Available online at www.sciencedirect.com ScienceDirect Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 www.elsevier.com/locate/cma Algebraic distance estimations for enriched isogeometric analysis K. Upreti, T. Song, A. Tambat, G. Subbarayan ∗ School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, United States Received 30 January 2014; received in revised form 10 July 2014; accepted 10 July 2014 Available online 19 July 2014 Abstract 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. c ⃝ 2014 Elsevier B.V. All rights reserved. Keywords: Algebraic geometry; NURBS; Implicitization; R-functions; Distance field; Enrichments 1. Introduction In general, in moving boundary problems, the motion of complex boundaries need to be tracked within the domain. Examples of such problems arise in many fields, including fluid mechanics, solid mechanics, optimal design, computer vision and image processing. Commonly, an Eulerian framework in which the geometry of the boundaries is inferred implicitly as the zero level set of an evolving field is used to numerically solve these class of problems [1]. Since the level sets implicitize both the geometry of the boundary as well as the distance from the boundary, in other computa- tional procedures for moving boundary problems, there is a common need for explicitly calculated distance fields. In general, in these problems, distance from the boundary or interface serves as a measure of influence of the behavior on the boundary at a point in the underlying domain. Computational procedures relying on distance fields are ∗ Corresponding author. Tel.: +1 765 494 9770; fax: +1 765 494 9770. E-mail address: ganeshs@purdue.edu (G. Subbarayan). http://dx.doi.org/10.1016/j.cma.2014.07.012 0045-7825/ c ⃝ 2014 Elsevier B.V. All rights reserved.