Executive Summary: Overlapping Schwarz Domain Decomposition Methods in Python with Applications in Structural Mechanics

This research aims to address the computational challenges inherent in solving large-scale nonlinear systems in structural mechanics, with particular focus on problems exhibiting material plasticity. The motivation is to harness advanced domain decomposition methods to improve the efficiency, scalability, and robustness of finite element analysis for complex engineering structures.

Methods:

The study employs the finite element method (FEM) as the foundational modeling framework, integrating overlapping Schwarz domain decomposition techniques as preconditioners for nonlinear iterative solvers. Algorithms are developed and tested using Python, utilizing both direct and iterative solvers for local subdomain problems. The workflow includes the use of automated mesh generation with GMsh and custom modules for subdomain partitioning, restriction, and extension operations. Special emphasis is placed on nonlinear material modeling using the von Mises yield criterion with isotropic hardening, alongside the application of Newton-Krylov-Schwarz solver strategies and advanced preconditioning.

Results:

The implementation demonstrates that overlapping Schwarz preconditioners significantly accelerate convergence and boost computational efficiency compared to traditional Newton-Raphson approaches, especially in large and highly nonlinear cases. Benchmarking across a series of test cases—including elastic and plastic materials, simple and complex geometries—confirms robustness, improved scalability, and maintained solution accuracy. Both direct and iterative solvers for subdomains are validated, with direct solvers showing particular effectiveness for moderate subdomain sizes. The Python-based toolkit proves highly adaptable and extensible for further research and industrial application.

Conclusions:

Overlapping Schwarz domain decomposition methods, when integrated with advanced nonlinear solvers, represent a transformative advancement for computational structural mechanics. They offer a scalable, robust pathway for efficiently solving complex, nonlinear finite element problems and open significant opportunities for future parallelization, high-performance computing integration, and application to a wider range of material models and multiphysics problems.

Keywords:

Finite Element Method, Overlapping Schwarz Method, Domain Decomposition, Nonlinear Structural Mechanics, ASPIN, Preconditioning, Plasticity, Python, Newton-Krylov-Schwarz

This series of articles is available in PDF format at pdf/ddm-nfem-summary.pdf.