Long short-term memory based modeling of heat treatment and trigger mechanism effect on thin-walled aluminum 6063 T5 for crashworthiness

ABSTRACT

Vehicle safety relies on the capacity of vehicle systems to decrease the likelihood of biomechanical injuries to both vehicle occupants and pedestrians. This can take the form of active or passive measures applied to the materials and their geometry in the event of an impact. Nevertheless, due to the nonlinear behaviour of materials and deformation, a distinct collapse process emerges that requires analysis; this analysis can be virtual or through experimental tests. While observed performance can be ascertained through experimental designs, executing such a design for every parameter combination can be time-consuming and costly. This study employs a Long Short-Term Memory (LSTM) to predict the energy absorbed and crushing force of thin-walled aluminium 6063 T5 tubes subject to different collapse triggers and heat treatments. The LSTM model is constructed using experimental data derived from experiments that consider trigger shape, area, trigger position, furnace duration, cooling temperature, and heat treatment soaking method. The LSTM model achieved a Root Mean Square Error (RSME) of 0.56 and 0.0025 for crushing force and energy absorption, respectively. LSTM proves to be a valuable tool for predicting results in nonlinear analysis, particularly in the context of crush behaviour. Also, a comparison of the LSTM and finite element analysis (FEA) predictive performance is presented.

KEYWORDS:

• Passive safety
• thin-walled
• heat treatment
• long short-term memory
• non-parametric modelling

1. Introduction

Passive safety innovations have spurred the creation of structures that can withstand heavy loads and protect occupants in the face of the most challenging and perilous impact scenarios that may arise in a crash accident. The concept of designing components specifically tailored to absorb kinetic energy and transform it into deformation energy is called crashworthiness (Yu et al. 2021). There has been a shift towards using lightweight materials to mitigate vehicle emissions. Consequently, structural components have been modified, moving from traditional steel structures to materials such as composites or lightweight alloys like aluminium.

Original Equipment Manufacturers (OEMs) and their suppliers have collaborated to design the vehicle’s body to meet various safety standards. To effectively handle energy absorption in the event of a frontal impact, a crash box is incorporated to dissipate kinetic energy and mitigate the sudden reduction in the vehicle’s velocity, thereby reducing the peak acceleration values Hou, He, Yang, and Sang (2023). Recent developments to achieve the desired performance involve enhancing the crash box’s energy absorption capacity and lowering acceleration peaks. These improvements are centred around adjustments to its mechanical properties and design characteristics and are collectively known as crashworthiness enhancement mechanisms.

One of the methods for crashworthiness enhancement in thin-walled tubes is to modify its strength by implementing a thickness variation (Sofuoğlu and Çam 2021). This strategy has been assessed previously, showing significant outcomes. Yu et al. (2021) studied the energy absorption capacity of tailor-rolled structures and determined that gradual thickness reduces peak crushing force, and subsequent peaks increase energy absorbed. Another method for increasing energy absorption capacity is filled tubes. Yi et al. (2019) implemented thin-walled tubes with gradual density foam filler, helping to reduce peak crushing force and keeping the benefits in energy absorption. Negative poisson filler structures (Gao and Liao 2021) and honeycomb designs have also been implemented (Nian et al. 2020) with promising results. Composite materials are an alternative to aluminium due to their higher specific strength, but their brittle behaviour limits their application (Patel, Vusa, and Soares 2019). The combination of aluminium tubes with composite material tubes has been studied; this configuration combined the plasticity of metal with the high strength of composites (Hwang, Wu, and Liu 2021). The interaction of both materials increases friction, changing the deformation mode and resulting in a higher energy absorption performance (Z. Wang et al. 2020).

Aluminium alloy 6063-T5 has been used before to promote crashworthiness enhancement. Nevertheless, the composition of this alloy requires some modifications to adjust its properties. The alloy composition and its mechanical characteristics can be enhanced by artificial ageing (Baganis, Bouzouni, and Papaefthymiou 2021). This method consists of a solid solution, fast cooling to generate an oversaturated phase, and heat for precipitation of the second solid phase. The heating time determines the quantity of the precipitated phase (Dilrukshi and De Silva 2020). Generally, more ageing time increases hardness and strength, but an excess of ageing can reduce mechanical properties due to the formation of precipitates in more extensive areas (Kumar et al. 2022).

The potential variations of mechanical properties with ageing require a reliable, repetitive, and objective strategy to characterise the alloy application in crash box energy adsorption. This assessment can be complemented using a trigger mechanism. The trigger mechanism’s main purpose in the tubular sections of the box is to generate progressive collapse, enhance energy absorption, and simultaneously reduce peak crushing force (Khan and Mahdi 2023). Generally, adding triggers increases crashworthiness, but exceeded triggers reduce the energy absorbed and could lead to fractures (Rogala and Gajewski 2023). Regarding the location of the trigger in the alloy-based tube, a positioning closer to the upper border is commonly used (Özbek, Bozkurt, and Erkliğ 2022). Still, it is closer to the base and has also shown relevant results, depending on the tube’s cross-section (Hussain et al. 2022). Triggers could be holes and slots (Williams and Cremaschi 2021) or chambers in the case of composite material (Chen et al. 2023).

A feasible methodology to determine the position of the trigger mechanisms is developing a mathematical model that uses the alloy properties as input. Determining the relationship between input and output in a system with uncertain descriptions must be asserted by algorithms and mathematical models. Surrogate models are designed to represent the input-output behaviour of real complex systems (Williams and Cremaschi 2021). Heuristic methods are used to model materials’ mechanical, physical, and chemical characteristics. Examples of heuristic methods are artificial neural networks (ANN), Support Vector Regression (SVR), Response Surface Method (RSM), Multiple Linear Regression (MLR) (Dadrasi et al. 2022), and the Kriging Method (KM). Optimisation models and algorithms like Genetic Algorithm (GA), Particle Swarm Optimisation (PSO), and Fruit Fly Optimisation Algorithm (FOA) are used to find the combination and value of input variables to obtain a desired response. For crashworthiness applications, the most common optimisation models used are the non-generic sorting genetic algorithm (NSGA-II) and multi-objective particle swarm optimisation (MPOSO) (W. Wang et al. 2022).

RSM is a mathematical model that allows data prediction according to experiments (Sarabia, Ortiz, and Sánchez 2020). Using RSM based on experimental data, Liang et al. optimised different topological distributions of foam-filled thin-walled structures. The resulting configuration increases specific energy absorption maximum displacement and reduces peak crushing force (Liang et al. 2023). Li et al. (2021) experimentally obtained data on the performance of different combinations of sectional dimension, material, and thickness of vehicle front rails to construct an RSM, using it to find a combination that minimises structural mass and increases mean crushing force. Booth et al. used RSM to perform multi-objective optimisation of mean crushing force and crush efficiency of multicellular extrusions subjected to dynamic axial and oblique load (Booth, Kohar, and Inal 2021). Song et al. implemented double arrow negative Poisson ratio structures to enhance energy absorption for impact load cases by modelling microstructure parameters with response surface and optimising with a non-dominated genetic algorithm (Song et al. 2023).

PSO is a single objective optimisation method for mathematical models that uses natural processes in living things like fish schooling and bird flocking (Dadrasi, Farzi, et al. 2020). Keshtegar et al. (2021) used a modified PSO model to predict the buckling load on laminated composite plates, depending on the number of layers, boundary conditions, aspect ratio, and load patterns. Reduce computational time and optimise processes. Dradasi et al. (2022) modelled the effect of nano-silica content and particle size on energy absorption, impact strength, Young modulus, and yield strength of thin-walled frusta of silica/epoxy nano-composites by using PSO Multivariable Nonlinear Regression. The obtained model and GEP identified the optimal combination of parameters. MOPSO is a variant of PSO but for multi-objective optimisation, handling numerous objective functions with contradictory responses (Bouchaala, Merroun, and Sakim 2022). Ali et al. (2022) modelled the effect of upper panel thickness, core thickness, lower panel thickness of steel, and polyurethane elastomer on sandwich plate structure crashworthiness by making a surrogate model from FEA data with RBF (radial basis function). Then, the best combination of parameters that optimise high specific energy absorbed and low peak crushing force was determined by MOPSO. Another surrogate model for the prediction of variables is GEP (gene expression programming), a modification of GP (Genetic Programming) that delivers an accurate empirical equation without requiring nonlinear configurations (Chu et al. 2021).

ANN is a preferred tool for studying the relationship between dependent variables and the output (Rogala, Gajewski, and Ferdynus 2020). ANN converts input into outputs by employing learning procedures that mimic biological neurons, having nonlinear computing agents working in parallel (Rahmanpanah et al. 2020). Neural networks consist of multiple neurons arranged in layers. Each neuron calculates a weighted sum of its inputs and the excitation level that calculates the output value of the neuron (Gajewski and Vališ 2021). Fredynus et al. used ANN to obtain determined energy absorption indicators in the function of thin-walled tube geometry and, simultaneously, determine which parameters had an effect or not (Ferdynus and Gajewski 2022). Rogala et al. (2020) also used ANN to determine the influence of hexagon trigger parameters on crushing behaviour, according to experimental data. Baykasoğlu et al. used ANN to model the effect of diameter lattice, number of lattices, and tube thickness on crashworthiness of lattice structure filled square tubes, then used Weighed Superposition Attraction (WSA) to minimise peak crushing force and maximise specific energy absorption (Baykasoğlu, Baykasoğlu, and Cetin 2020). Kazi, Eljack, and Mahdi (2022) built an ANN from experimental data of different load conditions on composite rectangular tubes with varying cross-sectional shapes. The model was used to find the design with optimum load-carrying capacity and energy absorption.

Fuzzy neural networks are a variation of neural networks that combine ANN with fuzzy logic, combining the neural networks that behave like the human mind and the fuzzy logic (where design variables are not completely true or false) for modelling natural parameters, improving the capacity for learning from nonlinear systems (Lin, Le, and Huynh 2018). Ponticelli et al. (2020) used FNN to determine the best combination of aluminium foam manufacturing parameters, considering uncertainties related to the model and statistical variability. The resulting design had the highest energy absorption with the lowest compressing deformation. Albak (2021). optimised the energy absorbed and peak crushing force of circular tubes with corrugated inside structures connected with ribs by determining the best combination of wall thickness. Although fuzzy algorithms are better for dealing with variance in experimental data, ANN requires less computational process (Kazi, Eljack, and Mahdi 2022).

Despite the advances in modelling the relationship between aluminium alloy manufacturing conditions and the trigger mechanism operation (centred on the energy absorption), the estimated outcomes have opportunity areas, especially considering the accuracy degrees. This work implemented a trigger mechanism and heat treatment on aluminium 6063 T5 square thin-walled components to enhance the energy absorption in crush. Experimentally, combinations of heat treatment and collapse initiators were performed. Design variables are trigger size (100 $\mathit{m}{m}^2$, 150 $\mathit{m}{m}^2$, 200 $\mathit{m}{m}^2$, 250 $\mathit{m}{m}^2$), shape (circle, square, diamond, rectangle), and distance from the upper border (40 mm, 80 mm, 120 mm, 160 mm). Variables are time in a furnace (1 h, 6 h, 12 h, 24 h, 48 h, 72 h), cooling method temperature (Water at 70^°C and 90^°C), and cooling method (half soaked, completely soaked). Heat treatments were performed on two electrical furnaces with a limit of 1000^°C and a ceramic internal layer. Water was selected as the cooling method. The water was heated to determine temperature using electrical resistance and a thermocouple to monitor the temperature. Using LSTM allowed us to analyse all the results in a single run, which allowed us to train the algorithm with the different variables. Due to the ability of LSTM to predict mechanical energy absorption behaviour, it is possible to analyse different design variables to predict behaviour with different design variables.

2. Energy absorption evaluation

The energy absorption process is developed by transforming mechanical energy into deformation energy; this energy (E) depends on the force generated over the mechanical properties of the component. Some indicators evaluate the performance, such as Specific Energy Absorption (SEA), which can be used to improve the absorption efficiency using light structures, increasing the absorption by reducing the mass. A relationship exists between the deceleration generated during impact and the absorption force. This absorption process responds through a history of force against time, in which there is a maximum force value known as the Peak Crushing Force (PCF). The energy process starts with the trigger of the first contact and develops during the reduction of the length of the component; this behaviour is analysed through the indicator of Mean Crushing Force (MCF).

To determine how much energy $\mathit{E},[\mathit{J}]$ is dissipated as a consequence of material deformation due to the presence of the crushing force $\mathit{F},[\mathit{N}]$ over the crushing distance $\mathit{\delta }$, it is necessary to measure this force and the deformation of the material during crushing given by:

(1) $\mathit{E}={\int }_0^{\delta }\mathit{F}\left(\mathit{\delta }\right)\cdot \mathit{ds}$(1)

Specific energy absorption (SEA) indicators can be obtained from the load-displacement curve. These indicators are used to evaluate the performance of the absorption device. The estimation of SEA is given as follows.

(2) $\mathit{SEA}=\frac{E}{m}$(2)

where $(\mathit{E})$ is given in (1) and $(\mathit{m})$ is the mass.

The mean crushing force $(\mathit{MCF})$ is the ratio of energy absorbed to length reduction (crushing distance):

(3) $\mathit{MCF}=\frac{E}{\delta }$(3)

The peak crushing force is the maximum crushing force in the load vs. displacement curve. This parameter approximates the average force of the material during collapse (Figure 1).

Figure 1. Crushing force history.

Graph showing the common behaviour of crushing force described in crushing distance and crushing force.

Energy absorption is achieved by transforming kinetic energy into deformation energy. The first deceleration peak generated by the peak crushing force allows it to absorb the greatest amount of energy. However, the Mean Crushing force is the result of deformation and energy dissipation, considering both the initial mechanical properties and the change of stiffness due to the compaction of the component. Although the first peak of absorption force absorbs energy, it also generates inertial forces due to the difference in velocities of the components impacting each other. Based on the difference in initial velocities during the impact, the Mean Crushing force is defined as the target value to make the energy absorption process more efficient. Meanwhile, the initial peak force depends on the component’s geometry and mechanical properties. The Mean Crushing Force is sought to modify through deformation initiators or defined geometries to generate a process of controlled deformation along the displacement; this is an evolutionary process by changing characteristics during the deformation process.

Looking for a component with high energy absorption and low peak crushing force is difficult due to the complex relationship of these characteristics with the material properties. Because energy absorption is integral to the crushing force vs displacement curve, the higher the force values, the higher the energy. However, an increase in force values commonly means an increase in peak crushing force (PCF). Energy efficiency ($E_e$) is calculated to compare components’ performance and determine the one that has the best combination of these parameters. $E_e$ can be calculated as the absorbed energy over peak crushing force.

(4) $E_e=\frac{E}{PCF\ast L_0}$(4)

The variable $L_0$ corresponds to initial longitude.

3. Artificial neural networks for energy absorption forecasting

Due to the complicated and generally nonlinear relationship observed in experimental designs between shape, material, and energy absorption enhancement methods parameters on the final crashworthiness, researchers have been using machine learning algorithms as a part of the design phase to identify functional relations and optimal material configuration (Kohar et al. 2016). Due to the nonlinear relationships between parameters, they used ANN to predict crashworthiness indicators. To find the optimal configuration and deal with the complicated relationship between SEA and PCF, they used NSGA-II (Pirmohammad and Esmaeili-Marzdashti 2019). Dradasi followed a similar procedure to model the effect of one-sided holes and rectangular cross-section aspect ratios of thin-walled columns on crashworthiness indicators and to find the optimal configuration, with good results compared with experimental tests (Dadrasi, Albooyeh, et al. 2020).

Kazi et al. also used ANN to model the effect of the rectangular cross-aspect ratio of cotton fibre epoxy thin-walled tubes, finding the optimal value for each load condition among predicted values (Kazi, Eljack, and Mahdi 2022). Sofuoğlu et al. used feed-forward neural networks to model the effect of rolling and forming manufacture procedures of gradual thickness square cross-section tubes on energy absorption and peak crushing force reduction. They built an ANN to model the effect of each method and find the optimal configuration to reduce force and increase absorption (Sofuoğlu and Çam 2021). Di Benedetto et al. (2021) used the ANN backpropagation to model and predict the energy absorption capacity of components manufactured with commingled thermoplastic composites, considering the effect of manufacturing procedures on mechanical properties.

Recently, the recurrent forms of ANN, namely recurrent neural networks (RNNs), have produced more complex representations of input-output relationships, especially if they are multivariable and highly nonlinear. Most existing RNN-based models usually do not consider the long and short-term memories among the input information and the output values. Nowadays, these associations have shown to be effective modelling methodologies to obtain approximate representation of complicated multivariable input-output relationships. The advantages shown by RNN can be considered part of the modelling alternatives for the kind of system considered in this study.

3.1. Long short-term memory

As an alternative to RNNs, long-short-term memories (LSTMs) were created in 1997 to address problems with long-term reliance and gradient vanishing. The advantage of having feedback connections makes them different from traditional feed-forward ANNs. Because of this property, they are primarily used for tasks involving time series data, where there may be erratic delays between essential events. LSTMs were created primarily to solve the vanishing gradient issue frequently arising when training conventional RNNs fails. The performance of RNNs, hidden Markov models, and other sequence-based learning techniques is frequently outperformed by LSTMs because of their superior ability to handle sequence gaps of various lengths.

In an LSTM, the current temporal input vector, $x_t$ ($x_t\in R^n$, where $\mathit{n}$ is the input vector’s dimension), the hidden states of the previous instant, $h_{\mathit{t}-1}$ ($h_{\mathit{t}-1}\in R^h$, where $\mathit{h}$ represents number of hidden cells), and the cell state vectors, $\mathit{Ct}-1$, are all sent to the cell. It is significant to notice that the cell’s vector and the hidden cell sizes must match.

The hyperbolic tangent (tanh) type and the sigmoid function are two activation functions incorporated into the LSTM network (Sak, Senior, and Beaufays 2014). The tanh type is the activation function that is most frequently used, which satisfies

(5) $\mathit{tanh}(\mathit{x})=\frac{e^x-e^{-\mathit{x}}}{e^x+e^{-\mathit{x}}}$(5)

The range of $[-1,1]$ produced by the tanh aids the network in adjusting the data flow switch and preventing the gradient explosion. The sigmoid activation function equation (6)(6) $\mathit{\delta }={\left(1+{\mathit{e}}^{-\mathit{x}}\right)}^{-1}$(6) is furthermore implemented in LSTM, and it has the following definition:

(6) $\mathit{\delta }={\left(1+{\mathit{e}}^{-\mathit{x}}\right)}^{-1}$(6)

The sigmoid function’s output range is $[0,1]$, and the neural network will discard irrelevant data. According to Bengio, Simard, and Frasconi (1994), if the output is close to zero, it is irrelevant information and should be discarded; if it is close to one, it should be kept. The four memory units of the LSTM neural networks are the input $\mathit{gate}-\mathit{i}$, forget $\mathit{gate}-\mathit{f}$, cell candidate $\mathit{gate}-\mathit{g}$, and output $\mathit{gate}-\mathit{o}$, and the following expressions control them:

(7) $\begin{array}{c}\mathit{f}={\delta }_g(R_{\mathit{hf}}{h}_{\mathit{t}-1}+b_{\mathit{hf}}+W_f{x}_t+b_f)\\ \mathit{g}={\delta }_g(R_{\mathit{hf}}{h}_{\mathit{t}-1}+b_{\mathit{hf}}+W_f{x}_t+b_g)\\ \mathit{i}={\delta }_c(R_{\mathit{hi}}{h}_{\mathit{t}-1}+b_{\mathit{hi}}+W_i{x}_i+b_i)\\ \mathit{o}={\delta }_g(R_{\mathit{ho}}{h}_{\mathit{t}-1}+b_{\mathit{ho}}+W_o{x}_i+b_o)\end{array}$(7)

The weights and biases matrix at the present instant is $\mathbf{W}=[W_f,W_g,W_i,W_o{]}^T$, $\mathbf{b}=[b_f,b_g,b_i,b_o{]}^T$, respectively; $\mathbf{R}=[R_{\mathit{hf}},R_{\mathit{hg}},R_{\mathit{hi}},R_{\mathit{ho}}{]}^T$, $\mathbf{b}=[b_{\mathit{hf}},b_{\mathit{hg}},b_{\mathit{hi}},b_{\mathit{ho}}{]}^T$ donate the so-called recursive weights, biases matrix at the previous time; $h_{\mathit{t}-1}$ shows the hidden cells of the previous instant; $h_t$ stands for the hidden cells of the current moment; and $x_t$ is the input parameter vector at the current moment. $C_{\mathit{t}-1}$ shows the cell state vectors at the previous moment; $C_t$ is the cell state vectors at the present moment. Regarding information processing, LSTM is primarily separated into three stages: the forgetting stage, the selecting memory stage, and the output stage. The primary goal of the forgetting stage is to selectively forget the input vectors from the previous moment. This is accomplished by controlling, via the forget gate-f, the information that will be remembered and forgotten in $C_{\mathit{t}-1}$ from the previous moment. The main goal of the selective memory stage is to calculate the output through the input $\mathit{gate}-\mathit{i}$ and pick the output through the cell candidate gate-g while selectively remembering the input vector $x_t$ at the current time. The primary goal of the output stage is to ascertain the current state’s output, which is mostly managed by output $\mathit{gate}-\mathit{g}$. From there, $C_t$ is scaled using the tanh activation function.

4. Methodology

Specimens are manufactured from aluminium 6063-T5 square tubes. 6063-T5 aluminium alloy is obtained by hot forging and cooling with air. Trigger’s design followed the geometry shown in Figure 2. Design variables are the distance from the upper border of the thin-walled square tube (40,80,120 and 160 mm), geometry (Circle, Square, diamond, and Rectangle), and area of the trigger (78.54, 100, 150, 491, 625, and 937 mm²). Initially, 32 combinations were configured, resulting from all the combinations of 4 geometric shapes, four levels of distance from the upper border, and two levels of area size. Combinations of oven temperature, cooling temperature, and soaking method for the temperature treatment are summarised in Table 1, where $\mathit{HS}$- means Half Soaked and $\mathit{CM}$-Completely Soaked.

Figure 2. Geometrical parameters of the trigger on first experimental phase.

Four figures that describe the geometrical parameters.

Table 1. Heat treatment parameters and combinations.

Furnace	Cooling method	Soaking method
time (h)	temperature (^°C)
1	70, 90	HS,CS
6	70, 90	HS,CS
12	70, 90	HS,CS
24	70, 90	HS,CS
48	70, 90	HS,CS
72	70, 90	HS,CS

4.1. Quasi static test configuration

Quasi-static axial crushing is performed on the INSTRON Universal testing machine (Figure 3), using a compression rate of 70 mm/min. The machine controls the movement of the moving plate, and the crushing force value is a result of the induced deformation.

Figure 3. Crushing test.

Picture of the machine employed for the crushing test.

4.2. Implementation of the long short term memory

In this work, two LSTMs ($\mathit{LST}{M}_P$ and $\mathit{LST}{M}_E$) were implemented: the first was for forecasting for $\mathit{F}$, and the second was for forecasting simulated energy (Figure 4). Based on experimental design for both heat treatment and trigger mechanisms, inputs and outputs of the LSTM are selected in the following way:

Figure 4. Schematic representation of the inputs (in red) and outputs (in green) of the LSTM. It can be seen that mechanical properties, treatment, and triggers are considered for forecasting the crushing force and energy.

Diagram that illustrates the inputs and outputs of the LSTM.

• Inputs: Vickers hardness (HV), Young modulus (E), ultimate tensile stress (UTS), yield strength ($\mathit{\sigma }$y), trigger geometric shape (shape), trigger distance from the upper (distance), trigger area (area), the ratio of maximum stress modified over unmodified (stress max/stress 0), the temperature in the furnace (furnace t), cooling temperature (cooling temp), cooling method (cooling method), time (t), d (contraction distance), unitary deformation (epsilon).

• Output: For $\mathit{LST}{M}_P$ was crushing force (F), and for $\mathit{LST}{M}_E$ was energy absorbed (energy).

The state activation function was a tanh for the input layer, and the gate activation function was a sigmoid. The inputs for the $\mathit{LST}{M}_P$ were Vickers hardness (HV), Young modulus (E), ultimate tensile stress (UTS), yield strength ($\mathit{\sigma }$y), trigger geometric shape (shape), trigger distance from the upper (distance), trigger area (area), the ratio of maximum stress modified over unmodified (stress max/stress 0), the temperature in the furnace (furnace t), cooling temperature (cooling temp), cooling method (cooling method), time (t), and d (contraction distance). For the $\mathit{LST}{M}_E$ were Vickers hardness (HV), Young modulus (E), ultimate tensile stress (UTS), yield strength ($\mathit{\sigma }$y), trigger geometric shape (shape), trigger distance from the upper (distance), trigger area (area), the ratio of maximum stress modified over unmodified (stress max/stress 0), the temperature in furnace (furnace t), cooling temperature (cooling temp), cooling method (cooling method), time (t), d (contraction distance), and unitary deformation (epsilon), as is shown in Figure 5.

Figure 5. Schematic LSTM.

Illustration of the different parts of the LSTM, with inputs and outputs depicted.

The number of hidden layers was 500, the input weight initialiser method was ’glorot’, and an ’orthogonal’ method was employed for the recurrent weight initialiser in both cases.

4.2.1. Adam optimizer

Based on adaptive estimations of lower-order moments, an Adam method was used to optimise stochastic objective functions using first-order gradients. The technique re-scales the gradient’s diagonal, is easy to use, efficient in terms of computing, needs little memory, and works well for issues with a lot of information or parameters.

Adam employs estimations of a gradient’s first and second moments to change the learning rate for each weight of the neural network. Adam is an adaptive learning rate method that computes individual learning rates for various parameters. The values of the properties employed for the Adam optimiser for both LSTM are shown in Table 2.

Table 2. Values employed in the Adam optimiser’s different parameters for both LSTMs.

Property	Value
GradientDecayFactor	0.90
SquaredGradientDecayFactor	0.99
Epsilon	1.00e-08
InitialLearnRate	0.005
LearnRateSchedule	’piecewise’
LearnRateDropFactor	0.20
LearnRateDropPeriod	125
L2Regularization	1.00e-04
GradientThresholdMethod	’l2norm’
GradientThreshold	1
MaxEpochs	500
MiniBatchSize	128
Verbose	0
VerboseFrequency	50

4.2.2. Quality metrics for time series analysis

Time series prediction performance metrics are concisely assessed to calculate the difference between two series (actual value vs. forecast value), which is employed to determine the performance of ML algorithms. There are many different performance measurements available, such as R-squared, Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Normalised Root Mean Squared Error (NRMSE), among others. Deciding which metric to use and how to interpret the data could be challenging.

Since the errors are squared before averaged, the RMSE lends a relatively high weight to significant errors, making it one of the most popular techniques among ML performance metrics. In situations when significant errors are most unwelcome, the RMSE is thus most helpful. The root-mean-squared error (RMSE) measures the discrepancies between expected and actual values (Singla, Duhan, and Saroha 2022), RSME is defined in equation (8)(8) $\mathit{RMSE}=\sqrt{\frac{1}{N}\sum _{\mathit{i}=1}^N{(x_i-m_i)}^2}$(8) .

(8) $\mathit{RMSE}=\sqrt{\frac{1}{N}\sum _{\mathit{i}=1}^N{(x_i-m_i)}^2}$(8)

Calculated on N discrepancies between the actual output and the forecasted output, where the actual output is represented by $m_i$ and the predicted output by $x_i$.

5. Results and discussion

Energy absorption from impacts is highly nonlinear due to the contacts, material nonlinearities, and high deformation rates involved. The performance of components with different geometric shapes and materials is evaluated to develop the crash box model. For crashworthiness in thin-walled tubes, the material, geometry, and parameters of the energy absorption enhancement mechanism affect the behaviour of the components. Therefore, determining the optimal combination of its parameters is crucial when implementing an energy absorption enhancement method.

Different combinations of design parameters of each method are manufactured and tested experimentally by axial quasi-static crushing in a universal testing machine to determine the effect of heat treatment and trigger mechanism design variables. The results from specimens are crushing force vs tube length reduction curves; with this data, crashworthiness indicators are evaluated (Figure 6).

Figure 6. Stress strain curve of heat-treated aluminum, in comparison with an original 6063 T5 aluminum (a) 70 °C and (b) 90 °C.

Pictures of the different stages of the crushing test.

The combination of parameters that results in the best crashworthiness indicators (higher energy efficiency) is a rectangle shape, 250 $\mathit{m}{m}^2$ area, and 160 mm from the upper border.

For the effects of heat treatment parameters on specific energy absorption, more time in the furnace reduces energy absorbed. Coolant temperature and soaking methods exhibit the same behaviour as peak crushing force.

The plastic folds are formed by the displacement of the hinge in two parallel faces to the inside and two to the outside in asymmetric folding. Square-shaped, thin-walled tubes dissipate mechanical energy into deformation by forming successive plastic folds until the length is reduced. Asymmetric folding always leads to a progressive collapse when the tube remains aligned to the longitudinal axis. On the contrary, if the number of symmetric folds increases, the collapse mechanism can pass from axial progressive to global.

Forecasting of energy and crushing force with LSTM

As mentioned in the past sections, two LSTMs with an Adam optimisation were developed to forecast crushing force and energy. Regarding the accumulated energy, the error is more significant at the beginning of the energy absorption process, as shown in Figure 7. In this figure, a zoom-in allows the reader to see how the LSTM approximates the desired energy; for the LSTM, the total $\mathit{RSM}{E}_E$ for all the experiments was 0.0024.

Figure 7. Zoom in to the graph of the experimental energy (orange solid line) against the LSTM approximation (blue dotted line). The whole graph can be seen on the bottom right.

With energy absorption indicators, the best configuration of parameters for each enhancement method was determined and combined on a single tube. The same experimental data surrogate model was built to predict the crushing force vs. displacement curve with energy enhancement parameters as inputs. The fitting of the surrogate model with experimental data was determined. An experimentally determined optimised model is compared with the LSTM-predicted results to determine how the model approximates the optimal result. As a complementary, the absolute error was obtained for each time sample between the prediction of the LSTM and the FEA against the experimental results. Table 3 shows the accumulative absolute error effect of the methods for each experiment presented in this work. This particular performance measure was considered to assess the effect of approximate quality over the number of samples simultaneously.

Table 3. Accumulated absolute error of the predicting achieved by FEA vs LSTM.

	Accumulated Absolute Error
Experimental name	FEA	LSTM
32probetas_2	7504.020	6483.121
32probetas_3	1589.745	1117.908
32probetas_4	1325.589	643.455
32probetas_9	1641.510	1394.562
32probetas_10	2538.430	1655.723
32probetas_11	1784.189	997.240
32probetas_17	1325.270	663.527
32probetas_18	19694.311	9396.855
32probetas_19	9743.273	6260.520
32probetas_20	11371.870	4013.842
32probetas_25	2073.580	1830.447
32probetas_26	1329.653	842.259
32probetas_27	2525.831	2041.809
32probetas_28	1141.232	908.489
geometria_1_1	21729.106	14530.559
geometria_1_2	41507.827	7788.697
geometria_1_3	2114.275	1779.860
geometria_2_1	1131.95	698.127
geometria_2_3	48001.196	27847.641
geometria_3_1	10183.876	9561.823
geometria_3_2	7460.105	7347.433
geometria_3_3	119084.520	21631.629
geometria_2_2	2538.076	1764.269
geometria_4_2	947177.320	784342.661
geometria_4_3	12350.881	1180.607
area_1_1	42542.561	30745.119
area_1_2	14652.324	11860.335
area_1_3	24718.725	23868.543
area_2_1	41800.252	41569.025
area_2_2	10332.032	10025.569
area_2_3	7528.701	212857.803
area_3_1	96491.600	23435.559
area_3_2	14192.812	1932.957
area_3_3	7723.679	6984.136
area_4_1	4388.271	3870.358
area_4_2	95905.960	43091.170
area_4_3	5287.207	5129.791
175C1h70C complete	5791.825	4135.139
175C1h90C complete	23271.697	9017.025
175C6h70C complete	2281.078	1102.548
175C6h90C complete	5273.094	5264.967
175C12h70C complete	1928.541	835.101
175C12h90C complete	10468.379	7527.035
175C24h70C complete	6817.063	3856.362
175C24h90C complete	12282.620	8123.877
175C48h70C complete	9988.308	6240.703
175C48h90C complete	17469.547	16797.335
175C72h70C complete	633874.038	347422.271
175C72h90C complete	12152.488	10337.981

Figure 8 presents the model performance for forecasting crushing force. In the beginning, it can be seen by a zoom into the figure how the LSTM is trying to stabilise in a blue dotted line. The first maximum absorption peak, where the peak crushing force is presented, is where the prediction referring to the experimental data is separated. However, in the complete history of the crushing force, the $\mathit{RSM}{E}_{\mathit{Cf}}$ was 0.56 for all the experimental samples.

Figure 8. Zoom in to the graph of the experimental crushing force (orange solid line) against the LSTM approximation (blue dotted line). The whole graph can be seen on the bottom right.

The dynamic behaviour for energy absorption processes in the case of impacts can be performed numerically through finite element programmes. Based on the previous work of the authors, the different variants were analysed through finite element simulation (Jiménez-Armendáriz et al. 2023). A result with a low prediction of behaviour was taken. Figures 9 and 10 show crushing force and energy results, respectively. For the design and analysis of mechanical energy absorbers, nonlinear finite element analyses are usually used. This is because large displacements develop, and there are nonlinearities of contact. Although there are constitutive models of materials that help improve the correlation of experimental results, the use of LSTM improves the correlation. Due to the evolutionary process of changing the mechanical properties of the component, the aim is to improve the energy prediction complemented by the design and analysis of the components by virtual tools such as the nonlinear finite element simulation to improve the energy absorption performance along the reduction of the component. This is because adding heat treatments modifies the constitutive models of materials using LSTM, considering the mechanical performance based on the experimental response. The comparison of the prediction among finite element simulation, LSTM prediction, and experimental data history is also presented.

Figure 9. The experimental crushing force (orange solid line) against the LSTM (blue dotted line) and FEA (purple dotted line) approximations.

Figure 10. The experimental energy (orange solid line) against the LSTM (blue dotted line) and FEA (purple dotted line) approximations.

The absorption process depends on the number of bends and the interaction of the same material, which depends on the characteristics and mechanical properties of the material and heat treatments. Because of this, it is necessary to adjust the constituent models of the material through nonlinear finite element simulations.

6. Conclusions

Thin-walled structures are implemented on crashworthiness applications due to their characteristic collapse behaviour, resulting in high plastic deformation rates in a low-mass component. Also, developing these structures with aluminium has additional advantages: this material can withstand a sizeable amount of weight.

In this specific case, it represents the base to ensure occupant safety by dissipating energy and, at the same time, generating low mass structures; means to enhance strength by changing component’s behaviour have been analysed without losing safety.

For the development of this low-mass structure, a trigger mechanism and heat treatment were applied on the same thin-walled 6063 aluminium tube, taking advantage of the characteristic modification to the component’s performance from each method. LSTM was developed to develop a method to reduce experimentation. The advantage of using machine learning for forecasting material properties is well-known in this specific case; it was the first to analyse if an LSTM could predict a thin-walled structure’s behaviour more precisely than simulation and in further work to employ this technique to reduce the cost of experimentation. To prove this, predicted data from LSTM were compared with experimental data, and as a result, the LSTM achieved an RSME of 0.0024 and 0.56 for energy and crushing force, respectively.

The use of LSTM requires the use of data to perform energy prediction through experimental data or numerical simulations. Unlike finite element, LSTM adjusts the model to predict the process of energy absorption, while finite element reproduces the whole process of deformation; through both tools, it is possible to analyse improvements in the components for energy absorption.

Finally, the experimentally obtained optimal design performance was compared with the predicted value from LSTM with this combination of parameters, showing that optimal configuration can be obtained with LSTM, reducing the need for experimental tests.

Acknowledgments

The authors would like to thank the Tecnologico de Monterrey Challenge-Based Research Program project ID IJXT070-22TE60001 and COMECyT project ID IKXT025-22UI60001.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data available on request to the authors

Additional information

Funding

This work was supported by the Tecnologico de Monterrey [IJXT070-22TE60001].

Notes on contributors

Moises Jimenez-Martinez

Moises Jimenez-Martinez is a PhD in Mechanical Engineering with extensive experience in Lightweight structures. His current research interests include Artificial Neural Network in Mechanical design. Finite Element Analysis, Nonlinear analysis and Mechanical Fatigue. Analysis. Currently, he is a Professor SNI level 1 at Tecnologico de Monterrey Campus Puebla.

Jorge Jiménez Armendáriz

Jorge Jiménez-Armendáriz currently studying PhD in Engineering Sciences at Tec de Monterrey Campus Puebla, since 2023. He is currently researching on fatigue of composite materiales at Escuela de Ingeniería y Ciencias of Tec de Monterrey. His research interests include finite element analysis, non-linear deformation, energy absorption and fatigue. He can be contacted at email: [email protected]

Isaac Chairez

Isaac Chairez received the B.S. degree in biomedical engineering from the National Polytechnic Institute (IPN), Mexico City, Mexico, in 2002, and the master’s and Ph.D. degrees from the Department of Automatic Control, Center of Investigation and Advanced Researching (CINVESTAV), IPN, in 2004 and 2007, respectively. He is currently with the Institute of Advanced Materials for Sustainable Manufacturing, Tecnologico de Monterrey. He has published about 210 papers in journals indexed in the JCR and two patent applications. His research interests include neural networks, fuzzy control theory, nonlinear control, adaptive control, and game theory.

Mariel Alfaro-Ponce

Mariel Alfaro Ponce is an assistant professor in the Biomedical Engineering Program at Tecnológico de Monterrey Ciudad de Mexico; she received a bachelor’s degree in Biomedical Engineering, a Master of Science in Microelectronic Engineering, and a Ph.D. in Computer Science from the Instituto Politecnico Nacional, Mexico. From 2022 until now, is the head of the Manufacturing Processes for Advanced Materials CDMX research unit.