A Non-Overlapping Schwarz Hybrid Finite Element-Neural Operator Framework for Solid Mechanics on Irregular Domains

2026-06-07

AI Summary

This work addresses the high computational cost of traditional finite element methods in problems involving local nonlinearities, fine-scale features, or long-time dynamics, as well as the limited geometric flexibility of existing hybrid approaches. To overcome these challenges, the authors propose a non-overlapping Schwarz alternating coupling framework that efficiently integrates finite elements with neural operators. Information exchange between subdomains is achieved through Neumann–Dirichlet interface conditions, eliminating overlapping regions to avoid redundant computation. The framework employs Point-DeepONet to directly handle unstructured point clouds, while strain and stress operators are analytically derived from the displacement operator, ensuring mechanical consistency and reducing model parameters. Benchmark tests in static linear elasticity, quasi-static hyperelasticity, and elastoplastic dynamics demonstrate the method’s superior geometric adaptability, parameter efficiency, and convergence stability.

Abstract

Finite element (FE) methods are the benchmark for solid mechanics simulations, yet their computational cost becomes prohibitive for problems with localised nonlinearities, fine-scale features, or long-time dynamic evolution. In our earlier FE-neural operator (FE-NO) hybrid framework [1], physics-informed deep operator networks were coupled with FE solvers through overlapping domain decomposition with Dirichlet-Dirichlet interface exchange, accelerating intensive subdomains while preserving FE fidelity elsewhere. Two limitations remained: the overlapping formulation required redundant interface computations that increased inner Schwarz iteration counts, and the convolutional feature extractor restricted the NO subdomain to structured grids, precluding irregular geometries. A non-overlapping Schwarz alternating method with Neumann-Dirichlet interface exchange replaces it, transmitting traction from the NO to FE rather than displacement. This eliminates the overlap layer and reduces inner Schwarz iterations while maintaining bounded error accumulation across all tested time horizons. For arbitrarily shaped subdomains, a Point-DeepONet operates on unstructured FE point clouds without interpolation, extending it to non-convex and irregular geometries. Strain and stress operators are derived analytically from the displacement operators via kinematic equations, rather than as independent networks, reducing trainable parameter sets while enforcing mechanical consistency by construction. The framework is validated on three benchmarks: static linear elasticity, quasi-static hyperelasticity, and elastodynamics with regular and irregular geometries. These results establish a non-overlapping FE-NO coupling paradigm that is geometry-flexible, parameter-efficient, and convergence-stable, providing a pathway for hybrid physics-based and operator-learning solvers in large-scale dynamic solid mechanics.

Problem

• solid mechanics
• irregular domains
• computational cost
• nonlinearities
• geometry flexibility

Innovation

• Non-overlapping Schwarz method
• Point-DeepONet
• Hybrid FE-Neural Operator
• Irregular domains
• Mechanical consistency