Triangulations Triangulations Triangulations Situations not admitted in triangulations

Triangulations

Triangulations

Triangulations Situations not admitted in triangulations. If two triangles have some intersection, it is either on common vertex or a common full edge. In particular, two different triangles do not overlap.

Triangulations This mesh is stored in 3 matrices 1) Matrix P which store the coordinates of the nodes 2) Matrix e which store the boundary nodes 3) Matrix t which stores local labeling. vs. global labeling

Element Labeling 6 3 14 2 5 10 15 11 9 13 12 7 4 16 triangles 16 elements 16 1 8

Node Labeling 2 (global labeling) 6 1 11 10 7 13 nodes 5 9 12 13 3 8 4 X-coordinate and y-coordinate Matrix p(2, #nodes) 1 2 3 4 5 6 7 8 9 10 11 12 13 x 1 0 0 1 0. 5 0 0. 5 1 0. 75 0. 25 0. 75 y 1 1 0 0 0. 5 1 0. 5 0. 75 0. 25

2 6 1 11 10 Boundary node 7 5 vector e(#boundary node) 9 12 13 3 8 4 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 start 1 2 3 4 6 7 8 9 end 6 7 8 9 2 3 4 1

Node Label (local labeling) Each triangle has 3 nodes. Label them locally inside the triangle 3 1 2

2 6 6 3 7 11 1 2 14 10 10 15 11 7 5 13 12 16 12 4 3 9 5 13 Matrix t(3, #elements) 1 8 8 Local label. vs. global label 9 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 4 1 2 3 4 5 5 10 11 12 13 2 9 6 7 8 10 11 12 13 13 10 11 12 9 6 7 8 10 11 12 13 9 6 7 8

Export 3 matrices

delaunay >> x=[0 1 0. 5] >> y=[0 0 1 1 0. 5] >> TRI = delaunay(x, y); >> triplot(TRI, x, y)

Triangulations Exercise 1 The adjacent Figure shows a small triangulation of an L-shape domain. The mesh has eight nodes and six triangles. Find the matrices p , e , t

Triangulations Exercise 2 The adjacent Figure shows a set of point in the domain (0, 3)X(0, 3). Use delaunay Matlab command to generate a triangulation with these points as a nodes. Then find the matrix p and t. 4 What are the nodes of the triangle 11 11 7 3 8 9 10 5 1 6 12 2