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Daniel Burkhart Subdivision for Volumetric Finite Elements

176 Seiten, Dissertation Technische Universität Kaiserslautern (2011), Softcover, A5

In the engineering design-analysis process exists a bottleneck which results from different toolsets typically used for the geometric design on the one side and for numerical analysis on the other side. While the design is usually realized with software tools describing geometries by means of exact methods like B-splines, NURBS, or CSG, for numerical analysis approximate mesh representations are required.

A novel solution to open up this bottleneck is iso-geometric analysis. This method employs the exact geometries in the analysis applications directly, which means that the same basis functions are used to describe the geometry and to represent unknown field variables of subsequent simulations. Iso-geometric analysis has been successfully tested for B-splines, NURBS, and subdivision surfaces. However, none of these approaches allows one to solve 3-dimensional problems based on solids described by control meshes of arbitrary topology. To fill this gap we have developed two approaches for combining volumetric subdivision schemes and finite element analysis.

The first approach is based on an adaptive and feature-preserving subdivision scheme for unstructured tetrahedral meshes inspired by the meshes. Existing tetrahedral subdivision schemes do not support both, adaptive refine ment and preservation of sharp features, and have traditionally been driven by the need to generate smooth 3-dimensional deformations of solids. We propose a new refinement operator which consists of a split and a flip operation, thus, adaptive refinement is implic itly supported. This subdivision algorithm is motivated primarily by the need to generate high-quality adaptive tetrahedral meshes for finite element simulations. However, due to the used refinement operator it is not possible to define its underlying basis functions and it cannot be used in an iso-geometric framework. Standard Lagrangian shape functions must be used instead.

In a second approach, well known Catmull-Clark subdivision solids are used for iso geometric finite element analysis. Here, the boundary of the solid is modeled as a Catmull- Clark subdivision surface with optional corners and creases to support the design phase. For the simulation phase – where efficient integration over the elements is required – we propose a method similar to the standard subdivision surface evaluation technique, such that numerical quadrature can be used. This iso-geometric method is analyzed in detail and applied to several problems from mechanical engineering and fluid dynamics. Experiments show that our approach converges faster than methods based on tri-linear and tri-quadratic Lagrangian elements and that it is less susceptible to bad shaped elements. Furthermore, control meshes are not limited to be all-hexahedral, as Catmull-Clark subdivision can also process meshes consisting of tetrahedra and prism as well as mixed meshes.