An efficient parallel computing method for random vibration analysis of a three-dimensional train-track-soil coupled model using Seed-PCG algorithm

Abstract

This study proposes an efficient parallel computation method based on Seed-preconditioned Conjugate Gradient (Seed-PCG) algorithm, to address the issue of computational inefficiency of random multi-sample in three-dimensional (3D) finite element (FE) model of train-track-soil. A 3D train-track-soil coupled random vibration analysis model is established using the finite element method (FEM) and the pseudo-excitation method (PEM) under track irregularity excitation. The Seed-PCG method is utilized to solve the system of linear equations with multiple right-hand sides arising from the random analysis of the vehicle-induced ground vibration. Furthermore, by projecting the Krylov subspace obtained from solving the seed system by the PCG method, the initial solution of the remaining linear equation systems and the corresponding initial residuals are improved, leading to an effective enhancement of the convergence speed of the PCG method. Finally, the parallel computing program is developed on a hybrid MATLAB-Compute Unified Device Architecture (CUDA) platform. Numerical examples demonstrate the effectiveness of the proposed method. It achieves 104.2 times acceleration compared with the multi-point synchronization algorithm (MPSA) proposed by author ZHU under the same computing platform. Moreover, compared with the PCG method, the number of iterations is reduced by 18 % and the acceleration is increased by 1.21 times.

Keywords

Seed-PCG method / linear equations with multiple right-hand sides / random vibration / GPU parallel computing / train-track-soil coupled model

References

[[1]] Hu J, Bian X. Experimental and numerical studies on dynamic responses of tunnel and soils due to train traffic loads [J]. Tunnelling and Underground Space Technology, 2022, 128: 104628,

[[2]] Farahani M V, Sadeghi J, Jahromi S G, et al.. Modal based method to predict subway train-induced vibration in buildings [J]. Structures, 2023, 47: 557-572,

[[3]] Cao Z, Xu Y, Yuan Z, et al.. Nonstationary vibration responses of a three-dimensional tunnel-soil system excited by moving stochastic loads [J]. Computers and Geotechnics, 2020, 125: 103658,

[[4]] Yu H, Wang B, Li Y, et al.. A two-step framework for stochastic dynamic analysis of uncertain vehicle-bridge system subjected to random track irregularity [J]. Computers & Structures, 2021, 253: 106583,

[[5]] Wu B, Zeng Y, Zhou Z, et al.. Vibration prediction based on the coupling method of half-train model and 3D refined finite element ground model [J]. Computers and Geotechnics, 2021, 134: 104133,

[[6]] Wang L, Zhu Z, Costa P A, et al.. A framework combining pseudo-excitation method and two-and-a-half-dimensional finite element method for random ground vibrations induced by high-speed trains [J]. Advances in Structural Engineering, 2020, 23(15): 3263-3277,

[[7]] Wang L, Zhu Z, Bai Y, et al.. A fast random method for three-dimensional analysis of train-track-soil dynamic interaction [J]. Soil Dynamics and Earthquake Engineering, 2018, 115: 252-262,

[[8]] Jomo J N, De Prenter F, Elhaddad M, et al.. Robust and parallel scalable iterative solutions for large-scale finite cell analyses [J]. Finite Elements in Analysis and Design, 2019, 163: 14-30,

[[9]] Zhu Z, Xia Y, Wang L, et al.. A parallel computing method for three-dimensional random vibration of train-track-soil dynamic interaction based on GPU [J]. Journal of Hunan University(Natural Sciences), 2021, 48(7): 79-88 (in Chinese)

[[10]] Jelich C, Karimi M, Kessissoglou N, et al.. Efficient solution of block Toeplitz systems with multiple right-hand sides arising from a periodic boundary element formulation [J]. Engineering Analysis with Boundary Elements, 2021, 130: 135-144,

[[11]] Amini S, Toutounian F, Gachpazan M. The block CMRH method for solving nonsymmetric linear systems with multiple right-hand sides [J]. Journal of Computational and Applied Mathematics, 2018, 337: 166-174,

[[12]] Heyouni M, Essai A. Matrix Krylov subspace methods for linear systems with multiple right-hand sides [J]. Numerical Algorithms, 2005, 40: 137-156,

[[13]] Chan T F, Wan W L. Analysis of projection methods for solving linear systems with multiple right-hand sides [J]. SIAM Journal on Scientific Computing, 1997, 18(6): 1698-1721,

[[14]] Gu G D. A seed method for solving nonsymmetric linear systems with multiple right-hand sides [J]. International Journal of Computer Mathematics, 2002, 79(3): 307-326,

[[15]] Mojarrab M, Toutounian F. Global LSMR (Gl-LSMR) method for solving general linear systems with several right-hand sides [J]. Journal of Computational and Applied Mathematics, 2017, 321: 78-89,

[[16]] Smith C F, Peterson A F, Mittra R. A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields [J]. IEEE Transactions on Antennas and Propagation, 1989, 37(11): 1490-1493,

[[17]] Sun D L, Huang T Z, Jing Y F, et al.. A block GMRES method with deflated restarting for solving linear systems with multiple shifts and multiple right-hand sides [J]. Numerical Linear Algebra with Applications, 2018, 25(5): e2148,

[[18]] Elbouyahyaoui L, Heyouni M. On applying weighted seed techniques to GMRES algorithm for solving multiple linear systems [J]. Boletim da Sociedade Paranaense de Matemática, 2018, 36(3): 155-172,

[[19]] Abdel-Rehim A M, Morgan R B, Wilcox W. Improved seed methods for symmetric positive definite linear equations with multiple right-hand sides [J]. Numerical Linear Algebra with Applications, 2014, 21(3): 453-471,

[[20]] Kalantzis V, Bekas C, Curioni A, et al.. Accelerating data uncertainty quantification by solving linear systems with multiple right-hand sides [J]. Numerical Algorithms, 2013, 62: 637-653,

[[21]] LI X, LIU H, ZHU J. MINRES seed projection methods for solving symmetric linear systems with multiple right-hand sides [J]. Mathematical Problems in Engineering, 2014, 2014. DOI: https://doi.org/10.1155/2014/357874.

[[22]] Li C, Xiong B, Qiang J, et al.. Multiple linear system techniques for 3D finite element method modeling of direct current resistivity [J]. Journal of Central South University, 2012, 19(2): 424-432,

[[23]] ABDEL-REHIM A, MORGAN R B, WILCOX W. Seed methods for linear equations in lattice qcd problems with multiple right-hand sides [J]. PoS-Proceedings of Science, 2009, lattice 2008. DOI: https://doi.org/10.48550/arXiv.0901.3512.

[[24]] Chen X, Wang D, Ren J, et al.. Application of hybrid CPU-GPU computing platform in large-scale geotechnical finite element analysis [J]. China Civil Engineering Journal, 2016, 49(6): 105-112 (in Chinese)

[[25]] Liu J, Xian Z, Zhou Y, et al.. A marker-and-cell method for large-scale flow-based topology optimization on GPU [J]. Structural and Multidisciplinary Optimization, 2022, 65(4): 125,

[[26]] Lopes P C F, Pereira A M B, Clua E W G, et al.. A GPU implementation of the PCG method for large-scale image-based finite element analysis in heterogeneous periodic media [J]. Computer Methods in Applied Mechanics and Engineering, 2022, 399: 115276,

[[27]] Liu J, Du Y, Du X, et al.. 3D viscous-spring artificial boundary in time domain [J]. Earthquake Engineering and Engineering Vibration, 2006, 5(1): 93-102,

[[28]] Zhu Z, Wang L, Gong W, et al.. Study on vertical random vibration of train-bridge coupled system based on improved iteration model [J]. Journal of Hunan University (Natural Sciences), 2016, 43(11): 120-130 (in Chinese)

[[29]] Lu F, Lin J H, Kennedy D, et al.. An algorithm to study non-stationary random vibrations of vehicle-bridge systems [J]. Computers & Structures, 2009, 87(3–4): 177-185,

[[30]] Fan W, Sheng X, Li Z, et al.. The higher-order analysis method of statistics analysis for response of linear structure under stationary non-Gaussian excitation [J]. Mechanical Systems and Signal Processing, 2022, 166: 108430,

[[31]] He X, Shi K, Wu T. An efficient analysis framework for high-speed train-bridge coupled vibration under non-stationary winds [J]. Structure and Infrastructure Engineering, 2020, 16(9): 1326-1346,