Adaptive refinement algorithm for the RBF-FD method in solid mechanics

This work proposes and evaluates refinement strategies for the radial basis function-generated finite difference method (RBF-FD) which are investigated using solid mechanics problems. The RBF-FD formulation is based on a combination of polyharmonic splines (PHS) with polynomials. We focus on structured point distributions and introduce a method that enables their application to arbitrary geometries. In addition, we present an adaptive refinement technique based on hierarchical domain decomposition, which enables simple control of local point density. The adaptive refinement process is guided by two error estimators, which are both based on the comparison of solutions obtained by varying the order of the polynomial extension. While the main focus of this paper lies on the proposed refinement strategy, we also investigate key factors affecting the accuracy of our simulations, for example the role of ghost nodes in enforcing Neumann-type boundary conditions. The results demonstrate both the benefits and challenges of applying RBF-FD to solid mechanics problems. In particular, we identify difficulties in handling Neumann-type boundary conditions near singularities, highlighting the need for careful boundary treatment. This is not related to the refinement strategy itself, but might be of interest, when simulating such problems with the RBF-FD method. Overall, this study illustrates the effectiveness of structured, adaptively refined RBF-FD discretizations for high-accuracy simulations in computational solid mechanics.