Global existence of weak solutions to unsaturated poroelasticity

IF 1.9 3区 数学 Q2 Mathematics
J. Both, Iuliu Sorin Pop, I. Yotov
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引用次数: 6

Abstract

We study unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in variably saturated porous media, here modeled by a non-linear extension of Biot's well-known quasi-static consolidation model. The coupled elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media, obtained under similar assumptions as usually considered for Richards' equation. In this work, existence of weak solutions is established in several steps involving a numerical approximation of the problem using a physically-motivated regularization and a finite element/finite volume discretization. Eventually, solvability of the original problem is proved by a combination of the Rothe and Galerkin methods, and further compactness arguments. This approach in particular provides the convergence of the numerical discretization to a regularized model for unsaturated poroelasticity. The final existence result holds under non-degeneracy conditions and natural continuity properties for the constitutive relations. The assumptions are demonstrated to be reasonable in view of geotechnical applications.
非饱和孔隙弹性问题弱解的整体存在性
我们研究了非饱和孔隙弹性,即变饱和多孔介质中的水-力耦合过程,这里通过Biot著名的准静态固结模型的非线性扩展来建模。耦合椭圆-抛物型偏微分方程组是可变形多孔介质中多相流一般模型的简化版本,它是在类似理查兹方程的假设下得到的。在这项工作中,弱解的存在性被建立在几个步骤中,这些步骤涉及使用物理动机正则化和有限元/有限体积离散化的问题的数值近似。最后,结合Rothe方法和Galerkin方法以及进一步的紧性论证,证明了原问题的可解性。这种方法特别提供了数值离散化的收敛到非饱和孔隙弹性的正则化模型。最后的存在结果在本构关系的非简并条件和自然连续性条件下成立。从岩土工程应用的角度来看,这些假设是合理的。
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来源期刊
CiteScore
2.70
自引率
5.30%
发文量
27
审稿时长
6-12 weeks
期刊介绍: M2AN publishes original research papers of high scientific quality in two areas: Mathematical Modelling, and Numerical Analysis. Mathematical Modelling comprises the development and study of a mathematical formulation of a problem. Numerical Analysis comprises the formulation and study of a numerical approximation or solution approach to a mathematically formulated problem. Papers should be of interest to researchers and practitioners that value both rigorous theoretical analysis and solid evidence of computational relevance.
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