A novel surrogate model to approximate microscopic behaviour and accelerate concurrent multiscale finite element simulations is proposed. The study serves as a proof of concept, focusing exclusively on 2D, geometric non-linear lattice materials. Despite numerous successful implementations of surrogate modelling techniques in literature, challenges remain, mainly with the black-box nature of most of these models, suffering from lack of interpretability. To tackle these issues, this study reintroduces physics into the model through the use of beam theory in so-called Beam Neural Networks. These networks are tested against a benchmark feed-forward neural network in both interpolation and extrapolation. Although the findings do not satisfy the requirements for practical application, they do indicate that the introduction of beam theory to the model has improved the model's extrapolation ability, suggesting that the proposal has improved robustness and interpretability of the model. Given further optimization, there is promise of Beam Neural Networks to become an useful tool to accelerate concurrent multiscale modelling in the future. ... Read more

In the past decade, the application of Neural Networks (NNs) has received increasing interest due to the growth in computing power. In the field of computational mechanics, this has led to numerous publications presenting surrogate models to assist or replace conventional simulation methods. A subset of these networks, referred to as Graph Neural Networks (GNNs) impose the graph-like structure of many physical problems as a relational inductive bias. Several time-stepper implementations of these GNNs are reported to be able to simulate the dynamic behaviour of various physical objects. Within this work, it is investigated whether such GNN-based surrogate models can be applied to simulate the dynamic behaviour of lattice structures.
Upon inference of such a GNN surrogate model, the computational time required for studying lattice behaviour could be considerably reduced, thus advancing research into lattice structures as metamaterials. In addition, many large-scale structures also form a composition of beams, which could be modelled with a similar GNN.
To this end the following research question was defined: ”To what extent can GNNs be applied to simulate the dynamic behaviour of lattice structures using time-stepper methods?”. To answer this question, several GNN architectures were constructed and subsequently analyzed.

In this research, it was found that the complexity of lattice structures could not be modelled in such a way as to obtain reliable, generalisable and stable behaviour, using a time-stepper method with an architecture similar to that of Pfaff et al., 2020. It was found that due to the existence of three physically different coupled Degrees of Freedom (DOF) per node the behaviour was too complex to learn for the proposed surrogate.
At the time of writing, there is no publication presenting an effective surrogate model to simulate the dynamic behaviour of a Timoshenko beam using time-stepper methods. It is concluded that the need to capture both bending and shear behaviour using a Timoshenko beam formulation is the bottleneck for successfully modelling lattice structures. ... Read more