Isogeometric analysis, as a computational tool, has gained widespread recognition due to its superiority over traditional approaches. The goal of the isogeometric analysis (IGA) is to deal with a single geometric definition of an object in its design and analysis stages. Moreover, IGA provides a platform in which basis functions with higher degrees of continuity can be conveniently accessed as needed in the analysis stage. The classic IGA method makes use of non-uniform rational B-splines (NURBS) or B-splines as its core basis function. While the NURBS-based IGA possesses the general merits of the classic technique, it suffers from a major drawback: the topology of control points used to create NURBS surfaces must be rectangular. This poses the limitation that local refinements in any subdomain invariably lead to global refinements. This problem has motivated researchers, including this study’s authors, to seek out more versatile basis functions that can preserve the advantages of the classic ones while providing local refinement capabilities; notably, hierarchical B-splines (HB-splines), which have been used for surface reconstruction, have been found to be suitable candidates for this purpose. However, to be able to work as a core of IGA methods the basis functions must be linearly independent.

The present study is an attempt to develop a framework for computing a new family of linearly independent set of basis functions from a set of multi-level B-spline functions. In this process the hierarchical basis functions are first picked based on their local support ranges and the hierarchical level they belong to. The support of the chosen basis functions of different levels may overlap in certain regions, and since basis functions of lower levels can be represented by those of higher levels this can lead to the issue of local linear dependencies between hierarchical basis functions. In the final step, the common contributions of basis functions of different levels in the areas where this overlap happens are extracted using the existing relations between basis functions of different levels. The eXtracted Hierarchical B-spline (XHB) basis can then be used as an analysis-suitable basis.

Two popular structural problems have been chosen for the case studies. The first study investigated the bending, free vibration, and buckling behaviors of Carbon Nano-tube Reinforced Composite (CNTRCs) skew plates. The concept of functionally grading CNTRCs has been used. To account for imperfect bonding between Carbon Nano-tube (CNT) fibers and base matrix material, an estimated set of CNT efficiency parameters is utilized. The displacement field of the plate is based on Reddy’s higher-order shear deformation theory (HSDT). As opposed to the classic plate theory, or first-order shear deformation theory, HSDT can be used for all ranges of width-to-thickness ratios. Moreover, shear correction factor is not needed for HSDT as shear stresses at top and bottom edges are calculated as zero.

Due to the existence of stress singularities at the obtuse corners of skew plates, use of the richer interpolating functions is unavoidable. This requirement can be readily satisfied by the so-called k-refinement technique in the framework of IGA to produce the basis functions of a higher order of continuity. The study also makes use of the affine covariance property, meaning that any affine transformation applied to NURBS objects can be performed by directly applying it to the control polygon. Moreover, the imposition of boundary conditions along the oblique edges of the plate are specially treated by introducing transformation matrices for 7 degrees of freedom using geometrical considerations and relations in HSDT. Using the unique capability of the method, i.e. increasing the degree of continuity of the basis functions, it has been demonstrated that the XHB-based isogeometric method can be effectively used for the analysis of skew plates.

The second study considers the fracture of the brittle materials using the phase field model. The phase field model is essentially an approximation of the variational formulation for brittle fracture. The formulation states that at any moment of time, the total potential energy of a system should be minimized. This approach frees itself from the limitations of classic Griffith's theory of linear elastic fracture in that the assumption of pre-existing cracks in the problem domain is no longer needed. Moreover, the nucleation, evolution, and branching of the cracks are inherently handled by the global energy minimization. The phase field model, however, is known to be computationally expensive due to the fact that the global energy functional to be minimized is neither linear nor convex. Consequently, a high number of iterations attempts for solving the nonlinear system of equations is inevitable. Adaptive, locally refinable XHB-based isogeometric analysis has been shown to greatly increase the efficiency of computational efforts by allowing the mesh refinements only in regions where cracks are evolving. Overall, the XHB-based isogeometric analysis has been proven to be a reliable, flexible, and accurate method for solving a variety of mechanical problems.