[Paper] Parallel simulation and adaptive mesh refinement for 3D elastostatic contact mechanics problems between deformable bodies

Source: arXiv - 2511.20142v1

Overview

This paper presents a scalable parallel algorithm for solving 3‑D elastostatic contact problems between deformable bodies using adaptive mesh refinement (AMR) within a finite‑element framework. By tightly coupling a node‑to‑node contact formulation with a parallel h‑adaptive mesh strategy, the authors achieve high performance on up to 1 024 CPU cores, opening the door to realistic, high‑resolution simulations that were previously out of reach.

Key Contributions

• Hybrid contact‑AMR solver: Combines a node‑to‑node penalized contact algorithm with a non‑conforming h‑adaptive refinement loop (estimate‑mark‑refine).
• MPI‑aware mesh partitioning: Guarantees that paired contact nodes reside on the same MPI rank, drastically cutting inter‑process communication for the contact operator.
• Super‑parametric elements: Uses higher‑order geometric mapping during refinement to preserve curved domain boundaries, even on coarse meshes.
• Robust parallel scalability: Demonstrates strong scaling up to 1 024 cores for both 2‑D and 3‑D Hertzian contact benchmarks.
• Comprehensive performance analysis: Includes memory footprint, preconditioner behavior, and a study of several AMR error indicators.

Methodology

1. Finite‑Element Discretization

   • The deformable bodies are meshed with quadrilateral (2‑D) or hexahedral (3‑D) elements.
   • A penalization term enforces contact constraints, while a node‑to‑node pairing identifies which surface nodes may come into contact.

2. Adaptive Mesh Refinement Loop

   • Error Estimation: Local a‑posteriori error indicators (e.g., strain energy jump, residual‑based) are computed on each element.
   • Marking: Elements exceeding a user‑defined threshold are flagged for refinement.
   • Refinement: Non‑conforming h‑refinement is performed, creating hanging nodes that are handled by the solver. Super‑parametric mapping keeps the geometry accurate.

3. Parallelization Strategy

   • The mesh is partitioned across MPI ranks with the equidistribution rule (equal number of elements per rank) and the extra constraint that any two nodes that may contact are placed on the same rank.
   • This eliminates the need for costly global exchanges when assembling the contact contribution.
   • Standard domain‑decomposition preconditioners (e.g., AMG, block‑Jacobi) are employed, and their performance under AMR is evaluated.

4. Benchmark Problems

   • Classical Hertzian contact (sphere‑on‑plane) in both 2‑D and 3‑D, with varying load levels and material properties, serves as the testbed.

Results & Findings

Overall, the combined contact‑AMR approach delivers high fidelity results with substantially less computational resources than traditional uniform‑mesh strategies.

Practical Implications

• Engineering Simulations: Designers of mechanical components (e.g., gear teeth, tire‑road interaction, biomedical implants) can run high‑resolution contact analyses on commodity clusters without resorting to expensive supercomputers.
• Real‑Time or Near‑Real‑Time Applications: The reduced DOF count and efficient MPI layout make it feasible to embed contact‑aware physics into interactive design tools or digital twins that require frequent re‑solves.
• Software Integration: The algorithm fits naturally into existing finite‑element libraries that support MPI and AMR (e.g., PETSc, deal.II, libMesh). The node‑pairing rule can be implemented as a lightweight preprocessing step.
• GPU/Hybrid Portability: While the paper focuses on CPU clusters, the communication‑aware partitioning and element‑wise refinement logic are amenable to GPU offloading, offering a clear migration path for future performance gains.

Limitations & Future Work

• Contact Model Simplicity: The study uses a penalization method; more sophisticated friction or adhesion models are not explored and may require additional communication handling.
• Load Balancing: Equidistributing elements does not guarantee perfect load balance once refinement becomes highly localized; dynamic repartitioning strategies could further improve scalability.
• Extension to Large Deformations: The current formulation targets elastostatic (small‑strain) problems. Adapting the framework to fully nonlinear or dynamic contact scenarios remains an open challenge.
• GPU Evaluation: No performance data on heterogeneous architectures are presented; future work could benchmark the algorithm on GPU‑accelerated clusters.

Bottom line: By marrying a contact‑aware node pairing scheme with a parallel‑friendly adaptive mesh workflow, the authors deliver a practical, high‑performance tool for 3‑D elastostatic contact analysis—one that can be adopted by developers building next‑generation simulation platforms.

Authors

• Alexandre Epalle
• Isabelle Ramière
• Guillaume Latu
• Frédéric Lebon

Paper Information

• arXiv ID: 2511.20142v1
• Categories: math.NA, cs.DC
• Published: November 25, 2025
• PDF: Download PDF