Keywords: finite element method, mesh generation, hexahedral elements, sweep, node projection, affine mapping.

Nowadays, mesh generation is the most time consuming part of numerical simulations in industry. Although a wide range of fast and robust algorithms are available to generate unstructured tetrahedral meshes [1], fully automatic unstructured hexahedral mesh generation algorithms are still not available. Therefore, special attention has been focused on existing algorithms that decompose the entire geometry into several simpler pieces that can be considered as the union of one-to-one extrusion volumes. Sweeping is one of the most robust and efficient algorithms to mesh these simpler volumes with hexahedral elements. Several algorithms have been devised to generate hexahedral meshes by projecting the cap surfaces along the sweep path [2,3,4]. In all of them the crucial step is the placement of the inner nodes. From the computational point of view, sweep methods based on a least-squares approximation of an affine mapping are the faster alternative to compute these projections [5]. Two main functionals have been introduced to perform the least-squares approximation [2,6]. Although the computational efficiency of these projection methods (both in terms of cpu time and memory) they present several drawbacks. For instance, the minimization of these functionals may lead to a set of normal equations with a singular system matrix for very usual geometrical configurations. In addition, the obtained mesh may present well known undesired effects such as flattening and skewness, see [6] for details.

In order to overcome these shortcomings in reference [6] we have introduced a new functional that depends on two vector parameters that can be selected by the user. However, only a feasible selection of these parameters, based on the experience of the authors, was provided. In this paper we first prove the relationship between the optimal solution of the classical functional and the optimal solution of the functional proposed by the authors. Based on this relationship we analyze two simple sweep geometries and we report two new undesired effects that can be observed on hexahedral meshes that are obtained by minimizing the previous functionals. We denote the first one as flipping. It appears when an inner loop is curved towards one direction of the sweep path and it is projected to another loop that is curved on the opposite direction to the first one. In this case tangled and distorted hexahedral elements may be generated. The second one is called offset scaling. It appears when a mesh delimited by a non-planar loop of nodes is projected over a non-planar loop of nodes with different thickness. In this case, the inner part of the projected mesh, i.e. the offset data, may be scaled along the sweep path leading to an unacceptable hexahedral mesh.

We argue that these four effects are due to the inability of functionals to preserve offset data. In order to overcome these drawbacks, we propose a definition of a measure of the area enclosed by a given loop of nodes and a measure of the normal vector to this loop. We denote these vectors as pseudo-area and pseudo-normal, respectively. Note that, in sweeping applications, the loops of nodes that define the inner layers are not supported by an underlying surface. Hence, the area defined by a loop of nodes or the normal to a loop of nodes cannot be defined in the classical geometrical sense. In addition, we also prove several properties related to these vectors. Taking into account the relationship, that has been proved, between the classical functional and the new functional, and based on the definition of the pseudo-normal vector, we detail a new algorithm that automatically selects the functional parameters. These parameters are selected in order to preserve the offset data. It is important to point out that in our algorithm the geometrical cases that lead to a set of normal equations with a singular system matrix are identified from the singular value decomposition (SVD) of the optimal solution of the classical functional. Moreover, to increase the computational efficiency of the proposed algorithm, the minimization of the new functional adequately reuses the optimal solution of the classical functional and its singular values. Finally, we present two simple examples that show the robustness and the reliability of the proposed algorithm.

1

J.F. Thompson, B. Soni , N. Weatherill "Handbook of Grid Generation". CRC Press, 1999.

2

P.M. Knupp, "Next-generation sweep tool: a method for generating all-hex meshes on two-and-one-half dimensional geometries", In: 7th International Meshing Roundtable 505-513, 1998.

3

M.L. Staten, S.A. Canan, S. J. Owen, "BMSweep: Locating interior nodes during sweeping, Engineering with Computers 15, 212-218, 1999. doi:10.1007/s003660050016

4

D.R. White, S. Saigal, S.J. Owen, "CCSweep: automatic decomposition of multi-sweep volumes, Engineering with Computers 20, 222-236, 2004. doi:10.1007/s00366-004-0290-6

5

M.A. Scott, M.A. Earp, S. E. Benzley, M.B. Stephenson, "Adaptive Sweeping Techniques, In: 14th International Meshing Roundtable 417-432, 2004.

6

X. Roca, J. Sarrate, A. Huerta, "A new least-squares approximation of affine mappings for sweep algorithms, In: 14th International Meshing Roundtable 433-448, 2005.

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