Civil Engineering Reference
In-Depth Information
Similarly, the estimated relative error of the FE solution
η
is given by
e
u
η=
u
The expected number of elements needed to achieve the target relative error (η
o
), NE′, is
calculated by assuming that the effect of singularity can be eliminated by an even error dis-
tribution, and the convergence rate implied by Equation 8.7 holds. We have
3
η
η
p
NE
=
NE
o
where NE is the number of elements in the current FE mesh. As the refinement will only be
optimal if the error of the mesh is uniformly distributed among all the elements, the local
allowable error norm (e
a
) is defined for each element.
u
e
=η
a
o
NE
The local refinement indicator for the i
th
element, ρ
i
, is thus given by
e
u
ρ
i
=
i
e
a
The error norm for a particular finite element converges in a rate (Onate and Bugeda
1993)
(
)
=
(
)
1
2
1
2
dim
2
3
2
=
() ()
=
()
=
β
+
β
+
∫
β
−
1
β
β
e
u
OCO
h
h
d
O
h
O
h
O
h
(8.9)
i
i
i
i
i
i
i
i
where h
i
is the size of the i
th
element. β is the convergence rate of the element error norm,
which is equal either to the strength of singularity ϕ if the element is located near a sin-
gularity (i.e. connected to a singularity in the actual implementation) or to the order of
interpolation p of the finite element. Following Equation 8.9, the refined element size hi′
i
′
is given by
2
23
−
i
ρ
β
+
hh
i
=
(8.10)
The FE mesh will
b
e refined in cycles of adaptive analysis until the FE solution falls below
the target error, i.e.
η≤
o
. If the error estimator is asymptotically exact, the estimated rela-
tive error will be pretty close to the actual error.
