Smart stiffness computation of one-dimensional Finite Elements

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In the present study, three sophisticated methods are introduced to compute tangent stiffness matrices in Finite Element Analysis by Artificial Neural Networks (ANNs). We propose a modified training procedure by adding a loss term corresponding to the element tangent stiffness matrix in the optimization criteria. This training procedure is referred to as Sobolev training and it ensures that the ANN learns the functional relationship to approximate the converged internal force and its corresponding first derivative to compute the converged tangent stiffness matrix. This new development eliminates the need to perform iterations in the nonlinear solver and leads to a significant acceleration of the Finite Element simulation. We then propose a simple scaling strategy that uses the trained network to predict the stiffness information of elements with different geometrical properties. Thus, the focus of this work is to establish a neural network-based FEM framework (independent of NN topology) to introduce an enhanced stiffness replacement for truss and beam elements taking physical non-linearities into account. The performance of the proposed methods is demonstrated on academic examples showing a maximum of 41% boost in simulation speed.