Solving plane crack problems via enriched holomorphic neural networks

Abstract

An efficient and accurate method to solve crack problems in plane linear elasticity via physics-informed neural networks is proposed. The method leverages holomorphic neural networks to learn the complex Kolosov–Muskhelishvili potentials that fulfill the problem boundary conditions. The use of the complex potentials implies that the governing differential equations are satisfied a priori. Therefore, only training points on the domain boundary are needed, leading to superior efficiency compared to analogous approaches based on real-valued networks. To accurately capture the stress singularities at the crack tips, two enrichment strategies are introduced. The first consists in enriching the holomorphic neural networks with the square root term from Williams’ series that provides the correct asymptotic profile near the crack tip. The second leverages Rice’s exact global representation of the solution for a straight crack, which effectively decouples the holomorphic part of the solution from the singular, non-holomorphic terms. The integration of the holomorphic neural network representation with the proposed enrichments significantly enhances the accuracy of the learned solution while maintaining a compact network size and reduced training time. Moreover, both enrichment strategies demonstrate stability and are potentially well-suited for crack detection analyses and simulating crack propagation through the use of transfer learning.