Geometric Modeling for Numerical Simulation
This project will address the challenging geometric problems which arise in the connection with Isogeometric Analysis (IGA, see ). IGA is a very promising concept which will potentially lead to major improvements of the product design process in industry. However, it was soon realized that the models produced by the current CAD technology are not directly usable for IGA.
First, CAD models provide only a description of the domain boundary, while the numerical simulation often requires a parametrization of the volume. Consequently, computational techniques for creating NURBS volume parametrizations are needed. The existing techniques for NURBS volume parametrizations are mainly restricted to simple objects, where they were mainly used for shape editing via free-form volume deformations.
Second, the spline spaces used by CAD models do not provide the possibility of local re-finement, which is essential for an adaptive numerical simulation. While algorithms for h- and p-refinement are readily available (called knot insertion and degree elevation, respectively), the construction of locally refined spline spaces requires additional research and is currently being studied by several researchers.
Third, the technology is not directly applicable to situations where the geometry is not directly available in a CAD-type format, e.g., in medical applications.
The planned work in this subproject is organized in four activities.
- Activity 7.1 will investigate advanced techniques for automatically constructing single-patch parametrizations of three-dimensional domains by non-uniform rational B-splines (NURBS) and box splines. This is to provide techniques that are applicable to a wide variety of geometries, ranging from medical data to industrial CAD models.
- Activity 7.2 will be devoted to coarse domain segmentation and generalized segmentation. By decomposing general domains into (possibly overlapping) subdomains, each of which can be parametrized using the techniques from Activity 7.1, we will extend the applicability of the isogeometric framework.
- Activity 7.3 will study quality measures for domain parametrizations and methods for optimizing parametrizations with respect to these measures. Here, we are interested in quality measures that capture the suitability of the parametrization and the test functions for numerical simulation, such as condition numbers.
- Activity 7.4 will explore new hierarchical spline spaces for IGA with local adaptivity, based on generalizations of tensor-product splines and box splines.
 T.J.R. Hughes, J.A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comp. Meth. in Appl. Mech. Engrg., 194:4135-4195, 2005. doi:10.1016/j.cma.2004.10.008.