Simulation of multivariate ergodic stochastic processes using adaptive spectral sampling and non-uniform fast Fourier transform基于自适应谱抽样与非均匀快速 Fourier 变换的多变量各态历经随机过程模拟
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Tao TY, Wang H, 2024. Simulation of multivariate ergodic stochastic processes using adaptive spectral sampling and non-uniform fast Fourier transform. Probabilistic Engineering Mechanics, 77: 103669.DOI: 10.1016/j.probengmech.2024.103669
多变量各态历经随机过程模拟对结构动力分析和可靠性评估至关重要。尽管传统谱表达方法 (spectral representation method, SRM) 在许多领域已有广泛应用，但对于具有大量模拟点或长持时的随机过程，其模拟效率仍然较低。这主要是因为与频率相关的互谱密度矩阵分解需耗费巨大的计算成本。针对这一挑战，本文提出了一种各态历经随机过程高效模拟方法，该方法将有限频率自适应谱抽样与非均匀快速 Fourier 变换 (non-uniform fast Fourier transform, NUFFT) 技术相结合。自适应抽样是在均匀区间内对包络谱进行非均匀抽样，以确定有限的非均匀分布频率。因此，该方法仅需在有限特定频率处进行 Cholesky 分解，从而大幅降低了矩阵分解的计算成本。由于随机抽样频率并非均匀分布，故无法采用快速 Fourier 变换加速三角函数求和的过程。为此，采用了适用于非均匀抽样点的 NUFFT 技术加快计算过程，其中非均匀增量通过插值近似而得。以某大跨度悬索桥的风场模拟为例，研究了随机频率对该方法模拟误差和谱收敛速率的影响。最后，围绕模拟风速样本的功率谱密度和概率密度函数，对所提出方法的有效性进行了验证，并将其与传统方法进行了对比。分析结果表明，该方法在模拟各态历经随机过程的效率和精度方面均具有卓越表现。关键词: 多变量随机过程, 各态历经过程, 模拟, 自适应谱抽样, 非均匀快速 Fourier 变换The simulation of multivariate ergodic stochastic processes is critical for structural dynamic analysis and reliability evaluation. Although the traditional spectral representation method (SRM) has a wide application in many areas, it is highly inefficient in simulating stochastic processes with many simulation points or long durations due to the significant computational cost associated with matrix factorizations concerning frequency. To address the encountered challenge, this paper presents an efficient approach for simulating ergodic stochastic processes with limited frequencies. Central to this approach is a fusion of the adaptive spectral sampling and the non-uniform fast Fourier transform (NUFFT) techniques. The adaptive spectral sampling of the envelope spectrum enables the determination of limited non-equispaced frequencies, which are randomly sampled according to a uniform distribution. Thus, the Cholesky decomposition is only required at limited specific frequencies, which dramatically reduces the computational cost of matrix factorizations. Since the randomly sampled frequencies are not equispaced, utilizing FFT to accelerate the summation of trigonometric functions becomes impractical. Then, the NUFFT that adapts the non-equispaced sampling points is employed instead to expedite this process with the non-uniform increment approximated through reduced interpolation. By taking the wind field simulation of a long-span suspension bridge as an example, a parametric analysis is conducted to investigate the effect of random frequencies on the simulation error of the developed approach and the convergence of spectra. Finally, the developed approach is further validated by focusing on the spectra and probabilistic density functions of the simulated wind samples, and the simulation performance is compared with that of the traditional approach. The analytical results demonstrate the efficiency and accuracy of the developed approach in simulating ergodic stochastic processes.
Keywords: Multivariate stochastic process; Ergodic process; Simulation; Adaptive spectral sampling; Non-uniform fast Fourier transform.![]()
Fig. 1. Envelope of the spectra of stochastic processes
Fig. 2. Sampling points obtained via adaptive spectral sampling
图 3: 基于自适应谱抽样与非均匀快速 Fourier 变换发展的方法实现流程Fig. 3. Workflow of the developed approach using adaptive spectral sampling and NUFFT
Fig. 4. Layout of simulation points on the suspension bridge
Fig. 5. Variation of simulation error versus the number of samples
Fig. 6. Variation of simulation error versus the number of random frequencies
Fig. 7. Typical wind speed samples generated using different number of random frequencies
Fig. 8. Wind speed samples simulated using the developed approach
Fig. 9. Spectral verification of the simulated wind samples using the developed approach
Fig. 10. PDFs of generated wind speeds with respect to time and samples
Fig. 11. Comparison of the mean PDF with respect to time and samples
图 12: 传统方法与所提方法随样本数的误差变化对比Fig. 12. Error comparison of traditional and developed approaches versus the number of samples
作者信息 | Authors
东南大学 (Southeast University) 混凝土及预应力混凝土结构教育部重点实验室
王浩 Hao Wang, 通讯作者 (Corresp.)东南大学 (Southeast University) 混凝土及预应力混凝土结构教育部重点实验室Email: wanghao1980@seu.edu.cn
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
