Technical Summary: Non-Overlapping Schwarz Hybrid Finite Element–Neural Operator Framework

Problem Statement

Finite Element (FE) methods remain the benchmark for solid mechanics simulations but face prohibitive computational costs in problems characterized by localized nonlinearities, fine-scale features, or long-time dynamic evolution. While Neural Operators (NOs) offer efficient alternatives for approximating PDE solution operators, standalone NOs struggle with large computational domains containing localized nonlinearities and often require high-fidelity training data. Previous hybrid approaches, such as the authors' earlier work [1], coupled FE solvers with Physics-Informed Deep Operator Networks (PI-DeepONets) using an overlapping Schwarz alternating method with Dirichlet-Dirichlet interface exchange. However, this prior framework suffered from two critical limitations:

1. Computational Redundancy: The overlapping formulation required redundant interface computations, increasing the number of inner Schwarz iterations needed for convergence.
2. Geometric Rigidity: The use of convolutional branch networks restricted the NO subdomain to structured, regular grids, precluding simulations on irregular or non-convex geometries.

Methodology

This work proposes a Non-Overlapping Schwarz Hybrid FE-NO Framework that addresses the aforementioned limitations through three primary methodological advancements:

1. Non-Overlapping Neumann-Dirichlet Coupling

The framework replaces the overlapping Dirichlet-Dirichlet scheme with a non-overlapping Schwarz alternating method.

• Domain Partitioning: The computational domain $\Omega$ is split into two non-overlapping regions: $\Omega_{FEM}$ (governed by a conventional FE solver) and $\Omega_{NO}$ (governed by a PI-DeepONet).
• Interface Exchange: Instead of exchanging displacements, the method enforces Neumann-Dirichlet conditions at the interface $\Gamma$. The NO subdomain transmits traction (stress) boundary conditions to the FE solver, while the FE solver provides displacement boundary conditions to the NO.
• Algorithm: The coupling utilizes a relaxation scheme (Algorithm 1) where the traction is updated iteratively until the $L_2$-norm of displacement increments between successive Schwarz iterations falls below a threshold.

2. Point-DeepONet for Unstructured Geometries

To enable simulations on arbitrarily shaped subdomains, the architecture integrates PointNet into the DeepONet branch network.

• Input Handling: Unlike the previous CNN-based branch which required regular grids, the PointNet branch directly processes unstructured point clouds (FE mesh nodes) without interpolation.
• Kinetic State Encoding: For dynamic problems, Branch 2 encodes the kinetic state (displacement and velocity from the previous time step) and reference coordinates. This allows the NO to handle non-convex and irregular geometries, such as filleted L-shapes.
• Time-Marching: The Newmark-$\beta$ scheme is embedded directly into the neural operator architecture, enabling stable long-time dynamic predictions.

3. Analytically Derived Strain and Stress Operators

To ensure physical consistency and reduce parameter counts, strain and stress operators are not trained independently.

• Derivation: Strain operators ($\mathcal{G}_{\epsilon}$) are derived analytically from the displacement operators ($\mathcal{G}_{u}$) via kinematic equations (e.g., $\epsilon_{xx} = \partial u_x / \partial x$). For hyperelasticity, Cauchy stress operators are derived from displacement operators using the constitutive laws.
• Benefit: This reduces the number of trainable parameter sets from five (in prior work) to two (for $u_x$ and $u_y$), significantly lowering GPU memory costs and enforcing mechanical consistency by construction.

Key Results

The framework was validated on four benchmarks: static linear elasticity, quasi-static hyperelasticity, and elastodynamics on both regular (circular) and irregular (filleted L-shaped) subdomains.

• Convergence Efficiency: The non-overlapping Neumann-Dirichlet approach drastically reduced the number of inner Schwarz iterations. In elastodynamic simulations, iterations dropped from 9–10 (in the prior overlapping framework) to 2–3 per time step.
• Computational Speed:
  • Static Linear Elasticity: The FE-NO solver was 2.72× faster than the equivalent FE-FE solver (12.8s vs. 34.8s).
  • Quasi-Static Hyperelasticity: The FE-NO solver was 1.75× faster than FE-FE (56s vs. 98s).
  • Elastodynamics: The non-overlapping scheme reduced wall-clock time from 188.8s (overlapping) to 1.8s (non-overlapping) for FE-NO, primarily due to the elimination of interpolation costs and reduced iteration counts.
• Accuracy and Stability:
  • Error Bounds: Maximum relative displacement errors remained below 0.2% in static cases and 2.5% in dynamic cases.
  • Auto-Regresive Stability: In long-time dynamic simulations (up to 169 time steps), strain errors remained bounded, preventing unbounded error accumulation. The framework demonstrated rapid self-correction when switching between pretrained NOs or handling geometric irregularities.
  • Geometric Flexibility: The Point-DeepONet successfully handled a filleted L-shaped domain, a geometry impossible for the previous CNN-based framework, with errors comparable to the regular disk case after initial stabilization.

Significance and Claims

The paper claims to establish a principled pathway for hybrid physics-based and operator-learning solvers in large-scale dynamic solid mechanics. The significance of this work is summarized as follows:

1. Geometry Flexibility: By integrating PointNet, the framework extends hybrid FE-NO coupling to arbitrarily shaped and non-convex subdomains, removing the restriction to structured grids.
2. Parameter Efficiency and Consistency: The analytical derivation of strain and stress operators enforces mechanical consistency by construction and reduces the trainable parameter space, making the model more efficient and robust.
3. Convergence Stability: The shift to a non-overlapping Neumann-Dirichlet formulation significantly accelerates convergence (reducing iterations by a factor of 3 in dynamic cases) while maintaining bounded error accumulation over long time horizons.

The authors conclude that this framework provides a geometry-flexible, parameter-efficient, and convergence-stable paradigm, offering a viable solution for accelerating computationally intensive subdomains in complex solid mechanics problems without sacrificing global solution fidelity. Future work is identified as extending the framework to 3D unstructured meshes, developing adaptive subdomain decomposition strategies, and implementing online transfer learning to further reduce errors in complex geometries.