Isogeometric analysis for unsaturated flow problems |
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| Institution: | 1. Institute for Structural Mechanics, Ruhr-University Bochum, 44780 Bochum, Germany;2. Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8552, Japan;3. Department of Civil Engineering, University of Siegen, Germany;4. Department of Engineering Mechanics, Hohai University, Nanjing, PR China;1. School of Civil Engineering, Zhengzhou University, Zhengzhou 450001, China;2. Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, W355 Nebraska Hall, Lincoln, NE 68588-0526, USA;1. Department of Mathematics, Oklahoma State University, Stillwater, OK 74078-1058, USA;2. Department of Mathematics, Texas A&M University, College Station, TX 77843, USA;3. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China;1. Institute for Mathematics, Mechanics and Informatics, Kuban State University, Krasnodar 350040, Russia;2. Chair of Structural Mechanics, Department of Civil Engineering, University of Siegen, Paul-Bonatz Strasse 9-11, D-57076 Siegen, Germany;3. Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, PR China;1. Department of Civil Engineering, University of Siegen, Paul-Bonatz-Straße 9-11, 57076 Siegen, Germany;2. Piping Department, Petrovietnam Engineering Company, Ho Chi Minh, Vietnam;3. Department of Civil Engineering, Lac Hong University, Dong Nai Province, Vietnam |
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| Abstract: | Unsaturated flow problems in porous media often described by Richards’ equation are of great importance in many engineering applications. In this contribution, we propose a new numerical flow approach based on isogeometric analysis (IGA) for modeling the unsaturated flow problems. The non-uniform rational B-spline (NURBS) basis is utilized for spatial discretization whereas the stable implicit backward Euler method for time discretization. The nonlinear Richards’ equation is iteratively solved with the aid of the Newton–Raphson scheme. Owing to some desirable features of an efficient numerical flow approach, major advantages of the present formulation involve: (a) numerical oscillation at the wetting front can be avoided or facilitated, simply by using either an h-refinement or a lumped mass matrix technique; (b) higher-order exactness can be obtained due to the nature of the IGA features; (c) the approach is straightforward to implement and it does not need any transformation, e.g., Kirchhoff transformation or filter algorithm; and (d) in contrast to the Picard iteration scheme, which forms linear convergences, the proposed approach can however yield quadratic convergences by using the Newton–Raphson method for solving resultant nonlinear equations. Numerical model validation is analyzed by solving a three-dimensional unsaturated flow problem in soil, and its derived results are verified against analytical solutions. Numerical applications are then studied by considering three extensive examples with simple and complex configurations to further show the accuracy and applicability of the present IGA. |
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| Keywords: | Unsaturated flow Porous media Isogeometric analysis NURBS Richards’ equations |
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