The natural radial element method

Abstract

In this work an innovative numerical approach is proposed, which combines the simplicity of low-order finite elements connectivity with the geometric flexibility of meshless methods. The natural neighbour concept is applied to enforce the nodal connectivity. Resorting to the Delaunay triangulation a background integration mesh is constructed, completely dependent on the nodal mesh. The nodal connectivity is imposed through nodal sets with reduce size, reducing significantly the test function construction cost. The interpolations functions, constructed using Euclidian norms, are easily obtained. To prove the good behaviour of the proposed interpolation function several data-fitting examples and first-order partial differential equations are solved. The proposed numerical method is also extended to the elastostatic analysis, where classic solid mechanics benchmark examples are solved.

Crack path prediction using the natural neighbour radial point interpolation method

“…1(a) the obtained Voronoï cells are represented in dashed lines. In the literature, it is possible to find several works addressing properly the Voronoï construction procedure [18,50].…”

“…Combining the natural neighbour concept and the radial point interpolator technique, it was possible to enhance the RPIM and develop the Natural Neighbour Radial Point Interpolation Method (NNRPIM) [15][16][17]. More recently, another highly efficient radial point interpolation meshless method was developed and applied to computational mechanics, the Natural Radial Element Method (NREM) [18][19][20].…”

Meshless Methods: The Future of Computational Biomechanical Simulation

“…In Figure 1c is represented the nodal discretization of a half human head. Truly meshless methods [5][6][7][8][9][10][11] allow to acquire the nodal cloud directly from the CAT scan or the MRI by considering the pixels (or voxels) position and then obtain the nodal connectivity, the integration points and the shape functions using only the nodal spatial information [5]. Using the grey tones of medical images, truly meshless methods are even capable of recognizing distinct biomaterial and then affecting directly to the nodes the corresponding material properties, Figure 1c.…”

The Mandible Remodeling Induced by Dental Implants: A Meshless Approach

“…Generally, meshless methods can be divided in approximation meshless methods [34][35][36][37][38][39] and interpolation meshless methods. [40][41][42][43][44][45][46][47] The major advantage of using interpolator meshless methods is the possibility to impose directly the essential and natural boundary conditions, since the constructed test functions possess the delta Kronecker property, ' i ðx j Þ ¼ ij .…”