chapter  5
14 Pages

Geometry Modeling

Formulation of any boundary-value problem starts with defining the geometry of a domain in which the differential equations are to be solved. The classical notion of a differential manifold with overlapping maps does not fit well into the logic of conforming finite elements. In our work, we have assumed that the domain of interest can always be partitioned into blocks that form an FE-like regular mesh. The concept is frequently referred to as the Mesh Based Geometry (MBG) description. No hanging nodes are allowed, and in the presented implementation we restrict ourselves to hexagonal blocks only. The concept is illustrated in Figure 5.1. Each of the blocks of the GMP (Geometry Modeling Package) manifold (as we will call it) is the image of the reference hexahedron, a unit cube, under a map x = xb(η). We shall denote the reference coordinates always with η = (η1, η2, η3). Recall that master element coordinates have been denoted with ξi , i = 1, 2, 3. The parametrizations define local curvilinear coordinates in each GMP block but, contrary to the classical notion of a differential manifold, the individual maps∗ do not overlap. More precisely, they overlap only on the block boundaries. The principal assumption about our notion of manifold is that the parametrizations must be compatible. The notion has already been discussed in the first volume. We will explain it precisely in the 3D setting in the following section. In Section 5.2 we recall the classical concept of transfinite interpolation of Gordon and Hall [80] for a hexahedron, and for a rectangle conforming to a surface. Preparation of geometry data by hand is time-consuming, and we have attempted to accelerate it by interfacing with existing mesh generators. In Section 5.3 we present an example of such an interface with Sandia’s CUBIT and the Ansys mesher ICEM CFD Hexa. Finally, in Section 5.4 we discuss mesh generation and compare the concepts of exact geometry and isoparametric finite elements. We conclude with remarks concerning a precise definition of the FE error in presence of geometry approximation.