Isogeometric enriched field approximations

Boundaries with specified behavior, phase boundaries, crack surfaces or singular points are, geometrically speaking, lower-dimensional features relative to two- or three-dimensional geometrical domains. Often, the distinguishing characteristics of the behavior at these features are known a priori and may be exploited to enrich isogeometric models. Explicit geometrical representations possess parametrically computable tangents, normals and curvature, while in implicit strategies, the geometric “exactness” of enriching lower-dimensional features is not exploited or retrieved only in the limit of mesh refinement. In the present work, CAD-inspired hierarchical partition of unity field compositions are extended to modeling explicitly defined enrichments within the isogeometric framework. The base approximations are “enriched” isogeometrically on parametrically defined lower-dimensional geometrical features of the base entity and by constructing distance fields from them. The efficiency and robustness of distance computations is significantly improved by composing monotonic distance measures, defined piecewise on the enriching geometric entity, using R-Functions. The procedure allows both the behavioral approximation as well as the material description to be enriched enabling the modeling of material damage (or, alternatively, local stiffening). Further, the enrichments may ensure known function value or its derivative. Function value enrichments are demonstrated to model Dirichlet boundary conditions and propagating cracks. The derivative enrichments are used to model Neumann boundary conditions as well as strain jumps across material interfaces. The material enrichments are demonstrated through the use of a cohesive damage law to model arbitrary crack initiation and propagation within the domain.