Mesh generation on planar domain - Finite Element Mesh Generation - page 143

Proof: Define an ordering relation for the set  ,



CC , ∈ ,C CCC ≥ ∈ ∠ ACB ≥∠ AC B

i j i j i BCj i j

i. Exclusivity
Given CC
i

, ∈  , we have either ∠AC i B ≥ ∠AC j B or ∠AC j B ≥ ∠AC i B

Hence, any two points of  can be compared.

ii. Transitivity

CCCCCand CC , , ,∈≥  ≥ ∠≥∠ ∠≥ ∠ AC B CBandACB AC B

i j k i j j k i j j k

∠≥ AC BACB CC ≥

i k i k

iii. Non-uniqueness

CC , ∈≥  ,C Cand CC CC ≥ =

i j i j j i i j

With (i) and (ii), since  is a non-empty finite set, there exists C m ∈  such that ∠AC m B is
a maximum, or CCC
mi

≥∀∈  .

i

Remarks: For the set of candidate nodes  , nodes having intersection with the generation
front are excluded from the set. However, this will not affect the setting up of an ordering
relationship for the points in the set with respect to the base line segment AB, such that an
optimal point C m can always be identified in  for which ∠AC m B is the largest.

3.7.2.7 Delaunay property of triangulation
By the lemma of Delaunay, the triangulation will be Delaunay if for any diagonal AC shared
by triangle ABC and ACD, D lies outside or on the circumference of the circumcircle of
triangle ABC and vice versa. Consider quadrilateral ABCD composed of two triangles ABC
and ACD of the mesh, as shown in Figure 3.59. If D′ lies inside the circumcircle of triangle
ABC, ∠AD′ B is greater than ∠ACB, and point D′ would have been chosen rather than point
C in the construction of triangle ABC. Similarly, if B lies at the interior of the circumcircle
of triangle ACD, ∠ABD is greater than ∠ACD, and in the construction of triangle ACD,
node B would have been selected. In both cases, the Delaunay property of the triangulation
is established by selecting the point making the largest angle possible with the base segment
for each triangular element constructed throughout the MG process.

D

D

A

C

B

Figure 3.59 Delaunay property of triangulation.