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G-Pre: A Graph-Theory-Based Matrix Preconditioning Algorithm for Finite Element Simulation Solutions

Min Chen, Jingyan Li, Lijie Li Orcid Logo, Kang Liang, Gai Wu Orcid Logo, Wei Shen Orcid Logo

Applied Sciences, Volume: 15, Issue: 9, Start page: 5130

Swansea University Author: Lijie Li Orcid Logo

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DOI (Published version): 10.3390/app15095130

Abstract

In finite element simulation and analysis, increasing simulation scales place high demands on the preconditioning and solution process of linear matrices. However, the most commonly used preconditioning methods for incomplete LU factorization usually increase data access and computation due to data...

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Published in: Applied Sciences
ISSN: 2076-3417
Published: MDPI AG 2025
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URI: https://cronfa.swan.ac.uk/Record/cronfa69671
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To address these challenges, this study proposes a graph&#x2013;theory-based matrix preconditioning algorithm called G-Pre. In this method, by introducing a graph partitioning algorithm and a graph rearrangement algorithm before ILU factorization, the matrix is partitioned into regions and the elements are rearranged, which improves the ease of data access for matrix computation and facilitates parallel computation. The results of numerical experiments show that in terms of solution efficiency, the solver based on the G-Pre preconditioning algorithm achieved an average speedup ratio of 2.1 and 4.3 times that of the solver based on ILU factorization and the direct solver, respectively. At the same time, the algorithm computed the results with an error of no more than 2%. 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spelling 2025-06-10T12:19:36.1239466 v2 69671 2025-06-10 G-Pre: A Graph-Theory-Based Matrix Preconditioning Algorithm for Finite Element Simulation Solutions ed2c658b77679a28e4c1dcf95af06bd6 0000-0003-4630-7692 Lijie Li Lijie Li true false 2025-06-10 ACEM In finite element simulation and analysis, increasing simulation scales place high demands on the preconditioning and solution process of linear matrices. However, the most commonly used preconditioning methods for incomplete LU factorization usually increase data access and computation due to data padding and forward/backward operations, as well as affect parallel computing design. To address these challenges, this study proposes a graph–theory-based matrix preconditioning algorithm called G-Pre. In this method, by introducing a graph partitioning algorithm and a graph rearrangement algorithm before ILU factorization, the matrix is partitioned into regions and the elements are rearranged, which improves the ease of data access for matrix computation and facilitates parallel computation. The results of numerical experiments show that in terms of solution efficiency, the solver based on the G-Pre preconditioning algorithm achieved an average speedup ratio of 2.1 and 4.3 times that of the solver based on ILU factorization and the direct solver, respectively. At the same time, the algorithm computed the results with an error of no more than 2%. This method is a novel technique for the matrix preconditioning of finite element solvers and a powerful algorithmic tool to cope with the increasing computational demands of finite element simulations. Journal Article Applied Sciences 15 9 5130 MDPI AG 2076-3417 graph reordering; factorization; preconditioning; matrix solving; finite element simulation 5 5 2025 2025-05-05 10.3390/app15095130 COLLEGE NANME Aerospace, Civil, Electrical, and Mechanical Engineering COLLEGE CODE ACEM Swansea University Another institution paid the OA fee This work was financially supported by the National Key Research and Development Program of China (No. 2024YFF0504903), the Knowledge Innovation Program of Wuhan-Shuguang (Grant Nos. 2023010201020255 and 2023010201020243), the Natural Science Foundation of Wuhan (Grant No. 2024040801020222), the National Natural Science Foundation of China (Grant Nos. 52202045 and 92473102), the Shenzhen Science and Technology Program (Grant No. JCYJ20240813175906008), the China Scholarship Council (Grant No. 202206275005), and the State Key Laboratory of Intelligent Vehicle Safety Technology. 2025-06-10T12:19:36.1239466 2025-06-10T11:41:30.8465885 Faculty of Science and Engineering School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Electronic and Electrical Engineering Min Chen 1 Jingyan Li 2 Lijie Li 0000-0003-4630-7692 3 Kang Liang 4 Gai Wu 0000-0002-9726-6328 5 Wei Shen 0000-0003-4389-3112 6 69671__34447__df8203eb7f5742ffb7b16c4e85023cd1.pdf applsci-15-05130-v2.pdf 2025-06-10T11:41:30.8428302 Output 3630624 application/pdf Version of Record true © 2025 by the authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. true eng https://creativecommons.org/licenses/by/4.0/
title G-Pre: A Graph-Theory-Based Matrix Preconditioning Algorithm for Finite Element Simulation Solutions
spellingShingle G-Pre: A Graph-Theory-Based Matrix Preconditioning Algorithm for Finite Element Simulation Solutions
Lijie Li
title_short G-Pre: A Graph-Theory-Based Matrix Preconditioning Algorithm for Finite Element Simulation Solutions
title_full G-Pre: A Graph-Theory-Based Matrix Preconditioning Algorithm for Finite Element Simulation Solutions
title_fullStr G-Pre: A Graph-Theory-Based Matrix Preconditioning Algorithm for Finite Element Simulation Solutions
title_full_unstemmed G-Pre: A Graph-Theory-Based Matrix Preconditioning Algorithm for Finite Element Simulation Solutions
title_sort G-Pre: A Graph-Theory-Based Matrix Preconditioning Algorithm for Finite Element Simulation Solutions
author_id_str_mv ed2c658b77679a28e4c1dcf95af06bd6
author_id_fullname_str_mv ed2c658b77679a28e4c1dcf95af06bd6_***_Lijie Li
author Lijie Li
author2 Min Chen
Jingyan Li
Lijie Li
Kang Liang
Gai Wu
Wei Shen
format Journal article
container_title Applied Sciences
container_volume 15
container_issue 9
container_start_page 5130
publishDate 2025
institution Swansea University
issn 2076-3417
doi_str_mv 10.3390/app15095130
publisher MDPI AG
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Electronic and Electrical Engineering{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Electronic and Electrical Engineering
document_store_str 1
active_str 0
description In finite element simulation and analysis, increasing simulation scales place high demands on the preconditioning and solution process of linear matrices. However, the most commonly used preconditioning methods for incomplete LU factorization usually increase data access and computation due to data padding and forward/backward operations, as well as affect parallel computing design. To address these challenges, this study proposes a graph–theory-based matrix preconditioning algorithm called G-Pre. In this method, by introducing a graph partitioning algorithm and a graph rearrangement algorithm before ILU factorization, the matrix is partitioned into regions and the elements are rearranged, which improves the ease of data access for matrix computation and facilitates parallel computation. The results of numerical experiments show that in terms of solution efficiency, the solver based on the G-Pre preconditioning algorithm achieved an average speedup ratio of 2.1 and 4.3 times that of the solver based on ILU factorization and the direct solver, respectively. At the same time, the algorithm computed the results with an error of no more than 2%. This method is a novel technique for the matrix preconditioning of finite element solvers and a powerful algorithmic tool to cope with the increasing computational demands of finite element simulations.
published_date 2025-05-05T05:28:43Z
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