Isogeometric Analysis: Fundamentals and details of implementation. From first steps to two-dimensional non-linear problems
Isogeometric analysis (IGA) is a computational analysis technique that can serve as an alternative to the traditional finite element method (FEM) in approximating solutions to differential equations. IGA is not necessarily more efficient that traditional FEM, but because of its nature, can naturally handle a greater variety of complex geometries. IGA is based on the use of NURBS (non-uniform rational B-splines), mathematical descriptions of geometry which are the standard of representing geometry in computer aided design (CAD) modeling software. IGA therefore links the CAD world to the world of analysis. Traditional FEM was developed before NURBS, in the 1950s and therefore developed quite separately. This project focuses on the fundamentals and implementation of IGA for problems, including one-dimensional, two-dimensional scalar, two-dimensional vector-valued and simple non-linear problems. For each new problem, the underlying mathematics is developed and the implementation is discussed in detail. One of the major contributions of this project is considered to be the detail in which the implementation of the Neumann boundary condition is described. There is none of this level of detail in any of the available literature. All problems solved are demonstrative and was written in a modular way that is easy to read and understand. Furthermore, how to extract NURBS data from CAD software is discussed, which would prove useful for future problems with more complex geometry. While the work done in this project is not considered novel, the thoroughness in which the project was approached is hoped to be useful for future projects. From this project, the work can be expanded to more complex geometries, multi-patch problems with the help of CAD programs or more complex non-linear problems.
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