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determined by the array. This sequence of operations must be done for each of the Q
elements.
A more common and somewhat more appealing technique is to perform the assembly
of the matrix. All the elements are scanned one by one and the nine associated contribu
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tions , added to the corresponding positions in the global
“stiffness” matrix. The assembled matrix must now be stored but the element matrices
may be discarded. The structure of the assembled matrix depends on the ordering of the
nodes. To facilitate the computations, a widely used strategy transforms all triangles into a
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reference triangle with vertices . The area of the triangle is then simply
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the determinant of the Jacobian of the transformation that allows passage from one set of
axes to the other.
Simple boundary conditions such as Neumann or Dirichlet do not cause any dif¬culty.
The simplest way to handle Dirichlet conditions is to include boundary values as unknowns
and modify the assembled system to incorporate the boundary values. Thus, each equation
associated with the boundary point in the assembled system is replaced by the equation
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. This yields a small identity block hidden within the linear system. For Neumann
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conditions, Green™s formula will give rise to the equations
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which will involve the Neumann data over the boundary. Since the Neumann data is
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typically given at some points only (the boundary nodes), linear interpolation (trapezoidal
rule) or the midline value (midpoint rule) can be used to approximate the integral. Note
that (2.36) can be viewed as the th equation of the linear system. Another important point
is that if the boundary conditions are only of Neumann type, then the resulting system is
singular. An equation must be removed, or the linear system must be solved by taking this
singularity into account.
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Generating a ¬nite element triangulation can be done quite easily by exploiting some initial
grid and then re¬ning the mesh a few times either uniformly or in speci¬c areas. The
simplest re¬nement technique consists of taking the three midpoints of a triangle, thus
creating four smaller triangles from a larger triangle and losing one triangle, namely, the
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original one. A systematic use of one level of this strategy is illustrated for the mesh in
Figure 2.8, and is shown in Figure 2.10.
Finite element mesh
6
16
15 14
Assembled matrix
4
14 15
4 12 5
13 12
3 10
13 11 10
11 2
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The simple ¬nite element mesh of Figure 2.8 af
ter one level of re¬nement and the corresponding matrix.
One advantage of this approach is that it preserves the angles of the original triangu
lation. This is an important property since the angles on a good quality triangulation must
satisfy certain bounds. On the other hand, the indiscriminate use of the uniform re¬nement
strategy may lead to some inef¬ciencies. Indeed, it is desirable to introduce more triangles
in areas where the solution is likely to have large variations. In terms of vertices, midpoints
should be introduced only where needed. To obtain standard ¬nite element triangles, the
points that have been created on the edges of a triangle must be linked to existing vertices in
the triangle. This is because no vertex of a triangle is allowed to lie on the edge of another
triangle.
Figure 2.11 shows three possible cases that can arise. The original triangle is (a). In
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(b), only one new vertex (numbered ) has appeared on one edge of the triangle and it
is joined to the vertex opposite to it. In (c), two new vertices appear inside the original
triangle. There is no alternative but to join vertices (4) and (5). However, after this is done,
either vertices (4) and (3) or vertices (1) and (5) must be joined. If angles are desired that
will not become too small with further re¬nements, the second choice is clearly better in
this case. In fact, various strategies for improving the quality of the triangles have been
devised. The ¬nal case (d) corresponds to the “uniform re¬nement” case where all edges
have been split in two. There are three new vertices and four new elements, and the larger
initial element is removed.
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(a) (b)