IsoGeometric Analysis (IGA) was proposed by Hughes et al. (2005) in order to overcome the various bottle necks in Computer Aided Design and analysis. IGA has shown considerable improvement in computational efficiency compared to standard Finite Element Methods (FEM) for problems with smooth solution. But IGA when employed for fracture mechanics problems, the smooth NURBS bases in IGA cannot capture the discontinuities and singularities in solution. To overcome these problems, the standard NURBS bases in IGA are usually enriched with functions containing solution nature and the method is termed as Generalized/eXtended IsoGeometric Analysis (GIGA/XIGA). But enriching the bases functions in IGA comes with extra problems of blending elements and ill conditioning. The presence of blending elements reduce the overall convergence rates and ill conditioning makes the inversion of stiffness matrices cost intensive and prone to errors.

Babuska and Banerjee (2012) proposed Stable Generalized Finite Element Method (SGFEM) to solve the problems of blending elements and ill conditioning in Generalized Finite Element Method (GFEM). The main idea of SGFEM is to achieve optimal convergence rates and at the same time maintaining the conditioning of system matrices close to the conditioning of standard FEM. In SGFEM, the enrichment functions are modified by subtracting linear interpolant of the enrichment function from the enrichment function itself. However the implementation of SGFEM in XIGA is not straightforward due to the higher order continuity and non-interpolatory nature of NURBS bases. Hence in the present thesis three variants of SGIGA viz., SGIGA-LS, SGIGA-LS-C0 and SGIGA-L-C0 are proposed based on basis functions employed for iii stabilizing the enrichment function and employed for problems with various discontinuities and singularities. The results obtained from proposed SGIGA methods are compared with the results obtained from GIGA in terms of accuracy and ill conditioning of system matrices.

SGIGA is employed for linear elastic fracture mechanics problems, and how SGIGA reduces the problems of blending elements and ill conditioning when enriched with multiple enrichment functions is studied. How the reduction in blending errors effects the errors in evaluating stress intensity factors is also studied. Various orthogonalization procedures available in literature to overcome the problem of ill conditioning have limitations in terms of computational cost or applicability to some class of problems. In order to overcome the issues in existing methods, Orthogonalized Generalized IsoGeometric Analysis (OGIGA) is proposed based on novel way of orthogonalization. Also in order to quantify the linear dependency between the basis functions, a novel way of finding smallest angle among all the bases is also proposed. The proposed OGIGA method is employed for several bench mark problems in fracture mechanics like plate with a center crack, bi-material interface crack and inclined crack problems. In addition to that, various performance studies are conducted in order to optimize the performance of proposed OGIGA method.