Simplified Integration of Triangular NURBS Finite Elements

Abstract

In CAD packages, solid objects are represented by a collection of NURBS bounding surfaces and curves. Traditional finite element analysis employs curved elements of limited geometric capability, such as quadratic finite elements whose boundaries are represented by quadratic curves and surfaces that cannot reproduce exactly real life geometries. This complicates the adaptive analysis in curved geometries where there is a need to do a lot of queries to the geometry model to refine the mesh. To overcome this problem, several formulations that consider the exact geometry in the finite element formulation, such as the isogeometric formulation have appeared. In general, these formulations are not simple and requires complex integration rules to obtain the finite element matrices. An alternative representation based on triangular NURBS finite elements, that are triangular elements whose boundaries are NURBS curves is used in this paper. A simplified analytical integration on these elements for planar stress analysis that does not require numerical integration is presented. It and can be easily extended to 3-D nonlinear problems, including contact and plasticity.