Method of fundamental solutions without fictitious boundary for three dimensional elasticity problems based on force-balance desingularization

Abstract

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.