朱志辉 夏禹涛 王力东 刘禹兵

摘要：針对轨道不平顺随机特征导致车辆-轨道-地基土耦合系统随机分析计算效率低的问题，采用虚拟激励法降低大样本分析的计算量；针对耦合系统等效刚度矩阵的稀疏特性，采用行压缩（Compressed Sparse Row，CSR）格式存储大型稀疏矩阵，采用预处理共轭梯度法（Preconditioned Conjugate Gradient，PCG）求解对称正定的等效静力平衡方程，最后通过MAT-LAB-CUDA（Compute Unified Device Architecture）混合平台开发基于GPU的并行计算程序.数值算例表明：基于MATLAB-CUDA混合平台求解等效静力平衡方程的效率是串行多点同步算法的86.13倍，大大缩短了随机振动分析的总计算时间，且内存占用小、易于在个人计算机上实施；采用PCG法求解车辆-轨道-地基土耦合系统形成的大型稀疏线性方程组时，建议以加速度指标作为迭代收敛精度的控制指标；可通过选取适当的迭代收敛精度，以达到计算精度和计算效率的平衡.

关键词：随机振动；GPU并行计算；3D有限元法；虚拟激励法；车辆-轨道-地基土耦合模型

中图分类号：U211.3；TP399文献标志码：A

基金项目：国家自然科学基金资助项目（51678576，52078498），National Natural Science Foundation of China（51678576，52078498）；国家重点研发计划项目（2017YFB1201204），National Key R&D Program of China（2017YFB1201204）

A Parallel Computing Method for Three-dimensional Random Vibration of Train-track-soil Dynamic Interaction Based on GPU

ZHU Zhihui1，2，XIA Yutao1，WANG Lidong3，LIU Yubing1

（1. School of Civil Engineering，Central South University，Changsha 410075，China；2. National Engineering Laboratory for High Speed Railway Construction，Central South University，Changsha 410075，China；3. School of Civil Engineering，Changsha University of Science & Technology，Changsha 410114，China）

Abstract：Aiming at the issue of low efficiency in the random analysis and calculation of train-track-soil cou-pled system due to the random characteristics of track irregularity，the pseudo-excitation method（PEM）is used to re-duce the computing cost of large samples. Considering the sparsity of equivalent stiffness matrix of train-track-soil coupling system，the large-scale sparse matrix was stored in Compressed Sparse Row（CSR）format. The symmetric positive definite equation of equivalent static equilibrium is solved by the preconditioned conjugate gradient（PCG）method，and the parallel program is developed by the MATLAB-CUDA（Compute Unified Device Architecture）hy-brid platform. The numerical example shows that the efficiency of solving equivalent static equilibrium equation based on MATLAB-CUDA hybrid platform is 86.13 times that of the multi-point synchronization algorithm，which greatly reduces the total calculation time of the random vibration analysis，and the memory usage is small and it is easy to im-plement on a personal computer. When using PCG method to solve the large sparse linear equations of vehicle-trackfoundation soil coupling system，it is suggested to take the acceleration as the control index of convergence accuracy of iteration. By choosing the appropriate convergence accuracy of iteration，the balance of calculation accuracy and effi-ciency is achieved.

Key words：random vibration；GPU parallel computing；three dimensional finite element method；pseudo-exci-tation method；train-track-soil couple model

随着列车速度、轴重、路网密度和行车密度的提高，铁路运输引起的振动对周围环境的影响问题日渐突出.计算机性能的不断提升使得有限元法[1-2]、有限元/无限元法[3]、边界元法[4]等数值计算方法能够广泛用于研究车辆引起的周围环境振动问题；根据所建立的模型维度可将数值模型分为2D模型[5]、2.5D模型[3]和3D模型[2].与3D模型相比，2D和2.5D模型假设轨道-地基土模型……