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Crushing analysis of energy absorbing materials using artificial neural networks

This article presents the use of artificial neural networks in data analysis. The subject of the research were energy-absorbing materials under oblique loading. The forces obtained during the analysis were used to determine the crushing indicators. The numerical analysis was performed using the FEM Abaqus software. The specimens were loaded with the same force at different angles, i.e. 15, 30, 45, 60 degrees. During the numerical analyses, the normal and shear forces were measured. The tests were carried out under both static and dynamic load. On the basis of the MLP and RBF networks, analyses were carried out to study the relationship between the foam properties and the crushing efficiency indicators.


Introduction
The phenomenon of energy absorption, as well as passive safety has been explored since the mid-20th century. The first structural elements in the form of a flat steel bar attached to the longitudinal bars were proposed in Cadillac vehicles. Subsequently, with the development of motor vehicles and the speeds achieved by them, passive safety played a major role [1]. In addition, with the increasing restrictions in the European Union, the manufacturers supplying vehicles to this area were obliged to improve the structural elements of the vehicle responsible for energy dissipation [2,3]. The energyabsorbing structure in a frontal impact is a crash-box. It is made up of thin-walled elements of a certain stiffness attached to the front beam and stringers. It provides a completely safe crushing to a speed of about 15 km/h. The energy-absorbent profiles in cars most often assume a rectangular cross-section; however, round, omega and hexagonal cross-sections are also well described in the literature and are used in various types of constructions [4,5]. During crushing, two types of local and global buckling can be distinguished [6,7]. Depending on the width/diameter, wall thickness and profile height, crushing can be done differently. Most often these are local buckling along the entire length of the profile. In this way, a thin-walled profile is able to absorb the most energy [8,9]. With the development of technology, the demands placed on motor vehicles have been increasing. Therefore, the ways to improve the energy efficiency of the absorbers were invented. The basic way to control the crushing process is to use a trigger that initiates the crush in the right place. The initiator can take various forms, such as notches or dents on walls or edges [2,10,11]. The first type of reinforcement used in columns are the honeycomb structures. The hexagonal shape increases the 2 stiffness in the direction of impact and resists the collapse of the walls into the profile, thus absorbing an additional portion of energy [12,13]. Another way to increase the energy efficiency of the absorber is to use a filling with a porous structure. Thorough research on foamed materials and the characteristics of their destruction began at the turn of the 20th and 21st centuries [14]. With the development of technology, in addition to the use of conventional materials for steel and aluminum profiles, both GFRP and CFRP composite materials have become increasingly popular. There are many publications in the literature describing limit states and propagation of destruction of the columns made of composite materials [15][16][17]. Nowadays, when analyzing data, a common solution is to use artificial neural networks [18][19][20]. A statistical method called regression is used in the study of the relationships between the variables. With the use of a function it allows determining the interdependence of data. In the data analysis, two types of networks are used, namely multi-layer perceptron (MLP) and radial base function (RBF). Moreover, the CAD software used in the analysis presented in this article is used to model various types of issues [21][22][23]. Stress distribution testing is used not only in mechanical engineering applications but also in biomedical engineering [24,25].

Subject of the research
In this article, a 20x20x40 mm foam element was numerically tested. The foam model was loaded in a static and dynamic way. The boundary conditions for restraint were the same for both analyses ( Figure 1). The base had all translation degrees blocked and the foam material had frictional contact. The tup had the translation degrees locked in the X and Z direction so that it could only move vertically. The coefficient of friction was specified as 0.3. The energy during dynamic load was assigned by mass m=50 kg and initial speed V=1.5 m/s. In the static analysis, the load was applied by displacement of the tup insert on the Z direction.

Research methodology
The tests were carried out for different inclination of the base from 15-60 degrees. The element used as a base and the tup insert were modeled as a discrete rigid i. e. R3D4. Owing to this type of model, the elements are not deformed; however, during dynamic analysis, it is possible to detect the 3 displacement values and force reactions. In the case of porous material, the type of element used is C3D8R, i.e. a fully deformable element. The output value for neural networks presented in Table 2 was the CLE indicator. The data presented above were used to model the foam material issue. The analysis was carried out using the Abaqus software. In the field of plasticity, the Crushable Foam option was used. It allows giving the element properties of porous materials. Moreover, quantities such as Young's module and Poisson's coefficient were used to model the flexible range.  The graphic above shows an example of a Force-Shortening characteristic. The graph was obtained during dynamic crushing and the foam was on a base inclined at 15 degrees. The five figures correspond to the points marked on the diagram, providing a comprehensive view of the dynamic crushing of the aluminum foam.

Results of FEM
This chapter presents the results obtained during two types of analysis. The static and dynamic compression of foamed materials was described in Table 1. For both analyses, the samples were placed on a base inclined at an angle of 15-60 degrees, every 15 degrees so that the distribution of forces over normal and tangential could be measured. Then, on the basis of the quantities measured during numerical analyses, the crushing efficiency coefficients i.e. EA (energy absorbed), CLE (crush load efficiency) were determined and characteristic forces such as PCF (peak crushing force) and MCF (mean crushing force) were determined. The input data was the type of foam and the angle of base inclination, and the output was EA or CLE indicator.   The quantities obtained using the finite element method shown in Figures 3 and 4 show the effect of the base angle inclination on the absorbed energy and the crushing performance determined by the CLE indicator, respectively. The crush load efficiency shows a significantly higher performance under static load. This is due to the lack of initial impact, which can occur in case of kinetic energy assignment.

Neural networks
Numerically acquired quantities were used to create the neural networks and perform the regression analyses. The neural networks were created on the basis of the PET foam data, which is characterized by better energy absorption capacity. The MLP type networks were used to study the dependencies and their description is presented in the table 2. The networks have achieved a high quality of learning as well as testing and validation and low error rate, which makes the results obtained from these networks highly probable.
The table below shows the parameters of neural networks used to analyze the data obtained during a static load test. The presented data show the high quality of the neural network, which allows confirming that the relationships determined by it are correct. Due to the divergent nature of the static and dynamic analyses, the analysis of the data from both tests was performed using separate neural networks.  Figure 5. Predicted CLE values for the static analysis, linearly approximated.
Observing the predicted values in Figure 5 show a value match. For the CLE values, the accuracy is very high and almost perfectly matches the linear approximation shown in Figure 6. Table 2 shows a description of the neural networks used for data analysis. Out of a hundred, five were selected for the best quality of training, testing, validation and error, which ensures the best results. The neural networks achieved a quality of learning, testing and validation higher than 85%. The lower value of test quality is due to the small number of samples; however, observing all the parameters of the neural network, the appropriate quality is found for further analysis. The learning error in both cases was very low and did not exceed 1%. The table 2 also shows the functions that use the neural networks.   Figure 6 show the prediction of artificial neural networks for data obtained during dynamic analysis. In the case of prediction of the CLE (Crush Load Efficiency) values two, main data clusters can be seen, but both of them are very accurate in their approximate linear function. In addition, global sensitivity analyses was performed for the neural networks, which show that the type of porous material filling used has the greatest influence on CLE. The inclination of the base also has lesser influence on stress distribution.

Conclusions
In the presented studies, the subject matter was compressed foams. The foam elements were modeled as Crushable Foam in order to test their energy-absorbing properties in the elastic-plastic range. Two types of foams were numerically analyzed: metal (aluminum ALPORAS) and polymer (PET). Observing the behavior of the foams, one can see that the energy absorbed by the PET foam increases significantly, i.e. up to 50%, along with the angle of inclination. In the case of the AL foam, the increase of absorbed energy is max. 15%. This indicates very good energy-absorbing properties of the PET foam in the tangential direction. During shearing, the foam shows good properties which increase its efficiency. The quality of neural networks is very high, the learning error reached a low value. The value adjustment for CLE reaches high accuracy and oscillates around the approximate function indicated in the graph. The approximation of the linear function is good due to the exact distribution of values around. There is a much higher CLE in the static analysis, due to the absence of impact during the first crushing stage. It can be clearly stated that both foams have good energy-absorbing properties, and the use of neural networks is a very good tool to analyze its behavior.