GPU-parallelized implicit discontinuous Galerkin finite element algorithm

GAO Huanqin; CHEN Hongquan; JIA Xuesong; XU Shengguan

doi:10.7638/kqdlxxb-2023.0033

English Version Journal Login

您当前的位置：

首页 >

文章列表页 >

GPU-parallelized implicit discontinuous Galerkin finite element algorithm

更新时间：2026-05-28

- GPU-parallelized implicit discontinuous Galerkin finite element algorithm
- Vol. 42, Issue 2, Pages: 21-33(2024)
- 作者机构：
  
  1. 南京航空航天大学航空学院，非定常空气动力学与流动控制工信部重点实验室,南京,210016
  2. 南京工业大学机械与动力工程学院,南京,211816
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2023.0033
  CLC： V211.3
- Published：2024
- 稿件说明：
移动端阅览
高缓钦, 陈红全, 贾雪松, 等. 间断Galerkin有限元隐式算法GPU并行化研究[J]. 空气动力学学报, 2024, 42(2): 21−33.

GAO Huanqin, CHEN Hongquan, JIA Xuesong, et al. GPU-parallelized implicit discontinuous Galerkin finite element algorithm[J]. 2024, 42(2): 21-33.
高缓钦, 陈红全, 贾雪松, 等. 间断Galerkin有限元隐式算法GPU并行化研究[J]. 空气动力学学报, 2024, 42(2): 21−33. DOI： 10.7638/kqdlxxb-2023.0033

GAO Huanqin, CHEN Hongquan, JIA Xuesong, et al. GPU-parallelized implicit discontinuous Galerkin finite element algorithm[J]. 2024, 42(2): 21-33. DOI： 10.7638/kqdlxxb-2023.0033.

摘要

为了提高间断伽辽金（discontinuous Galerkin

DG）有限元方法的计算效率，围绕求解Euler方程，构建了基于图形处理器（graphics processing unit

GPU）并行加速的隐式DG算法。算法结合Roe格式进行空间离散，采用人工黏性法处理激波等间断问题，时间推进选用下上对称高斯-赛德尔（lower-upper symmetric Gauss-Seidel

LU-SGS）隐式格式。为了克服传统隐式格式固有的数据关联依赖问题，借助于本文提出的面向任意网格的单元着色分组技术，先给出了LU-SGS隐式格式的并行化改造，使得隐式时间推进能按颜色组别依次并行，由于同一颜色组内算法已不存在数据关联，可以据此实现并行化。在此基础上，再结合DG算法局部紧致等特点，基于统一计算设备架构（compute unified device architecture

CUDA）编程模型，设计了依据单元的核函数，并构建了对应的线程与数据结构，给出了DG有限元隐式GPU并行算法。最后，发展的算法通过了多个二维和三维典型流动算例考核与性能测试，展示出隐式算法GPU加速的效果，且获得的计算结果能与现有的文献或实验数据接近。

Abstract

In order to improve the parallel efficiency of the discontinuous Galerkin (DG) finite element method

a graphics processing unit (GPU) parallelized implicit DG algorithm is developed for solving Euler equations with additional artificial viscosity terms. The classic Roe scheme is adopted to treat the numerical flux involved in the spatial discretization

and the implicit lower-upper symmetric Gauss-Seidel (LU-SGS) scheme is selected for time marching. In order to resolve the inherent data dependency of the traditional LU-SGS algorithm

which causes thread-racing conditions destabilizing numerical computation

a coloring method is presented for arbitrary meshes and applied to organize the computational elements into different color groups by painting neighboring elements with different colors. Algebraic operations of the elements in the same color group are independent in the algorithm and thus can be easily parallelized. Based on the presented coloring technique

the traditional LU-SGS algorithm is modified to be parallelized accordingly by performing calculations in a color-by-color manner. By taking advantage of the local compactness of the DG finite element method

a GPU-parallelized implicit DG algorithm based on the modified LU-SGS algorithm is then implemented under the compute unified device architecture (CUDA) programming model. The time marching procedure

which is the most time-consuming part of the algorithm

is assigned to be computed on GPU. The computational task is split into a set of small tasks

and element-based kernels are designed for these tasks with corresponding thread hierarchies and data structures. The resultant algorithm is verified by a set of typical two- and three-dimensional flow test cases and performance analysis

which shows that implicit GPU speedups can be achieved

and the obtained solutions agree well with experimental data or other computed results reported in the literature.

关键词

Keywords

references

REED W H, HILL T R. Triangular mesh methods for the neutron transport equation: Report LA-UR-73-479[R]. Los Alamos: Los Alamos Scientific Laboratory, 1973.

COCKBURN B, SHU C W. The Runge-Kutta local projection P1-discontinuous-Galerkin finite element method for scalar conservation laws[C]// 1st National Fluid Dynamics Conference, Cincinnati, OH, USA. Reston, Virginia: AIAA, 1988. doi: 10.2514/6.1988-3797

COCKBURN B, HOU S C, SHU C W. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: The multidimensional case[J]. Mathematics of Computation, 1990, 54(190): 545−581. doi: 10.2307/2008501

JADOUI M, BLONDEAU C, MARTIN E, et al. Comparative study of inner-outer Krylov solvers for linear systems in structured and high-order unstructured CFD problems[J]. Computers Fluids, 2022, 244: 105575. doi: 10.1016/j.compfluid.2022.105575

DUAN Z J, ZHANG T, QIN M M, et al. Implicit discontinuous finite element algorithm for solving three-dimensional Euler equations[J]. Application Research of Computers, 2019, 36(6): 1665−1667,1672 (in Chinese). doi: 10.19734/j.issn.1001-3695.2017.12.0801

KERCHER A D, CORRIGAN A, KESSLER D A. The moving discontinuous Galerkin finite element method with interface condition enforcement for compressible viscous flows[J]. International Journal for Numerical Methods in Fluids, 2021, 93(5): 1490−1519. doi: 10.1002/fld.4939

PARSANI M, BOUKHARFANE R, NOLASCO I R, et al. High-order accurate entropy-stable discontinuous collocated Galerkin methods with the summation-by-parts property for compressible CFD frameworks: Scalable SSDC algorithms and flow solver[J]. Journal of Computational Physics, 2021, 424: 109844. doi: 10.1016/j.jcp.2020.109844

DEMIDOV D, MU L, WANG B. Accelerating linear solvers for Stokes problems with C++ metaprogramming[J]. Journal of Computational Science, 2021, 49: 101285. doi: 10.1016/j.jocs.2020.101285

CAO C, CHEN H Q, ZHANG J L, et al. A multi-layered point reordering study of GPU-based meshless method for compressible flow simulations[J]. Journal of Computational Science, 2019, 33: 45−60. doi: 10.1016/j.jocs.2019.04.001

FUHRY M, GIULIANI A, KRIVODONOVA L. Discontinuous Galerkin methods on graphics processing units for nonlinear hyperbolic conservation laws[J]. International Journal for Numerical Methods in Fluids, 2014, 76(12): 982−1003. doi: 10.1002/fld.3963

CASTRO M D, OSORIO R R, VILARIÑO D L, et al. Implementation of a motion estimation algorithm for Intel FPGAs using OpenCL[J]. The Journal of Supercomputing, 2023, 79: 9866−9888. doi: 10.1007/s11227-023-05051-3

XUE W C, JACKSON C W, ROY C J. An improved framework of GPU computing for CFD applications on structured grids using OpenACC[J]. Journal of Parallel and Distributed Computing, 2021, 156: 64−85. doi: 10.1016/j.jpdc.2021.05.010

SIEBENBORN M, SCHULZ V, SCHMIDT S. A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations[J]. Computing and Visualization in Science, 2012, 15(2): 61−73. doi: 10.1007/s00791-013-0197-0

KLÖCKNER A, WARBURTON T, BRIDGE J, et al. Nodal discontinuous Galerkin methods on graphics processors[J]. Journal of Computational Physics, 2009, 228(21): 7863−7882. doi: 10.1016/j.jcp.2009.06.041

GAO H Q, CHEN H Q, ZHANG J L, et al. A GPU-accelerated discontinuous Galerkin method for solving two-dimensional laminar flows[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2022, 39(04): 450−466. doi: 10.16356/j.1005-1120.2022.04.007

ZHANG J L, MA Z H, CHEN H Q, et al. A GPU-accelerated implicit meshless method for compressible flows[J]. Journal of Computational Physics, 2018, 360: 39−56. doi: 10.1016/j.jcp.2018.01.037

PERSSON P O, PERAIRE J. Sub-cell shock capturing for discontinuous Galerkin methods[C]// 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. Reston, Virginia: AIAA, 2006. doi: 10.2514/6.2006-112

YU X J, ZHOU T. Discontinuous finite element methods for solving hydrodynamic equations[J]. Chinese Journal of Computational Physics, 2005, 22(2): 108−116 (in Chinese). doi: 10.3969/j.issn.1001-246X.2005.02.002

ROE P L. Approximate Riemann solvers, parameter vectors, and difference schemes[J]. Journal of Computational Physics, 1997, 135(2): 250−258. doi: 10.1006/jcph.1997.5705

XIONG W, ZHANG W G, DING J, et al. Methods for two-dimensional laminar and turbulent numerical simulations[J]. Journal of Shanghai University (Natural Science Edition), 2022, 28(4): 619−531 (in Chinese). doi: 10.12066/j.issn.1007-2861.2242

ECONOMON T D, MUDIGERE D, BANSAL G, et al. Performance optimizations for scalable implicit RANS calculations with SU2[J]. Computers Fluids, 2016, 129: 146−158. doi: 10.1016/j.compfluid.2016.02.003

DANG Y B, LIU K L, SUN Y F, et al. Implicit high order discontinuous Galerkin method for compressible laminar and turbulent flow simulation[J]. Acta Aerodynamica Sinica, 2018, 36(3): 535−541 (in Chinese). doi: 10.7638/kqdlxxb-2017.0183

XIA Y D, LUO H, FRISBEY M, et al. A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids[J]. Computers Fluids, 2014, 98: 134−151. doi: 10.1016/j.compfluid.2014.01.023

CAO C, CHEN H Q. An implicit preconditioned gridless method for solving Euler equations at low Mach numbers[J]. Journal of Harbin Institute of Technology, 2020, 52(7): 161−169 (in Chinese). doi: 10.11918/201911068

LIU J T, MA Z S, LI S H, et al. A GPU accelerated Red-Black SOR algorithm for computational fluid dynamics problems[J]. Advanced Materials Research, 2011, 320: 335−340. doi: 10.4028/www.scientific.net/AMR.320.335

HOFFMANN M, MUNZ C, WANG Z J. Efficient implementation of the CPR formulation for the Navier-Stokes equations on GPUs[C]// Seventh International Conference on Computational Fluid Dynamics (ICCFD7), Big Island, Hawaii, 2012.

MICHALAK K, OLLIVIER-GOOCH C. Limiters for unstructured higher-order accurate solutions of the Euler equations[C]// 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. Reston, Virginia: AIAA, 2008. doi: 10.2514/6.2008-776

CHOI J H, ALONSO J J, VAN DER WEIDE E. An analysis of inviscid transonic flows over three-dimensional wings using the discontinuous Galerkin solver in SU2[C]// AIAA Scitech 2020 Forum, Orlando, FL. Reston, Virginia: AIAA, 2020. doi: 10.2514/6.2020-1564

BERG B, OSKAM B. Boundary layer measurements on a two-dimensional wing with flap and a comparison with calculations: NLR MP 79034 U[R]. National Aerospace Laboratory NLR, 1979.

ECONOMON T D, PALACIOS F, COPELAND S R, et al. SU2: An open-source suite for multiphysics simulation and design[J]. AIAA Journal, 2016, 54(3): 828−846. doi: 10.2514/1.j053813

LI W A, LIU Y. The p-weighted limiter for the discontinuous Galerkin method in solving compressible flows on tetrahedral grids[J]. International Journal of Computational Fluid Dynamics, 2021, 35(7): 510−533. doi: 10.1080/10618562.2021.2003789

SCHMITT V, CHARPIN F. Pressure distributions on the ONERA M6 wing at transonic Mach numbers: AGARD Advisory Report No. 138[R]. Neuilly sur Seine, France: Advisory Group for Aerospace Research and Development North Atlantic Treaty Organization, 1979.

ZHANG J L, CHEN H Q, CAO C. A graphics processing unit-accelerated meshless method for two-dimensional compressible flows[J]. Engineering Applications of Computational Fluid Mechanics, 2017, 11(1): 526−543. doi: 10.1080/19942060.2017.1317027

Views

下载量

CSCD

Alert me when the article has been cited

提交

Tools

Publicity Resources

Dynamic ablation morphology measurement of thermal protection materials using structured light vision

Noise prediction and reduction methods for LAGOON landing gear based on Lattice Boltzmann method

Numerical simulation on gust responses of high aspect ratio aircraft

Research progress on noise reduction of centrifugal fans

Positivity-preserving strategies and their comparision based on staggered CPR methods

Related Author

No data

Related Institution

No data

AI问答

⁰