On the solution of 3D problems in physics: From the geometry definition in CAD to the solution by a meshless method

Abstract

This paper presents a simple meshfree approach, from the grid generation to the final solution, for the simulation of 3D problems geometrically defined by CAD. First, the domain grid for 3D problems is generated through a discrete searching algorithm. Non-Uniform Rational B-Splines (NURBS) are employed, just as a choice among similar tools, to define the boundaries through geometrical control points obtained by the CAD program. A predefined regular grid of nodes, embedding the whole geometry, is then trimmed to follow the constructed boundaries. The spatial solution is performed by construction of shape functions satisfying the governing differential equation, through using exponential basis functions (EBFs). A straightforward strategy is proposed for choosing appropriate EBFs via their shape-parameters making the method efficient for solution of 3D problems. Several 3D Laplace and Helmholtz problems are solved and the results are compared with those of commercial programs to show the efficiency of the method.