An improved non-singular method of fundamental solutions (INMFS), where the sources are located on the domain boundary, is developed for 3D linear elasticity problems. To circumvent the singularities of the kernel in Fundamental Solution (FS) these are substituted by the volume integral of the FS over a small sphere. The normalized volume integral is used when the sources and collocation points are coincident. The desingularization of the fundamental traction is achieved by assuming the balance of the forces for complying with the mechanical equilibrium, calculated through meshless boundary patches coinciding with the boundary nodes. This improved approach avoids any need to solve the problem three times, as in the recently developed Non-singular Method of Fundamental Solutions (NMFS) by Liu and Šarler in 2018. The INMFS, NMFS and MFS solutions as well as the analytical solutions for a pair of single and one bi-material elasticity problems are employed to evaluate the viability and correctness of the new method in 3D. The INMFS results are reasonably accurate and converge uniformly to the analytical solution. The absence of an artificial boundary, trivial coding, and the straightforward use of the INMFS in problems with different materials in contact are demonstrated in this paper.

COBISS.SI-ID: 1509546

In this paper a new criterion for stability of strong-form meshless discretizations of advection-dominated problems is developed. The criterion follows from the analysis of the equivalent PDE. It incorporates the influence of the positions of the nodes in the local subdomain, the influence of free parameters of local approximation, such as the weight function and the order of local approximation, and the time stepping formula used. We use the developed criterion in an adaptive upwinding scheme and test the developed scheme on a simple two-dimensional advection problem for regular and irregular node arrangement by using second-, third- and fourth-order accurate explicit diffuse approximate method. The proposed upwinding algorithm significantly improves the accuracy of the resulting solution in comparison with the existing approaches which do not consider the parameters of the local approximation and the irregular positioning of nodes, which is the usual setting for meshless methods.

COBISS.SI-ID: 17037339

The present paper develops two new techniques, namely additive correction multicloud (ACMC) and smoothed restriction multicloud (SRMC), for the efficient solution of systems of equations arising from Radial Basis Function-generated Finite Difference (RBF-FD) meshless discretizations of partial differential equations (PDEs). RBF-FD meshless methods employ arbitrary distributed nodes, without the need to generate a mesh, for the numerical solution of PDEs. The proposed techniques are specifically designed for the RBF-FD data structure and employ simple restriction and interpolation strategies in order to obtain a hierarchy of coarse-level node distributions and the corresponding correction equations. Both techniques are kept as simple as possible in terms of code implementation, which is an important feature of meshless methods. The techniques are verified on 2D and 3D Poisson equations, defined on non-trivial domains, showing very high benefits in terms of both time consumption and work to convergence when comparing the present techniques to the most common solver approaches. These benefits make the RBF-FD approach competitive with standard grid-based approaches when the number of nodes is very high, allowing large size problems to be tackled by the RBF-FD method.

COBISS.SI-ID: 16607003