In recent years, artificial neural networks were included in the prediction of deformations of structural elements, such as pipes or tensile specimens. Following this method, classical mechanical calculations were replaced by a set of matrix multiplications by means of artificial intelligence. This was also continued in finite element approaches, wherein constitutive equations were substituted by an artificial neural network (ANN). However, little is known about predicting complex non-linear structural deformations with artificial intelligence. The aim of the present study is to make ANN accessible to complicated structural deformations. Here, shock-wave loaded plates are chosen, which lead to a boundary value problem taking geometrical and physical non-linearities into account. A wide range of strain-rates and highly dynamic deformations are covered in this type of deformation. One ANN is proposed for the entire structural model and another ANN is developed for replacing viscoplastic constitutive equations, integrated into a finite element code, leading to an intelligent finite element. All calculated results are verified by experiments with a shock tube and short-time measurement techniques.

Introduction

Artificial neural networks have been applied in engineering problems as an alternative approach compared to classical methods based on continuum mechanical modelling. Promising results were achieved by investigating stress-strain curves of metal specimens under hightemperature [1], design of steel structures [2], vibrations of structures [3,4], or stability problems of structures [5]. Reliability studies of structures were reported in [6] and influences of welding on material properties are investigated in [7]. An ANN can lead to much lower computational time and can replace the mechanical model completely. It can be trained by experimental data, only, and needs therefore no identification of material parameters. Consequently, a mathematical model is generated by means of an algebraic system of equations. Following this approach, the ANN approximates to the trained data. The learning procedure of the ANN is based on the examples, which are provided by the user [8]. However, weaknesses of ANNs have been reported in [9] due to the difficulties of interpreting parameters in neural networks, e.g. the number of hidden layers or neurons. Also the components of the synapse matrices of a trained ANN can hardly be interpreted as it can be done with material parameters in a constitutive law. In several studies, the problem of a so-called black box is described [10,11]. Consequently, it is difficult to find reasons to explain discorrelations between predictions using ANN and experimental data. Once, the ANN has been trained well with input and output data sets, it can recalculate the provided data very accurately. However, predictions beyond that data can lead to uncertain results, which is documented in literature [12]. An additional approach using the advantages of ANN together with well-established numerical methods is the development of intelligent finite elements. These elements have been proposed in literature, leading to a combination of classical finite elements with an ANN and are used only for a part of the entire mechanical model. Studies substituting the constitutive model by means of an ANN have been published in [13]. A beam element, based on a neural network, is proposed in [14] and leads to lower computational costs than a classical approach. This benefit is even more pronounced since multiscale approaches are concerned [15]. In literature several neural network constitutive models (NNCM) were discussed [16]. However, it was reported that the choice of the provided training data is essential for a reliable intelligent finite element [17].