Civil Engineering Reference
In-Depth Information
s
h
δ
d
ρ
Figure 4.11
Compare lengths of chord and curve.
In the limiting case,
2
4
sh
s
−
22
2
ρθ ρθ
ρθ
−
sin
sin
θ
θθ
ε
=
=
=− =− −+−
1
1
1
!
θ
35
!
Taking one term in the expansion, we have
2
θ
θε
2
ε
=
=
6
or
θ
=
6
ε
3
!
h
=−=− =− =
(
1
ε
)
s
2 1
(
ε ρθ ε ρβρ
)
2 16
1
(
)
Taking two terms in the expansion, we have
2
4
ε
θθ
ε θθ
2
4
=− =
35
120
20
−
!
!
Let μ = θ
2
. We have
μ
2
− 20μ + 120ε = 0
Solving for μ and taking the smaller root, we get
=− −
10
100
120
εθ
= =− −
10
100
120
ε
h
=−=− =− −
(
1
ε
)
s
2 1
(
ε ρθ
)
2 1 0
(
ε
)
100
−
120
ερβρ
=
2
For instance, set ε = 1%; we have β
1
= 0.485 and β
2
= 0.48573, i.e. h cannot exceed 0.485ρ.
Whichever criterion we would like to apply for a close approximation of a linear FE mesh
of triangular facets to an analytical curved surface, we can simply modify the metric tensor
by replacing the principal curvatures with the reduced principal curvatures according to the
chosen criterion, such that
2
κ
0
v
v
ˆ
ˆ
κ
λ
κ
λ
1
1
M
= ˆˆ
where
κ
=
1
and
κ
=
2
vv
1
2
1
2
2
0
κ
2
2
