Mesh generation over curved surfaces - Finite Element Mesh Generation - page 214

poor parameterisation and/or at singular points, it is still possible to define the orientation
of the surface by considering a close neighbourhood around a singular point (Section 2.4.6).

4.4.2 Forming triangular elements on a surface
In the mesh generation process, a line segment AB is taken randomly from the generation
front Γ. A node C Γ on the generation front is selected, which forms the best triangle with
the base segment AB. The possibility of forming a better triangle with the introduction of
an interior point C I is then explored. If the quality of triangle ABC I is superior to that of
triangle ABC Γ judging from a number of factors including the surface curvature, the ele-
ment shape and the size requirement, triangle ABC I will be formed with interior node C I
to replace triangle ABC Γ . Otherwise, the best triangle ABC Γ formed with a node C Γ on the
generation front will be taken as the element for base segment AB.
Unlike the case of the two-dimensional domain where discretisation error only occurs at
the domain boundary, discretisation error will also occur at the interior of the curved surface
due to the surface curvature. In order to control the geometrical discrepancy induced by the
surface curvature, the angle between the triangle connected to segment AB, say AOB, and
the new triangle ABC (C = C Γ or C I ) is checked to ensure that it is smaller than the allowable
tolerance ϕ ε before it is accepted. This can be done fairly easily by computing the cosine of the
angle between the normals to the triangles AOB and ABC, as shown in Figure 4.59.

nn
nn

⋅

12

> cos( φ ε ) where n = OABand × n = AB × AC

1 2

12

By setting a small φ ε , there is a limit to the size of the largest triangle that can be formed
with a base segment, thus resulting in a better approximation of the curved surface. For the
generation of elements with relatively small sizes over a curved surface with a large radius
of curvature, such a test might not be necessary. However, for graded meshes on general
curved surfaces, this element-to-element curvature control is essential.
To facilitate the search for an interior node, a reference frame is constructed at the mid-
point M of base segment AB, as shown in Figure 4.60. The orthonormal base vectors n , e 1
and e 2 are defined as follows:

1. e 1 is the unit vector along AB, e 1 = AB
AB

.

2. n is the surface normal at M perpendicular to e 1 .
3. e 2 is the unit normal vector to the plane spanned by n and e 1 , i.e. e 2 = n × e 1 .

< φ ε

B

n 1

n 2

O

A

C

Figure 4.59 Angle constraint between elements.