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Non-intrusive polynomial chaos methods for uncertainty quantification in wave problems at high frequencies
Journal of Computational Science ( IF 3.3 ) Pub Date : 2021-03-27 , DOI: 10.1016/j.jocs.2021.101344
Nabil El Mocayd , M Shadi Mohamed , Mohammed Seaid

Numerical solutions of wave problems are often influenced by uncertainties generated by a lack of knowledge of the input values related to the domain data and/or boundary conditions in the mathematical equations used in the modeling. Conventional methods for uncertainty quantification in modeling waves constitute severe challenges due to the high computational costs especially at high frequencies/wavenumbers. For a given accuracy and a high wavenumber it is necessary to perform a mesh convergence study by refining the discretization of the computational domain with an increased resolution, which leads to increasing the number of degrees of freedom at a much higher rate than the wavenumber. This effect also known as the pollution error often limits the computations to relatively small values of the wavenumber. To estimate the uncertainties, many model evaluations are required, but only a single surrogate model is created in the process. In the present work, we propose the use of a non-intrusive spectral projection applied to a finite element framework with enriched basis functions for the uncertainty quantification of waves at high frequencies. The method integrates (i) the partition of unity finite element method for effectively computing the solutions of waves at high frequencies; and (ii) a non-intrusive spectral projection for effectively propagating random wavenumbers that encode uncertainties in the wave problems. Compared to the conventional finite element methods, the proposed method is demonstrated to reduce the total cost of accurately computing uncertainties in waves with high values of the wavenumber. Numerical results are presented for two sets of numerical tests. First, the interference of plane waves in a squared domain and then a wave scattering by a circular cylinder are studied at high wavenumbers. Comparisons to the Monte Carlo simulations and the regression based polynomial chaos expansion confirm the computational effectiveness of the proposed approach.



中文翻译:

高频波动问题不确定性量化的非侵入式多项式混沌方法

波浪问题的数值解通常受到不确定性的影响,这些不确定性是由于缺乏对与域数据和/或建模中使用的数学方程中的边界条件相关的输入值的了解而产生的。由于计算成本高,尤其是在高频/波数下,用于建模波中不确定性量化的传统方法构成了严峻的挑战。对于给定的精度和高波数,有必要通过以更高的分辨率细化计算域的离散化来执行网格收敛研究,这导致以远高于波数的速率增加自由度数。这种效应也称为污染误差,通常将计算限制为相对较小的波数值。为了估计不确定性,需要进行许多模型评估,但在此过程中只创建了一个替代模型。在目前的工作中,我们建议将非侵入式光谱投影应用于具有丰富基函数的有限元框架,用于高频波的不确定性量化。该方法集成了(i)用于有效计算高频波解的统一有限元方法的划分;(ii) 用于有效传播随机波数的非侵入式频谱投影,该随机波数编码波问题中的不确定性。与传统的有限元方法相比,所提出的方法被证明可以降低准确计算具有高波数值的波浪中的不确定性的总成本。给出了两组数值试验的数值结果。第一的,在高波数下研究了平面波在平方域中的干涉,然后是圆柱体的波散射。与蒙特卡罗模拟和基于回归的多项式混沌展开的比较证实了所提出方法的计算有效性。

更新日期:2021-03-27
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