Abstract:

Efficient and accurate evaluation of shape functions is critical for element stiffness assembly and the global solution in finite element (FE) computations. A general framework was proposed for approximating shape functions was proposed to handle multiple element types and variable node counts. First, learnable type embeddings were used to map discrete element classes to continuous vectors, enabling parameter sharing across element types. Second, geometric information and query-point features were processed by a node encoder; graph convolutions were used to propagated local geometric constraints, while global pooling was applied provided element-level context. Finally, shape function values were produced by a decoder from node- and element-level features, and the model was trained with a loss combining mean square error (MSE) and a physics-inspired sum constraint to enforce shape-function properties. Experiments on synthetic datasets of linear 4-node tetrahedra and quadratic 10-node showed demonstrated that the model achieved a test MSE of approximately 0.001 8 and that shape function values were computed accurately across the test set. Moreover, in a large-scale computing environment, the neural-network inference attained roughly three times the throughput of a conventional interpolation-based implementation. These results suggested a promising route to accelerate or replace classical shape-function evaluations using learning methods.

Key words: graph neural networks, finite element, shape function, CAE, deep learning