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* (This study is at the final stage of submission. Therefore, I cannot share more specific details of this study until my paper is published. However, after submission, I will share the entire manuscript of the paper.)*

Granular materials are widely
used in various fields such as sintering, tablet manufacturing, and
geotechnical engineering.

To predict their constitutive behavior in the finite
element method (FEM), the modified Drucker–Prager Cap model (DPC model) is
commonly adopted.

However, the conventional
method requires multiple triaxial compression experiments and isotropic
compression experiments to determine a great number of parameters in the DPC
model.

**This study**** proposes an alternative method for the DPC model that
utilizes the entire stress–strain curves from triaxial compression tests and
the gradient descent algorithm to extract the parameters. The method significantly
reduced the total optimization time by replacing FEM simulation with analytical
calculations during the optimization process.**

The method was applied to acquire the constitutive relation for unsaturated sand-bentonite mixtures, and it successfully predicted experimental depth-sensing indentation results using FEM simulation.

**Figure 1. Overall Algorithm**

** Fig. 1** describes the **overall methodology
used in this study. **

In the beginning, the parameters are randomly selected.

After selecting random parameters, the analytical expression calculates the stress–strain
curve—the result of the triaxial compression test—from the selected parameters.

Afterward, the error between the experiment results and the calculation is
computed. This study used mean squared error (MSE).

If
MSE is larger than an objective value, the gradient descent algorithm updates
the parameters and repeats the whole process until MSE becomes lower than the
objective value.

**Figure 2.** Result of optimization. (a) mean square error
during the optimization (b) fitted curve at the last stage of the optimization (c)
optimized parameters

**Fig. 2** is a result of
optimization using the customized algorithm displayed in **Fig. 1**.

**Fig. 2(a)** shows the changes of MSE during the iterations. MSE converged after 500
iterations, and the final value was less than 10. It means that the difference
between the experiment and the simulation is about 3 N, so it can be inferred
that **the DPC model parameters were calibrated with high precision.**

Next, **Fig. 2(b)** compares the final curve to the triaxial compression test result. The
red line is the experiment result, and the blue line is the calculated curve from
the calibrated parameters. As mentioned above, the** two curves were well matched
to each other. **

Finally,

**Table 1**. Mean (\(\mu\)) and standard
deviation (\(\sigma\) ) of the optimized DPC model parameters.

**Table. 1** shows the mean and
the standard deviation of the DPC model parameters calibrated from ten
different initial points.

As indicated in the table, most of the parameters did not show significant
variances.

It means that** the triaxial
compression test and the gradient descent algorithm were sufficient to
characterize these parameters.**

**Figure 3. **Load–depth curve of the
indentation test and corresponding result of FE simulation which was implemented
with the parameters obtained by the gradient descent algorithm.

The parameters shown in **Fig. 2(c)**
were optimized to reproduce the triaxial compression test result.

Hence, we need to check whether the parameters are valid in other
circumstances. **This study used an indentation
test to substantiate the parameters’ validity. **

**Fig. 3** compares the result of
the indentation test and its FE simulation result, which was implemented with
the optimized parameters.

The indentation test was repeated nine times, and the
blue line is the mean value of the experiment result.

The red line is the
result of the FE simulation with the parameters shown in **Fig. 2(c)**.

The entire
red line is inside the error bar of the blue line. This result means that **the
parameters fitted from the triaxial compression test results can describe the
constitutive behavior of the material in the general case.**

**Figure 4**. Optimization result of
gradient descent algorithm, Bayesian optimization, and genetic algorithm. (a) MSE
vs iteration (b) final MSE

**The analytical solution proposed in this study enabled the gradient descent algorithm to use because error gradients were computed from the solution. (We can not compute the gradients from FEM simulation!) **

Here, the result of the gradient
descent algorithm to that of Bayesian optimization (BO) and genetic algorithm (GA) to check the performances of the
gradient descent algorithm. (BO and GA are representative derivative-free optimization method)

**Fig. 4** compares the results of
each optimization algorithm.

First, **Fig. 4(a)** shows the MSE changes during the
optimization process. According to the figure, the
gradient descent algorithm decreased the MSE faster than other algorithms did.

Second, **Fig. 4(b)** shows the
final MSE of each optimization method. In **Fig. 4(b)**, the final MSE of
the gradient descent algorithm was lower than that of other algorithms.

In summary,** the gradient
descent algorithm was able to obtain more precise DPC model parameters in fewer
iterations than derivative-free optimization algorithms.**

**Figure 5**. Load-depth
curves of the indentation test and FE simulation results with the parameters
optimized by different algorithm.

**Fig. 5** compares FE simulation
results to the indentation experiment result. Each FE simulation was implemented
with the DPC model parameters calibrated by different methods.

**FE simulation with the
parameters optimized by BO falls outside the error range of the experiment in
the early displacement region.** This was because the parameters fitted by BO
showed bigger MSE than the parameters evaluated by the gradient descent
algorithm. (**Fig. 4(b)**)

**The parameters fitted with the
GA caused a significant deviation from the experiment results in the FE
simulation.** The updating rules of the
method may have caused this massive error. In a GA, parameters having
considerable deviation in small domains may survive in a generation if the
overall MSE is lower than others. Therefore, the parameters that only affect
small domains may not be fitted correctly by this algorithm.

** This study developed a novel
characterization method by utilizing the triaxial compression test, the
algorithm that calculates the material's constitutive behavior during the
triaxial compression test, and the gradient descent algorithm. **The proposed method
was possible because the triaxial compression test results contain information
about shearing and hardening.

The algorithm for the constitutive behavior of the material replaced the time-consuming FE simulation in the optimization process, so the total optimization time decreased.

In addition, the algorithm made it possible to use the gradient descent algorithm to calibrate the DPC model parameters.

From this study, we expect that people may characterize any granular materials more easily than before.

Also, the study may contribute to applying granular materials to fields that require high accuracies, such as micro-transfer printing or nanotechnology.