Algebraic distance estimations for enriched isogeometric analysis

In problems with evolving boundaries, interfaces or cracks, blending functions are used to enrich the underlying domain with the known behavior on the enriching entity. The blending functions are typically dependent on the distance from the propagating boundaries. For boundaries defined by free form curves or surfaces, the distance fields have to be constructed numerically. This may require either a polytope approximation to the boundary and/or an iterative solution to determine the exact distance to the boundary. In this paper a purely algebraic, and computationally efficient technique is described for constructing distance measures from Non-Uniform Rational B-Splines (NURBS) boundaries that retain the geometric exactness of the boundaries while eliminating the need for iterative and non-robust distance calculation. The constructed distance measures are level sets of the implicitized constituent Bezier patches of the NURBS surfaces that are obtained purely algebraically. Since, in general, the implicitized functions extend beyond the parametric range of the generating Bezier patch, algorithmic procedures are developed to trim these global implicit functions to the boundaries of the Bezier patch. Boolean compositions are then carried out between adjoining Bezier patches to construct a composite distance field over the domain. The compositions rely on R-functions that are also algebraic in nature. The developed technique is demonstrated by constructing algebraic distance field for complex geometries and by solving a variety of examples culminating in the analysis of steady state heat conduction in a solid with arbitrary shaped three-dimensional cracks.