Volume 44 Issue 1
Jan.  2023
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ZHEN Yujie, XU Kang, JIANG Tao, REN Jinlian. GPU Parallelization Computation of High-Dimensional Multi-Phase Separation in Complex Domains Based on the Corrected FPM[J]. Applied Mathematics and Mechanics, 2023, 44(1): 93-104. doi: 10.21656/1000-0887.430147
Citation: ZHEN Yujie, XU Kang, JIANG Tao, REN Jinlian. GPU Parallelization Computation of High-Dimensional Multi-Phase Separation in Complex Domains Based on the Corrected FPM[J]. Applied Mathematics and Mechanics, 2023, 44(1): 93-104. doi: 10.21656/1000-0887.430147

GPU Parallelization Computation of High-Dimensional Multi-Phase Separation in Complex Domains Based on the Corrected FPM

doi: 10.21656/1000-0887.430147
  • Received Date: 2022-04-26
  • Rev Recd Date: 2022-06-13
  • Available Online: 2023-01-06
  • Publish Date: 2023-01-15
  • Based on the corrected finite pointset method (CFPM) with CPU-GPU heteroid parallelization (CFPM-GPU), a high-efficiency, accurate and fast parallel algorithm was developed for the high-dimensional phase separation phenomena governed by the multi-component Cahn-Hilliard (C-H) equation in complex domains. The proposed parallel algorithm with the CFPM-GPU was built in a process like: ① introduce the Wendland weight function into the discretization of the finite pointset method (FPM) scheme for the 1st/2nd spatial derivatives, based on the Taylor series and the weighted least square concept; ② use the above FPM scheme twice to approximate the 4th spatial derivative in the C-H equation, which is called the CFPM method; ③ for the first time establish an accelerating parallel algorithm for the CFPM with local matrices by means of a single GPU card based on the CUDA programming. Two benchmark problems of 2D radially and 3D spherically symmetric C-H equations were first solved to test the accuracy and high-efficiency of the proposed CFPM-GPU, and the acceleration ratio of the GPU parallelization to the single CPU computation is about 160. Subsequently, the proposed CFPM-GPU was used to predict the 2D/3D multi-phase separation phenomena in complex domains, and the prediction was compared with other numerical results. The numerical results show that, the proposed CFPM-GPU is valid and high-efficiency to simulate the 2D/3D multi-phase separation cases in complex domains.

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  • [1]
    CHOKSI R, PELETIER M, WILLIAMS J F. On the phase diagram for microphase separation of diblock copolymers: an approach via a nonlocal Cahn-Hilliard functional[J]. SIAM Journal on Applied Mathematics, 2009, 69(6): 1712-1738. doi: 10.1137/080728809
    [2]
    LI Y B, KIM J. Multiphase image segmentation using a phase-field model[J]. Computers and Mathematics With Applications, 2011, 62(2): 737-745. doi: 10.1016/j.camwa.2011.05.054
    [3]
    汪精英, 翟术英. 分数阶Cahn-Hilliard方程的高效数值算法[J]. 应用数学和力学, 2021, 42(8): 832-840

    WANG Jingying, ZHAI Shuying. An efficient numerical algorithm for fractional Cahn-Hilliard equations[J]. Applied Mathematics and Mechanics, 2021, 42(8): 832-840.(in Chinese)
    [4]
    CHARLES M E, HARALD G. Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix[J]. Physica D: Nonlinear Phenomena, 1997, 109(3/4): 242-256.
    [5]
    DIEGEL A, FENG X B, WISE S M. Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system[J]. SIAM Journal on Numerical Analysis, 2015, 53: 127-152. doi: 10.1137/130950628
    [6]
    GUO Z, LIN P, LOWENGRUB J, et al. Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier-Stokes-Cahn-Hilliard system: primitive variable and projection type schemes[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 326: 144-174. doi: 10.1016/j.cma.2017.08.011
    [7]
    ZHANG Z G, QIAO Z H. An adaptive time-stepping strategy for the Cahn-Hilliard equation[J]. Communications in Computational Physics, 2012, 11(4): 1261-1278. doi: 10.4208/cicp.300810.140411s
    [8]
    JEONG D, CHOI Y, KIM J. A benchmark problem for the two- and three-dimensional Cahn-Hilliard equation[J]. Communications in Nonlinear Science and Numerical Simulation, 2018, 61: 149-159. doi: 10.1016/j.cnsns.2018.02.006
    [9]
    KIM J. A numerical method for the Cahn-Hilliard equation with a variable mobility[J]. Communications in Nonlinear Science and Numerical Simulation, 2007, 12: 1560-1571. doi: 10.1016/j.cnsns.2006.02.010
    [10]
    JEONG D, YANG J X, KIM J. A practical and efficient numerical method for the Cahn-Hilliard equation in complex domains[J]. Communications in Nonlinear Science and Numerical Simulation, 2019, 73: 217-228. doi: 10.1016/j.cnsns.2019.02.009
    [11]
    CHENG R J, CHENG Y M. Solving unsteady Schrödinger equation using the improved element-free Galerkin method[J]. Chinese Physics B, 2016, 25: 020203. doi: 10.1088/1674-1056/25/2/020203
    [12]
    DEHGHAN M, ABBASZADEH M. The meshless local collocation method for solving multi-dimensional Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations[J]. Engineering Analysis With Boundary Elements, 2017, 78: 49-64. doi: 10.1016/j.enganabound.2017.02.005
    [13]
    王红, 李小林. 二维瞬态热传导问题的无单元Galerkin法分析[J]. 应用数学和力学, 2021, 42(5): 460-469

    WANG Hong, LI Xiaolin. Analysis of 2D transient heat conduction problems with the element-free Galerkin method[J]. Applied Mathematics and Mechanics, 2021, 42(5): 460-469.(in Chinese)
    [14]
    曹维鸿, 傅卓佳, 汤卓超. 水槽动力特性数值模拟的新型局部无网格配点法[J]. 应用数学和力学, 2022, 43(4): 392-400

    CAO Weihong, FU Zhuojia, TANG Zhuochao. A novel localized meshless collocation method for numerical simulation of flume dynamic characteristics[J]. Applied Mathematics and Mechanics, 2022, 43(4): 392-400.(in Chinese)
    [15]
    LIU G R, LIU M B, LI S. Smoothed particle hydrodynamics: a mesh-free particle method[J]. Computational Mechanics, 2003, 33: 491.
    [16]
    林晨森, 陈硕, 李启良, 等. 耗散粒子动力学GPU并行计算研究[J]. 物理学报, 2014, 63(10): 104702 doi: 10.7498/aps.63.104702

    LIN Chensen, CHEN Shuo, LI Qiliang, et al. Accelerating dissipative particle dynamics with graphic processing unit[J]. Acta Physica Sinica, 2014, 63(10): 104702.(in Chinese) doi: 10.7498/aps.63.104702
    [17]
    JIANG T, CHEN Z C, LU W G, et al. An efficient split-step and implicit pure mesh-free method for the 2D/3D nonlinear Gross-Pitaevskii equations[J]. Computer Physics Communications, 2018, 231: 19-30. doi: 10.1016/j.cpc.2018.05.007
    [18]
    CRESPO A J C, DOMÍNGUEZ J M, ROGERS B D, et al. DualSPHysics: open-source parallel CFD solver based on smoothed particle hydrodynamics(SPH)[J]. Computer Physics Communications, 2015, 187: 204-216. doi: 10.1016/j.cpc.2014.10.004
    [19]
    KING J R, POGORELOV I V, AMYX K M, et al. GPU acceleration and performance of the particle-beam-dynamics code Elegant[J]. Computer Physics Communications, 2019, 235: 346-355. doi: 10.1016/j.cpc.2018.09.022
    [20]
    HE F, ZHANG H S, HUANG C, et al. A stable SPH model with large CFL numbers for multi-phase flows with large density ratios[J]. Journal of Computational Physics, 2022, 453: 110944. doi: 10.1016/j.jcp.2022.110944
    [21]
    杨秀峰, 刘谋斌. 瑞利-泰勒不稳定问题的光滑粒子法模拟研究[J]. 物理学报, 2017, 66(16): 164071

    YANG Xiufeng, LIU Moubin. Numerical study of Rayleigh-Taylor instability by using smoothed particle hydrodynamics[J]. Acta Physica Sinica, 2017, 66(16): 164071.(in Chinese)
    [22]
    HE F, ZHANG H S, HUANG C, et al. Numerical investigation of the solitary wave breaking over a slope by using the finite particle method[J]. Coastal Engineering, 2020, 156: 103617. doi: 10.1016/j.coastaleng.2019.103617
    [23]
    DANIEL W, MASSOUD R, WOLFGANG R. Neighbour lists for smoothed particle hydrodynamics on GPUs[J]. Computer Physics Communications, 2018, 225: 140-148. doi: 10.1016/j.cpc.2017.12.014
    [24]
    任金莲, 蒋戎戎, 陆伟刚, 等. 基于局部加密纯无网格法非线性Cahn-Hilliard方程的模拟[J]. 物理学报, 2020, 69(8): 080202 doi: 10.7498/aps.69.20191829

    REN Jinlian, JIANG Rongrong, LU Weigang, et al. Simulation of nonlinear Cahn-Hilliard equation based on local refinement pure meshless method[J]. Acta Physics Sinica, 2020, 69(8): 080202.(in Chinese) doi: 10.7498/aps.69.20191829
    [25]
    SUCHDE P, KUHNERT J, TIWARI S. On meshfree GFDM solvers for the incompressible Navier-Stokes equations[J]. Computers & Fluids, 2018, 165: 1-12.
    [26]
    SUDARSHAN T, JÖRG K. Modeling of two-phase flows with surface tension by finite pointset method (FPM)[J]. Journal of Computational and Applied Mathematics, 2007, 203: 376-386. doi: 10.1016/j.cam.2006.04.048
    [27]
    EDGAR O R F, IRMA D G C. Application of the finite pointset method to non-stationary heat conduction problems[J]. International Journal of Heat and Mass Transfer, 2014, 71: 720-723. doi: 10.1016/j.ijheatmasstransfer.2013.12.077
    [28]
    LI Y B, CHOI J I, KIM J. Multi-component Cahn-Hilliard system with different boundary conditions in complex domains[J]. Journal of Computational Physics, 2016, 323: 1-16. doi: 10.1016/j.jcp.2016.07.017
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