BiLoGo: Reducing the smoothing of localised responses in Transformer-based neural operators for sheet forming

Trial-and-error FE simulation of sheet forming could be computationally expensive, neural-network-based learned simulators are used to accelerate design. These simulators should accurately capture localised responses, such as thinning around sharp radii or near ribs, which determine formability. Transformer-based neural operators propagate information efficiently across the mesh but tend to smooth such localised responses. This study develops BiLoGo, a Bipartite Local-Global learned simulator with an FE-guided design that connects local node-element message passing with a Transformer-based neural operator for long-range information exchange. BiLoGo has three modules: a bidirectional node-element GraphSAGE model that learns nodal-elemental coupling on a bipartite graph and predicts each quantity on its native FE entity without interpolation; a dual-stream Transformer-based neural operator that processes blank deformation and tool contact with separate token sets, preventing localised contact information from being averaged with the whole-blank state; and a fusion and element update module that combines global and local features via gated residual fusion and updates element features conditioned on element-node kinematics. Evaluated on a dome-shaped cold forming case and two corner-shaped hot stamping cases (one with ribs), BiLoGo achieves the lowest positional and threshold-based thinning errors on all three cases compared with single-level and hierarchical graph-based models and Transformer-based neural operators. Near the ribs, the smoothing index of predicted thinning increases from 0.41-0.56 for Transformer-based neural operators to 0.94 for BiLoGo-U. Compared with the hierarchical graph network CAtt-BiUGNN, BiLoGo-T reduces mean positional error by 15.5% on the Corner case and BiLoGo-U by 8.3% on the Corner+Rib case, with 66-69% less training memory.