A meshless neural network approach for the dynamic analysis of elastic systems

A meshless approach is presented for the computation of the approximated solution of static and dynamic problems in linear elasticity in terms of displacement fields. The displacement field is modeled by means of two different kinds of Artificial Neural Networks (ANN). This task is accomplished by means of a meshless approach coupled to a net training based on the weak formulation of the differential problem, related to Hu-Washizu principle. A common benchmark, namely the Timoshenko cantilever beam, is analyzed and discussed in detail; several researches have shown that severe difficulties are encountered with the Galerkin and the collocation approach since the neural networks never satisfy essential boundary conditions (EBC): in the proposed meshless approach, the trial functions can be modified in order to satisfy EBC. A possibility to overcome such difficulty in elasticity is to employ an energy-based training, that is, to employ an approach (such as the Hu-Washizu functional) which can take into account EBC in the error function to be minimized. An example is given in one dimension, analyzing the deflection of a horizontal beam subject to transverse loads. The presented examples clearly show the importance of the optimization of the non-linear pa-rameters of the network, which control the shape and location of the activation functions. It is shown that such parameters can be optimized for a static problem and subsequently employed for a dynamic problem. The paper in fact aims to extend the results investigating the bench-mark problem in the dynamic field.