The Method of Fundamental Solutions for Three-Dimensional Nonlinear Free Surface Flows Using the Iterative Scheme

Cheng-Yu Ku,Chih-Yu Liu,Jing-En Xiao

doi:10.3390/app9081715

Cheng-Yu Ku, Chih-Yu Liu + Show 1 more

Open Access

https://doi.org/10.3390/app9081715

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Journal: Applied Sciences	Publication Date: Apr 25, 2019
Citations: 3	License type: CC BY 4.0

Affiliation: National Taiwan Ocean University

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.

Highlights

Accurate determination of the unknown phreatic line is regarded as one of the most important considerations for affecting the safety of an embankment dam or weirs, since failure of the earth–filled structure occurs because of piping and internal erosion mainly from seepage [1]
The determination of the phreatic line in seepage flow is a nonlinear problem which needs to find the location of the movable surface from the nonlinear boundary conditions [2]
With the advantage of the boundary-type meshless method, only the collocation points on the moving surface have to be renewed during iteration for the computation of the position of the three-dimensional nonlinear free surface [1]

Summary

Introduction

Accurate determination of the unknown phreatic line is regarded as one of the most important considerations for affecting the safety of an embankment dam or weirs, since failure of the earth–filled structure occurs because of piping and internal erosion mainly from seepage [1]. The determination of the phreatic line in seepage flow is a nonlinear problem which needs to find the location of the movable surface from the nonlinear boundary conditions [2]. The free surface problems can be solved using mesh–based methods with an adaptive mesh [3,4,5,6] or a fixed mesh [7,8,9,10,11,12]. The extended pressure method [13] based on finite differences is probably the simplest one for free surface calculation. Solving three-dimensional free surface flow problems needs to deal with three-dimensional geometric complexity. Mesh-based methods for handling the complexity of three-dimensional boundary conditions require sophisticated remeshing scheme. To have a successful three-dimensional mesh generation algorithm is, quite a challenging task

Methods

Discussion

Conclusion