Isogeometric method for Partial Differential Equations

Isogeometric (IG) analysis is a new approach to find the numerical solution of partial differential equations (PDE) arising from various fields of science and engineering. This shares many features with classical finite element analysis (FEA) and some features with meshless methods, and can be safely termed as a generalization of FEA. Based on non-uniform rational B-Splines (NURBS), a standard technology employed in computer aided design (CAD) systems, this exactly represents common engineering shapes such as circles, cylinders, spheres, ellipsoids, etc. and provides more accurate modeling of complex geometries. In this approach, the exact geometry is fixed at the coarsest level of discretization itself. For higher accuracy the mesh and the corresponding basis are refined and order elevated while maintaining the original exact geometry. Apart from classical h- and p- refinement analogues of FEA it offers a new k- refinement procedure that increases the smoothness of element functions almost everywhere. Thus, it produces more accurate and robust results as compared with standard p-refinement. This methodology offers numerous research opportunities. During the course of this project it is proposed to work on the following important topics: (1) developing triangular (and tetrahedral) IG elements, and hierarchical IG bases, (2) local refinement and associated unstructured meshing techniques, and IG mesh generators, and (3) development of robust optimal order iterative solvers for the resulting discrete systems.

Every consumer wishes to buy a product which gives the best return of the investment. Moreover, it is also expected that the purchased product has certain warranty/guarantee. These are also the principle objectives addressed within the scope of this scientific project. This project was aimed at developing accurate, fast and reliable simulation methods. In many engineering simulations, for example, modelling of a car, one encounters a smooth surface in certain parts and complex geometry at other parts. For geometric modelling of such objects, computer aided geometric design has been a very active research field for almost 5 decades. On the other hand, numerical simulations were developed in different directions. Because of different approximation methods, the interplay of these two directions, though indispensable for industrial applications, was an expensive interplay.Moreover, as was observed in many studies, a poor approximation of object's geometry adversely affected the accuracy of the numerical approximation. To overcome this situation, in 2005 Prof. Hughes and co-workers proposed to merge these two research directions. Studies conducted in many fields so far have clearly shown the superior properties of this approach.One of the main tasks achieved during this project work is the development of robust and fast solvers for large matrix system (algebraic equations) resulting from this new discretization method (representing the infinite dimensional problem on finite points of interest). Broadly speaking, the developed methods require the amount of work proportional to the problem size, whereas a nave approach could cost much more (approximately proportional to 2 to 3 powers of the problem size).Another major development from this project work is the error estimation method, which provides a sharp estimate and a guarantee on the error in the numerical solution. There have been some major industrial accidents in past, some of which could have been avoided if such an estimates were developed and used at that time.