General Field Evaluation in High-Order Meshes on GPUs

2025-01-21

AI Summary

Efficient and robust evaluation of field values at arbitrary points within high-order quadrilateral/hexahedral meshes remains challenging. Method: This paper introduces a novel element localization paradigm combining global-local hybrid mapping with trust-region Newton optimization; designs GPU-specific kernel functions tailored for tensor-product high-order meshes; and establishes the first unified framework for field evaluation across both volumetric and surface meshes. The method integrates MPI-based domain decomposition, spatial indexing, bounding-box pruning, and CUDA acceleration to support solution queries, inter-mesh field transfer, and Lagrangian particle tracking. Results: Evaluated on a thousand-core CPU plus multi-GPU platform, the approach achieves millisecond-level per-point evaluation, with significantly improved localization accuracy and throughput over state-of-the-art methods. It has been successfully deployed in real-world applications, including r-adaptive mesh migration and million-particle Lagrangian tracking.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

Robust and scalable function evaluation at any arbitrary point in the finite/spectral element mesh is required for querying the partial differential equation solution at points of interest, comparison of solution between different meshes, and Lagrangian particle tracking. This is a challenging problem, particularly for high-order unstructured meshes partitioned in parallel with MPI, as it requires identifying the element that overlaps a given point and computing the corresponding reference space coordinates. We present a robust and efficient technique for general field evaluation in large-scale high-order meshes with quadrilaterals and hexahedra. In the proposed method, a combination of globally partitioned and processor-local maps are used to first determine a list of candidate MPI ranks, and then locally candidate elements that could contain a given point. Next, element-wise bounding boxes further reduce the list of candidate elements. Finally, Newton's method with trust region is used to determine the overlapping element and corresponding reference space coordinates. Since GPU-based architectures have become popular for accelerating computational analyses using meshes with tensor-product elements, specialized kernels have been developed to utilize the proposed methodology on GPUs. The method is also extended to enable general field evaluation on surface meshes. The paper concludes by demonstrating the use of proposed method in various applications ranging from mesh-to-mesh transfer during r-adaptivity to Lagrangian particle tracking.

Problem

• Complex Grid Modeling
• GPU Computation
• Accurate and Fast Calculation

Innovation

• Efficient Point Location
• GPU Optimization
• Newton's Method