Lattice structures have become quite popular for their excellent energy absorption abilities and have applications in a wide range of fields such as automotive, aerospace, biomedical, and safety equipment [1]. One of the exciting developments is the use of Additive Manufacturing to customize these lattice designs and geometries, especially in improving vehicle crash safety and enhancing the protective features of sports and military equipment. Some big questions remain unanswered: which lattice design should be used? What happens if the parameters of this lattice are changed, and how will it affect the energy absorption?

In this article, we'll take a closer look at the key factors used to compare how different lattice structures absorb energy. We'll also explore how engineers can efficiently and quickly assess various designs by using a Design of Experiment approach combined with simulation.

## Energy Absorption Metrics

When the lattice structure is subject to a load, it experiences stress, which in turn leads to deformation. By examining the relationship between stress and strain, we can gather valuable insights. The compressive stress-strain diagram of lattice structures can be divided into three distinct regions: elastic, plastic, and densification as shown in Figure 1.

In the elastic region, the lattice behaves like a spring, exhibiting a linear relationship between stress and strain. This region contributes only a small amount to the overall energy absorption capability of the lattice. However, once the lattice surpasses the yield strain εy, it enters the plateau region. This is where significant amounts of energy absorption occur, as the lattice struts buckle, yield, or crush. The plateau region is essential for energy absorption and enhancing the lattice's ability to withstand external forces. As the strain approaches the onset of densification ε_{cd}, more struts within the lattice come into contact with each other, causing the lattice to act as a solid part. At this point, the stress-strain curve experiences a steep increase in slope at the densification strain εd, indicating a rapid increase in resistance to deformation.

The specific energy absorption (SEA) is a crucial indicator for evaluating a material's energy absorption capacity. SEA is determined by dividing the absorbed energy (AE) by the mass of the compressed lattice. It is represented by the area under the load-displacement curve on the stress-strain diagram. This measure provides essential insights into the lattice's ability to absorb energy effectively. The EA and SEA can be calculated by the equations below

The majority of energy is absorbed during the plateau region. Many scholars tend to overlook the energy absorbed during densification, as the lattice transforms into a rigid body, leading to higher stresses that may exceed the capacity of the testing machine. Thus, determining the onset of densification strain is crucial for understanding when the plateau region concludes and for accurately calculating the absorbed energy when comparing various designs.

There are three commonly used methods for determining the onset of densification strain:

- The first method involves identifying the point of intersection between the tangents drawn from the stress plateau and densification regions on the stress-strain curve.
- The second method defines the densification strain as the strain value at the last local minimum before a significant increase in stress is observed.
- The third method sets the densification strain as the point where the slope of the tangent line matches that of the elastic regime.

These methods, although extensively studied, can introduce uncertainties and errors when comparing different designs. QM Li et al [2] proposed a novel approach. They suggest using an energy efficiency parameter to assess the optimal energy absorption condition. This parameter correlates the absorbed energy with the mass of the compressed lattice to provide a more robust measure. The energy efficiency parameter is calculated using the following equation:

## Case Study: Assessing a lattice dataset energy absorption using Metafold

Design of Experiments (DoE) is an effective and organized approach for understanding the relationship between factors at different levels and their impact on one or more responses. In this case study, we will utilize DoE to analyze a lattice dataset generated using Metafold. By systematically varying the parameters, we will design a set of experiments to evaluate the energy absorption of these lattice structures using Metafold's meshless quasi-static simulation module.

By performing statistical analysis on the results obtained from these experiments, we will be able to determine which factors have the most significant influence on the energy absorption. This process will guide us in selecting a lattice design that is most suitable for any application, ensuring efficient and accurate decision-making. Through this case study, we will highlight the effectiveness of DoE in quickly and efficiently guiding the selection process, combined with the capabilities of the Metafold platform to generate lattice structures and simulation them rapidly on the cloud.

### Step 1: Design of Experiment

In this case study, we focused on investigating the impact of two factors, namely lattice design and graded relative density, on two key response metrics: SAE (specific absorption energy) and ε_{cd} (onset densification strain). We conducted the simulations using four different lattice designs: Gyroids, Diamonds, Kelvin, and IWP. Seven levels of graded relative density maps were tested as illustrated in Figure 2. Uniform, linear density in the y-direction, thicker and thinner configurations at the center of the y-direction, and linear density in the x-direction, as well as thicker and thinner configurations at the center of the x-direction. A full factorial design resulted in the comparison of a total of 28 distinct lattice designs.

### Step 2: Simulating the lattice structures

The material model used in this simulation was Carbon EPU 45, it was calibrated using Metafold team in collaboration with Carbon^{(R)}. The simulation is a quasi static compression simulation, where a rigid body compacts the lattice structure by a displacement control. The simulation can be performed using a one button click on Metafold app. The results in the app is a von Mises and displacement stresses distribution as shown in Figure 3.

All parts were compressed to 70 % of their original height. The wall time for each simulation would take around 20 minutes using the cloud based platform. To make the process faster, multiple simulations can be run in parallel. The stress strain chart were then plotted as shown in Figure 4 , and the efficiency-strain chart was plotted. SAE and ε_{cd} were calculated for the 27 designs.

### Step 3: Statistical Analysis:

In the analysis of the results, a box and whisker chart was utilized to visualize the outcomes. It was observed that the lattice designs derived using the Diamond configuration exhibited relatively higher SAE values compared to the other designs. However, it remains difficult to ascertain whether the design itself has a significant impact on the ε_{cd}. Furthermore, it is also challenging to determine whether any of the selected relative density grading schemes have a discernible effect on both the SAE and ε_{cd}. To provide a more robust conclusion, a statistical test is necessary to evaluate whether these observed differences are statistically significant or simply due to random variation.

The aim of the study was to assess whether there were any differences in the mean response of the lattice designs and relative density groups. To achieve this, an ANOVA test was conducted. The null hypothesis assumed that there were no differences among the group means, i.e. the design and relative density grading will not affect the amount of energy absorbed by the lattice. In the ANOVA test, if any group showed a significant difference from the overall group mean, the test would report a statistically significant result. A small p-value would lead to the rejection of the null hypothesis, indicating a significant difference between the groups.

Based on the ANOVA tables analyzed, it was observed that only the lattice design had a notable impact on the SAE response, with a p-value of 0.005. However, neither the design nor the graded relative density exerted a significant effect on the ε_{cd}.

## Conclusion

Using a robust methodological DoE approach is indeed critical for obtaining reliable and valid results, allowing engineers to make data-driven decisions. However, choosing the right software tool to navigate the data exploration phase is equally important. That's where Metafold can be extremely helpful. Metafold has several capabilities that can enhance the lattice exploration analysis process, including:

- Creating various lattice designs from Metafold’s library quickly and efficiently or import your custom lattice design. This allows the evaluation of many factors and configurations.
- Using meshless quasi-static simulation with various validated material models from vendors, providing accurate predictions of the lattice mechanical behavior.
- Run parallel simulations to shorten the time required to perform a Design-of-Experiments (DoE) analysis.
- Offers an API and SDK tools to extract all the data you need, and statistically analyze and interpret this data, providing reliable and objective answers to your research questions.

By leveraging these capabilities, Metafold can help you optimize your research and development processes by identifying critical factors, developing predictive models, and making informed, data-driven decisions.

[1] Mueller, J., Matlack, K. H., Shea, K., & Daraio, C. (2019). Energy absorption properties of periodic and stochastic 3D lattice materials. *Advanced Theory and Simulations*, *2*(10), 1900081.

[2] Li, Q. M., Magkiriadis, I., & Harrigan, J. J. (2006). Compressive strain at the onset of densification of cellular solids. *Journal of cellular plastics*, *42*(5), 371-392.