This project aims to combine numerical and symbolic methods of scientific computing in order to develop advanced methods for the design and construction, the analysis, and for the simulation and optimization of free-form shapes. The free-form curves and surfaces will mainly be described by their implicit representations, as the zero contours of bivariate and trivariate spline functions, i.e., by algebraic spline curves and surfaces. Traditionally, most techniques of Computer Aided Design (CAD) rely on parametric representations (Non-Uniform Rational B-Spline (NURBS) curves and surfaces) in order to describe free form shapes. The use of algebraic spline curves and surfaces, however, offers several computational advantages. For instance, the problem of fitting curves and surfaces to scattered data, which is a fundamental task in various applications (e.g., in reverse engineering of geometric objects), can be solved without generating auxiliary triangulations and parameterizations of the data. In addition, algorithms for basic geometric operations, including intersections with lines and foot point generation, are much easier to implement.
ObjectivesThis project consists of four parts:
- Part 1 will be devoted to computational methods for surface fitting. Here we will generalize a novel approach to curve fitting to surfaces, leading to reduction of the computational complexity. It is planned to explore several applications, including reverse engineering (cooperation with Holometric Technologies GmbH, Aalen, Germany), and optimal design problems (cooperation with Subproject 9). In order to obtain results which are compatible with existing CAD standards, such as NURBS curves and surfaces, exact and approximate techniques for parameterizing algebraic spline surfaces are needed.
- In Part 2 of the subproject, we plan to develop the existing parameterization techniques into numerically stable ones, by investigating methods for controlling the distribution of parametric speed and the polynomial degrees.
- Parts 3 and 4 will be devoted to the problem of FEM/BEM mesh generation, and to hierarchical description of free-form geometry. With the help of suitable implicitization techniques (exact or approximate ones) we plan to convert free-form curves and surfaces into implicit form.
See www.ag.jku.at for further information.
|Prof. Dr. Josef Schicho||5231 mail|
|Prof. Dr. Bert Jüttler||9178 mail|
|Mag. Szilvia Béla|
|DI Mario Kapl||7175 mail|