Three-dimensional Coons macroelements in Laplace and acoustic problems

This paper introduces a new global functional set for the FEM solution of three-dimensional boundary-value problems. The main idea is to construct large isoparametric finite elements based on the interpolation formula, which was developed in 1960s by S.A. Coons for the numerical representation of arbitrary solid CAD regions bounded by six curvilinear surfaces. In this way, besides the geometry, Coons interpolation formula is used here for the global interpolation of the unknown potential within the whole solid region (problem area), a procedure that leads to large elements, called “macroelements”. For adequately smooth regions, the degrees of freedom appear only at the 12 boundary edges of the macroelement and can be used in the solution of both static (Laplace) and eigenvalue (acoustic) problems. The proposed approach is sustained by five numerical results where it is successfully compared with conventional finite elements and exact analytical solutions.