Shifting the Sweet Spot: High-Performance Matrix-Free Method for High-Order Elasticity

2026-01-13

AI Summary

This work addresses the performance bottleneck of conventional matrix-free (PA) methods in high-order elastic finite element analysis, where the optimal polynomial degree is limited to low orders (p≈2) due to underutilization of modern CPU architectures and tensor-product element structure. Within the MFEM framework, we design and implement a highly optimized matrix-free operator coupled with a geometric multigrid (GMG) preconditioner. Through multi-level optimizations—including O(p⁴) tensor decomposition, exploitation of Voigt symmetry, and macro-kernel fusion—we significantly alleviate memory bandwidth constraints and computational redundancy at high orders. Our approach shifts the performance sweet spot to p≥6, achieving 7–83× speedup in core operators and 3.6–16.8× end-to-end solver acceleration over the MFEM baseline, demonstrating efficient large-scale high-order elastic simulations on both x86 and ARM architectures.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

In high-order finite element analysis for elasticity, matrix-free (PA) methods are a key technology for overcoming the memory bottleneck of traditional Full Assembly (FA). However, existing implementations fail to fully exploit the special structure of modern CPU architectures and tensor-product elements, causing their performance"sweet spot"to anomalously remain at the low order of $p \approx 2$, which severely limits the potential of high-order methods. To address this challenge, we design and implement a highly optimized PA operator within the MFEM framework, deeply integrated with a Geometric Multigrid (GMG) preconditioner. Our multi-level optimization strategy includes replacing the original $O(p^6)$ generic algorithm with an efficient $O(p^4)$ one based on tensor factorization, exploiting Voigt symmetry to reduce redundant computations for the elasticity problem, and employing macro-kernel fusion to enhance data locality and break the memory bandwidth bottleneck. Extensive experiments on mainstream x86 and ARM architectures demonstrate that our method successfully shifts the performance"sweet spot"to the higher-order region of $p \ge 6$. Compared to the MFEM baseline, the optimized core operator (kernel) achieves speedups of 7x to 83x, which translates to a 3.6x to 16.8x end-to-end performance improvement in the complete solution process. This paper provides a validated and efficient practical path for conducting large-scale, high-order elasticity simulations on mainstream CPU hardware.

Problem

• matrix-free
• high-order elasticity
• performance sweet spot
• finite element analysis
• CPU architecture

Innovation

• matrix-free
• high-order finite elements
• tensor factorization
• macro-kernel fusion
• geometric multigrid