Abstract
As the intersection curve between two cylinders cannot be represented exactly using a rational parametrization, CAD-systems have to resort to approximations when representing pipe junctions.In the early days of CAD when curves were plotted on paper this did not represent a problem, as the plotted curve itself was inaccurate as the plotting process involved approximation of the intersection curve by straight lines or circle segments. When the intersection curve was used for production purposes (e.g., numerically controlled machining) there was no problem as long as the tolerances of the curve approximation were kept within production tolerances.When boundary structured (B-rep) based CAD with trimming was introduced from the end of the 1980s CAD models still were only used for design and production purposes so pipe joint approximations represented no large problem. B-rep based CAD allows that adjacent surface patches are not exactly matched; small gaps between surfaces are allowed provided they are within controlled tolerances. When such models are used in Finite Elements Analysis they have to go through a conversion process named gridding. In this process the gaps are removed, making the model watertight. Traditional Finite Elements are of low degree (typically degree 2 or lower), the cylinders and their intersection curves have consequently to be approximated. So also in this traditional analysis scenario the inaccurate pipe junctions of CAD poses no special problem. However, the introduction of NURBS based isogeometric analysis in 2005 by Tom Hughes once more put the pipe junction challenge on the agenda. As the general intersection curve between two surfaces do not have a rational parameterization, some approximation strategy has to be employed for modelling pipe junctions for NURBS based isogeometric analysis. An additional complication is that B-rep type trimming is not employed in isogeometric analysis, so the pipe junction has to be built from regular volumet
