Civil Engineering Reference
In-Depth Information
element density, there is no guarantee that γ = 1 for all elements and all edges, and thus, any
value above 90% should be considered satisfactory. Here, the required element size is taken
as the average of the element size density at the three vertices of the triangle. From Table 4.1,
the element size distribution of the meshes is in good agreement with the specified one with
a mean δ value greater than 88%. Such a difference is probably due to the rather strict
element-to-element curvature control, which prevents some relatively large elements of the
required size from being formed at places of rapid change of curvature.
Remarks:
Direct mesh generation on curved surfaces offers additional flexibility in the size
and shape control of the elements and in the definition or modelling of the curved surface.
The amount of discretisation error induced by the surface curvature can be easily con-
trolled by limiting the maximum allowable element to an element-turning angle. As the
mesh generation is carried out in the Euclidean space, the shape of triangular elements can
be accurately specified and measured. For simple analytical surfaces, maximum efficiency
can be attained by using the explicit formula for the closest-point projection. For arbitrary
curved surfaces, the general projection algorithm provides a practical means to produce the
required projected triangle.
4.5 MESH GENERATION BY SURFACE INTERSECTION
4.5.1 Introduction
The primary objective is to merge two meshed surfaces into a single FE mesh in a fully
automatic and robust manner. The types of surfaces to be dealt with are surfaces already
discretised into triangular facets or elements. This definition of surface intersection is broad
enough to cover the intersection of artificial surfaces that can be discretised into triangular
facets and FE meshes made up of surfaces of triangular and quadrilateral elements.
The intersection lines between surfaces are represented in terms of line segments, which
are to be determined by considering a pair of triangular facets at a time. Intersection line
segments are determined one by one in sequence to form structural loops and chains. An
intersection line will be formed progressively by connecting line segments end to end. Two
situations can be identified: (i) the line segment terminates on the boundary of the intersec-
tion surfaces to form an open chain, and (ii) it goes back to the starting point to form a
closed loop, as shown in Figure 4.68.
An accurate and efficient process for the determination of intersection chains and loops
is of utmost importance to the robustness of the intersection of discretised surfaces. A
neighbour-tracing technique proposed in the work of Lo and Wang (2003, 2004, 2005a) is
perhaps the most effective for this purpose, which greatly enhances the reliability and the
Loop
Chain
Figure 4.68
Intersection between surfaces.
