The Whisker Weaving Algorithm: A Connectivity-Based Method for Constructing All-Hexahedral Finite Element Meshes

Abstract

SUMMARYThis paper introduces a new algorithm called whisker weaving for constructing unstructured, all-hexahedral finite element meshes. Whisker weaving is based on the Spatial Twist Continuum (STC), a global interpretation of the geometric dual of an all-hexahedral mesh. Whisker weaving begins with a closed, all-quadrilateral surface mesh bounding a solid geometry, then constructs hexahedral element connectivity advancing into the solid. The result of the whisker weaving algorithm is a complete representation of hex mesh connectivity only: Actual mesh node locations are determined afterwards.The basic step of whisker weaving is to form a hexahedral element by crossing or intersecting dual entities. This operation, combined with seaming or joining operations in dual space, is sufficient to mesh simple block problems. When meshing more complex geometries, certain other dual entities appear such as blind chords, merged sheets, and self-intersecting chords. Occasionally specific types of invalid connectivity arise. These are detected by a general method based on repeated STC edges. This leads into a strategy for resolving some cases of invalidities immediately.The whisker weaving implementation has so far been successful at generating meshes for simple block-type geometries and for some non-block geometries. Mesh sizes are currently limited to a few hundred elements. While the size and complexity of meshes generated by whisker weaving are currently limited, the algorithm shows promise for extension to much more general problems.

A characterization of the quadrilateral meshes of a surface which admit a compatible hexahedral mesh of the enclosed volume

“…Currently, Gasilov et al [6] is developing a practical algorithm for CUBIT, along the lines of the proof of Theorem 6, to reduce the problem of constructing a compatible hexahedral mesh for a topological sphere with n-handles to the problem of constructing a compatible hexahedral mesh for a topological sphere. The sphere will then be meshed using the Whisker Weaving algorithm [3], or perhaps one of the other techniques available in CUBIT, such mapping identifiable subregions[ 141. Whisker Weaving is based on the STC, and meshes a topological sphere by creating the arrangement of pseudo-manifolds in an advancing-front manner (in contrast to Section 5).…”

“…Of those that do, all either change the surface mesh in some way, or add non-hexahedral elements. For example, Plastering[l6], the current version of Whisker Weaving [3], and Algor's Hexagen coupled with Houdini[lS], all allow the user to chose between changing the surface mesh, and having all hexahedral elements. This paper shows that these caveats are not usually necessary, so I hope developers will be inspired to remove them from their codes.…”

“…This paper shows that these caveats are not usually necessary, so I hope developers will be inspired to remove them from their codes. In particular, Whisker Weaving [3] is a heuristic algorithm that attempts to create a valid STC, which is then dualized to a hexahedral mesh: The fix-up rules given in Section 6 should remove one of the few remaining difficulties for this algorithm.…”

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AbstractA popular three-dimensional mesh generation scheme is to start with a quadrilateral mesh of the surface of a volume, and then attempt to fill the interior of volume with hexahedra, so that the hexahedra touch the surface in exactly the given quadrilaterals [3]. Folklore has maintained that there are many quadrilateral meshes for which no such compatible hexahedral mesh exists.

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3D finite element meshing from imaging data

“…Plastering places elements on boundaries first and advances towards the center of the volume [10] [8]. Whisker weaving first construct the spatial twist continuum (STC) or dual of the hex mesh, then the hex elements can be fitted into the volume using the STC as a guide [59].…”