Nonlinear two-scale beam simulations accelerated by thermodynamics-informed neural networks

We introduce an efficient computational framework for the simulation of complex beam networks and architected materials. At its core stands a thermodynamics-informed neural network, which serves as a surrogate material model for the cross-sectional response of hyperelastic, slender beams with varying crosssectional sizes and geometries. The beam description relies on a formal asymptotic expansion from 3D elasticity, which decomposes the problem into an efficient macroscale simulation of the beam's centerline and a finite elasticity problem on the cross-section (microscale) at each point along the beam. From the solution on the microscale, an effective energy is passed to the macroscale simulation, where it serves as the material model. We introduce a Sobolev-trained neural network as a surrogate model to approximate the effective energy of the microscale. We compare three different neural network architectures, viz. two well established Multi-Layer Perceptron based approaches - a simple feedforward neural network (FNN) and a partially input convex neural network (PICNN) - as well as a recently proposed Kolmogorov-Arnold (KAN) network, and we evaluate their suitability. The models are trained on varying cross-sectional geometries, particularly interpolating between square, circular, and triangular cross-sections, all of varying sizes and degrees of hollowness. Based on its smooth and accurate prediction of the energy landscape, which allows for automatic differentiation, the KAN model was chosen as the surrogate material model, whose effectiveness we demonstrate in a suite of examples, ranging from cantilever beams to 3D beam networks and architected materials. The surrogate model also shows excellent extrapolation capabilities to load cases outside the training dataset.