Civil Engineering Reference
In-Depth Information
Auxiliary meshing techniques
No algorithm is perfect, and there are always alternatives, complements and supplements.
There are many techniques that are not directly applied to generate finite elements but are
extremely useful in serving various aspects to facilitate finite element (FE) mesh generation
(MG). The surface verification and preparation presented in Section 8.1 aims at analysing
whether a given surface is closed and constitutes a well-defined boundary for MG within
a volume. The multi-grid insertion introduced in Section 8.2 is a recursive application of a
simple regular grid for Delaunay triangulation (DT) of highly non-uniform point distribu-
tions. Multi-grid insertion is a generic concept that can be readily applied also in 3D, and
such a procedure is presented in detail in Section 8.3. Mesh refinement, especially in 3D, is
a rapid and reliable method in producing valid FE mesh in compliance with an element size
specification. Guided by a shape quality measure along with optimisation by geometrical
and topological means, high-quality FE meshes can be generated for adaptive refinement
analysis. Mesh refinement algorithms over surface and within volume according to an ele-
ment size function are presented in Section 8.4. Following mesh refinement in Section 8.4,
volume bounded by analytical surfaces can be easily meshed into tetrahedral elements in
two steps: in the first step, an adaptive refinement is applied with respect to the curvature
of the boundary surface, and in the second step, nodal points close to the boundary are
snapped on to the boundary surface to obtain the final mesh as detailed in Section 8.5.
Similar to the merging of surface meshes introduced in Section 4.5, many objects can only be
conveniently defined through an intersection process, and meshes of various characteristics can
be created simply by putting one or several solid FE meshes together. A generic mesh merging
scheme for tetrahedral elements will be discussed in Section 8.6, in which arbitrary tetrahe-
dral meshes can be automatically combined into one single FE mesh with full compatibility for
numerical analysis. Based on the merging of tetrahedral elements, merging of hexahedral ele-
ments is also possible by first converting each hexahedral element into five or six tetrahedral ele-
ments, most of which can be recovered after mesh merging as shown in Section 8.7. A brief note
on the generation of higher-order curvilinear FE meshes is given in Section 8.8. A simple way is
to first generate a p1 (linear) mesh, which can then be converted to a p2 mesh by adding mid-side
nodes; however, a more sophisticated scheme is to generate the p2 mesh directly following the
curved boundary of the object. However, there are a lot of difficulties in the direct generation
of the curved edges for p2 elements. As an alternative, the associated p1 mesh is further refined,
and by means of the optimisation techniques developed in Chapter 6, the refined p1 mesh can be
optimised to produce the curved edges and surfaces of a curvilinear mesh quite automatically in
a natural formation process. Adaptive refinement is an effective automatic procedure to control
the error of an FE analysis in which element sizes have to adapt to the error of the FE solutions
so as to achieve the optimal rate of convergence. Based on the current FE solution, how the size
of the FEs can be estimated in practical applications will be given in Section 8.9.
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