Physical Space is the already known Cartesian Space, where the real model is represented. Simple orthogonal shapes from Parameter Space are transformed into complex entities in the Physical Space. Physical coordinates of the Control Points play a major role in the aforementioned mapping, but an equally drastic role is set upon basis functions. In fact, for a given set of Control Points, only a single set of basis functions can lead to the same geometry. We will examine this thoroughly later.

Control Points can often be seen outside the model in Physical Space in contrast to FEM’s nodes which always belong to the mesh. It is one of the reasons NURBS and SPLine entities in general can accurately represent multiple types of geometries and the understanding of this peculiarity is one of the many challenges of Isogeometric Analysis.

The usual design process requires both exact geometrical representation of the model and accurate engineering results. Unfortunately, computational geometry and Finite Element Method are represented in different file types and are not compatible to each other. The engineer has to create a model for FEM solution and the designer a model for CAD representation. Moreover, the typical design process is not straightforward. The designer produces CAD designs, which are transformed into FEM-compatible forms of representation by the analyst. After generating mesh and obtaining results, the analyst informs the designer of the appropriate changes in geometry. The designer then gives the new CAD model to the engineer, who has to regenerate the FEM model and the new mesh. This cycle of CAD/CAE interaction can go on multiple times. In complex projects each design consists of numerous CAD entities combined together and the integration process is estimated to take up at around 80% of the whole design time. Researchers around the world have been trying to achieve automatic CAD/CAE integration.

The main problem is that CAD and FEM, even though they refer to the same object, evolved differently. This incompatibility drove researchers into separate roads, building a wall between the two methods. Thomas J.R. Hughes, a Professor of Aerospace Engineering and Engineering Mechanics at the University of Texas at Austin, came up with a different point of view. Instead of trying to connect present CAD and CAE formulas, we should reinvent them in ways that enable the integration. This is the scope of Isogeometric Analysis.

The basic idea is to exploit the functions used for the exact geometrical representation in order to describe the solution field. Isogeometric Analysis extends, in essence, isoparametric elements, but the process of altering geometry for the sake of the solution approximation is reversed.

This leads to the creation of a single model, capable both of exact representation and analysis. Designers and engineers will be working on the same platform. Time for meshing and entity translation will be eliminated in an instant. This direct contact between analysis and geometry means that every single change can be integrated as soon as it happens, with no risk of errors or timely tasks involved. Most importantly, the designer has to follow the engineer’s perspective and vice versa; the modern designer has to learn how to help the engineer and the modern engineer has to learn the methods the designer is using.

Isogeometric Analysis brings together two very different technologies, combining their best points to one. This leads to a better adaptation both from engineers and designers. In order to understand and improve Isogeometric Analysis, it simply needs to improve its counterparts. Finite element and computational geometry codes need not change drastically. This makes the new technology even more attractive. Understanding the basics of an innovation and implementation in the daily routine is usually a difficult and time-consuming task.

There are many geometrical forms of representation suitable for analysis, such as NURBS, T-SPLines, Polycube SPLines and Subdivision surfaces. Each entity has its own advantages and drawbacks, but the variety provided ensures a vast number of alternatives to use, depending on the case. This ensures the generalization of Isogeometric Analysis method to even more complex geometries.

FEM’s shape functions are defined only in the interior of the element. Each element has 1 C^-1 continuity in the edges. IGA’s shape functions are not contained in one element. Most of the times, they are defined through many elements. This ensures a greater continuity and interconnectivity. This different approximation works better and leads to greater convergence than the classical methods.

Refinement by order elevation or knot insertion has always been important for computational geometry. Hierarchical adaptation has been developed for a vast number of entities. All these technologies can be exploited by IGA. Hierarchical structures can be easily developed, straight from the geometrical model. Meshing and refinement is also immediately accomplished.

Until recently, the majority of CAD software users had not realized that by designing a model, they simultaneously created its corresponding mesh of finite elements. This information, although redundant for designers devoted to computational geometry, is a revolutionary remark for the engineering community. Before Thomas J.R. Hughes’ idea, known as IsoGeometric Analysis (IGA), engineers used to create a new approximate mesh instead of taking advantage of the existing accurate one. The additional geometry error makes the process less accurate, though more timeconsuming. This observation seems now very obvious, but it took years of research until 2003, when Thomas J.R. Hughes and his research team succeeded to cut the Gordian Knot of CAD – CAE integration.

The main idea is the elimination of the node mesh in the analysis process. The role and properties of the node mesh are inherited by two separate meshes, obtained directly from the geometrical representation:

- The Control Point mesh, which defines geometry and the finite number of degrees of freedom that form the problem equation.
- The Knot mesh, which provides appropriate discretization for numerical integration and boundaries for Shape function influence in the model.

For the scope of this thesis, I have worked exclusively with Non-Uniform Rational B-SPLines (NURBS), as they are the most commonly used computational geometry technology. Despite the fact that quite more advanced SPLines have emerged, CAD industry still invests in NURBS. Since 1970, billions of dollars have been directed towards the outspread and evolution of NURBS, establishing them as a common tool for graphic representation around the globe. Both professionals and amateurs still use NURBS despite their disadvantages, such as difficulties in Patch connection and local Refinement. The reason for this is that NURBS are not only much more simple in their definition and use, but also able to represent with accuracy smooth curves and all conic sections.