The Design of Experiment (DoE) was created using Latin Hypercube Sampling (LHS) methodology. On each one-dimensional projection, LHS is based on stratification and provides good homogeneity (Hou, et al., 2012).

The geometric parameters will highly influence the DoE’s parameters, so new ones must be developed to accommodate all geometric shapes and simplify the optimization process. Five parameters were chosen: length (L), width (W), bending’s height (H), cross section’s thickness (t), and the number of layers (n). The number of layers explains how many auxetic cells were stacked on top of each other in the “L” direction. An illustration of the new geometric parameters is shown in Figure 2.

Five parameters were then chosen for the DoE’s parameters, consisting of three continuous and two categorical parameters, shown in Table 1.

The value for each level was selected based on several considerations. The minimum and maximum lengths were chosen to keep the whole system small but still can protect the battery. The minimum and maximum ratio of width/length were chosen to provide enough variety to the shape of individual auxetic honeycomb while not taking too much space. The minimum and maximum cross section’s thickness were chosen to keep within the geometrical constraint while stable enough during the numerical simulation. The number of layers was chosen to observe the significance of the number of layers. The bending’s height (h) was set at a fixed ratio of 0.3 of the length for all geometry. Each parameter was rounded to two decimal places to ease the data input. The 3D plot of the first three parameters (all continuous parameters) is shown in Supplementary Figure S1.

A total of 103 different models were generated, 100 models using the LHS method, and three models (model no. 101, 102, 103) were generated from failed optimization attempts to fill the three failed numerical analyses from the original design of experiment (model no. 9, 24, and 88).

Pouch batteries are employed as the test subject. A homogenized macro model by Kermani et al. (Kermani, et al., 2021) serves as the foundation for the numerical model for pouch batteries.

The battery cell was axially squeezed with a flat impactor that moved uniformly downward, much like in the reference (see Figure 3). A rigid wall (brown) was used to restrict the cell in the machine (in-plane) direction. Two rigid walls (turquoise and violet) were utilized to confine the cell in the through-thickness (out-of-plane) direction. The impactor was sized to be twice the length and thickness of the battery to compensate for any deformation during the compression.

The battery’s dimension in this study is 60 × 34 × 5.35   m m with an element size of 1 × 1 × 0.486   m m . The battery’s cell was modeled using constant stress solid element (ELFORM = 1), Flanagan-Belytschko stiffness hourglass form, and anisotropic honeycomb (MAT_126) material model with a material property that is summarized in Table 2 and Supplementary Figure S2.

The simulation is carried out in a quasistatic mode, with the impactor’s velocity applied as a prescribed motion velocity, with a final velocity of 0.1   m / s in the direction of Y − D O F = 2 . Every node of impactor elements’ movement is restricted in all directions bar the Y direction to ensure that the impactor moves without deflection.

There are three types of contact in battery simulation: Automatic Single Surface applied to the battery, One Way Surface to Surface applied between the battery and the impactor, and Interior applied to the battery.

The displacement of the battery was measured by the distance between the node right in the center of the battery. The load was measured by measuring the load between the battery and the rigidwall behind the battery. This was done because, despite the writer’s best efforts, there is still small penetration between the battery and the impactor, so no measurement was taken on the impactor.

It was found that the risk of a short circuit is much higher on this pouch battery when it is compressed at about 12   m m (See Figure 4). However, to add more challenges to the optimization process, we have decided to limit the battery deformation to about 10   m m .

Battery’s von Mises stress was used to measure exactly when the battery in the system model failed. This is done because the load condition in system modeling is dynamic load, where the stress distribution during deformation will not be as uniform as in the validation model. Furthermore, multiple battery configurations were used in the system modeling, complicating the measuring process. The von Mises stress when the battery is compressed at about 10   m m is 67.97   M P a (shown in Figure 5).

Two kinds of codes were used to develop the ANN model. The first kind is called optimization code, and the second kind is called validation code. The optimization code was used to help find the rough hyperparameter for the validation code using the Keras Tuner package (Keras Tuner, 2019). The validation code was used to fine-tune and develop the final ANN model for each output using the Tensorflow package with a callback mechanism (Abadi et al., 2015). Both were written in Google Colab.

The loss function was evaluated using the mean square error (MSE) equation, as written in Eq. 2. M S E = 1 m ∑ y p r e d − y a c t 2 (2)

The model’s performance is also assessed using maximum absolute percentage error (MAPE), which is written in Eq. 3. M A P E = 1 m ∑ y p r e d − y a c t y a c t (3)

The optimization code was written in Google Colab using the Platypus package (Platypus, n.d.) with TOPSIS.

The NSGA-II part of the optimization code was set with 11 decision variables, 2 number of objectives, and 4 number of constraints. The decision variables are L, Wr, t, 4 geometrical shapes (DA, DU, RE-A, and RE-B), and 4 options of layers ( 1 to 4 ). The objectives are SEAinv and Stress. The constraints are only one kind of geometrical shape allowed, only one number of layers allowed, and battery stress must be a positive number.

The TOPSIS part of the optimization code was set with the assumption that the weight of SEA inversed and battery stress are 50 % and 50 % , respectively, the minimum amount of SEA inversed preferred for the optimum design, and the battery stress is preferred to be as close to the limit of the battery stress according to Chapter 2.2. Those weights were chosen because the vehicle safety (represented by Battery Stress) cannot be compromised. However, the maximum amount of SEA (or, in this case, the minimum amount of SEA inversed) is preferred for two things: to maximize the energy absorption from the impact and to minimize the structure’s mass.

The NSGA-II algorithm processed the combination of SEAinv and Stress predictor to obtain the Pareto Front curve, which shows the relation between SEAinv and Stress. Data points were generated by the NSGA-II algorithm and then fitted with regression to obtain the relation function between SEAinv and Stress.

The Pareto Front curve shows the relation between SEAinv and Stress, shown in Figure 16. Data points in Pareto Front were fitted with regression function as shown in the black curve in Eq. 4. The regression fitted the data points with R 2 = 0.9447 . S t r e s s = 6 × 10 10 S E A i n v 2 − 5 × 10 6 S E A i n v + 127.77 (4)

Figure 16 also shows the optimum point obtained by the TOPSIS method, with SEA of 37,037.04   J / K g (inversed of the obtained data) and Stress of 38.30   M P a . The input parameter of the optimum configuration is shown in Table 5. The CAD of the optimum configuration is shown in Figure 17.

This result is better than every model simulated in the design of experiment in Chapter 2.1. Model 71, which has the highest SEA over all other models that successfully protect the battery, has a SEA of 37,074.20   J / K g and Stress of 52.77   M P a , which is worse in battery stress and equal in SEA than the optimum design.

The optimum configuration was then numerically simulated with the same configuration in Chapter 2.3 to validate the optimization result, as shown in Supplementary Figure S6.

The comparison between prediction and numerical simulation results is shown in Table 6. The comparison shows that the error between prediction and validation results is still pretty big. It was found that the SEA of the optimum design is even higher than the prediction, but the maximum battery stress is lower than the prediction. However, the maximum battery stress of the optimum design is well within the safe limit of 67.91   M P a . The comparison between the internal energy of the auxetic honeycomb and the impactor’s kinetic energy of optimum design is shown in Figure 18 and battery’s von Mises stress vs time of system with optimum design is shown in Figure 19.

Supplementary Figure S7 shows that the crash mechanics of the optimum design follows classical crash mechanics during vehicle impact, where there are zero crossing time of 1.39   m s and rebound velocity of 2.50   m / s . Mean crushing force ( P m ) of the whole system was calculated in Eq. 5. P m = E A δ = 427.84   J 10.68   m m = 40.05   k N (5)

Just as with every other numerical model done for the design of experiment, the energy state and the detailed sum slave/master sliding energy were checked on the optimum design’s validation result to ensure the accuracy of the obtained data, shown in Supplementary Figure S8. The total energy is almost in constant value over time, with the value ranging between 917.10   J to 917.17   J , which is almost exactly the same as the initial kinetic energy of the impactor ( 917.13   J ). The hourglass energy is very small compared to other forms of energy, and the sliding energy is below 10% of the internal energy. The sum salve/master sliding energy for each contact is almost always positive throughout the simulation. These indications show that the simulation is accurate.