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.