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Unformatted text preview: ce, let's also place a Cartesian coordinate system
at Vertex 2.
We proceed as usual, constructing shape functions on each element. Although not
really needed for our present development, those bilinear shape functions that are nonzero
on Edge 1-2 follow. 4 Adaptive Finite Element Techniques 1 1
2 x 1111
0000 Figure 8.2.2: Irregular node at the intersection of a re ned element.
On Element 1: y
N11 = ( h + x )( h )
h N21 = ( h + x )( h ; y ):
h On Element 2: 2
N12 = ( h=h=; x )( y ; h=2 )
N32 = ( h=h=; x )( hh=2y ):
2 On Element 3: 2
N23 = ( h=h=; x )( h=h=; y )
N33 = ( h=h=; x )( h=2 ):
2 As in Chapter 2, the second subscript on Nje denotes the element index.
The restriction of U on Element 1 to Edge 1-2 is U (x y) = c1N11 (x y) + c2N21 (x y):
Evaluating this at Node 3 yields U (x3 y3) = c1 + c2
2 x < 0: The restriction of U on Elements 2 and 3 to Edge 1-2 is
U (x y) = c1N12 (x y) + c3N32 (x y) iif y < h=2 :
c2N23 (x y) + c3N33 (x y) f y
In either case, we have U (x3 y3) = c3
x > 0:
Equating the two expressions for U (x3 y3) yields the constraint condition
c = c1 + c2 :
3 2 (8.2.1) 8.2. h-Re nement 5 Figure 8.2.3: The one-irregular rule: the intended re nement of an element to create two
irregular nodes on an edge (left) necessitates re nement of a neighboring element to have
no more than one irregular node per element edge (right).
Thus, instead of determining c3 by Galerkin's method, we constrain it to be determined
as the average of the solutions at the two vertices at the ends of the edge. With the
piecewise-bilinear basis used for this illustration, the solution along an edge containing
an irregular node is a linear function rather than a piecewise-linear function.
Software based on this form of adaptive re nement has been implemented for elliptic
27] and parabolic 1] systems. One could guess that di culties arise when there are too
many irregular nodes on an edge. To overcome this, software developers typically use
Bank's 9, 10] \one-irregular" and \three-neighbor" rules. The one-irregular rule limits
the number of irregular nodes on an element edge to one. The impending introduction
of a second irregular node on an edge requires re nement of a neighboring element as
shown in Figure 8.2.3. The three-neighbor rule states that any element having irregular
nodes on three of its four edges must be re ned.
A modi ed quadtree (Section 5.2) can be used to store the mesh and solution data.
Thus, let the root of a tree structure denote the original domain . With a structured
grid, we'll assume that is square, although it could be obtained by a mapping of a
distorted region to a square (Section 5.2). The elements of the original mesh are regarded
as o spring of the root (Figure 8.2.4). Elements introduced by adaptive re nement are
obtained by bisection and are regarded as o spring of the elements of the original mesh.
This structure is de...
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- Spring '14