[Paper] Smoothed aggregation algebraic multigrid for problems with heterogeneous and anisotropic materials

Source: arXiv - 2602.05686v1

Overview

The paper presents a material‑aware strength‑of‑connection (SoC) measure for smoothed‑aggregation algebraic multigrid (SA‑AMG). By feeding the actual material tensors (e.g., conductivity, diffusivity) into the coarsening stage, the authors eliminate the “blind spots” of classic AMG that often mis‑interpret weak links across sharp material jumps or anisotropic directions. The result is a more reliable, scalable solver for scalar PDEs that appear in high‑performance simulations of batteries, solar cells, and other heterogeneous devices.

Key Contributions

• Material‑driven SoC metric – a novel formulation that blends matrix entries with the underlying material tensor to decide which degrees of freedom are strongly coupled.
• Robust coarsening for heterogeneous & anisotropic problems – the method automatically respects material interfaces and directional anisotropies, preserving smooth error components on coarse levels.
• Extensive validation – benchmark tests (including high‑contrast diffusion, rotated anisotropy, and random mesh refinements) and two real‑world case studies (thermally‑activated battery packs and thin‑film solar cells).
• Scalable parallel implementation – integration into the open‑source AMG library hypre (and a prototype in PETSc) showing near‑linear weak scaling up to thousands of MPI ranks.
• Open‑source release – code and test harnesses are made publicly available, enabling immediate adoption by the HPC community.

Methodology

1. Problem setting – scalar PDE of the form (-\nabla!\cdot(\mathbf{K}(\mathbf{x})\nabla u)=f) discretized with standard finite elements, where (\mathbf{K}) is a spatially varying, possibly highly anisotropic material tensor.

2. Classic SA‑AMG recap – builds a hierarchy of coarse grids by grouping fine‑level unknowns based on a strength‑of‑connection predicate that looks only at matrix entries (or geometric distance). This works well for smooth coefficients but fails when (\mathbf{K}) jumps abruptly.

3. Material‑aware SoC – the authors compute a local metric

   [ s_{ij}= \frac{|a_{ij}|}{\sqrt{a_{ii}a_{jj}}}; \cdot ; \phi(\mathbf{K}_i,\mathbf{K}_j) ]

   where (a_{ij}) are stiffness matrix entries and (\phi) measures alignment between the material tensors at nodes (i) and (j) (e.g., cosine similarity of principal diffusion directions). A threshold (\theta) then decides if the edge ((i,j)) is “strong”.

4. Coarsening & interpolation – using the new strong‑edge graph, standard aggregation (max‑weight matching) is performed, followed by the usual smoothing of tentative prolongators. Because the graph now respects material physics, the resulting coarse operators retain the anisotropic/heterogeneous structure.

5. Implementation details – the tensor‑aware metric is computed on‑the‑fly during the AMG setup phase; the extra cost is linear in the number of non‑zeros and negligible compared with the overall solve time. The authors plug the routine into hypre’s BoomerAMG and expose a few extra parameters (material_metric, anisotropy_threshold).

Results & Findings

• Convergence: iteration counts become mesh‑independent and contrast‑independent for the material‑aware variant.
• Performance: the extra SoC computation adds < 5 % to AMG setup time; overall time‑to‑solution improves by 30 %–70 % on the tested applications.
• Scalability: weak scaling tests show ~ 90 % parallel efficiency up to 4096 MPI ranks, confirming that the new metric does not hinder the communication pattern of standard SA‑AMG.

Practical Implications

• Plug‑and‑play robustness – developers can drop the new SoC into existing AMG‑based solvers (e.g., PETSc, Trilinos, hypre) and immediately gain resilience against material jumps that previously caused solver stalls.
• Accelerated design cycles – battery‑management or photovoltaic‑cell simulations often require many parametric runs; fewer AMG iterations translate directly into faster design‑space exploration.
• Enabling larger‑scale multiphysics – coupling heat, mass, and electro‑chemical transport in heterogeneous media becomes feasible on current HPC clusters without hand‑tuned preconditioners.
• Reduced need for manual mesh refinement – because the coarse operators now respect anisotropy, developers can rely on coarser meshes without sacrificing accuracy, saving memory and compute resources.

Limitations & Future Work

• Scalar PDE focus – the current formulation assumes a single scalar field; extending to vector‑valued systems (e.g., elasticity, Navier‑Stokes) will require a more sophisticated tensor metric.
• Parameter sensitivity – the threshold (\theta) and the anisotropy weighting function (\phi) still need modest tuning for extreme cases (e.g., ultra‑high contrast > 10⁸).
• GPU adoption – while the algorithm is linear‑complexity, the authors note that the tensor‑aware SoC computation has not yet been ported to GPU kernels; future work will explore CUDA/ROCm implementations.
• Adaptive refinement – integrating the material‑aware metric with adaptive mesh refinement loops is an open research direction, potentially yielding even tighter error control.

Bottom line: By letting the material physics speak directly to the AMG coarsening process, this work bridges a long‑standing gap between robust linear solvers and the messy reality of heterogeneous, anisotropic simulations—making high‑performance scientific computing a bit more “plug‑and‑play” for developers.

Authors

• Max Firmbach
• Malachi Phillips
• Christian Glusa
• Alexander Popp
• Christopher M. Siefert
• Matthias Mayr

Paper Information

• arXiv ID: 2602.05686v1
• Categories: cs.CE, cs.DC
• Published: February 5, 2026
• PDF: Download PDF