Isogeometric Finite Element Analysis based on Bézier Extraction of NURBS and T-Splines

Abstract

Data transmission between finite element analysis (FEA) and computer-aided design (CAD) is ahuge bottle-neck today. Therefore, isogeometric analysis has been introduced with aim to merge these fields. While FEA utilizes Lagrange polynomials to approximate both the geometry and the solution field, isogeometric analysis employs non-uniform rational B-splines (NURBS) from CAD technology to this objective. Isogeometric analysis will therefore have the advantage in nogeometric error in the sense that the model is exact.

T-splines are a recently introduced generalization of NURBS which allow local refinement, handling complex geometry in a subtle way with fewer degrees of freedom. Increasing the order of the elements in isogeometric analysis is easy and gives higher continuous basis functions than FEA, while also maintaining few degrees of freedom.

In conventional isogeometric analysis the basis functions are not confined to one single element, but span a global domain, complicating implementation. The Bézier extraction operator decomposes a set of NURBS or T-spline basis functions to linear combinations of Bernstein polynomials. These polynomials bear a close resemblance to the Lagrange polynomials as they allow for generation of C0 continuous Bézier elements. A local data structure for isogeometric analysis close to traditional FEA is provided.

Codes are developed to illustrate conventional isogeometric data structures as well as structures based on Bézier extraction of NURBS. Modifications are made to the latter to be able to run analysis of T-splines modelled in the CAD system Rhino, and numerical studies are performed.Generally, NURBS elements display the same convergence rate as Lagrange elements of equal order, but higher accuracy. The reasons are a smooth solution field and exact geometrical representation.