Unformatted text preview: picted in Figure 8.2.4. Coarsening can be done by \pruning" re ned
quadrants. It's customary, but not essential, to assume that elements cannot be removed
(by coarsening) from the original mesh 3].
Irregular nodes can be avoided by using transition elements as shown in Figure 8.2.5.
The strategy on the right uses triangular elements as a transition between the coarse and
ne elements. If triangular elements are not desirable, the transition element on the left
uses rectangles but only adds a midedge shape functions at Node 3. There is no node
at the midpoint of Edge 45. The shape functions on the transition element are N11 = ( h + x )( y ; h=2 )
h
h=2 2
N21 = ( h + x )( h=h=; y )
h
2 6 Adaptive Finite Element Techniques
11
00
11
00
11 11 11 11 11 11
00 00 00 00 00 00
11 11 11 11 11 11
00 00 00 00 00 00
11 11 11 11 11 11
00 00 00 00 00 00
11 11 11 11 11 11
00 00 00 00 00 00
11 11 11 11
00 00 00 00
11 11 11 11
00 00 00 00
11 11 11 11
00 00 00 00
11 11 11 11
00 00 00 00 Figure 8.2.4: Original structured mesh and the bisection of two elements (left). The tree
structure used to represent this mesh (right).
40
1 1
0
1
0 1
0
1
0
50
1 1 1
0y
1
0
1
0
1
0
1
10
1
0
1
0
1
0
2
30
1
1
0
1
0
3
1
0
1
0
1
20 1
0
1
0
1
0 1111 1
0000 0
1
0
1
0
x 1
0
1
0
1
0
1
0
1
0
1
0
1
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1
0
1
0 Figure 8.2.5: Transition elements between coarse and ne elements using rectangles (left)
and triangles (right). (y
h + x ) ( h=2 ) if 0 y h=2
N31 = ( h
( h;2y ) if h=2 y h
h=
y
N41 = ( ;x )( h )
N51 = ( ;x )( h ; y ):
h
h
h Again, the origin of the coordinate system is at Node 2. Those shape functions associated
with nodes on the right edge are piecewisebilinear on Element 1, whereas those associated
with nodes on the left edge are linear.
Berger and Oliger 12] considered structured meshes with structured mesh re nement,
but allowed elements of ner meshes to overlap those of coarser ones (Figure 8.2.6). This
method has principally used with adaptive nite di erence computation, but it has had
some use with nite element methods 29]. 8.2.2 Unstructured Meshes
Computation with triangularelement meshes has been done since the beginning of adaptive methods. Bank 9, 11] developed the rst software system PLTMG, which solves 8.2. hRe nement 7 Figure 8.2.6: Composite grid construction where ner grids overlap elements of coarser
ones.
our model problem with a piecewiselinear polynomial basis. It uses a multigrid iterative procedure to solve the resulting linear algebraic system on the sequence of adaptive
meshes. Bank uses uniform bisection of a triangular element into four smaller elements.
Irregular nodes are eliminated by dividing adjacent triangles sharing a bisected edge
in two (Figure 8.2.7). Triangles divided to eliminate irregular nodes are called \green
triangles" 10]. Bank imposes oneirregular and threeneighbor rules relative to green
triangles. Thus, e.g., an intended second bisection of a vertex angle of a green triangle
would not be done. Instead, the green triangle would be uniformly re ned (Figure 8.2.8)
to keep angles bounded aw...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Finite Element Method, coarsening, Adaptive Finite Element, Finite Element Techniques, J.E. Flaherty

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