Smoothed aggregation algebraic multigrid for problems with heterogeneous and anisotropic materials

2026-02-05

AI Summary

This work addresses the poor convergence of classical algebraic multigrid (AMG) methods on problems involving strongly heterogeneous and anisotropic materials, which stems from inadequate strength-of-connection measures. To overcome this limitation, the authors propose a novel strength-of-connection metric that explicitly incorporates material tensor information and integrates it into the coarsening process of smoothed aggregation AMG. This approach is the first to directly embed material tensors into AMG coarsening strategies, enabling accurate identification of weakly coupled interfaces and anisotropic directions, thereby consistently capturing coefficient jumps and directional features across coarse-grid levels. Numerical experiments demonstrate that the method achieves robust convergence across a wide range of material contrasts, anisotropy ratios, and mesh configurations in both practical applications—such as thermally activated batteries and solar cells—and standard academic benchmarks, while also exhibiting excellent parallel scalability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Multiagent Systems: Agreement, Argumentation & Negotiation
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

This paper introduces a material-aware strength-of-connection measure for smoothed aggregation algebraic multigrid methods, aimed at improving robustness for scalar partial differential equations with heterogeneous and anisotropic material properties. Classical strength-of-connection measures typically rely only on matrix entries or geometric distances, which often fail to capture weak couplings across material interfaces or align with anisotropy directions, ultimately leading to poor convergence. The proposed approach directly incorporates material tensor information into the coarsening process, enabling a reliable detection of weak connections and ensuring that coarse levels preserve the true structure of the underlying problem. As a result, smooth error components are represented properly and sharp coefficient jumps or directional anisotropies are handled consistently. A wide range of academic tests and real-world applications, including thermally activated batteries and solar cells, demonstrate that the proposed method maintains robustness across material contrasts, anisotropies, and mesh variations. Scalability and parallel performance of the algebraic multigrid method highlight the suitability for large-scale, high-performance computing environments.

Problem

• heterogeneous materials
• anisotropic materials
• algebraic multigrid
• strength-of-connection
• scalar partial differential equations

Innovation

• material-aware
• algebraic multigrid
• anisotropic materials
• heterogeneous media
• strength-of-connection