Quasi-Optimal Local Refinements for Isogeometric Analysis in Two and Three Dimensions

B-Spline and NURBS techniques have already been successfully used in Isogeometric Analysis which is a method for directly integrating CAD models in numerical simulations. Our purpose is to improve existing techniques to enhance the efficiency. First, we use local B-Spline subdivisions and knot insertions for the goal of achieving better accuracy in simulations where we concentrate on two and three dimensions. Our main emphasis is to keep the curved geometry describing the physical CAD domain intact during the whole simulation process. In order to avoid unnecessary global refinements, grids are allowed to be non-conforming. The treatment of nonmatching grids is done with the help of the interior penalty methods. Only local refinements are required during the adaptivity. To achieve that, an a-posteriori error indicator is introduced in order to dynamically evaluate the errors. That is, we use spline error gauge with the help of the de Boor-Fix functional. On the other hand, we allow mesh coarsenings at regions where a sparse mesh density is sufficient to achieve a prescribed accuracy. To obtain an optimal mesh, some method is described to choose the types of refinement which are likely to reduce the error most. That is done by accurately determining the bases of the enrichment spaces using non-uniform B-splines enhanced with discrete B-splines. That is, the space of approximation is hierarchically decomposed into a coarse space and an enrichment space. Finally, we report on some practical results from our implementations. Some adaptive grid refinements in 2D and 3D from problems such as internal layers are reported. Besides, we briefly describe the problems to encounter when handling real CAD models for IGA simulations. We address the problem of decomposing a CAD object into parametrized curved hexahedral blocks which can be subsequently used in mesh-free simulations. Some problems and extensions related to Boundary Element Method (BEM) which is treated on CAD or molecular surfaces are equally discussed.