Mat-1.3656 Numeerisen analyysin ja laskennallisen tieteen seminaari.

Ma 5.11. 2007

Lourenco Beirao, University of Milano

Isogeometric analysis with NURBS

In a large range of engineering applications, the bodies/domains of interest are often described usign CAD, which means using Non Uniform Rational B-Splines to define the problem geometry instead of piecewise polinomials. This leads to a series of difficulties when classical finite element methods are adopted, mainly the need of approximating the geometry and the computational time spent in designing and refining the element mesh.

A new methodology for solving applied problems in the realms of PDE's is presented. The idea is based on NURBS instead of piecewise polynomials, and on an isoparametric mapping which exactly describes the CAD computational domain. As a consequence, all the aforementioned difficulties are avoided. In the present talk, we first describe the method and make a brief introduction to NURBS functions. Afterwards, in the main part of the talk, we address the theoretical analysis of various schemes stemming from this idea. We treat the interpolation properties of NURBS, inverse estimates, stabilized methods and inf-sup stable methods. As a particular consequence, we can derive optimal error bounds when the NURBS isogeometric methodology is applied to classical linear elliptic problems as well as saddle point problems. Finally, extensive numerical tests are presented.