Orthogonalized Generalized Iso-Geometric Analysis (OGIGA) and its applications to problems of fracture mechanics

Ill conditioning of system matrices is one of the drawbacks of Generalized/eXtended IsoGeometric Analysis (GIGA/XIGA), which results in increase in computational cost in the case of iterative solvers or erroneous results in the case of direct solvers. Linear dependency between the basis functions which can be measured in terms of smallest angle among all the bases is one of the attributes of ill conditioning in XIGA. In the present study, a simplified method to find the smallest angle among all the bases is introduced. Also to reduce the problem of ill conditioning, Orthogonalized Generalized IsoGeometric Analysis (OGIGA) is proposed in which the standard and the corresponding enriched basis functions are orthogonalized. In order to study the performance of present method, various fracture mechanics problems like 2-D homogenous cracks, bi-material interface cracks, inclined cracks and 3-D crack problems are chosen. The effect of crack position and the radius of enrichment zone on accuracy, scaled condition number and smallest angle among all the bases is studied. The results obtained from the proposed OGIGA method are compared with results obtained from standard XIGA and XIGA with enrichment function orthogonalization (OE-XIGA) (Agathos et al., 2019). In all the cases considered, the conditioning of system matrices is significantly improved by employing present methodology and it can be easily extended to wide variety of problems.