The Triangulation.

 

In this section, we define the data structures used to specify the domain as a triangulation.   Let denote the triangulation, consisting of NTF  triangles . Let the triangulation have NVF  vertices and NBF  edges . Triangles may have curved edges, which are approximated by arcs of circles. The centers of the circles are given by .  Curved edges may lie either on the boundary or interior of the region .

  
Figure: Clockwise, from upper left: example domain; triangle numbers; vertex numbers; curved edges; edge numbers.

For example, consider the circle of radius one with a crack along the positive x-axis. This domain can be triangulated using 8 triangles, 10 vertices, and 10 edges, 8 of which are curved, as illustrated in Figure . Vertices and have the same (x,y) coordinates, but is ``above'' the crack, and ``below.'' Similarly, edge is the top of the crack, while edge is the bottom. The ordering of vertices, triangles, and edges is arbitrary.

The arrays VX  and VY  are of length NVF, and contain as their I-th entries the x and y coordinates of , illustrated for this example in Table . If a triangle has a curved edge,   that edge is approximated by a circular arc passing through the endpoints of the edge, with the center of the circle located at one of the points . Because there are generally two such arcs for every pair of endpoints, the shorter arc is taken to be the correct edge; therefore, one should specify curved edges that subtend (strictly) less than of arc; is a reasonable upper bound. The centers for the various circles used in specifying curved edges are given in the arrays XM  and YM  of length NCF, which contain as their I-th entries the x and y coordinates of the center . The data for our example is shown in Table .

 

I 1 2 3 4 5 6 7 8 9 10

VX(I) 0 1 0 -1 0 1

VY(I) 0 0 1 0 -1 0

XM(I) 0

YM(I) 0

The VX, VY, XM and YM arrays. NVF=10 and NCF=1.

I 1 2 3 4 5 6 7 8 9 10

IBNDRY(1,I) 1 2 3 4 5 6 7 8 9 10

IBNDRY(2,I) 2 3 4 5 6 7 8 9 10 1

IBNDRY(3,I) 0 1 1 1 1 1 1 1 1 0

IBNDRY(4,I) 2 2 2 2 2 2 2 2 2 1

IBNDRY(5,I) 2 0 0 0 0 0 0 0 0 1

The IBNDRY array. NBF=10.

I 1 2 3 4 5 6 7 8

ITNODE(1,I) 1 1 1 1 1 1 1 1

ITNODE(2,I) 2 3 4 5 6 7 8 9

ITNODE(3,I) 3 4 5 6 7 8 9 10

ITNODE(4,I) 1 1 2 2 3 3 4 4

The ITNODE array. NTF=8.

 

To simplify data entry, we provide the routine CENTRE for computing the center of a circle given three points on its boundary. CENTRE    is called using the statement:

Call CENTRE( X1, Y1, X2, Y2, X3, Y3, XC, YC )

Here (X1,Y1) and (X2,Y2) are the endpoints of an arc of the circle, and (X3,Y3) is a third point on the arc (e.g., the midpoint). CENTRE returns the center of the circle in (XC,YC).

A given triangle is specified by giving an accounting of its three vertices, and by specifying an integer label or tag. Such labels are provided strictly for the convenience of the user, and can be used to identify differing regions or material properties associated with the element. The array ITNODE  is a integer array whose I-th column contains information about . The first three entries of ITNODE contain the three vertex numbers of triangle . ITNODE(J,I)=K, for , means (VX(K),VY(K)) is the J-th vertex of . The ordering of the vertices of a given triangle is arbitrary and independent of the other triangles. The fourth entry of column I of ITNODE contains the label for triangle I. In our example, we choose to label each element by the quadrant in the Euclidean plane in which it lies. The ITNODE array for our example is shown in Table .

The array IBNDRY  is a integer array whose I-th column contains information about . The first two entries of IBNDRY contain the indices of the endpoints of the interval (pointers to the VX and VY arrays). IBNDRY(J,I)=K, , means (VX(K),VY(K)) is an endpoint of . Ordering of vertices is arbitrary. The third entry of the I-th column contains the index for the circle center for the edge (pointer to the XM and YM arrays). IBNDRY(3,I)=K means (XM(K),YM(K)) is the circle center for edge . If the edge is straight, IBNDRY(3,K) should be set to 0.

The fourth entry defines the boundary type for edge . There are four possibilities for edge type. If IBNDRY(4,I)=2, then is a boundary edge on which Dirichlet boundary conditions are imposed. If IBNDRY(4,I)=1, then is a boundary edge on which natural boundary conditions are imposed. If IBNDRY(4,I)=0, then is an internal edge. There are two reasons to include internal edges in a triangulation. First, if the internal edge is curved, it must be specified in IBNDRY in order to be treated properly. Second, the set in equation () is taken as the set edges defined in IBNDRY. Thus internal edges which are part of must be defined in IBNDRY. An important restriction on internal edges of a triangulation is that they must lie on an internal interface. That is, the two triangles sharing must have different labels as their fourth entries in ITNODE.

The fourth type of edge is a linked edge. Linked edges occur only in pairs. If and are a pair of linked edges, then IBNDRY(4,I)=-J and IBNDRY(4,J)=-I. Linked edges and must be geometrically congruent. That is, must be mapped to using only a translation, an orthogonal rotation, and a refection. Continuity of the solution and weak continuity of is imposed on linked edge pairs. Thus if and are boundary edges, this is equivalent to imposing periodic boundary conditions.

The fifth entry in column I of IBNDRY contains an integer label for the edge, similar to the fourth entry of ITNODE; this label can be used to distinguish uniquely a particular edge, or to associate some property with the edge. The IBNDRY array for our example is shown in Table .

In our example, we impose Dirichlet boundary conditions on the outer boundary of the circle, and also along the top of the crack, and Neumann boundary conditions on the bottom of the crack. The outer boundary of the circle is labeled 0, the top of the crack 2, and the bottom of the crack 1.

In the case of a singular Neumann problem (e.g., , , , and in ()), the solution u is determined only up to an arbitrary constant. To make such a solution unique, the user may select a distinguished vertex where u will be specified. The parameter IDBC , where , denotes the vertex number. In other situations, one should set IDBC=0.