Richards方程Laplace变换解中几个遗留问题的讨论  

Discussion on Several Remaining Problems in the Laplace Transform Solution of Richards Equation

作　　者：朱悦璐 李光灿 吴帅兵[1] ZHU Yue-lu;LI Guang-can;WU Shuai-bing(College of Water Conservancy and Ecological Engineering,Nanchang Institute of Technology,Nanchang 330099,China)

机构地区：[1]南昌工程学院水利与生态工程学院,南昌330099

出　　处：《长江科学院院报》2024年第12期126-132,共7页Journal of Changjiang River Scientific Research Institute

基　　金：国家自然科学基金项目(52069014)。

摘　　要：对Richards方程的Laplace变换解在原函数敛散性、逆变换求解步骤、方程适用性等3个方面的问题进行了补充和讨论。通过列举理论反例、使用积分变换和换元方案、结合实际工程算例等方法,证明了直接默认原函数收敛的传统做法有可能出现较大误差,因此在Laplace正变换前,需对入渗函数的敛散性进行判断;针对性补全了Laplace逆变换的求解步骤,从理论层面闭合了该解在土工类文献中缺失的一环;揭示了Laplace解在应用中可能出现土体下层含水率大于上层的反常现象,并从物理背景和数学背景对该现象进行分析,指出该解所存在的天然缺陷。研究结果可从理论上进一步巩固非饱和土理论体系。In this paper we addressed and elaborated on three critical issues concerning the Laplace transform solution of the Richards equation:the convergence of the original function,the steps for inverse transform,and the applicability of the equation.By presenting theoretical counterexamples,employing integral transformations and substitution schemes as well as real engineering cases,we demonstrated that assuming the convergence of the original function can lead to significant errors.Therefore,it is necessary to assess the convergence of the infiltration function before applying the Laplace transform.We also elucidated the steps for solving the Laplace inverse transform,theoretically addressing a gap in geotechnical literature.Furthermore,we reveal an anomaly where,in certain applications,the Laplace solution may show higher moisture content in lower soil layers than in upper layers.We analyzed this phenomenon from both physical and mathematical perspectives,highlighting inherent limitations in this solution.The findings of this study may help to theoretically strengthen the framework of unsaturated soil theory.

关 键 词：土力学 RICHARDS方程 LAPLACE逆变换 非饱和入渗 微分方程求解 

分 类 号：TU43[建筑科学—岩土工程]