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Model Geometry and Topology

Similarly as in the sequential mesh generator (Section Model Geometry and Topology (Sequential Mesh Generation)), the model is used in a passive context only. It is described by a boundary representation and consists of the following model entities - vertices, curves, surfaces, and regions. This time, however, each model entity (except vertices) is based on a free-form representation in terms of tensor-product polynomial entities, which essentially restricts the model topology. In this approach, each region is bounded by six surfaces, boundary each of which is formed by four curves, each of which is in turn given by two vertices. The advantage of this model representation consists in the existence of a unique mapping between the parametric and real spaces of each model entity. This significantly contributes to the unified handling of individual model entities. On the other hand, the restriction on the model topology results in some reduction of modelling flexibility. To enhance the modelling capability, curves and surfaces are allowed to be degenerated into entity of a lower dimension. Also a simple entity-to-entity fixation concept is employed. Each model entity is keeping the list of entities which are bounding and sharing that entity. No further topological information is required for the description of a valid domain of considerable complexity.

In the presented work, the rational Bezier entities have been employed for the geometrical representation of free-form curves, surfaces, and regions. The rational Bezier curves and surfaces have been described in Section Model Geometry and Topology (Sequential Mesh Generation). The rational Bezier region has the form


where is the positional vector of a point in the region, are Bezier control points, are weights of Bezier control points, , , and stand for Bernstein polynomials, , , and denote parameters ranging from to , , , and are region degrees (orders are greater by ) in , , and  parametric directions, respectively, and stands for the rational Bernstein polynomial in form


If the control points are arranged in a matrix of type then the corner points correspond to model vertices, the edge points correspond to control polygons of model curves bounding the region, the face points correspond to control polygons of model surfaces bounding the region, and the remaining points form the control polygon of the region, affecting the profile of curvilinear coordinates , , and  inside the region.

The hierarchy of the topological data structure and geometrical representation of individual model entities used in the current implementation of the parallel mesh generator is schematically presented in Table 3.1. Note that the complexity of this topological data structure in terms of the number of individual model entities is considerably larger than the complexity of topological data structure described in Section Model Geometry and Topology (Sequential Mesh Generation), especially when 3D objects are considered.

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Daniel Rypl