of experiment. Provided that
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** KL-divergences of the calibrated and exact joint distribution $S_{11}^0$ and $\nu_{12}$ for the 20 repetitions in the two stages. Remember that the KL-divergence expresses a notion of distance between two distributions and the lower the more accurate the calibration of $f_{\Theta|\Pi}$ is.</figcaption></figure>

mainly comes from the fact that, with the compression experiments, the estimation of the population parameters that concern the joint distribution of $S_{11}^0$ and $\nu_{12}$ shows higher levels of error compared to the estimate of the population parameters from the tension experiments, as it arises from Tables 4 and 7. Thus, choosing this order of calibration implies an improvement when the other was synonymous of a deterioration. It is also worth noticing that, with the compression experiments, the joint distribution of the $S_{11}^0$ and $\nu_{12}$ is estimated with increased accuracy when its calibration is carried out in the second phase rather than in the first phase, as show the KL-divergences in Figures 7 and 8. Indeed, though their medians are comparable in both situations, the range of variation of the KL-divergences (without consideration of the outliers) in the second phase of the calibration process is about half of their range of variation in the first phase of the calibration process when starting with the compression experiments. This illustrates the interest to use a sequential scheme. Indeed, the MLE calibrated in the first phase (conducted with the tension experiments and described in Sect. 4.5.2) is used to define a trust region, which reduces the size of the search space by focusing the optimization within a region that concentrates proposals of population parameters that seems to be close to the value of MLE of the second phase. This eases the optimization and seems to limit the deterioration of the population parameters estimates, contrary to the situation in which the population parameters are learnt from scratch (that is to say when the compression experiments are used first). Still, this assumption would demand to be consolidated with uncertainty quantification techniques and confirmed by appropriate convergence studies. It can be interesting to know whether this is a global improvement (i.e., the estimation of all parameters was improved)

or if it is concentrated on one specific parameter. To study this particular point, the averaged error on the population parameters of the joint distribution of $S_{11}^0$ and $\nu_{12}$ whose values are reported in Tables 10 and 11 can be investigated.

From these tables, it follows that the estimation of all the population parameters is slightly improved from the first to the second phase except $\sigma_{S_{11}^0}$, whose averaged estimation error is reduced by about 30%. This mainly comes from the fact that the linear compliances are identified with better accuracy with the tension experiments than with the compression experiments, as show Tables 5 and 8. Still, it can be noticed that the calibration of the standard-deviation of $\nu_{12}$ and of the correlation is not improved as for the other parameters given that the coefficient of variation of their corresponding averaged error is increased, indicating that this improvement is not uniform among the repetitions. However, globally, with this particular model, provided the modeling assumptions and the values chosen for population parameters, it seems better to identify first the joint distribution of the population parameters in compression, before a refinement with the tension experiments. Of course, this conclusion does not apply in all cases and is only specific to our model of interest, parameters value and modeling choices.

So far, the results provided by a sequential strategy have only been discussed. However, in this particular case, it is possible to use classical mixed-effects to estimate, with the help of design matrices, the 13 parameters of the joint distribution of $S_1^T, S_1^C, S_{11}^0$ and $\nu_{12}$ (recall that $\text{Cov}(S_1^T, S_1^C)$ cannot be estimated with the actual data). Furthermore, with the two different kind of experiments considered here, the tension and compression experiments, a reasonable assumption is to consider different residual models for each kind of experiment. Provided that
<table><caption>**Table 10.** Averaged marginal parameters ($\Pi_{1,mean}$ from the first phase and $\Pi_{2,mean}$ from the second phase), alongside the averaged error ($\mathcal{E}_{\text{mean}}(\hat{\Pi}_2)$ and $\mathcal{E}_{\text{mean}}(\hat{\Pi}_2)$) over 20 repetitions of the calibration process with samples of 50 individuals each. The first two columns refer to the error made on the population parameters estimated in the first stage $\Pi_1$ (with the compression experiment) and the two left to parameters $\Pi_2$ estimated in the second phase (with the tension experiment). The coefficients of variation COV in % are indicated between brackets.</caption><thead><tr><th></th><th>S<sup>0</sup><sub>11</sub> [×10<sup>-6</sup>MPa<sup>-1</sup>] (1<sup>st</sup> phase)</th><th>ν<sub>12</sub> [×10<sup>-2</sup>] (1<sup>st</sup> phase)</th><th>S<sup>0</sup><sub>11</sub> [×10<sup>-6</sup>MPa<sup>-1</sup>] (2<sup>nd</sup> phase)</th><th>ν<sub>12</sub> [×10<sup>-2</sup>] (2<sup>nd</sup> phase)</th></tr></thead><tbody><tr><td>Exact Means</td><td>85.2</td><td>33.1</td><td>85.2</td><td>33.1</td></tr><tr><td>Averaged Calibrated Means</td><td>85.4(0.438)</td><td>33.1(0.343)</td><td>85.3(0.306)</td><td>33.1(0.366)</td></tr><tr><td>Averaged Errors On Calibrated Means (%)</td><td>0.364(78.9)</td><td>0.277(73.4)</td><td>0.251(73.1)</td><td>0.309(64.8)</td></tr><tr><td>Exact Standard-Deviations</td><td>1.31</td><td>0.660</td><td>1.31</td><td>0.660</td></tr><tr><td>Averaged Calibrated-Standard Deviations</td><td>1.07(30.1)</td><td>0.675(10.5)</td><td>1.20(21.1)</td><td>0.669(10.7)</td></tr><tr><td>Averaged Errors On Calibrated Standard-Deviations (%)</td><td>25.5(73.9)</td><td>9.44(59.5)</td><td>17.7(64.7)</td><td>7.48(108)</td></tr></tbody></table>

<table><caption>**Table 11.** Averaged marginal parameters correlation parameter $\hat{\rho}(S_{11}^0, \nu_{12})_{\text{mean},1/2}$ in both phases and averaged error $\mathcal{E}_{\text{mean}}(\hat{\rho}(S_{11}^0, \nu_{12}))_{\text{mean},1/2}$ over 20 repetitions of the calibration process with samples of 50 individuals each. The first column refers to the error made on the population parameters in the second stages of the sequential methodology (with the compression experiment on the left and with the tension experiment on the right). The coefficients of variation COV in % are indicated between brackets.</caption><thead><tr><th></th><th>ρ(S<sup>0</sup><sub>11</sub>, ν<sub>12</sub>)</th><th>ρ(S<sup>0</sup><sub>11</sub>, ν<sub>12</sub>)</th></tr></thead><tbody><tr><td>Exact Correlation</td><td>-0.741</td><td>-0.741</td></tr><tr><td>Averaged Calibrated Correlations</td><td>-0.771(20.8)</td><td>-0.715(18.5)</td></tr><tr><td>Averaged Errors On Calibrated Correlations (%)</td><td>18.2(68.2)</td><td>13.8(85.8)</td></tr></tbody></table>

there are two different curves to identify (longitudinal and transverse measurements), that makes 4 parameters for the two residual models. In total, $p = 17$ parameters have to be estimated. This illustrates a first interest to use a sequential scheme as with such techniques, only $p = 11$ parameters have to be calibrated in the different phases. The three calibration techniques are studied here. They respectively stand for the calibrated distribution in the second phase of the first implementation of the sequential strategy (the one obtained after the calibration with the compression experiments) labeled $C_1$, the calibrated distribution in the second phase of the second implementation (the one obtained after the calibration with the tension experiments) $C_2$ and finally, the one that comes from the complete calibration the joint distribution of $S_1^T, S_1^C, S_{11}^0$ and $\nu_{12}$, labeled $C_3$. Several items can be investigated among which the error on all population parameters in the three cases, as reported in Table 12.

In Table 12, it is noticeable that the estimation of the marginals of the asymptotic compliance in tension and compression are slightly downgraded in case $C_3$ compared to cases $C_1$ and $C_2$. This could come from the fact in case $C_3$, all variables are considered together and may be more difficult to tune than when the population parameters corresponding to a single type of experiment are estimated. Still, the differences remain small and may also originate from numerical issues. Furthermore, it appears that, surprisingly, the calibration that considers both compression and tension does not necessarily provide the best results for the calibration of the joint distribution of $S_{11}^0$ and $\nu_{12}$. Indeed, as these parameters express on both types

of experiments with 50 individuals each (so 100 individuals in total), we could expect it to be the best solution among the three possibilities given that this distribution is estimated with the highest number of individual possible. Obviously, it is not true in this particular case. In fact, it seems that the calibration process makes trade-offs between the tension and compression experiments, and provided that the linear compliances are identified with less accuracy on the compression than the tension experiments. The phenomenon of trade-off can be illustrated in the parameters space by depicting the PDFs of the elastic linear parameters, estimated with either the tension or compression experiments in Figure 9.

In Figure 9, on the left, it can be noticed that, if the calibration is carried out only from the compression experiments, because of the higher levels of error, the marginal parameters of $S_{11}^0$ are estimated less accurately than when they are estimated from only the tension experiment or when both types of experiments are jointly considered, which is consistent with previous results. For this repetition, adding the compression experiments to the tension one slightly decreases the error on the marginal estimation. This seems to be linked to the level of errors on the individual parameters as discussed earlier. The calibration of the other marginals and correlations do not exhibit specific problems provided the limitations of the population parameters values and the model issues. The KL-divergence is computed for two distributions: the joint distribution of $S_{11}^0$ and $\nu_{12}$ on the one hand and on the other hand for the full joint distribution, and illustrated in Figure 10.
<table><caption>**Table 10.** Averaged marginal parameters ($\Pi_{1,mean}$ from the first phase and $\Pi_{2,mean}$ from the second phase), alongside the averaged error ($\mathcal{E}_{\text{mean}}(\hat{\Pi}_2)$ and $\mathcal{E}_{\text{mean}}(\hat{\Pi}_2)$) over 20 repetitions of the calibration process with samples of 50 individuals each. The first two columns refer to the error made on the population parameters estimated in the first stage $\Pi_1$ (with the compression experiment) and the two left to parameters $\Pi_2$ estimated in the second phase (with the tension experiment). The coefficients of variation COV in % are indicated between brackets.</caption><thead><tr><th></th><th>S<sup>0</sup><sub>11</sub> [×10<sup>-6</sup>MPa<sup>-1</sup>] (1<sup>st</sup> phase)</th><th>ν<sub>12</sub> [×10<sup>-2</sup>] (1<sup>st</sup> phase)</th><th>S<sup>0</sup><sub>11</sub> [×10<sup>-6</sup>MPa<sup>-1</sup>] (2<sup>nd</sup> phase)</th><th>ν<sub>12</sub> [×10<sup>-2</sup>] (2<sup>nd</sup> phase)</th></tr></thead><tbody><tr><td>Exact Means</td><td>85.2</td><td>33.1</td><td>85.2</td><td>33.1</td></tr><tr><td>Averaged Calibrated Means</td><td>85.4(0.438)</td><td>33.1(0.343)</td><td>85.3(0.306)</td><td>33.1(0.366)</td></tr><tr><td>Averaged Errors On Calibrated Means (%)</td><td>0.364(78.9)</td><td>0.277(73.4)</td><td>0.251(73.1)</td><td>0.309(64.8)</td></tr><tr><td>Exact Standard-Deviations</td><td>1.31</td><td>0.660</td><td>1.31</td><td>0.660</td></tr><tr><td>Averaged Calibrated-Standard Deviations</td><td>1.07(30.1)</td><td>0.675(10.5)</td><td>1.20(21.1)</td><td>0.669(10.7)</td></tr><tr><td>Averaged Errors On Calibrated Standard-Deviations (%)</td><td>25.5(73.9)</td><td>9.44(59.5)</td><td>17.7(64.7)</td><td>7.48(108)</td></tr></tbody></table>

<table><caption>**Table 11.** Averaged marginal parameters correlation parameter $\hat{\rho}(S_{11}^0, \nu_{12})_{\text{mean},1/2}$ in both phases and averaged error $\mathcal{E}_{\text{mean}}(\hat{\rho}(S_{11}^0, \nu_{12}))_{\text{mean},1/2}$ over 20 repetitions of the calibration process with samples of 50 individuals each. The first column refers to the error made on the population parameters in the second stages of the sequential methodology (with the compression experiment on the left and with the tension experiment on the right). The coefficients of variation COV in % are indicated between brackets.</caption><thead><tr><th></th><th>ρ(S<sup>0</sup><sub>11</sub>, ν<sub>12</sub>)</th><th>ρ(S<sup>0</sup><sub>11</sub>, ν<sub>12</sub>)</th></tr></thead><tbody><tr><td>Exact Correlation</td><td>-0.741</td><td>-0.741</td></tr><tr><td>Averaged Calibrated Correlations</td><td>-0.771(20.8)</td><td>-0.715(18.5)</td></tr><tr><td>Averaged Errors On Calibrated Correlations (%)</td><td>18.2(68.2)</td><td>13.8(85.8)</td></tr></tbody></table>

there are two different curves to identify (longitudinal and transverse measurements), that makes 4 parameters for the two residual models. In total, $p = 17$ parameters have to be estimated. This illustrates a first interest to use a sequential scheme as with such techniques, only $p = 11$ parameters have to be calibrated in the different phases. The three calibration techniques are studied here. They respectively stand for the calibrated distribution in the second phase of the first implementation of the sequential strategy (the one obtained after the calibration with the compression experiments) labeled $C_1$, the calibrated distribution in the second phase of the second implementation (the one obtained after the calibration with the tension experiments) $C_2$ and finally, the one that comes from the complete calibration the joint distribution of $S_1^T, S_1^C, S_{11}^0$ and $\nu_{12}$, labeled $C_3$. Several items can be investigated among which the error on all population parameters in the three cases, as reported in Table 12.

In Table 12, it is noticeable that the estimation of the marginals of the asymptotic compliance in tension and compression are slightly downgraded in case $C_3$ compared to cases $C_1$ and $C_2$. This could come from the fact in case $C_3$, all variables are considered together and may be more difficult to tune than when the population parameters corresponding to a single type of experiment are estimated. Still, the differences remain small and may also originate from numerical issues. Furthermore, it appears that, surprisingly, the calibration that considers both compression and tension does not necessarily provide the best results for the calibration of the joint distribution of $S_{11}^0$ and $\nu_{12}$. Indeed, as these parameters express on both types

of experiments with 50 individuals each (so 100 individuals in total), we could expect it to be the best solution among the three possibilities given that this distribution is estimated with the highest number of individual possible. Obviously, it is not true in this particular case. In fact, it seems that the calibration process makes trade-offs between the tension and compression experiments, and provided that the linear compliances are identified with less accuracy on the compression than the tension experiments. The phenomenon of trade-off can be illustrated in the parameters space by depicting the PDFs of the elastic linear parameters, estimated with either the tension or compression experiments in Figure 9.

In Figure 9, on the left, it can be noticed that, if the calibration is carried out only from the compression experiments, because of the higher levels of error, the marginal parameters of $S_{11}^0$ are estimated less accurately than when they are estimated from only the tension experiment or when both types of experiments are jointly considered, which is consistent with previous results. For this repetition, adding the compression experiments to the tension one slightly decreases the error on the marginal estimation. This seems to be linked to the level of errors on the individual parameters as discussed earlier. The calibration of the other marginals and correlations do not exhibit specific problems provided the limitations of the population parameters values and the model issues. The KL-divergence is computed for two distributions: the joint distribution of $S_{11}^0$ and $\nu_{12}$ on the one hand and on the other hand for the full joint distribution, and illustrated in Figure 10.
<table><caption>**Table 12.** Averaged error $E_{\text{mean}}(\Pi)$ on the population parameters calibrated in the three cases over 20 repetitions of the calibration process with samples of 50 individuals each. For cases $C_1$ and $C_2$, the error on the parameters of the joint distribution of $S_{11}^0$ and $\nu_{12}$ correspond to those obtained after completion of the sequential scheme, that is to say after the second step. The coefficients of variation COV in % are indicated between brackets.</caption><thead><tr><th>Parameter</th><th>μ<sub>S<sub>11</sub><sup>0</sup></sub></th><th>μ<sub>ν<sub>12</sub></sub></th><th>μ<sub>S<sub>1</sub><sup>T</sup></sub></th><th>μ<sub>S<sub>1</sub><sup>C</sup></sub></th></tr></thead><tbody><tr><td>Averaged Errors in Case C<sub>1</sub> (%)</td><td>0.376(76.5)</td><td>0.273(74.6)</td><td>0.738(68.7)</td><td>1.52(74.0)</td></tr><tr><td>Averaged Errors in Case C<sub>2</sub> (%)</td><td>0.251(73.1)</td><td>0.309(64.8)</td><td>0.736(66.7)</td><td>1.60(72.2)</td></tr><tr><td>Averaged Errors in Case C<sub>3</sub> (%)</td><td>0.259(78.1)</td><td>0.281(68.2)</td><td>0.683(77.9)</td><td>1.61(72.6)</td></tr></tbody><thead><tr><th>Parameter</th><th>σ<sub>S<sub>11</sub><sup>0</sup></sub></th><th>σ<sub>ν<sub>12</sub></sub></th><th>σ<sub>S<sub>1</sub><sup>T</sup></sub></th><th>σ<sub>S<sub>1</sub><sup>C</sup></sub></th></tr></thead><tbody><tr><td>Averaged Errors in Case C<sub>1</sub> (%)</td><td>26.3(67.5)</td><td>9.41(54.8)</td><td>7.82(68.5)</td><td>6.68(84.9)</td></tr><tr><td>Averaged Errors in Case C<sub>2</sub> (%)</td><td>17.7(64.7)</td><td>7.48(108)</td><td>7.85(67.4)</td><td>7.11(75.7)</td></tr><tr><td>Averaged Errors in Case C<sub>3</sub> (%)</td><td>18.1(69.5)</td><td>7.96(87.3)</td><td>7.90(78.5)</td><td>7.43(73.3)</td></tr></tbody><thead><tr><th>Parameter</th><th>ρ(ν<sub>12</sub>, S<sub>1</sub><sup>T</sup>)</th><th>ρ(ν<sub>12</sub>, S<sub>1</sub><sup>C</sup>)</th><th>ρ(S<sub>1</sub><sup>C</sup>, S<sub>11</sub><sup>0</sup>)</th><th>ρ(S<sub>1</sub><sup>T</sup>, S<sub>11</sub><sup>0</sup>)</th><th>ρ(S<sub>11</sub><sup>0</sup>, ν<sub>12</sub>)</th></tr></thead><tbody><tr><td>Averaged Errors in Case C<sub>1</sub> (%)</td><td>18.8(83.0)</td><td>4.91(90.4)</td><td>41.1(69.0)</td><td>17.9(81.4)</td><td>14.7(74.6)</td></tr><tr><td>Averaged Errors in Case C<sub>2</sub> (%)</td><td>26.7(97.8)</td><td>5.44(90.3)</td><td>53.9(72.8)</td><td>16.9(69.4)</td><td>13.8(85.8)</td></tr><tr><td>Averaged Errors in Case C<sub>3</sub> (%)</td><td>26.3(110.6)</td><td>8.79(168)</td><td>68.0(121)</td><td>25.8(76.9)</td><td>16.4(165)</td></tr></tbody></table>

<caption>**Fig. 9.** For the 17th repetition, PDF of $S_{11}^0$ and $\nu_{12}$ and the corresponding individual parameters for the calibration from case $C_1$ (with the compression experiments in the second phase), from case $C_2$ (with the tension experiments in the second phase) and considering both types of experiments (case $C_3$).</caption>

In Figure 10, it is interesting to notice that the KL-divergences are similar in terms of values and range of variation, either for the joint distribution of $S_{11}^0$ and $\nu_{12}$ or for the complete joint distribution. Still, it is possible to observe the KL-divergences have significant range of variation (from $10^{-1}$ to over $10^8$). This can be partly explained by the fact that the complete distribution is parameterized by 13 parameters (against 5 for the joint distribution of $S_{11}^0$ and $\nu_{12}$) whose 2 correlations exhibit large level of errors (in particular $\rho(S_1^C, S_{11}^0)$). These results show that

the three different cases provide similar results in terms of accuracy of the calibrated distribution, suggesting that implementing a sequential approach does not downgrade much the estimation of the population parameters. A final comparison can be made to study the computational behavior of the sequential scheme with respect to the method that uses design matrices. Two indicators are studied here: the number of (material) model evaluations and the number of likelihood evaluations, all reported in Figure 11 for the three cases over the twenty repetitions.
<table><caption>**Table 12.** Averaged error $E_{\text{mean}}(\Pi)$ on the population parameters calibrated in the three cases over 20 repetitions of the calibration process with samples of 50 individuals each. For cases $C_1$ and $C_2$, the error on the parameters of the joint distribution of $S_{11}^0$ and $\nu_{12}$ correspond to those obtained after completion of the sequential scheme, that is to say after the second step. The coefficients of variation COV in % are indicated between brackets.</caption><thead><tr><th>Parameter</th><th>μ<sub>S<sub>11</sub><sup>0</sup></sub></th><th>μ<sub>ν<sub>12</sub></sub></th><th>μ<sub>S<sub>1</sub><sup>T</sup></sub></th><th>μ<sub>S<sub>1</sub><sup>C</sup></sub></th></tr></thead><tbody><tr><td>Averaged Errors in Case C<sub>1</sub> (%)</td><td>0.376(76.5)</td><td>0.273(74.6)</td><td>0.738(68.7)</td><td>1.52(74.0)</td></tr><tr><td>Averaged Errors in Case C<sub>2</sub> (%)</td><td>0.251(73.1)</td><td>0.309(64.8)</td><td>0.736(66.7)</td><td>1.60(72.2)</td></tr><tr><td>Averaged Errors in Case C<sub>3</sub> (%)</td><td>0.259(78.1)</td><td>0.281(68.2)</td><td>0.683(77.9)</td><td>1.61(72.6)</td></tr></tbody><thead><tr><th>Parameter</th><th>σ<sub>S<sub>11</sub><sup>0</sup></sub></th><th>σ<sub>ν<sub>12</sub></sub></th><th>σ<sub>S<sub>1</sub><sup>T</sup></sub></th><th>σ<sub>S<sub>1</sub><sup>C</sup></sub></th></tr></thead><tbody><tr><td>Averaged Errors in Case C<sub>1</sub> (%)</td><td>26.3(67.5)</td><td>9.41(54.8)</td><td>7.82(68.5)</td><td>6.68(84.9)</td></tr><tr><td>Averaged Errors in Case C<sub>2</sub> (%)</td><td>17.7(64.7)</td><td>7.48(108)</td><td>7.85(67.4)</td><td>7.11(75.7)</td></tr><tr><td>Averaged Errors in Case C<sub>3</sub> (%)</td><td>18.1(69.5)</td><td>7.96(87.3)</td><td>7.90(78.5)</td><td>7.43(73.3)</td></tr></tbody><thead><tr><th>Parameter</th><th>ρ(ν<sub>12</sub>, S<sub>1</sub><sup>T</sup>)</th><th>ρ(ν<sub>12</sub>, S<sub>1</sub><sup>C</sup>)</th><th>ρ(S<sub>1</sub><sup>C</sup>, S<sub>11</sub><sup>0</sup>)</th><th>ρ(S<sub>1</sub><sup>T</sup>, S<sub>11</sub><sup>0</sup>)</th><th>ρ(S<sub>11</sub><sup>0</sup>, ν<sub>12</sub>)</th></tr></thead><tbody><tr><td>Averaged Errors in Case C<sub>1</sub> (%)</td><td>18.8(83.0)</td><td>4.91(90.4)</td><td>41.1(69.0)</td><td>17.9(81.4)</td><td>14.7(74.6)</td></tr><tr><td>Averaged Errors in Case C<sub>2</sub> (%)</td><td>26.7(97.8)</td><td>5.44(90.3)</td><td>53.9(72.8)</td><td>16.9(69.4)</td><td>13.8(85.8)</td></tr><tr><td>Averaged Errors in Case C<sub>3</sub> (%)</td><td>26.3(110.6)</td><td>8.79(168)</td><td>68.0(121)</td><td>25.8(76.9)</td><td>16.4(165)</td></tr></tbody></table>

<caption>**Fig. 9.** For the 17th repetition, PDF of $S_{11}^0$ and $\nu_{12}$ and the corresponding individual parameters for the calibration from case $C_1$ (with the compression experiments in the second phase), from case $C_2$ (with the tension experiments in the second phase) and considering both types of experiments (case $C_3$).</caption>

In Figure 10, it is interesting to notice that the KL-divergences are similar in terms of values and range of variation, either for the joint distribution of $S_{11}^0$ and $\nu_{12}$ or for the complete joint distribution. Still, it is possible to observe the KL-divergences have significant range of variation (from $10^{-1}$ to over $10^8$). This can be partly explained by the fact that the complete distribution is parameterized by 13 parameters (against 5 for the joint distribution of $S_{11}^0$ and $\nu_{12}$) whose 2 correlations exhibit large level of errors (in particular $\rho(S_1^C, S_{11}^0)$). These results show that

the three different cases provide similar results in terms of accuracy of the calibrated distribution, suggesting that implementing a sequential approach does not downgrade much the estimation of the population parameters. A final comparison can be made to study the computational behavior of the sequential scheme with respect to the method that uses design matrices. Two indicators are studied here: the number of (material) model evaluations and the number of likelihood evaluations, all reported in Figure 11 for the three cases over the twenty repetitions.
<figure><img src="image_10.png" /><figcaption>**Fig. 10.** On the left, KL-divergences between the calibrated and exact joint distribution of $S_{11}^0$ and $\nu_{12}$ and on the right, KL-divergences between the calibrated and exact joint full distributions for the 20 repetitions and 3 cases. Remember that the KL-divergence expresses a notion of distance between two distributions and the lower the more accurate the calibration of $f_{\Theta|\Pi}$ is.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 11.** On the left, number of model evaluations for the three cases and on the right, number of likelihood evaluations for the three cases. In cases $C_1$ and $C_2$, the costs of both stages are combined together.</figcaption></figure>

Figure 11 shows that with a sequential scheme, in total over the two phases, the likelihood function is evaluated a comparable number of times in each case, even if in Case $C_2$ (when ending with the tension experiments), a slight increase can be noticed. Still, the increase remains an within acceptable range. On the contrary, the interest of using a sequential approach arises with the study of number of model evaluations, which is of prime interest. Indeed, the costly part of the likelihood computation refers to the resolution of the minimization problem defined in equation (20). Focusing on the number of model evaluations demonstrate the interest of using a sequential scheme as over the two phases, the number of model evaluations is approximately divided by two from case $C_3$ to case $C_1$ and by 40% from case $C_3$ to case $C_2$, which can be considered as a significant gain in terms of computational cost. Still, the number of model evaluations remains high, about $10^8$. This number can be partly explained by the choices and settings of the optimization algorithms used for the minimization of both the opposite log-likelihood and function $g_i(\cdot)$. For more complex, it may be necessary to combine the sequential strategy with other approaches such as surrogate modeling to guarantee the applicability of the calibration method. If such solution is chosen, the surrogate model would be easier to train compared to the standard case. Indeed, throughout the sequential process, the search spaces over which population parameters evolve are limited to preserve consistency between the different steps. Therefore, the range of validity of the chosen surrogates can be smaller, making them easier to build, contrary to the standard case in which the

entire search space is considered. Note that, as in most procedures implementing surrogate models, it should be checked that the trained surrogate model does not introduce errors that could impair the calibration process. The observed reduction can be explained by different reasons. First, notice in Figure 11, for the three cases, the number of likelihood evaluations are (in total for cases $C_1$ and $C_2$) similar. In fact, the number of likelihood evaluations in each stage of the sequential approaches is about half of the number of likelihood evaluations when design matrices are used. This can be understood as there more parameters to estimate (17 instead of 11). However, for case $C_3$, for each likelihood evaluation, the models for both the compression and tension experiments are identified. This adds to the fact that, in the second step of the methodology, the individual calibrations are eased. Indeed, the mean parameters around which the individual deviations $b_i$ are searched (see Eq. (20)) are already estimated with few error (smaller than 1% in average). This is even more important here because of the concentration of the linear elastic parameters around their average values. This is illustrated in Figure 12, which compares for the twenty repetitions, the number of model evaluations for the second steps of the sequential strategy, i.e., case $C_1$ with the compression experiments and case $C_2$ with the tension experiments to their counterpart of case $C_3$.

Figure 12 confirms that thanks to the focus realized by the first identification for the mean parameters, it seems easier to identify the individual parameters. Indeed, the number of model evaluations is almost cut by half between the case $C_3$ and the second stages of both cases $C_1$ and
<figure><img src="image_10.png" /><figcaption>**Fig. 10.** On the left, KL-divergences between the calibrated and exact joint distribution of $S_{11}^0$ and $\nu_{12}$ and on the right, KL-divergences between the calibrated and exact joint full distributions for the 20 repetitions and 3 cases. Remember that the KL-divergence expresses a notion of distance between two distributions and the lower the more accurate the calibration of $f_{\Theta|\Pi}$ is.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 11.** On the left, number of model evaluations for the three cases and on the right, number of likelihood evaluations for the three cases. In cases $C_1$ and $C_2$, the costs of both stages are combined together.</figcaption></figure>

Figure 11 shows that with a sequential scheme, in total over the two phases, the likelihood function is evaluated a comparable number of times in each case, even if in Case $C_2$ (when ending with the tension experiments), a slight increase can be noticed. Still, the increase remains an within acceptable range. On the contrary, the interest of using a sequential approach arises with the study of number of model evaluations, which is of prime interest. Indeed, the costly part of the likelihood computation refers to the resolution of the minimization problem defined in equation (20). Focusing on the number of model evaluations demonstrate the interest of using a sequential scheme as over the two phases, the number of model evaluations is approximately divided by two from case $C_3$ to case $C_1$ and by 40% from case $C_3$ to case $C_2$, which can be considered as a significant gain in terms of computational cost. Still, the number of model evaluations remains high, about $10^8$. This number can be partly explained by the choices and settings of the optimization algorithms used for the minimization of both the opposite log-likelihood and function $g_i(\cdot)$. For more complex, it may be necessary to combine the sequential strategy with other approaches such as surrogate modeling to guarantee the applicability of the calibration method. If such solution is chosen, the surrogate model would be easier to train compared to the standard case. Indeed, throughout the sequential process, the search spaces over which population parameters evolve are limited to preserve consistency between the different steps. Therefore, the range of validity of the chosen surrogates can be smaller, making them easier to build, contrary to the standard case in which the

entire search space is considered. Note that, as in most procedures implementing surrogate models, it should be checked that the trained surrogate model does not introduce errors that could impair the calibration process. The observed reduction can be explained by different reasons. First, notice in Figure 11, for the three cases, the number of likelihood evaluations are (in total for cases $C_1$ and $C_2$) similar. In fact, the number of likelihood evaluations in each stage of the sequential approaches is about half of the number of likelihood evaluations when design matrices are used. This can be understood as there more parameters to estimate (17 instead of 11). However, for case $C_3$, for each likelihood evaluation, the models for both the compression and tension experiments are identified. This adds to the fact that, in the second step of the methodology, the individual calibrations are eased. Indeed, the mean parameters around which the individual deviations $b_i$ are searched (see Eq. (20)) are already estimated with few error (smaller than 1% in average). This is even more important here because of the concentration of the linear elastic parameters around their average values. This is illustrated in Figure 12, which compares for the twenty repetitions, the number of model evaluations for the second steps of the sequential strategy, i.e., case $C_1$ with the compression experiments and case $C_2$ with the tension experiments to their counterpart of case $C_3$.

Figure 12 confirms that thanks to the focus realized by the first identification for the mean parameters, it seems easier to identify the individual parameters. Indeed, the number of model evaluations is almost cut by half between the case $C_3$ and the second stages of both cases $C_1$ and
<figure><img src="image_12.png" /><figcaption>**Fig. 12.** On the left, total number of model evaluations for the estimation of the individual parameters with the compression experiments in case $C_3$ and on the second stage of case $C_1$. On the right, total number of model evaluations for the estimation of the individual parameters with the tension experiments in case $C_3$ and on the second stage of case $C_2$.</figcaption></figure>

$C_2$. This significant improvement partly explains the gain observed in terms of model evaluations. A further study on the convergence behavior should be carried out to demonstrate rigorously the interest of the sequential approach compared to classical mixed-effects methods with design matrices.

This study of the sequential scheme allows to illustrate some advantages and drawbacks. Among the benefits of such a strategy can be cited a proper estimation of the population parameters, at least as accurate as the one that can be obtained with the use of design matrices in this test-case. Furthermore, it allows to significantly reduce the number of model evaluations necessary to carry out the calibration of the population parameters. Finally, it also reduces the number of population parameters of estimated in each calibration problem, allowing to consider more complex phenomena in future works. However, the process exhibits its own limitations. For instance, it could be interesting to test different solutions for the trust regions instead of a simple hypercube. In addition, the routine could have been continued until reaching convergence as in a fixed-point approach at the expense of a larger computational costs. Still, this test-case is interesting as the results can be compared to the standard calibration process that sometimes cannot be carried out because of the computational costs and the dimension of the optimization problem.

## 5 Conclusion

This article presents a sequential calibration procedure that aims to calibrate complex material constitutive models compliant with the mixed-effects framework using multivariate data. The method consists in defining a sequence of calibration subproblems, each one corresponding to a subset of the available experimental data. The subdivision of the complete problem into subproblems can be based on the choice of the expert and on sensitivity analyses. Indeed, well-chosen tests can activate different parts of the model, i.e. different subsets of the model parameters. Thus, each type of test makes it possible to define a subproblem whose objective is to identify

the joint distribution of a subset of the model parameters. A parameter can be involved in several subproblems. In this case, it is identified once and the search space of the corresponding distribution parameters is reduced to a trust region in the subsequent calibrations, to ensure consistency between the different subproblems.

The method is applied to a virtual test case to the calibration of the orthotropic model in plane stress with nonlinear longitudinal behavior. This test-case is interesting as it allows to compare the sequential strategy to its usual counterpart using design matrices that is not necessary available in all configurations. A first implementation allows to split the identification of the joint distribution of the compliances with the usual elementary UD 0°, UD 90° and ±45° laminates. The second allows to divide the calibration of the joint distribution of the longitudinal and transverse nonlinear elasticity of UD0° laminates between the tension and compression experiments. The consistency of the results throughout the calibration process is ensured by bounding the parameters estimated twice. The results of the methodology applied on synthetic data generated with the exact model demonstrate the ability of the proposed procedure to estimate properly the model parameters distribution in the presence of significant material variability. The results obtained considering the different orders of calibration are investigated and compared to the straightforward calibration of the complete joint distribution using design matrices. It shows that in particular this calibration method is not necessarily the best in this test case, the one step calibration method is not necessarily the best, in particular because of the trade-offs made between the different type of experiments. An application (not shown in this paper) with experiments carried on a material with carbon and epoxy resin confirm this point and shows in particular that with a sequential strategy, the number of model evaluations to provide an estimation of the full joint distribution is significantly reduced. This is a key feature, and combined to the preservation of the calibrated distributions throughout the calibration process, it shows that sequential calibration is not only compatible with mixed-effects models but scales better to more complex model with a high number of parameters to be calibrated.
<figure><img src="image_12.png" /><figcaption>**Fig. 12.** On the left, total number of model evaluations for the estimation of the individual parameters with the compression experiments in case $C_3$ and on the second stage of case $C_1$. On the right, total number of model evaluations for the estimation of the individual parameters with the tension experiments in case $C_3$ and on the second stage of case $C_2$.</figcaption></figure>

$C_2$. This significant improvement partly explains the gain observed in terms of model evaluations. A further study on the convergence behavior should be carried out to demonstrate rigorously the interest of the sequential approach compared to classical mixed-effects methods with design matrices.

This study of the sequential scheme allows to illustrate some advantages and drawbacks. Among the benefits of such a strategy can be cited a proper estimation of the population parameters, at least as accurate as the one that can be obtained with the use of design matrices in this test-case. Furthermore, it allows to significantly reduce the number of model evaluations necessary to carry out the calibration of the population parameters. Finally, it also reduces the number of population parameters of estimated in each calibration problem, allowing to consider more complex phenomena in future works. However, the process exhibits its own limitations. For instance, it could be interesting to test different solutions for the trust regions instead of a simple hypercube. In addition, the routine could have been continued until reaching convergence as in a fixed-point approach at the expense of a larger computational costs. Still, this test-case is interesting as the results can be compared to the standard calibration process that sometimes cannot be carried out because of the computational costs and the dimension of the optimization problem.

## 5 Conclusion

This article presents a sequential calibration procedure that aims to calibrate complex material constitutive models compliant with the mixed-effects framework using multivariate data. The method consists in defining a sequence of calibration subproblems, each one corresponding to a subset of the available experimental data. The subdivision of the complete problem into subproblems can be based on the choice of the expert and on sensitivity analyses. Indeed, well-chosen tests can activate different parts of the model, i.e. different subsets of the model parameters. Thus, each type of test makes it possible to define a subproblem whose objective is to identify

the joint distribution of a subset of the model parameters. A parameter can be involved in several subproblems. In this case, it is identified once and the search space of the corresponding distribution parameters is reduced to a trust region in the subsequent calibrations, to ensure consistency between the different subproblems.

The method is applied to a virtual test case to the calibration of the orthotropic model in plane stress with nonlinear longitudinal behavior. This test-case is interesting as it allows to compare the sequential strategy to its usual counterpart using design matrices that is not necessary available in all configurations. A first implementation allows to split the identification of the joint distribution of the compliances with the usual elementary UD 0°, UD 90° and ±45° laminates. The second allows to divide the calibration of the joint distribution of the longitudinal and transverse nonlinear elasticity of UD0° laminates between the tension and compression experiments. The consistency of the results throughout the calibration process is ensured by bounding the parameters estimated twice. The results of the methodology applied on synthetic data generated with the exact model demonstrate the ability of the proposed procedure to estimate properly the model parameters distribution in the presence of significant material variability. The results obtained considering the different orders of calibration are investigated and compared to the straightforward calibration of the complete joint distribution using design matrices. It shows that in particular this calibration method is not necessarily the best in this test case, the one step calibration method is not necessarily the best, in particular because of the trade-offs made between the different type of experiments. An application (not shown in this paper) with experiments carried on a material with carbon and epoxy resin confirm this point and shows in particular that with a sequential strategy, the number of model evaluations to provide an estimation of the full joint distribution is significantly reduced. This is a key feature, and combined to the preservation of the calibrated distributions throughout the calibration process, it shows that sequential calibration is not only compatible with mixed-effects models but scales better to more complex model with a high number of parameters to be calibrated.
Though the proposed method seems promising, the proposed application case is simple and very fast to evaluate. More complex models usually require to solve differential equations which can be much more time consuming, making the overall computational cost of the method intractable with direct calls to the model. In that regard, combining the proposed approach with surrogate models falls as a direct perspective of this work.

## References

[1] U.D.O. Defense, Technomic Publishing Company, Materials Sciences Corporation, American Society for Testing, and Materials, Composite Materials Handbook-MIL 17, Vol. III: Materials Usage, Design, and Analysis, The Composite Materials Handbook-MIL 17. Taylor & Francis, 1999

[2] A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, Society for Industrial and Applied Mathematics, 2005.

[3] L. Wasserman, All of Statistics: A Concise Course in Statistical Inference, Springer Publishing Company, Incorporated, 2010

[4] A. Antoniadis, R. Carmona, Régression non linéaire et applications. Economie et statistiques avancées, Economica, 1992

[5] G.C. Ballesteros, P. Angelikopoulos, C. Papadimitriou, P. Koumoutsakos, Bayesian hierarchical models for uncertainty quantification in structural dynamics, Vulnerabil. Uncertain. Risk: Quantif. Mitig. Manag. **162**, 1615–1624 (2014)

[6] M. Song, I. Behmanesh, B. Moaveni, C. Papadimitriou, Accounting for modeling errors and inherent structural variability through a hierarchical bayesian model updating approach: an overview, Sensors **20**, 3874–3901 (2020)

[7] E. Demidenko, Mixed Models. Theory and Applications with R, John Wiley & Sons, 2013

[8] M. Lavielle, Mixed Effects Models for the Population Approach: Models, Tasks, Methods and Tools, CRC press, 2014

[9] P. Congdon, Bayesian Statistical Modelling, John Wiley & Sons, 2007

[10] C. Laboulfie, M Balesdent, L. Brevault, F.-X. Irisarri, J.-F. Maire, S. Da Veiga, R. Le Riche, Calibration of material model parameters using mixed-effects booktitle = 25ème Congrès Français de Mécanique, model, 2021, pp. 258–295

[11] J. Lemaitre, J.-L. Chaboche, Mechanics of solid materials, Cambridge University Press, 1994

[12] J. Lemaitre, R. Desmorat. Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures. Springer Science & Business Media, 2006

[13] S. Moreau, A. Chrysochoos, J.-M. Muracciole, B. Wattrisse, Analysis of thermoelastic effects accompanying the deformation of pmma and pc polymers, Comptes Rendus Méc. **333**, 648–653 (2005)

[14] R. Schapery, On the characterization of nonlinear viscoelastic materials, Polymer Eng. Sci. **9**, 295–310 (1969)

[15] J.-L. Chaboche, A review of some plasticity and viscoplasticity constitutive Theories, Int. J. Plasticity **24**, 1642–1693 (2008)

[16] B. Miled, I. Doghri, L. Delannay, Coupled viscoelastic-viscoplastic modeling of homogeneous and isotropic polymers: numerical algorithm and analytical solutions.

Comput. Methods Appl. Mech. Eng. **200**, 3381–3394 (2011).

[17] A. Krairi, I. Doghri, A thermodynamically-based constitutive model for thermoplastic polymers coupling viscoelasticity, viscoplasticity and ductile damage, Int. J. Plasticity **60**, 163–181 (2014)

[18] F. Praud, G. Chatzigeorgiou, J. Bikard, F. Meraghni, Phenomenological multi-mechanisms constitutive modelling for thermoplastic polymers, implicit implementation and experimental validation, Mech. Mater. **114**, 9–29 (2017)

[19] C. Hochard, P.-A. Aubourg, J.-P. Charles, Modelling of the mechanical behaviour of woven-fabric cfrp laminates up to failure, Composites Sci. Technol. **61**, 221–230 (2001)

[20] P. Ladeveze, E. LeDantec, Damage modelling of the elementary ply for laminated composites, Composites Sci. Technol. **43**, 257–267 (1992)

[21] A. Launay, M.H. Maitournam, Y. Marco, I. Raoult, F. Szmytka, Cyclic behaviour of short glass fibre reinforced polyamide: experimental study and constitutive equations, Int. J. Plasticity **27**, 1267–1293 (2011)

[22] F. Praud, G. Chatzigeorgiou, Y. Chemisky, F. Meraghni, Hybrid micromechanical-phenomenological modelling of anisotropic damage and anelasticity induced by microcracks in unidirectional composites, Composite Struct. **182**, 223–236 (2017)

[23] F. Lachaud, Delaminage de materiaux composites a fibres de carbone et a matrices organiques: étude numérique et experimentale, suivi par émission acoustique. PhD thesis, 1997

[24] A. Forrester and A. Sobester, A.and Keane. Engineering Design via Surrogate Modelling – A Practical Guide, Wiley, 2008

[25] J.-M. Berthelot, Matériaux Composites-Comportement Mécanique et Analyse des Structures, 2012.

[26] M. Gallagher, J. Doherty, Parameter estimation and uncertainty analysis for a watershed model, Environ. Model. Softw. **22**, 1000–1020 (2007)

[27] M. Kennedy, A. O’Hagan, Bayesian calibration of computer models, J. Roy. Stat. Soc. B (Stat. Methodol.) **63**, 425–464 (2001)

[28] K. Levenberg, A method for the solution of certain nonlinear problems in least Squares, Q. Appl. Math. **2**, 164–168 (1944)

[29] D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameter, J. Soc. Ind. Appl. Math. **11**, 431–441 (1963)

[30] R. Pintelon, J. Schoukens, System Identification, John Wiley & Sons, Ltd, 2012

[31] F. Meraghni, Y. Chemisky, B. Piotrowski, R. Echchorfi, N. Bourgeois, E. Patoor, Parameter identification of a thermodynamic model for superelastic shape memory alloys using analytical calculation of the sensitivity matrix, Eur. J. Mech. A/Solids **45**, 226–237 (2014)

[32] J. Pinheiro, D. Bates, Mixed-Effect Models in S and S-plus, Vol. 96, 2002

[33] J.B. Nagel, B. Sudret, A unified framework for multilevel uncertainty quantification in Bayesian inverse problems, Probab. Eng. Mech. **43** 68–84 (2016)

[34] S.L. Beal, L.B. Sheiner, Estimating population kinetics, Crit. Rev. Biomed. Eng. **8**, 195–222 (1982)

[35] D. Blei, A. Kucukelbir, J. McAuliff, Variational inference: a review for statisticians, J. Am. Stat. Assoc. **112**, 859–877 (2017)
Though the proposed method seems promising, the proposed application case is simple and very fast to evaluate. More complex models usually require to solve differential equations which can be much more time consuming, making the overall computational cost of the method intractable with direct calls to the model. In that regard, combining the proposed approach with surrogate models falls as a direct perspective of this work.

## References

[1] U.D.O. Defense, Technomic Publishing Company, Materials Sciences Corporation, American Society for Testing, and Materials, Composite Materials Handbook-MIL 17, Vol. III: Materials Usage, Design, and Analysis, The Composite Materials Handbook-MIL 17. Taylor & Francis, 1999

[2] A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, Society for Industrial and Applied Mathematics, 2005.

[3] L. Wasserman, All of Statistics: A Concise Course in Statistical Inference, Springer Publishing Company, Incorporated, 2010

[4] A. Antoniadis, R. Carmona, Régression non linéaire et applications. Economie et statistiques avancées, Economica, 1992

[5] G.C. Ballesteros, P. Angelikopoulos, C. Papadimitriou, P. Koumoutsakos, Bayesian hierarchical models for uncertainty quantification in structural dynamics, Vulnerabil. Uncertain. Risk: Quantif. Mitig. Manag. **162**, 1615–1624 (2014)

[6] M. Song, I. Behmanesh, B. Moaveni, C. Papadimitriou, Accounting for modeling errors and inherent structural variability through a hierarchical bayesian model updating approach: an overview, Sensors **20**, 3874–3901 (2020)

[7] E. Demidenko, Mixed Models. Theory and Applications with R, John Wiley & Sons, 2013

[8] M. Lavielle, Mixed Effects Models for the Population Approach: Models, Tasks, Methods and Tools, CRC press, 2014

[9] P. Congdon, Bayesian Statistical Modelling, John Wiley & Sons, 2007

[10] C. Laboulfie, M Balesdent, L. Brevault, F.-X. Irisarri, J.-F. Maire, S. Da Veiga, R. Le Riche, Calibration of material model parameters using mixed-effects booktitle = 25ème Congrès Français de Mécanique, model, 2021, pp. 258–295

[11] J. Lemaitre, J.-L. Chaboche, Mechanics of solid materials, Cambridge University Press, 1994

[12] J. Lemaitre, R. Desmorat. Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures. Springer Science & Business Media, 2006

[13] S. Moreau, A. Chryssochoos, J.-M. Muracciole, B. Wattrisse, Analysis of thermoelastic effects accompanying the deformation of pmma and pc polymers, Comptes Rendus Méc. **333**, 648–653 (2005)

[14] R. Schapery, On the characterization of nonlinear viscoelastic materials, Polymer Eng. Sci. **9**, 295–310 (1969)

[15] J.-L. Chaboche, A review of some plasticity and viscoplasticity constitutive Theories, Int. J. Plasticity **24**, 1642–1693 (2008)

[16] B. Miled, I. Doghri, L. Delannay, Coupled viscoelastic-viscoplastic modeling of homogeneous and isotropic polymers: numerical algorithm and analytical solutions.

Comput. Methods Appl. Mech. Eng. **200**, 3381–3394 (2011).

[17] A. Krairi, I. Doghri, A thermodynamically-based constitutive model for thermoplastic polymers coupling viscoelasticity, viscoplasticity and ductile damage, Int. J. Plasticity **60**, 163–181 (2014)

[18] F. Praud, G. Chatzigeorgiou, J. Bikard, F. Meraghni, Phenomenological multi-mechanisms constitutive modelling for thermoplastic polymers, implicit implementation and experimental validation, Mech. Mater. **114**, 9–29 (2017)

[19] C. Hochard, P.-A. Aubourg, J.-P. Charles, Modelling of the mechanical behaviour of woven-fabric cfrp laminates up to failure, Composites Sci. Technol. **61**, 221–230 (2001)

[20] P. Ladeveze, E. LeDantec, Damage modelling of the elementary ply for laminated composites, Composites Sci. Technol. **43**, 257–267 (1992)

[21] A. Launay, M.H. Maitournam, Y. Marco, I. Raoult, F. Szmytka, Cyclic behaviour of short glass fibre reinforced polyamide: experimental study and constitutive equations, Int. J. Plasticity **27**, 1267–1293 (2011)

[22] F. Praud, G. Chatzigeorgiou, Y. Chemisky, F. Meraghni, Hybrid micromechanical-phenomenological modelling of anisotropic damage and anelasticity induced by microcracks in unidirectional composites, Composite Struct. **182**, 223–236 (2017)

[23] F. Lachaud, Delaminage de materiaux composites a fibres de carbone et a matrices organiques: étude numérique et experimentale, suivi par émission acoustique. PhD thesis, 1997

[24] A. Forrester and A. Sobester, A.and Keane. Engineering Design via Surrogate Modelling – A Practical Guide, Wiley, 2008

[25] J.-M. Berthelot, Matériaux Composites-Comportement Mécanique et Analyse des Structures, 2012.

[26] M. Gallagher, J. Doherty, Parameter estimation and uncertainty analysis for a watershed model, Environ. Model. Softw. **22**, 1000–1020 (2007)

[27] M. Kennedy, A. O’Hagan, Bayesian calibration of computer models, J. Roy. Stat. Soc. B (Stat. Methodol.) **63**, 425–464 (2001)

[28] K. Levenberg, A method for the solution of certain nonlinear problems in least Squares, Q. Appl. Math. **2**, 164–168 (1944)

[29] D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameter, J. Soc. Ind. Appl. Math. **11**, 431–441 (1963)

[30] R. Pintelon, J. Schoukens, System Identification, John Wiley & Sons, Ltd, 2012

[31] F. Meraghni, Y. Chemisky, B. Piotrowski, R. Echchorfi, N. Bourgeois, E. Patoor, Parameter identification of a thermodynamic model for superelastic shape memory alloys using analytical calculation of the sensitivity matrix, Eur. J. Mech. A/Solids **45**, 226–237 (2014)

[32] J. Pinheiro, D. Bates, Mixed-Effect Models in S and S-plus, Vol. 96, 2002

[33] J.B. Nagel, B. Sudret, A unified framework for multilevel uncertainty quantification in Bayesian inverse problems, Probab. Eng. Mech. **43** 68–84 (2016)

[34] S.L. Beal, L.B. Sheiner, Estimating population kinetics, Crit. Rev. Biomed. Eng. **8**, 195–222 (1982)

[35] D. Blei, A. Kucukelbir, J. McAuliff, Variational inference: a review for statisticians, J. Am. Stat. Assoc. **112**, 859–877 (2017)
[36] J. Wakefield, The Bayesian analysis of population pharmacokinetic models, J. Am. Stat. Assoc. **91**, 62–75 (1996)

[37] P. Damlen, J. Wakefield, S. Walker, Gibbs sampling for Bayesian non-conjugate and hierarchical models by using auxiliary variables, J. Roy. Stat. Soc. B (Stat. Methodol.) **61**, 331–344 (1999)

[38] M. Davidian, D.M. Giltinan, Nonlinear Models for Repeated Measurement Data, Chapman & Hall/CRC Monographs on Statistics & Applied Probability, Taylor & Francis, 1995

[39] R. Fisher, K. Pearson, On an Absolute Criterion for Fitting Frequency Curves. Messenger of Mathematics, 1911

[40] D. Hall, M. Clutter, Multivariate multilevel nonlinear mixed-effects models for timber yield predictions, Biometrics **60**, 16–24 (2004)

[41] J. McFarland, S. Mahadevan, L. Romeroand, V. Swiler, Calibration and uncertainty analysis for computer simulations with multivariate output, AIAA J. **46**, 1253–1265 (2008)

[42] N. Metropolis, S. Ulam, The monte carlo method, J. Am. Stat. Assoc. **44**, 335–341 (1949)

[43] D. Bates, D. Hamilton, D. Watts, Calculation of intrinsic and parameter-effects curvatures for nonlinear regression models. Commun. Stat. Simul. Comput. **12**, 469–477 (1983)

[44] N. Hansen, S. Müller, P. Koumoutsakos, Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES), Evol. Comput. **11**, 1–18 (2003)

[45] W. Oberkampf, C. Roy, Verification and Validation in Scientific Computing, Cambridge University Press, 2010

[46] L. Schwer, Guide for Verification and Validation in Computational Solid Mechanics, ASME, 2006

[47] B. Jung, H. Yoon, H. Oh, G. Lee, M. Yoo, B. Youn, Y. Hu. Hierarchical model calibration for designing piezoelectric energy harvester in the presence of variability in material properties and geometry, Struct. Multidiscipl. Optim. **53**, 161–173 (2016)

[48] B.D. Youn, B. Jung, Z. Xi, S.B. Kim, W.R. Lee, A hierarchical framework for statistical model calibration in engineering product development, Comput. Methods Appl. Mech. Eng. **200**, 1421–1431 (2011)

[49] A. Urbina, T. Paez, T. Consulting, A bayes network approach to uncertainty quantification in hierarchically-developed computational models, Int. J. Uncertain. Quantif. **2**, 173–193 (2012)

[50] J. Ye, M. Mahmoudi, K. Karayagiz, L. Johnson, R. Seede, Y. Karaman, R. Arroyave, A. Elwany, Bayesian calibration of multiple coupled simulation models for metal additive manufacturing: a Bayesian network approach.

ASCE-ASME J. Risk Uncertain. Eng. Syst. B: Mech. Eng. **8**, 011111 (2022)

[51] T. Kimand, B. Youn, Identifiability-based model decomposition for hierarchical Calibration, Struct. Multidiscipl. Optim. **60**, 1801–1811 (2019)

[52] M. Sklar, Fonctions de répartition à n dimensions et leurs marges. in *Annales de l'ISUP*, Vol. 8, 1959, pp. 229–231

[53] G. Gilbert, Positive definite matrices and sylvester's criterion, Am. Math. Monthly **98**, 44–46 (1991)

[54] F. Mentré, A. Mallet, D. Baccar, Optimal design in random-effects regression models, Biometrika **84**, 429–442 (1997)

[55] C. Vong, S. Ueckert, J. Nyberg, A.C. Hooker, Handling below limit of quantification data in optimal trial design, J. Pharmacokinet. Pharmacodyn. (2014)

[56] F. Laurin, Approche Multiéchelle des Mécanismes de Ruine Progressive des Matériaux Stratifiés et Analyse de la Tenue de Structures Composites (in French) PhD thesis, 2005.

[57] J. Germain, Évaluation des capacités prédictives d'un modèle avancé pour la prévision de la tenue de plaques stratifiées perforées (in french), Theses, Université Paris-Saclay, January 2020

[58] I. Daniel, O. Ishai, Engineering Mechanics of Composite Materials, Oxford university press New York, 2006

[59] J. Tomblinand, Y Ng, S. Keshavanarayana, Material qualification and equivalency for polymer matrix composite material systems, Technical report, 2001

[60] S. Kullback, R. A. Leibler, On information and sufficiency, Ann. Math. Stat. **22**, 79–86 (1951)

[61] L. Pardo, Statistical Inference based on Divergence Measures, CRC Press, 2018

[62] Y. Rollet, Vers une Maîtrise des Incertitudes en Calcul des Structures Composites (in French), PhD thesis, 2007

[63] C.R. Harris, K. Jarrod Millman, S. van der Walt, R. Gommers, P0 Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N.J. Smith, R. Kern, M. Picus, S. Hoyer, M.H. van Kerkwijk, M. Brett, A. Haldane, J. Fernández del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Shepard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, T.E. Oliphant, Array programming with NumPy, Nature **585**, 357–362 (2020)

[64] M. Baudin, A. Dutfoy, B. Iooss, A.-L. Popelin, OPENTURNS: an industrial software for uncertainty quantification in simulation. in R. Ghanem, D. Higdon, H. Owhadi (Eds.), Handbook of Uncertainty Quantification, Springer International Publishing, 2016, pp. 1–38

[65] S.J. Johnson, The nlopt nonlinear-optimization package

[66] J. Pinheiro, D. Bates, Unconstrained parametrizations for variance-covariance matrices. Stat. Comput. **6**, 289–296 (1996)

**Cite this article as:** Clément Laboulief, Mathieu Balesdent, Loïc Brevault, François-Xavier Irisarri, Jean-François Maire, Sebastien Da Veiga, Rodolphe Le Riche, Sequential calibration of material constitutive model using mixed-effects calibration, Mechanics & Industry **24**, 32 (2023)
[36] J. Wakefield, The Bayesian analysis of population pharmacokinetic models, J. Am. Stat. Assoc. **91**, 62–75 (1996)

[37] P. Damlen, J. Wakefield, S. Walker, Gibbs sampling for Bayesian non-conjugate and hierarchical models by using auxiliary variables, J. Roy. Stat. Soc. B (Stat. Methodol.) **61**, 331–344 (1999)

[38] M. Davidian, D.M. Giltinan, Nonlinear Models for Repeated Measurement Data, Chapman & Hall/CRC Monographs on Statistics & Applied Probability, Taylor & Francis, 1995

[39] R. Fisher, K. Pearson, On an Absolute Criterion for Fitting Frequency Curves. Messenger of Mathematics, 1911

[40] D. Hall, M. Clutter, Multivariate multilevel nonlinear mixed-effects models for timber yield predictions, Biometrics **60**, 16–24 (2004)

[41] J. McFarland, S. Mahadevan, L. Romeroand, V. Swiler, Calibration and uncertainty analysis for computer simulations with multivariate output, AIAA J. **46**, 1253–1265 (2008)

[42] N. Metropolis, S. Ulam, The monte carlo method, J. Am. Stat. Assoc. **44**, 335–341 (1949)

[43] D. Bates, D. Hamilton, D. Watts, Calculation of intrinsic and parameter-effects curvatures for nonlinear regression models. Commun. Stat. Simul. Comput. **12**, 469–477 (1983)

[44] N. Hansen, S. Müller, P. Koumoutsakos, Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES), Evol. Comput. **11**, 1–18 (2003)

[45] W. Oberkampf, C. Roy, Verification and Validation in Scientific Computing, Cambridge University Press, 2010

[46] L. Schwer, Guide for Verification and Validation in Computational Solid Mechanics, ASME, 2006

[47] B. Jung, H. Yoon, H. Oh, G. Lee, M. Yoo, B. Youn, Y. Hu. Hierarchical model calibration for designing piezoelectric energy harvester in the presence of variability in material properties and geometry, Struct. Multidiscipl. Optim. **53**, 161–173 (2016)

[48] B.D. Youn, B. Jung, Z. Xi, S.B. Kim, W.R. Lee, A hierarchical framework for statistical model calibration in engineering product development, Comput. Methods Appl. Mech. Eng. **200**, 1421–1431 (2011)

[49] A. Urbina, T. Paez, T. Consulting, A bayes network approach to uncertainty quantification in hierarchically-developed computational models, Int. J. Uncertain. Quantif. **2**, 173–193 (2012)

[50] J. Ye, M. Mahmoudi, K. Karayagiz, L. Johnson, R. Seede, Y. Karaman, R. Arroyave, A. Elwany, Bayesian calibration of multiple coupled simulation models for metal additive manufacturing: a Bayesian network approach.

ASCE-ASME J. Risk Uncertain. Eng. Syst. B: Mech. Eng. **8**, 011111 (2022)

[51] T. Kimand, B. Youn, Identifiability-based model decomposition for hierarchical Calibration, Struct. Multidiscipl. Optim. **60**, 1801–1811 (2019)

[52] M. Sklar, Fonctions de répartition à n dimensions et leurs marges. in *Annales de l'ISUP*, Vol. 8, 1959, pp. 229–231

[53] G. Gilbert, Positive definite matrices and sylvester's criterion, Am. Math. Monthly **98**, 44–46 (1991)

[54] F. Mentré, A. Mallet, D. Baccar, Optimal design in random-effects regression models, Biometrika **84**, 429–442 (1997)

[55] C. Vong, S. Ueckert, J. Nyberg, A.C. Hooker, Handling below limit of quantification data in optimal trial design, J. Pharmacokinet. Pharmacodyn. (2014)

[56] F. Laurin, Approche Multiéchelle des Mécanismes de Ruine Progressive des Matériaux Stratifiés et Analyse de la Tenue de Structures Composites (in French) PhD thesis, 2005.

[57] J. Germain, Évaluation des capacités prédictives d'un modèle avancé pour la prévision de la tenue de plaques stratifiées perforées (in french), Theses, Université Paris-Saclay, January 2020

[58] I. Daniel, O. Ishai, Engineering Mechanics of Composite Materials, Oxford university press New York, 2006

[59] J. Tomblinand, Y Ng, S. Keshavanarayana, Material qualification and equivalency for polymer matrix composite material systems, Technical report, 2001

[60] S. Kullback, R. A. Leibler, On information and sufficiency, Ann. Math. Stat. **22**, 79–86 (1951)

[61] L. Pardo, Statistical Inference based on Divergence Measures, CRC Press, 2018

[62] Y. Rollet, Vers une Maîtrise des Incertitudes en Calcul des Structures Composites (in French), PhD thesis, 2007

[63] C.R. Harris, K. Jarrod Millman, S. van der Walt, R. Gommers, P0 Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N.J. Smith, R. Kern, M. Picus, S. Hoyer, M.H. van Kerkwijk, M. Brett, A. Haldane, J. Fernández del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Shepard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, T.E. Oliphant, Array programming with NumPy, Nature **585**, 357–362 (2020)

[64] M. Baudin, A. Dutfoy, B. Iooss, A.-L. Popelin, OPENTURNS: an industrial software for uncertainty quantification in simulation. in R. Ghanem, D. Higdon, H. Owhadi (Eds.), Handbook of Uncertainty Quantification, Springer International Publishing, 2016, pp. 1–38

[65] S.J. Johnson, The nlopt nonlinear-optimization package

[66] J. Pinheiro, D. Bates, Unconstrained parametrizations for variance=covariance matrices. Stat. Comput. **6**, 289–296 (1996)

**Cite this article as:** Clément Laboulief, Mathieu Balesdent, Loïc Brevault, François-Xavier Irisarri, Jean-François Maire, Sebastien Da Veiga, Rodolphe Le Riche, Sequential calibration of material constitutive model using mixed-effects calibration, Mechanics & Industry **24**, 32 (2023)