In this work, an Improved Non-singular Method of Fundamental Solutions (INMFS) is developed for the solution of two-dimensional linear elasticity problems. The source points and field points are collocated on the physical boundary, while the conventional MFS requires a troublesome fictitious boundary outside the physical domain. In INMFS, the desingularization is, for complying with the displacement boundary conditions, achieved by replacement of the concentrated points sources by distributed sources over circular discs around the singularity, and for complying with the traction boundary conditions by assuming the balance of the forces. This procedure is much more efficient than the previously proposed procedure that involves two reference solutions and at the same time enables INMFS for solving problems with internal voids and inclusions. The method has been assessed by comparison with MFS, analytical solutions and previous desingularization technique. The method is easy to code, accurate, efficient, and straightforwardly extendable to three dimensions.
COBISS.SI-ID: 1197994
A correction of the method of regularized sources is developed which can be used to properly calculate also the velocity and pressure derivatives at the boundary. This improvement is essential for solving free and moving boundary problems that are intended to be tackled in the perspective.
COBISS.SI-ID: 1263018
Purpose – The purpose of this paper is to find a solution of Stokes flow problems with Dirichlet and Neumann boundary conditions in axisymmetry using an efficient, non-singular, method of fundamental solutions that does not require an artificial boundary, i.e. source points of the fundamental solution coincide with the collocation points on the boundary. The fundamental solution of the Stokes pressure and velocity represents analytical solution of the flow due to a singular Dirac delta source in infinite space. Methodology – Instead of the singular source, a non-singular source with a regularization parameter is employed. Regularized axisymmetric sources were derived from the regularized three-dimensional sources by integrating over the symmetry coordinate. The involved analytical expressions for related Stokes flow pressure and velocity around such regularized axisymmetric sources have been derived. The solution to the problem is sought as a linear combination of the fields due to the regularized sources that coincide with the boundary. The intensities of the sources are chosen in such a way that the solution complies with the boundary conditions. Findings – An axisymmetric driven cavity numerical example, the flow in a hollow tube and the flow between two concentric tubes are chosen to assess the performance of the method. The results of the newly developed method of regularized sources in axisymmetry are compared with the results obtained by the fine-grid second-order classical finite difference method and analytical solution. The results converge with a finer discretization, however, as expected, they depend on the value of the regularization parameter. The method gives accurate results if the value of this parameter scales with the typical nodal distance on the boundary.
COBISS.SI-ID: 1206954