scholarly journals Free surface groundwater flow solution using boundary collocation methods

2018 ◽  
Vol 196 ◽  
pp. 03026 ◽  
Author(s):  
Juraj Mužík ◽  
Roman Bulko
Keyword(s):  
Diffusion Equation ◽  
Groundwater Flow ◽  
Moving Boundary ◽  
Numerical Algorithms ◽  
Type Equation ◽  
Fundamental Solutions ◽  
Collocation Methods ◽  
Method Of Fundamental Solutions ◽  
Singular Boundary ◽  
State Diffusion

In this paper, two meshless numerical algorithms are developed for the solution of two-dimensional steady-state diffusion equation that describes the stationary groundwater flow. The proposed numerical methods, which are truly meshless, quadrature-free and boundary only, are based on the method of fundamental solutions and singular boundary method respectively. The diffusion equation is transformed into a Poisson-type equation with a known fundamental solution. Numerical examples with moving boundary are presented and compared to the solutions obtained by the finite element method.

scholarly journals A Spacetime Meshless Method for Modeling Subsurface Flow with a Transient Moving Boundary

Water ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 2595 ◽  
Author(s):  
Cheng-Yu Ku ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Weichung Yeih ◽  
Chia-Ming Fan
Keyword(s):  
Diffusion Equation ◽  
Meshless Method ◽  
Moving Boundary ◽  
Iterative Scheme ◽  
Separation Of Variables ◽  
Subsurface Flow ◽  
Basis Functions ◽  
Method Of Fundamental Solutions ◽  
Trefftz Method ◽  
Flow Problems

In this paper, a spacetime meshless method utilizing Trefftz functions for modeling subsurface flow problems with a transient moving boundary is proposed. The subsurface flow problem with a transient moving boundary is governed by the two-dimensional diffusion equation, where the position of the moving boundary is previously unknown. We solve the subsurface flow problems based on the Trefftz method, in which the Trefftz basis functions are obtained from the general solutions using the separation of variables. The solutions of the governing equation are then approximated numerically by the superposition theorem using the basis functions, which match the data at the spacetime boundary collocation points. Because the proposed basis functions fully satisfy the diffusion equation, arbitrary nodes are collocated only on the spacetime boundaries for the discretization of the domain. The iterative scheme has to be used for solving the moving boundaries because the transient moving boundary problems exhibit nonlinear characteristics. Numerical examples, including harmonic and non-harmonic boundary conditions, are carried out to validate the method. Results illustrate that our method may acquire field solutions with high accuracy. It is also found that the method is advantageous for solving inverse problems as well. Finally, comparing with those obtained from the method of fundamental solutions, we may obtain the accurate location of the nonlinear moving boundary for transient problems using the spacetime meshless method with the iterative scheme.

An Improved Formulation of Singular Boundary Method

2012 ◽  
Vol 4 (5) ◽  
pp. 543-558 ◽  
Author(s):  
Wen Chen ◽  
Yan Gu
Keyword(s):  
Intensity Factor ◽  
Neumann Boundary ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Benchmark Problems ◽  
Singular Boundary ◽  
Boundary Method ◽  
New Formulation ◽  
Physical Boundary ◽  
Singular Boundary Method

AbstractThis study proposes a new formulation of singular boundary method (SBM) to solve the 2D potential problems, while retaining its original merits being free of integration and mesh, easy-to-program, accurate and mathematically simple without the requirement of a fictitious boundary as in the method of fundamental solutions (MFS). The key idea of the SBM is to introduce the concept of the origin intensity factor to isolate the singularity of fundamental solution so that the source points can be placed directly on the physical boundary. This paper presents a new approach to derive the analytical solution of the origin intensity factor based on the proposed subtracting and adding-back techniques. And the troublesome sample nodes in the ordinary SBM are avoided and the sample solution is also not necessary for the Neumann boundary condition. Three benchmark problems are tested to demonstrate the feasibility and accuracy of the new formulation through detailed comparisons with the boundary element method (BEM), MFS, regularized meshless method (RMM) and boundary distributed source (BDS) method.

Evaluating the Origin Intensity Factor in the Singular Boundary Method for Three-Dimensional Dirichlet Problems

2017 ◽  
Vol 9 (6) ◽  
pp. 1289-1311 ◽  
Author(s):  
Linlin Sun ◽  
Wen Chen ◽  
Alexander H.-D. Cheng
Keyword(s):  
Dirichlet Boundary ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Singular Boundary ◽  
Dirichlet Problems ◽  
Boundary Method ◽  
Interpolation Technique ◽  
Inverse Interpolation ◽  
New Formulation ◽  
Singular Boundary Method

AbstractIn this paper, a new formulation is proposed to evaluate the origin intensity factors (OIFs) in the singular boundary method (SBM) for solving 3D potential problems with Dirichlet boundary condition. The SBM is a strong-form boundary discretization collocation technique and is mathematically simple, easy-to-program, and free of mesh. The crucial step in the implementation of the SBM is to determine the OIFs which isolate the singularities of the fundamental solutions. Traditionally, the inverse interpolation technique (IIT) is adopted to calculate the OIFs on Dirichlet boundary, which is time consuming for large-scale simulation. In recent years, the new methodology has been developed to efficiently calculate the OIFs on Neumann boundary, but the Dirichlet problem remains an open issue. This study employs the subtracting and adding-back technique based on the integration of the fundamental solution over the whole boundary to develop a new formulation of the OIFs on 3D Dirichlet boundary. Several problems with varied domain shapes and boundary conditions are carried out to validate the effectiveness and feasibility of the proposed scheme in comparison with the SBM based on inverse interpolation technique, the method of fundamental solutions, and the boundary element method.

scholarly journals Stress analysis of elastic bi-materials by using the localized method of fundamental solutions

2021 ◽  
Vol 7 (1) ◽  
pp. 1257-1272
Author(s):  
Juan Wang ◽  
◽  
Wenzhen Qu ◽  
Xiao Wang ◽  
Rui-Ping Xu ◽  
...  
Keyword(s):  
Stress Analysis ◽  
Large Scale ◽  
Classical Method ◽  
Fundamental Solutions ◽  
Least Square ◽  
Collocation Methods ◽  
Method Of Fundamental Solutions ◽  
Computational Domain ◽  
Desktop Computer ◽  
Moving Least Square

<abstract> <p>The localized method of fundamental solutions belongs to the family of meshless collocation methods and now has been successfully tried for many kinds of engineering problems. In the method, the whole computational domain is divided into a set of overlapping local subdomains where the classical method of fundamental solutions and the moving least square method are applied. The method produces sparse and banded stiffness matrix which makes it possible to perform large-scale simulations on a desktop computer. In this paper, we document the first attempt to apply the method for the stress analysis of two-dimensional elastic bi-materials. The multi-domain technique is employed to handle the non-homogeneity of the bi-materials. Along the interface of the bi-material, the displacement continuity and traction equilibrium conditions are applied. Several representative numerical examples are presented and discussed to illustrate the accuracy and efficiency of the present approach.</p> </abstract>

NUMERICAL INVESTIGATION OF THREE-DIMENSIONAL HAUSDORFF DERIVATIVE ANOMALOUS DIFFUSION MODEL

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050020 ◽  
Author(s):  
WEI CAI ◽  
FAJIE WANG
Keyword(s):  
Diffusion Equation ◽  
Anomalous Diffusion ◽  
Numerical Experiments ◽  
Three Dimensional ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Efficient Tool ◽  
Scaling Transform ◽  
Diffusion Phenomena ◽  
Temporal And Spatial

The Hausdorff derivative has been recognized as an efficient tool to characterize anomalous diffusion phenomena. This paper makes the first attempt to numerically investigate three-dimensional Hausdorff derivative diffusion equation by the method of fundamental solutions. The fundamental solution of the three-dimensional Hausdorff derivative diffusion equation is closely related to scaling transform and non-Euclidean Hausdorff fractal distance. The used method, as a meshless technique, is simple, accurate and efficient for solving the partial differential equations with fundamental solutions. Three numerical experiments have been conducted to reveal the effectiveness and rationality of Hausdorff derivative anomalous diffusion models with various temporal and spatial fractal orders, as well as the accuracy of the developed methodology.

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