非饱和土壤水一维流动有限元法求解编程思路
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1.江西师范大学地理与环境学院 鄱阳湖湿地与流域研究教育部重点实验室;2.江西师范大学 地理与环境学院 鄱阳湖湿地与流域研究教育部重点实验室

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国家自然科学基金项目(42201035)和江西省急需紧缺海外人才项目(20232BCJ25048)资助


A Programming Idea for Solving One-dimensional Unsaturated Infiltration Equation by Finite Element Method
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Affiliation:

1.School of Geography and Environment, Jiangxi Normal University;2.Ministry of Education''s Key Laboratory of Poyang Lake Wetland and Watershed Research, School of Geography and Environment, Jiangxi Normal University, Nanchang 330022, China

Fund Project:

Supported by the National Natural Science Foundation of China (No.42201035) and the Jiangxi Province Overseas Talent Project (No. 20232BCJ25048)

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    摘要:

    《新时代中国土壤物理学主要领域进展与展望》指出我国土壤物理学研究存在的一个问题是原创性研究较少,其原因之一是从事我国土壤物理学研究的大部分学者数理基础不强,因此较难在土壤物理过程数值模拟上有突破。土壤物理过程数值模拟的关键方程是理查兹方程。目前关于有限元法求解理查兹方程的论文不少,但是大多理论性强且可操作性较弱,对数理基础不强的研究人员而言,理解与编程实现存在较大困难。因此,本文旨在提供一份包含详细推导步骤的有限元法求解一维理查兹方程的编程思路:首先通过加权余量方程的建立得到理查兹方程的弱形式,继而利用雅可比变换和高斯数值积分等方法将弱形式方程转化为非线性代数方程,最后采用牛顿-拉夫逊法并代入边界条件求解非线性代数方程。基于上述编程思路编写了相应代码,模拟结果得到了实测土壤入渗试验数据的验证。本文提供的编程思路及代码可让数理基础不强背景的初学研究人员能够较快实现一维理查兹方程有限元法数值模拟,以期对未来我国在土壤物理过程数值模拟上实现突破起到一些积极作用。

    Abstract:

    Some Key Research Fields of Chinese Soil Physics in the New Era: Progresses and Perspectives pointed out that one of the reasons for the lack of original research on soil physics in China is that Chinese scholars engaged in soil physics research lack a strong mathematical foundation. This makes it difficult to achieve breakthroughs in the numerical simulation of soil physical processes. The key equation for numerical simulation of soil physical processes is the Richards equation. Although there are many papers on solving the Richards equation using the finite element method, most are highly theoretical and lack practicality, posing significant challenges for researchers with limited mathematical and physical backgrounds in understanding and programming implementation. Therefore, this paper aims to present a programming framework incorporating detailed derivation steps for solving the one-dimensional Richards equation using the finite element method. The weak form of the Richards equation was derived by establishing the weighted residual equation. Subsequently, the weak form equation was transformed into a nonlinear algebraic equation by employing Jacobian transformation and Gaussian numerical integration. Finally, the nonlinear algebraic equation was solved using the Newton-Raphson method with boundary condition substitution. The corresponding code developed based on this programming framework demonstrated simulation results validated by experimental data from soil infiltration tests. The programming framework and code provided in this paper enable researchers with limited mathematical and physical backgrounds to efficiently implement numerical simulation of the one-dimensional Richards equation using the finite element method. This effort aims to facilitate potential breakthroughs in numerical modeling of soil physical processes in China, thereby contributing positively to future advancements in this field.

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张晴晴,李 想,王忠富.非饱和土壤水一维流动有限元法求解编程思路[J].土壤学报,DOI:10.11766/trxb202412010459,[待发表]
ZHANG Qingqing, LI Xiang, WANG Zhognfu. A Programming Idea for Solving One-dimensional Unsaturated Infiltration Equation by Finite Element Method[J]. Acta Pedologica Sinica, DOI:10.11766/trxb202412010459,[In Press]

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  • 收稿日期:2024-12-01
  • 最后修改日期:2025-04-01
  • 录用日期:2025-04-02
  • 在线发布日期: 2025-04-14
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