Abstract
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.
Highlights
Groundwater is one of the important aspects of the geotechnical design that the engineers have to deal with
The flagship of a fore above-mentioned family of numerical methods is the boundary element method (BEM). Main drawbacks of these methods are the need for a fundamental solution of the differential operator to know, integration of the fundamental solution which contains the singularity around the source point and from the computational point of view, a solution of resulting fully populated, non-symmetric characteristic matrix
The singularity of the fundamental solution in the method of fundamental solutions (MFS) is tackled using very simple approach - the source point is moved outside the computational domain - the fictitious boundary that doesn't need to have the same shape as the computational domain is created (See Fig.1)
Summary
Groundwater is one of the important aspects of the geotechnical design that the engineers have to deal with. The flagship of a fore above-mentioned family of numerical methods is the boundary element method (BEM) Main drawbacks of these methods are the need for a fundamental solution of the differential operator to know, integration of the fundamental solution which contains the singularity around the source point and from the computational point of view, a solution of resulting fully populated, non-symmetric characteristic matrix. The promising advantage of this method is very easy programming and the method can be developed using few lines of code in Matlab or Octave, while it still maintains the advantage of boundary only formulation Advantages of such approaches are even magnified when 3D problems come into play with complex boundaries; the creation of just boundary nodes is much more flexible and adaptive than creating the mesh of elements that have to obey some sort of tessellation. The MFS and SBM numerical schemes are presented for the solution of free-surface flow problems and the results are compared with the solutions of the same problems solved using FEM from the view computational resources and precision of results which are presented
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