Fast generation of variable density node distributions for mesh-free methods

Abstract

Problems with pronounced differences in intensity, such as appear in contact mechanics, where locally concentrated high stresses are present, are usually attacked with a spatially variable nodal distribution. In a meshless context, such distribution has to be quasi-smooth with minimal spacing requirements to produce satisfactory results. To that end, development of fast discretization procedures, which distribute the nodes according to given (non-constant) spacing function, has become of interest. We improve a recently published algorithm for fast 2D meshless discretizations, by lowering its time complexity from O(NS) to O(N log S) resulting in an algorithm that generates a million nodes per second. The proposed algorithm is independent of the dimensionality of the space, does not rely on a coordinate system and has provable minimal node spacing requirements. The original and new algorithms are compared in terms of node quality and execution time. The usability and robustness of the new algorithm is presented by solving PDE examples on irregular 3D domains with the RBF-FD method and by using it as a node generation algorithm in fully the automatic adaptive solver for linear elasticity.