Civil Engineering Reference
In-Depth Information
of a background grid are listed in Table 3.2. The gains in CPU time are 3.7 times and 5.3
times, respectively, for these two meshes, showing that the efficiency of the background
grid reduces for more complicated node spacing functions whose computation may take a
substantial portion of the overall MG time.
Upon further analysis on the use of the two background grids, it is found that identical
sets of potential segments susceptible to intersections adjacent to the base segment could be
accurately retrieved by both grids, verifying the reliability of the procedure in marking and
unmarking of cells intersected by line segments. Similar to the first series of meshes, the use
of background grid has no bearing on the characteristics of the resulting meshes, such that
the number of elements
N
e
, the number of nodes
N
n
and the mean shape factor α are almost
exactly the same.
As a practical example of adaptive refinement FE analysis, the electromagnetic wave dis-
tribution over a planar domain is considered. A uniform plane wave is scattered by a circu-
lar cylinder and propagates freely towards infinity, as shown in Figure 3.54. The analytical
solution expressed in cylindrical co-ordinates to this problem was given by Balanis (1989).
+∞
∑
Jkr
Hkr
()
()
−
n
n
0
()
2
e
jn
φ
E
(,)
ρφ
=
E
j
J
(
k
ρ
)
−
Hk
(
ρ
)
z
0
n
0
n
0
()
2
n
0
−∞
Linear triangular elements T3 were used in the adaptive refinement analysis, and the
required accuracy was set equal to 1%. Four analyses in three steps of refinements were
required to bring the energy error norm from 22.7% to about 1%, as shown in Table 3.3.
The refinement adaptive triangular meshes generated by the proposed dynamic grid tech-
nique are shown in Figure 3.55. It is seen
t
hat there is little difference between the exact
error norm and the estimate error norm
(
η/
, showing that the meshes are of very high
quality, and the required nodal spacing over the domain is well respected.
)
y
A - A
H
i
A
A
r
Incident wave
φ
x
z
E
i
Figure 3.54
Wave direction and the cylindrical domain and its cross section.
Table 3.3
Adaptive refinement analysis of the electromagnetic wave problem
Mesh DOF NN NE ||
e
||*
η
(%) ||
e
||
η
(%)
η
/
1 308 324 576 0.089047 24.10 0.084717 22.69 1.06
2 3695 3755 7338 0.013265 3.43 0.011950 3.09 1.11
3 7069 7135 14,092 0.009856 2.54 0.009569 2.47 1.03
4 13,872 13,985 27,350 0.004561 1.17 0.004472 1.15 1.02
Note: DOF = number of degrees of freedom; NN = nu
m
ber of nodes; NE = number of elements;
||
e
||* = estimated energy norm; ||
e
|| = exact energy norm;
η
= estimated % error;
η
= exact % error.
