2.1. Finite Element Model
A FE model of an unidirectional carbon/epoxy composite tube was developed using LS-DYNA software (Livemore Software Technology, Livermore, CA, USA) and it was validated against the experimental data published by Kim, Yoon [
37].
Figure 1 shows the dimensions of the [90/0]
7 tube that was utilized in that experimental study. The cylinder had a 30 mm internal diameter and a height of 100 mm with seven layers of carbon fibers with a thickness of 0.13 mm for each layer. The total mass of the tube was 28.3 g.
Table 1 lists the mechanical properties of the composite tube and the resin epoxy that was extracted from the experimental tests [
37].
The FE model has three main components: a cylinder using 19,900 shell elements with a size of 0.5 mm, a loading plate using solid elements, and a stationary plate using the Rigid_Wall option. The composite tube was modeled using 14 integration points. The 60 × 60 × 10 mm
3 rigid plate was modeled to apply the quasi-static load. The details of the geometry and FE model components are illustrated in
Figure 1. The details of the trigger mechanism are illustrated in this figure. The three layers of shell elements were used to obtain the closest possible angle in experimental model.
The failure mechanism of the composite tube was modeled using the Tsai-Wu theory [
38]. The MAT_ENHANCED_COMPOSITE_DAMAGE (MAT 54/55 keyword in LS-DYNA) was used to model the damage in the composite tube. Based on the Tsai-Wu theory the failure criterion for the compressive fiber mode, tensile matrix mode, and the combination of tensile and compressive matrix mode are given as following equations respectively:
If it fails, then Ea = Eb = Gab = vba = vab = 0
For the compressive longitudinal direction:
If failed, then E
a = v
ba = v
ab = 0; for the compressive transverse direction:
If failed, then Ea = Eb = Gab = vba = vab = 0; where the is stress in the (ij) direction, Ea is Young’s modulus in the (i) direction, Gij is in-plane shear modulus in the (ij) direction, vij is Poisson’s ratio in the (ij) direction, S is shear strength and X and Y is strength, respectively.
Other parameters that have a significant influence on failure occurs during the simulation cannot be extracted through the experimental tests. Therefore, their values were obtained through the trial and error process which are shown in
Table 2. These parameters are the maximum strain for the fibers in tensile (DFAILT), compression (DFAILC) mode, and the maximum strain for the matrix for the tensile/compression (DFAILM) and shear (DFAILS) modes. The load was applied as a quasi-static with a maximum velocity of 0.1 mm/s.
Figure 2 shows the velocity profile implemented in our simulation over time. Three different contacts were used for this simulation include NODE_TO_SURFACE between the loading plate and tube, AUTOMATIC_SINGLE_SURFACE between all components, and ERODING_SINGLE_SURFACE to prevent the penetration of the composite layers. Because the explicit solver was selected in our analysis, the mass scaling option was also used to improve the simulation time.
The strength reduction parameters were used to degrade the pristine fiber strength of a ply under a compressive load. The SOFT parameter, which cannot be measured experimentally, was selected by trial and error during several crash simulations. It should be noted that studies have shown that while the material strength parameters affect the stress-strain curve significantly, changing the material strength has a very small effect on the total energy of simulation [
27]. In addition, the failure strain was identified as the most critical parameter for the MAT 54 that affects the mechanical responses. Since for the entire numerical simulations in this study only one type of composite tube and load condition was selected to investigate the crashworthy behavior, the values of parameters may not be applicable when those parameters change.
Figure 3 shows the results of the FE simulation results at different moments and the first 80 mm of the predicted load-displacement profile for both experiment and simulation. The results show a good correlation between the FE and experimental results. The two failure mechanisms including the transverse shearing and lamina bending (delamination) were observed which are illustrated in
Figure 3.
Table 3 indicates that the primary evaluation parameters calculated from the experimental and numerical analysis are close to each other.
2.2. Modeling Using GMDH Neural Network.
The classical GMDH is a set of neurons in which different pairs of them in each layer are connected through a quadratic polynomial. Hence, this will produce new neurons for the next layer which can be used to map the inputs to output parameters. For example, consider a function f with multi-input X = (x
1, x
2, x
3,…, x
n) and a single output y
i and the
is the function that approximates the predicted output value:
The trained GMDH-type neural network that can predict the output values given
M observations are:
The goal is now to minimize the difference between the predicted value using GMDH-type and actual output:
Based on this algorithm, the general form of the connection between the inputs and output variables are expressed with the following equation which is known as Ivankhnenko polynomial [
32]:
In most applications, the quadratic form was used only for two variables which can be written as:
In Equation (8), all the coefficients
ai are calculated using the regression method [
35] to minimize the differences between the actual output y and approximated
for the input variables of
xi and
xj. This can be expressed with the following equation:
All the combinations of the two independent variables from n input variables are taken into account for the basic form of the GMDH to construct the regression polynomial similar to Equation (8). As a result,
neurons will be built in the first hidden layer for the feed-forward network structure from the observations which can be expressed in the following form:
Consequently, using the quadratic sub-expression in the form of Equation (8) the following matrix can be obtained:
where the variables are:
To calculate the coefficients, the least square method from multiple regression analyses can be used which leads to the Equation (15). This allows us to determine the vector of the best coefficients for the quadratic equations for the whole set of M data. This procedure is repeated for each neuron in the next hidden layer based on the topology of the neural network. To prevent the singularity and round errors for the solutions, the singular value decomposition (SVD) method was implemented [
32]:
In this study, the generalized structure GMDH method (GS-GMDH) using a genetic algorithm was used to train the model which was proposed by Nariman-Zadeh, Darvizeh [
31]. The genome or chromosome which presents the topology of the GMDH-type network consists of a symbolic string where the input variables are alphabetical names. This shows how the genetic algorithm was taken into account for the design of GMDH neural networks.
Figure 4 shows an example of this network with four input variables and a single output with two hidden layers which corresponds to the length of 2
2+1=8 genes. Further details of the mathematical model and the conditions of this method can be found in Atashkari, Nariman-Zadeh [
36].
These two main concepts include a hybrid GA and SVD are involved to optimally design such a polynomial neural network [
34]. The method that was used in that study was successfully used in this paper to obtain the polynomial models of the energy absorption, maximum force, and critical buckling force. The design variables in this study are the fiber angles (θ
1, θ
2) and the number of layers n. These variables range are from 0°–180° degrees for fiber angle and 5–9 for the number of layers. The GMDH-type models have shown a promising prediction of those outputs during the training process which will be presented in the following section.