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.
