Monte Carlo Methods of Optimization of Layers for High Speed Impacts Analytics and Classification
The construction of structures subjected to impact was traditionally car-ried out empirically, relying on real impact tests. The need for design tools to simulate this process triggered the development in recent years of a large number of models of different types. Taking into account the difficulties of these methods, poor precision and high computational cost, a neural network for the classification of the result of impacts on steel armours was designed. Furthermore, the numerical simulation method was used to obtain a set of input patterns to probe the capacity of themodel development. In the problem tackled with, the available data for the net- work designed include, the geometrical parameters of the solids involved — radius and length of the projectile, thickness of the steel armour — and the impact velocity, while the response of the system is the prediction about the plate perforation.
Introduction
There are a wide number of systems which, during their service life, can suffer the impact of objects moving at high speed (over 500 m/s). This phenomenon is named impact ballistic and the most characteristic examples are found in the military field. However over the last decades, this kind of problems has become of interest in civil applications. In them, the structural elements are required to absorb the projectile energy so that it does not damage critical parts of the global system.
Due to this, there are new possibilities in similar fields, among which passive safety in vehicles stands out. In this field, it is especially relevant the design of structures whose mission is to absorb energy in crashes of Crashworthiness type (200 m/s speed 500 m/s), as well as those that can appear in road or railway accidents, helicopters’ emergency landings, etc. Therefore, what is being sought is to design structures capable of absorbing energy to avoid or lessen the damages to the passengers of the concerned vehicles.
The construction of structures subjected to impact was traditionally carried out empirically, relying on real impact tests, each one using a given configuration of projectile and target. The mathematical complexity of solving the equations that rule the impact phenomenon, and the relative ignorance of the mechanical behaviour of the materials at high strain rates, discouraged any simulation of the problem.
The need for design tools to simulate this process triggered the development in recent years of a large number of models of different types; all of them belong to two families: analytical modelling and numerical simulation. Thus, the use of expensive experimental tests has been postponed to the final stage of the design. All the previous stages can be covered by the use of this kind of simulation tools.
Taking into account the difficulties of these methods, poor precision and high computational cost, a neural network for the classification of the result of impacts on steel armours was designed. Furthermore, the numerical simulation method was used to obtain a set of input patterns to probe the capacity of the model development. In the problem tackled with, the available data for the network designed include, the geometrical parameters of the solids involved — radius and length of the projectile, thickness of the steel armour — and the impact velocity, while the response of the system is the prediction about the plate perforation.
Ballistic Impact
The ballistic terminology is defined as the scientific study of everything related to the movement of the projectile. Within this science, the Terminal Ballistic or Effect Ballistic discipline stands out. It is in charge of studying the results produced in the body or object that suffers the impact of the bullet or projectile. Moreover it analyses the way the projectile behaves when it reaches its target, how the projectile ends up, how the transference of kinetic energy is carried out and what effects it has on the objective, etc.
The conditions in which this projectile impacts on the objective depend most of the times on different parameters, namely the materials, both of the projectile and the target, the speed, the impact angle, the thickness of the target, the diameter of the projectile and finally the shape and weight. The combination of these factors leads to the existence of a vast variety of possible behaviours once the contact between them is produced The model developed was simplified to reduce the computational complexity of the numeric simulations needed to recreate impact tests of this type. For this purpose the obliquity of the projectile was not taking into account. Therefore, a right angle between the axis of the projectile and the surface of the plate where it impacts against, has been considered.
From a purely physique point of view, the impact of a projectile on a protection can originate three different results: arrest, perforation or ricochet. Meanwhile the process of the projectile’s entrance inside the protection during an impact is called penetration. A perforation is produced when the projectile goes through the protec- tion completely. On the other hand, an arrest takes place when the projectile goes inside the protection but does not go through it. Finally if the angle of attack in the impact comes into play, the projectile can rebound, what is known as ricochet (Fig. 32.1).
The investigation carried out in this study analyses the behaviour of metallic projectiles when the impact against metallic plates, both recreated with the same SAE 1006 steel material. This material has been chosen because it is isotope, its parameters are known and it allows carrying out further tests in the laboratory at a low cost.
Use of ANN in Impacts Situations
Artificial Neural Networks (ANNs) are statistical models of real world systems which are built by tunning a set of parameters. These parameters, known as weights, describe a model which forms a mapping from a set of given values, the inputs, to an associated set of values, the outputs.
The mathematical model that the neural network builds is actually made up of a set of simple functions linked together by the weights. The weights describe the effect that each simple function (known as unit or neuron) will have on the overall model [1].
Within the field of research of this article, the ANNs have been applied success- fully within the limits of the fracture mechanic [2] to estimate material breakage parameters such as concrete [3], and in no-destructive tests to detect breaks in more fragile materials [4]. In all these applications, the input data needed for the training has been obtained by means of experimentation and numeric simulation.
Nevertheless there are not enough studies dedicated to the application of ANN to solve problems of ballistic impacts. At the present time the investigations made have focused on low speed impacts or situations where it is necessary an energy absorption (Crashworthiness) [5 — 7].
Therefore, and due to the mentioned experience on the mechanical problems and on the good results shown on the previous researches [8], a MultiLayer Perceptron (MLP) neural network with backpropagation algorithm has been defined. The most important attribute of this kind of ANN is that it can learn a mapping of any complexity or implement decision surfaces separating pattern classes.
An MLP has a set of inputs units whose function it is to take input values from the outside, a set of outputs units which report the final answer, and a set of processing hidden units which link the inputs to the outputs. The function of the hidden neurons is to extract useful features from the input data which are, in turn, used to predict the values of the output units.
MLPs are layered neural network, see Fig. 32.2, that means they are based on several layers of adaptive weights and neurons. There are three different layers: the input layer, at least one hidden layer and finally the output layer. Between the units or neurons that compose the network there are a set of connections that link one to each other. Each unit transmits signals to the ones connected with its output. Asso- ciated with each unit there is a output signal or transference signal, that transform current state of the unit into an output signal.
The feedforward connections between the units of the layers in a MLP represent the adaptive weights that permit the mapping between the input variables and the output variables [9].
Different authors have showed that MLP networks with as few as a single hidden layer are universal approximators, in other words ANN are capable to approximate, with accurate, arbitrary regions, if enough hidden units are available [10, 11].
Impact Scenario Parameters
Within the limits this article deals with, there are different variables or parameters that characterize the behaviour of the projectile when it impacts on an steel armour. Therefore, there are various parameters to shape the input data to the network, and these define the number of neurons of the input layer.
The available variables are: kinetic energy (K), velocity (V), radius (R), length (L), mass (M) and quotient L/R, being all of them related to the projectile, and on the other hand the thickness of the objective (H).
However, the use of all the available variables is not always necessary to carry out the training. In some cases, an information overload or the existing connections between the variables can saturate the prediction model that adjusts the output of the network, therefore complicating the learning and reducing the generalization rates. In this domain, K is correlated with V and M. On the other hand, the ratio L/R and M are correlated with R and L because density is constant. So for the neural network performance it is better to remove K and ratio L/R from the list of available variables.
Moreover, in order to support the latter assertion, a series of studies were designed to measure the influence that each variable has, separately and in con- nection with the remainder, on the learning and the network generalization ability.
Network Structure
For the network design, a three level MLP architecture was selected. According to Lippman’s studies, this type of structure allows to solve most of the problems, being able to form any complex random limit of decision [12].
The number of inputs in a MLP is determined by the number of available input parameters in the problem dealing with. The number of outputs is the same as the number of classes needed to solve the problem.
The number of hidden layers and the number of neurons of these layers have to be chosen by the designer. There is not a method or rule that determines the optimum number to solve a problem given. In some cases these parameters are determined by test and error. However, there are current techniques for obtaining automatic estimations [13], although this research follows the steps described by Tarassenko [14].
The neural network was created with the commercial software Neurosolutions for Excel v4 and its architecture is shown in Fig. 32.3.
- Five neurons in the input layer linked with the five input variables (R,L,H,M,V). The chosen transference function is the identity.
- Four neurons in the hidden layer with hyperbolic tangent transference function.
- Two neurons in the output layer associated to the output variable plate perforation (both outputs are complementary as it allows to improve the learning process). The chosen transference function in this case is the hyperbolic tangent.
Input Data
It is important to make an estimation of the necessary information to train the net- work in an appropriate way. For this purpose, it is essential to have a number of data large enough to ensure that. While the network is trained, it is essential to recog- nise and answer to the wide set of conditions that may appear during the testing process. If the number of data is too small, the complete range of connections that the neural network should learn will not be covered. In this sense, there are differ- ent approximations to calculate the optimum number of input data, like [9] or [15]. The designed system has followed the guidelines set by Tarassenko to calculate the number of examples needed [14].
Fifty impact examples were generated by numerical simulation. Some of them were used for the training and testing process and the other for the validation of the network after learning. The modelling and simulation were done with the commer- cial software ABAQUS/Explicit v6.4.1 [16], a finite element explicit code widely used in highly non-linear dynamic simulation.
The stopping criterion selected was to perform a specific number of epochs to train the network, in this case, 10,000 epochs. Consequently the neural network does not use validation data in the training process because the stopping criterion is set by the epochs and not by the cross validation error. Therefore the ANN only uses a train subset for the training process and a test set to validate the accurate of the network.
The set of training patterns is repeatedly presented in random order to the net- work and the weight update equations are applied after the presentation of each pattern. When all the patterns have been presented once to the network, one epoch of training is made. After the training process, the unseen patterns of the test set are presented to the network. This demonstrates the generalization performance of the trained network.
Solution Proposed
In the light of the results obtained by Garcia et al. in 2006 [8], the neural networks present themselves as an alternative to be borne in mind to recreate impact prob- lems. The network results reliable to predict the projectile arrest with the advantage, opposing to the use of simulation tools, that it boasts a computational cost very inferior once the training has been carried out.
However, these conclusions lead to another question. Could there be a more effi- cient ANN model? That is to say a model that would need less input variables and input data to be able to approximate the output function correctly.
The research carried out to solve this question established two fundamental goals, first, to minimize the number of training data without affecting the generalization ability of the model; and second, to analyse the influence of the different input variables on the network learning ability.
These input variables form the independent variables of the function that allows to approximate the perforation function. The result of this answer function is a dichotomic variable that depends on the input variables, and admits “Yes” or “No” values.
Hence it is possible to infer that the easier the function that describes the net- work’s behaviour is, the less costly will be generate the set of training data through numeric simulation.
The type of heuristic selection of input variables and training set size selected is well documented in the early literature of Neural Networks within other domains, and the results obtained certifies the suitability of this election [17].
Randomness Elimination
The results obtained in the former research [8], present the uncertainty if they depend on a possible randomness of the data intended for training and testing. In other words, if the assignment of the available data influence on the predictions of the network.
To carry out a research with conclusive results, it was established a number of trials by far superior to the previous work. The study carried out is broken out in a series of experiments, in which the number of patterns intended for training and test varies in each series. This way, the minimum number of data needed to ensure an acceptable percentage of correct answers will be found.
It has been accomplished 100 different trials in each experiment, exchanging in each of them and in a random way the data destined to train and test the network. Moreover, in two different trials of a same experiment the same training and test data will never coincide, therefore increasing the reliability on the results obtained. Thanks to this, the possible random data that are provided to the network, which could lead to obtain not very reliable results, are eliminated.
The catalogue of experiments to be done in the study is the following: the first one includes 100 trials with 50 training data and 10 testing data. From this point onwards, in each of them the number of data intended to training is reduced to 5. The last experiment consists of 100 trials with 15 training data and 10 testing data.
The result obtained in each one of these 100 trails is processed and the average error porcentage is showed for each experiment. In this way, the results obtained in each of the eight experiments can be compared. Thanks to this, it can be deter- mined how many training data are necessary to achieve good generalization rates;
and besides, it ensures that the possible random data of the input patterns does not influence on these results.
Determination of the Influence of the Independent Variables
The second part of this study is focused on the analysis of the influence of each variable within the training process and, therefore, of the mathematical model that regulates its output.
So the main question is to find the explanatory variables, in other words, to include in the model just those variables that can influence on the answer, rejecting those ones that do not provide information. It is also necessary look for the possible interactions between the independent variables that affect the answer variable. For example, this work will analyse the results obtained by including the thickness and the velocity separately, as well as when they are both included.
Evaluation
The data shown in Table 32.1 are the average error percentages that the network makes for each experiment. The different experiments are grouped in columns, where it is indicated the parameters or variables used, e.g the column “All-H” points out that the experiment uses all the variables except the parameter H. One result example in Table 32.1 is the following one: in the model with all the variables, the value 3,2 is the average error of the network for 100 trials with 50 data to train and 10 to testing.
Regarding the objective sought, to find the network’s architecture that makes the smallest error, when the data is analysed it could be established that the more input variables the network has, the better it learns.
However, it can occur that the networks saturation, that is to say to introduce all the available parameters, is not the best option. In this problem, this network architecture has the lowest error probability only for 50 and 45 train data. In addition to this, there are two designs that have lower average probability of error, the ones without mass and without length. Specifically the design that omits the mass as a network learning variable is the one that has the smallest average error.
On the other hand, for the average of the error obtained, it can be observed that the velocity is a very relevant magnitude for the network learning. Its omission leads the error probability to increase 342%, or what is the same, 4.4 times in relation to the network’s design considered as the most efficient (architecture of network without mass).
Finally as expected, the network with the worst results is the one with less infor- mation, that is, with less input variables. However, it is advisable to highlight that the network without the thickness and mass variables has quite acceptable results (12.91%).
Taking the best configuration for the most architectures, 50 train and 10 testing, the Figure 32.4 depicts the average error percentage made for each network archi- tecture in which some variable is removed with regard to the network that holds all of them.
Table 32.2 portrays the differences of the error percentages for each of the network designs with regard to the one with the minimum error. As it has been
mentioned, the network that includes all the available variables in the learning is the one that makes the smaller error for 50 and 45 data; whilst the network without mass is the one that boasts the smallest error for a smaller number of trials. Based on these results, it can be concluded that the influence of the mass on learning is not very relevant, and therefore the network that does not include it, it is considered the most efficient.
Conclusions
In the light of the results and taking into account that perceptron is one of the sim- plest topologies, this work shows clearly the possibilities of the neural networks in the prediction of the material behaviour at high deformation speeds. Furthermore, the architecture chosen presents a high reliability when predicting the result of a projectile impact. Moreover, its computational cost once the training has started, is smaller than the one of the simulations carried out with tools of finite elements.
It is crucial to highlight the small number of training data needed for the network to learn with a relative small error in its predictions. With only 40 numeric simula- tions of finite elements, the network obtains an error below 6% in both designs with the best results (with all the variables and without the mass variable). In spite of the small number of input data, the network does not present overlearning problems.
The experiments developed help to better understand the ballistic impact, analy- sing the influence of the parameters that appear in the penetration problem. The results obtained verify that the variable with the most influence is the velocity. Fur- thermore, any network architecture where the velocity variable is removed obtains error percentages quite high, what confirms this variable importance.
On the other hand, the research determines that the influence of the mass on learn- ing is not very relevant. Therefore and taking into account its numeric simulation cost, the network without this variable is considered the most efficient.
The network with the worst results is the one with less information, that is to say, with less input variables. However, it is advisable to highlight that the network without thickness and mass variables has quite acceptable results taking into account the little information it receives in comparison with the rest.
Finally, the knowledge acquired as a result of this research, can be spread out to other fields of great practical interest. Among these, the design of structural compo- nents of energy absorption stands out, being of great relevance in the field of passive safety in vehicles.
