ISOGEOMETRIC BOUNDARY ELEMENT ANALYSIS APPLIED TO ELASTIC PROBLEMS

Resumo

The main idea of this work is to solve elastic problems using Isogeometric Boundary Element
Formulation. A standard BEM with quadratic elements is also used in order to compare the efficiency of
both methods. In isogeometric method, instead of using polynomial shape functions, both geometry and
analysis use non-uniform rational B-splines (NURBS). NURBS are widely used for geometric modelling
in CAD software and, due to this, makes the discretization of the geometry unnecessary. One obvious
advantage of using this type of B-splines is that it can perfectly describe complex shapes, making results
more accurate. The most important feature, however, is the decrease in the amount of user’s work,
because the most time-consuming step – mesh generation – is reduced or even eliminated. In order to
easy implementation in existing boundary element codes, NURBS are transformed into Bezier curves ́
(Bezier decomposition). So, each B ́ ezier curve can be viewed as a boundary element in a conventional ́
boundary element implementation.
It is worth mentioning that displacement and tractions have their values solved at the control points

and NURBS curves do not necessarily touch them. For the definition of collocation points, Gauss-
Legendre collocation points are used in this study. Therefore, a transformation matrix, which uses basis

functions for relating values at control and at collocation points, is needed. The equation for isogeometric
BEM is defined in terms of the control points and, after applying the transformation, can be solved as the
standard BEM. Lastly, numerical and analytical solutions are compared in order to validate the method.