Boundary Mesh-Free Model for potential problems

Resumo

Within the engineering field, many problems are described by the Laplace Equation, which does not
have an analytical solution for complex geometries and boundary conditions. Therefore, numerical methods are
highly important for obtaining approximate solutions for problems of this nature. The most utilized numerical
methods nowadays involve a mesh for the discretization step, which lead to a high computational cost. This work
proposes a Boundary Mesh-Free Model (BMFM) that combines the Boundary Integral Equations and the shape
functions generated by the Moving Least Square (MLS) method. This way, the presented method minimizes
the need of discretization and eliminates the need of a mesh. The proposed formulation was then applied to a
benchmark potential problem governed by the Laplace equation. The results obtained using the BMFM had a
small error when compared to the analytical ones, showing how promising the method is.