Solving Elliptical Equations in 3D by Means of an Adaptive Refinement in Generalized Finite Differences

Abstract

We apply a 3D adaptive refinement procedure using meshless generalized finite difference method for solving elliptic partial differential equations. This adaptive refinement, based on an octree structure, allows adding nodes in a regular way in order to obtain smooth transitions with different nodal densities in the model. For this purpose, we define an error indicator as stop condition of the refinement, a criterion for choosing nodes with the highest errors, and a limit for the number of nodes to be added in each adaptive stage. This kind of equations often appears in engineering problems such as simulation of heat conduction, electrical potential, seepage through porous media, or irrotational flow of fluids. The numerical results show the high accuracy obtained.

Reduced-order strategy for meshless solution of plate bending problems with the generalized finite difference method

“…In the last two decades, the possibility of unstructured point clouds with the generalized version of the finite difference method (GFDM) achieved an important status, with possible applications to adaptive refinements in h and p versions, with some examples obtained by Benito et al (2002) and Gavete et al (2018), who explored the reduction of local errors by adding new degrees of freedoms to the existing cloud. An h-adaptive method using a frame decomposition approach with conventional finite differences was also proposed by Srinivasa (2006).…”

“…Nowadays, some of the most recent advances include: computational acoustics (Wei et al, 2015), the solution of nonlinear water waves using potential flow theory (Zhang et al, 2016 andFan et al, 2018), application to bidimensional shallow water equations (Li and Fan, 2017) and tridimensional adaptive cloud refinement (Gavete et al, 2018). As described by Zhang et al (2016), the GFDM is versatile enough for many engineering applications.…”