The Method of Fundamental Solutions for Three-Dimensional Nonlinear Free Surface Flows Using the Iterative Scheme
Abstract: In this article, we present a meshless method based on the method of fundamental solutions (MFS) capable of solving free surface flow in three dimensions. Since the basis function of the MFS satisfies the governing equation, the advantage of the MFS is that only the problem boundary
needs to be placed in the collocation points. For solving the three-dimensional free surface with nonlinear boundary conditions, the relaxation method in conjunction with the MFS is used, in which
the three-dimensional free surface is iterated as a movable boundary until the nonlinear boundary conditions are satisfied. The proposed method is verified and application examples are conducted. Comparing results with those from other methods shows that the method is robust and provides high accuracy and reliability. The effectiveness and ease of use for solving nonlinear free surface flows in three dimensions are also revealed.
Keywords: three–dimensional; tree surface; nonlinear; the method of fundamental solutions (MFS); meshless method
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.
Keywords: heat conduction problems; the collocation scheme; the meshless method; the domain decomposition method; layered materials.
A Novel Boundary-Type Meshless Method for Modeling Geofluid
Flow in Heterogeneous Geological Media
Jing-En Xiao, Cheng-Yu Ku , Chih-Yu Liu, and Wei-Chung Yeih
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.
Full paper link:
Modeling of Transient Flow in Unsaturated Geomaterials for Rainfall-Induced Landslides Using a Novel Spacetime Collocation Method
The modeling of transient flow in unsaturated soils for rainfall-induced landslides using a novel spacetime collocation method is presented. A numerical solution obtained in the spacetime coordinate system is approximated by superpositioning Trefftz basis functions satisfying the linearized Richards equation for collocation points on the spacetime domain boundary. The Gardner exponential model is adopted to derive the linearized Richards equation to describe the soil-water characteristic curve in unsaturated soils. To deal with the rainfall-induced landslides, the infinite slope stability analysis coupled with the proposed meshless method with the consideration of the fluctuation of time-dependent matric suction is developed. The proposed method is validated for several test problems. Application examples of transient modeling of flow for rainfall-induced landslides in homogenous unsaturated soils are also conducted. Numerical results demonstrate that the proposed method is highly accurate to deal with transient flow in unsaturated soils for rainfall-induced landslides. In addition, it is found that the numerical method using the Richards equation with the Gardner model may provide a promising solution for different soil textures.
This paper presents the numerical solutions of free surface seepage flow in layered soil using the method of fundamental solutions (MFS). The numerical solutions are approximated by a set of fundamental solutions of the two-dimensional Laplace equation which are expressed in terms of sources located outside the domain of the problem. The unknown coefficients in the linear combination of the fundamental solutions which are accomplished by collocation imposing the boundary condition at a finite number of points can then be solved. To deal with the seepage problems of layered soil profiles, 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, including seepage problems in a rectangular dam, a trapezoidal dam, and an earth dam, by comparing numerical results with those from other methods. Application examples were also carried out. The results reveal that the proposed method based on the MFS has great numerical stability for solving seepage flow with nonlinear free surface in layered heterogeneous soil even with large contrasts in the hydraulic conductivity.
Keywords：Seepage, Free surface, Layered soil, The method of fundamental solutions, The domain decomposition method.
Jing-En Xiao, Cheng-Yu Ku*, Chih-Yu Liu, Chia-Ming Fan, Weichung Yeih (2017), “On Solving Free Surface Problems in Layered Soil Using the Method of Fundamental Solutions “, Engineering Analysis with Boundary Elements, Vol. 83, pp. 96-106. (IF 1.721, Q2 in ENGINEERING, MULTIDISCIPLINARY, JCR 2016)
Full paper link: