Keywords: mesh generation, extrusion, sweeping, least-squares approximation.

Algorithms for the creation of extruded meshes are generally considered very attractive. There are several reasons for that. First of all, prismatic elements such as bricks and wedges, produced by the extrusion methods, are quite popular in the engineering analysis community because of their favourable features especially from the computational point of view. However, a general purpose fully automatic algorithm capable of discretizing complex geometrical domains of arbitrary topology into a valid high quality hexahedral mesh is still not available. Therefore, much attention has been focused on the development of decomposition methods. These methods are based on splitting the domain volume into simple parts that can be effectively discretized by specialized algorithms. One of them is the extrusion method, which has the potential to handle quite general shapes often referred to as generalized cylinders.

The most simple one-to-one extruded shape is formed by sweeping a trimmed surface along a control curve. This implies that the topology of such a volume is defined by a pair of cap surfaces on its top and bottom sides which are linked together by a set of lateral surfaces. In this work, a quite complex class of extruded volumes, operating with cap surfaces that are identical topologically but not geometrically is considered. This implies several significant features. First of all, the cap surfaces may be of different area, shape and curvature. Moreover, a single common control curve describing the extrusion does not exist. The sweep trajectory is uniquely defined only for those boundary vertices of the cap surfaces that correspond to appropriate sides (in the sweeping direction) of lateral surfaces.

The standard procedure to generate an extruded semi-structured mesh consists of the following steps: 1) identify the source and target cap surfaces, 2) generate an appropriate unstructured surface mesh over the source surface, 3) generate a valid surface mesh over the target surface that is topologically identical with the source mesh, 4) generate a valid structured quadrilateral mesh over lateral surfaces compatible with the cap surfaces, 5) generate a valid volume mesh inside the extruded volume. With respect to the sweeping methodology, the most specific algorithms are related to steps 1, 3, and 5. The mesh generation in steps 2 and 4 can be performed by ordinary structured and unstructured surface mesh generation techniques. However it is important to mention that the quality of source and lateral surface meshes has usually a great impact on the quality of the resulting volume mesh.

For steps 2 and 4, a mapping method based on Bernstein polynomials is adopted in this study. The mapping between the parametric spaces of source and target surfaces is approximated by a Bezier surface of appropriate order. Points of the control polygon of that surface are determined by means of a least-squares fit between the parametric coordinates of corresponding boundary nodes on the source and target surfaces. Since the intermediate layers of nodes are not described by a surface parameterization, this approach cannot be directly applied to the generation of volume interior nodes. Instead, the extruded volume is approximated by a Bezier volume of appropriate order. Its control polygon points are again determined by a least-squares fit but this time applied to all the boundary nodes of the extruded volume. Because neither from the mappings defined by the Bezier surface and by the Bezier volume is exact, the surface mesh on the target surface as well as the final volume mesh are subjected to appropriate smoothing.

The capabilities of the proposed and implemented sweeping algorithm are presented using a number of examples. One of them, a semi-structured mesh generated over a general extruded volume, is depicted in Figure 1.

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