Civil Engineering Reference
In-Depth Information
further divided into six tetrahedra to form an initial coarse tetrahedral mesh. The tetrahe-
dral mesh is then refined mainly based on the boundary characteristics and the geometri-
cal features of the physical object, though other mesh requirements can also be taken into
consideration in the refinement. Nodes on the boundary of the Meccano approximation
are mapped onto the boundary of the solid object by means of the admissible mapping
defined earlier. By this mapping, the refined tetrahedral mesh will undergo a severe defor-
mation such that many tetrahedral elements will be flattened, elongated or even inverted.
Interior nodes have to be relocated to untangle the invalid elements and to open up the
flattened or elongated elements. The final mesh has to be optimised by refinement and
de-refinement along with some other smoothing procedures involving geometrical and
topological operations.
The major difficulty with the Meccano method is the setting up of the admissible map-
ping for which manual intervention is necessary for objects with general geometry and/
or specific requirements. Poor quality elements will be generated by forcing the Meccano
approximation onto the physical boundary of the solid object. Inverted elements have
to be untangled, which is not always possible, and even though this could be done in a
majority of cases, inevitably, elements of lower quality have to be accepted. The method
relies heavily on the ability of the refinement and de-refinement techniques developed and
the available optimisation procedures to improve the overall quality of the final mesh.
The other restriction is that the method is limited to objects with the same topology of
a cube in the current implementation, though objects with more complicated topological
structures can be envisaged by the introduction of interior boundary surfaces; however,
the mapping process has not been formally studied, and how elements will be deformed
and corrected is still an open question.
With the introduction of a generic refinement algorithm in Section 8.4.3, high-quality
refinement meshes can be readily generated in compliance with a specified node spacing
function. Accordingly, a tetrahedral mesh can be generated with appropriate refinement
following the boundary of the given solid object. The boundary of the object can be recov-
ered simply by projecting nodes onto the boundary surface. Compared to the Meccano
method, by projecting nodes close to object boundary, elements will suffer much less
distortion, and meshes of much higher quality can be resulted. Based on the idea of refine-
ment over object boundaries, Sullivan et al. (1997) employed the DT to mesh volumes of
multiple material types, and Yu et al. (2008a,b) applied the mesh refinement and projec-
tion techniques for MG and adaptation in modelling molecular shapes and biomedical
parts. By means of the Octree refinement techniques, Su et al. (2004), Qian et al. (2009)
and Zhang et al. (2010) presented an MG scheme to produce hex and tetrahedral meshes
over a domain with multiple material types. Using a Delaunay-based surface meshing
algorithm, Oudot et al. (2010) generated tetrahedral meshes over volume bounded by
curved surfaces. Qian and Zhang (2012) proposed a scheme for the generation of unstruc-
tured hex meshes for an object defined by CAD B-Rep surfaces. Following a bottom-up
algorithm, Gosselin and Ollivier-Gooch (2011) triangulated volume bounded by piece-
wise smooth surfaces based on a controlled Delaunay point insertion on the boundary
surfaces. Sazonov and Nithiarasu (2012) generated surface and volume meshes for bio-
medical objects by means of the marching cube method.
8.5.2 MG algorithm by refinement and boundary fitting
As tetrahedral mesh is more adaptive to curved surfaces, MG based on the refinement of a
tetrahedral mesh rather than a hex mesh for objects bounded by analytical surfaces will be
presented. The essential steps are given as follows.
