Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady‐State Conditions

IF 3.4 2区工程技术 Q2 ENGINEERING, GEOLOGICAL

International Journal for Numerical and Analytical Methods in Geomechanics Pub Date : 2024-10-23 DOI:10.1002/nag.3876

Robert P. Chapuis, Coline Taveau, François Duhaime, Simon Weber, Vahid Marefat, Lu Zhang, Daniela Blessent, Najib Bouaanani, Dominique Pelletier

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引用次数: 0

Abstract

Unsaturated zones are important for geotechnical design, geochemical reactions, and microbial reactions. The numerical analysis of unsaturated seepage is complex because it involves highly nonlinear partial differential equations. The permeability can vary by orders of magnitude over short vertical distances. This article defines and uses H‐convergence tests to quantify numerical errors made by uniform meshes with element size (ES) for 1D steady‐state conditions. The quantitative H‐convergence should not be confused with a qualitative mesh sensitivity study. The difference between numerical and mathematical convergences is stated. A detailed affordable method for an H‐convergence test is presented. The true but unknown solution is defined as the asymptote of the numerical solutions for all solution components when ES decreases to zero. The numerical errors versus ES are then assessed with respect to the true solution, and using a log–log plot, which indicates whether a code is correct or incorrect. If a code is correct, its results follow the rules of mathematical convergence in a mathematical convergence domain (MCD) which is smaller than the numerical convergence domain (NCD). If a code is incorrect, it has an NCD but no MCD. Incorrect algorithms of incorrect codes need to be modified and repaired. Existing codes are shown to converge numerically within large NCDs but generate large errors, up to 500%, in the NCDs, a dangerous situation for designers.

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利用 H 融合计算稳态条件下一维非饱和渗流的数值误差

非饱和区对岩土工程设计、地球化学反应和微生物反应都很重要。非饱和渗流的数值分析非常复杂，因为它涉及高度非线性的偏微分方程。在很短的垂直距离内，渗透率可能会有数量级的变化。本文定义并使用 H- 收敛测试来量化一维稳态条件下元素尺寸（ES）均匀网格的数值误差。定量 H 趋同不应与定性网格敏感性研究相混淆。说明了数值收敛与数学收敛之间的区别。介绍了 H 收敛测试的详细实惠方法。真实但未知的解被定义为 ES 降为零时所有解成分的数值解的渐近线。然后，相对于真实解，使用对数-对数图评估数值误差与 ES 的关系，从而显示代码的正确与否。如果代码正确，则其结果遵循数学收敛域（MCD）中的数学收敛规则，该收敛域小于数值收敛域（NCD）。如果代码不正确，则有 NCD 而无 MCD。不正确代码的不正确算法需要修改和修复。现有的代码在较大的 NCD 范围内可以数值收敛，但在 NCD 范围内会产生较大的误差，误差率最高可达 500%，这对设计人员来说是一个危险的情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal for Numerical and Analytical Methods in Geomechanics 工程技术-材料科学：综合

CiteScore

6.40

自引率

12.50%

发文量

160

审稿时长

9 months

期刊介绍： The journal welcomes manuscripts that substantially contribute to the understanding of the complex mechanical behaviour of geomaterials (soils, rocks, concrete, ice, snow, and powders), through innovative experimental techniques, and/or through the development of novel numerical or hybrid experimental/numerical modelling concepts in geomechanics. Topics of interest include instabilities and localization, interface and surface phenomena, fracture and failure, multi-physics and other time-dependent phenomena, micromechanics and multi-scale methods, and inverse analysis and stochastic methods. Papers related to energy and environmental issues are particularly welcome. The illustration of the proposed methods and techniques to engineering problems is encouraged. However, manuscripts dealing with applications of existing methods, or proposing incremental improvements to existing methods – in particular marginal extensions of existing analytical solutions or numerical methods – will not be considered for review.