scholarly journals A Novel Boundary-Type Meshless Method for Modeling Geofluid Flow in Heterogeneous Geological Media

Geofluids ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Jing-En Xiao ◽  
Cheng-Yu Ku ◽  
Chih-Yu Liu ◽  
Wei-Chung Yeih
Keyword(s):  
Free Surface ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Domain Decomposition Method ◽  
Subsurface Flow ◽  
Test Problems ◽  
Pressure Potential ◽  
Trefftz Method ◽  
Flow Problems ◽  
Boundary Type

A novel boundary-type meshless method for modeling geofluid flow in heterogeneous geological media was developed. The numerical solutions of geofluid flow are approximated by a set of particular solutions of the subsurface flow equation which are expressed in terms of sources located outside the domain of the problem. This pioneering study is based on the collocation Trefftz method and provides a promising solution which integrates the T-Trefftz method and F-Trefftz method. To deal with the subsurface flow problems of heterogeneous geological media, the domain decomposition method was adopted so that flux conservation and the continuity of pressure potential at the interface between two consecutive layers can be considered in the numerical model. The validity of the model is established for a number of test problems. Application examples of subsurface flow problems with free surface in homogenous and layered heterogeneous geological media were also carried out. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stability for solving subsurface flow with nonlinear free surface in layered heterogeneous geological media even with large contrasts in the hydraulic conductivity.

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.

Numerical Solutions for Groundwater Flow in Unsaturated Layered Soil with Extreme Physical Property Contrasts

2015 ◽  
Vol 16 (7-8) ◽  
pp. 325-335 ◽  
Author(s):  
Chih-Yu Liu ◽  
Cheng-Yu Ku ◽  
Chi-Chao Huang ◽  
Der-Guey Lin ◽  
Wei-Chung Yeih
Keyword(s):  
Hydraulic Conductivity ◽  
Groundwater Flow ◽  
Numerical Solutions ◽  
Physical Property ◽  
Layered Soil ◽  
Richards Equation ◽  
Test Problems ◽  
Pressure Potential ◽  
Numerous Test ◽  
Novel Method

AbstractIn this paper, the numerical solutions for groundwater flow in unsaturated layered soil using the Richards equation are presented. A linearisation process for the nonlinear Richards equation to deal with groundwater flow in unsaturated layered soil is derived. To solve one-dimensional flow in the unsaturated zone of layered soil profiles, flux conservation and the continuity of pressure potential at the interface between two consecutive layers are considered in the numerical model. In addition, a novel method, named the dynamical Jacobian-inverse free method, incorporated with a two-side equilibration algorithm for solving ill-conditioned systems with extreme contrasts in hydraulic conductivity is proposed. The validity of the model is established in numerous test problems by comparing the numerical results with the analytical solutions. The results show that the proposed method can improve convergence and numerical stability for solving groundwater flow in unsaturated layered soil with extreme contrasts in hydraulic conductivity.

scholarly journals On Solving Two-Dimensional Inverse Heat Conduction Problems Using the Multiple Source Meshless Method

2019 ◽  
Vol 9 (13) ◽  
pp. 2629 ◽  
Author(s):  
Ku ◽  
Xiao ◽  
Huang ◽  
Yeih ◽  
Liu
Keyword(s):  
Heat Conduction ◽  
Meshless Method ◽  
Domain Decomposition Method ◽  
Domain Boundary ◽  
Addition Theorem ◽  
Two Dimensions ◽  
Multiple Source ◽  
Inverse Heat Conduction ◽  
Trefftz Method ◽  
Inverse Heat Conduction Problems

In this article, a newly developed multiple-source meshless method (MSMM) capable of solving inverse heat conduction problems in two dimensions is presented. Evolved from the collocation Trefftz method (CTM), the MSMM approximates the solution by using many source points through the addition theorem such that the ill-posedness is greatly reduced. The MSMM has the same superiorities as the CTM, such as the boundary discretization only, and is advantageous for solving inverse problems. Several numerical examples are conducted to validate the accuracy of solving inverse heat conduction problems using boundary conditions with different levels of noise. Moreover, the domain decomposition method is adopted for problems in the doubly-connected domain. The results demonstrate that the proposed method may recover the unknown data with remarkably high accuracy, even though the over-specified boundary data are assigned a portion that is less than 1/10 of the overall domain boundary.

scholarly journals A spacetime collocation Trefftz method for solving the inverse heat conduction problem

2019 ◽  
Vol 11 (7) ◽  
pp. 168781401986127 ◽  
Author(s):  
Cheng-Yu Ku ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Wei-Po Huang ◽  
Yan Su
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Heat Conduction Problem ◽  
Domain Boundary ◽  
Inverse Heat Conduction Problem ◽  
Test Problems ◽  
Inverse Heat Conduction ◽  
Trefftz Method ◽  
Base Functions

In this article, a novel spacetime collocation Trefftz method for solving the inverse heat conduction problem is presented. This pioneering work is based on the spacetime collocation Trefftz method; the method operates by collocating the boundary points in the spacetime coordinate system. In the spacetime domain, the initial and boundary conditions are both regarded as boundary conditions on the spacetime domain boundary. We may therefore rewrite an initial value problem (such as a heat conduction problem) as a boundary value problem. Hence, the spacetime collocation Trefftz method is adopted to solve the inverse heat conduction problem by approximating numerical solutions using Trefftz base functions satisfying the governing equation. The validity of the proposed method is established for a number of test problems. We compared the accuracy of the proposed method with that of the Trefftz method based on exponential basis functions. Results demonstrate that the proposed method obtains highly accurate numerical solutions and that the boundary data on the inaccessible boundary can be recovered even if the accessible data are specified at only one-fourth of the overall spacetime boundary.

scholarly journals On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method

Water ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 835 ◽  
Author(s):  
Cheng-Yu Ku ◽  
Jing-En Xiao ◽  
Chih-Yu Liu
Keyword(s):  
Boundary Conditions ◽  
Laplace Equation ◽  
Meshless Method ◽  
Moving Boundary ◽  
Domain Decomposition Method ◽  
Nonlinear Boundary Conditions ◽  
Surface Flow ◽  
Moving Boundary Problems ◽  
Boundary Problems ◽  
Trefftz Method

In this article, a solution to nonlinear moving boundary problems in heterogeneous geological media using the meshless method is proposed. The free surface flow is a moving boundary problem governed by Laplace equation but has nonlinear boundary conditions. We adopt the collocation Trefftz method (CTM) to approximate the solution using Trefftz base functions, satisfying the Laplace equation. An iterative scheme in conjunction with the CTM for finding the phreatic line with over–specified nonlinear moving boundary conditions is developed. To deal with flow in the layered heterogeneous soil, the domain decomposition method is used so that the hydraulic conductivity in each subdomain can be different. The method proposed in this study is verified by several numerical examples. The results indicate the advantages of the collocation meshless method such as high accuracy and that only the surface of the problem domain needs to be discretized. Moreover, it is advantageous for solving nonlinear moving boundary problems with heterogeneity with extreme contrasts in the permeability coefficient.

scholarly journals A Novel Hybrid Boundary-Type Meshless Method for Solving Heat Conduction Problems in Layered Materials

2018 ◽  
Vol 8 (10) ◽  
pp. 1887 ◽  
Author(s):  
Jing-En Xiao ◽  
Cheng-Yu Ku ◽  
Wei-Po Huang ◽  
Yan Su ◽  
Yung-Hsien Tsai
Keyword(s):  
Heat Conduction ◽  
Composite Materials ◽  
Meshless Method ◽  
Domain Decomposition Method ◽  
Superposition Principle ◽  
Layered Materials ◽  
Basis Functions ◽  
Method Of Fundamental Solutions ◽  
Boundary Type ◽  
Layer Composite

In this article, we propose a novel meshless method for solving two-dimensional stationary heat conduction problems in layered materials. The proposed method is a recently developed boundary-type meshless method which combines the collocation scheme from the method of fundamental solutions (MFS) with the collocation Trefftz method (CTM) to improve the applicability of the method for solving boundary value problems. Particular non-singular basis functions from cylindrical harmonics are adopted in which the numerical approximation is based on the superposition principle using the non-singular basis functions expressed in terms of many source points. For the modeling of multi-layer composite materials, we adopted the domain decomposition method (DDM), which splits the domain into smaller subdomains. The continuity of the flux and the temperature has to be satisfied at the interface of subdomains for the problem. The validity of the proposed method is investigated for several test problems. Numerical applications were also carried out. Comparison of the proposed method with other meshless methods showed that it is highly accurate and computationally efficient for modeling heat conduction problems, especially in heterogeneous multi-layer composite materials.

Nonlinear Sloshing in Fixed and Vertically Excited Containers

2002 ◽  
Author(s):  
Jannette B. Frandsen ◽  
Alistair G. L. Borthwick
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Small Perturbation ◽  
Numerical Solutions ◽  
Vertical Direction ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Surface Flow ◽  
Nonlinear Behaviour ◽  
Computational Domain

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.

A Meshless Method for Solving the Mathematical Model Associated the Leakage Problem in Gas Pipeline

2014 ◽  
Vol 986-987 ◽  
pp. 1418-1421
Author(s):  
Jun Shan Li
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Basis Function ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Gas Pipeline ◽  
The Mathematical Model

In this paper, we propose a meshless method for solving the mathematical model concerning the leakage problem when the pressure is tested in the gas pipeline. The method of radial basis function (RBF) can be used for solving partial differential equation by writing the solution in the form of linear combination of radius basis functions, that is, when integrating the definite conditions, one can find the combination coefficients and then the numerical solution. The leak problem is a kind of inverse problem that is focused by many engineers or mathematical researchers. The strength of the leak can find easily by the additional conditions and the numerical solutions.

Model coupling techniques for free-surface flow problems: Part II

2005 ◽  
Vol 63 (5-7) ◽  
pp. e1897-e1908 ◽  
Author(s):  
E. Miglio ◽  
S. Perotto ◽  
F. Saleri
Keyword(s):  
Free Surface ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Model Coupling ◽  
Flow Problems ◽  
Coupling Techniques

scholarly journals An accurate, stable and efficient domain-type meshless method for the solution of MHD flow problems

2009 ◽  
Vol 228 (21) ◽  
pp. 8135-8160 ◽  
Author(s):  
G.C. Bourantas ◽  
E.D. Skouras ◽  
V.C. Loukopoulos ◽  
G.C. Nikiforidis
Keyword(s):  
Meshless Method ◽  
Mhd Flow ◽  
Flow Problems ◽  
Domain Type
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