Efficient Robust Optimization of Structures Subjected to Earthquake Load and Characterized by Uncertain Bounded System Parameters - Structural Seismic Design Optimization and Earthquake Engineering - page 119

Figure 2. The building frame

The coefficient of determination ( R 2 ) is de-
fined as:

p p

∑

2

∑

2

( ) ( )

R

2

= y − y y − y (32) ˆ

i i i i

i = i =

1 1

where, y i is the mean value of the actual response.
The value of this coefficient close to one represents
a good metamodel for response approximation.

The average prediction error ( ε m ) is defined as,

100 ˆ

y − y

i i

ε m = (33)

y

i

Table 2 shows the results of the statistical test
for both the LSM and MLSM based metamodels.
It can be observed that the lesser RMSE and ε m
values and the higher R 2 value are attained by the
MLSM based RSM compared to the LSM based
RSM. This clearly indicates the accuracy of the
MLSM based RSM over the LSM based RSM.
The CPU time required for complete generation
of all y i by the direct MCS is about 2.5 hrs,
whereas the CPU time needed for computation
of y i is only 11 minutes by the LSM based RSM
and 12 minutes by the MLSM based RSM. The
MLSM based RSM needs more computational
time compare to the LSM based RSM due to the
fact that the MLSM based approach needs re-
peated evaluation of the response surface as it is
required to be generated afresh for each updated
DV set. Whereas, the LSM based approach may
need more rigorous DOE compare to the MLSM
based RSM to obtain a comparable accuracy
level. It may be noted here that one needs to
evaluate the structural responses analysis at all
these training points involving the solution of a
system having number of unknowns in the order
of few thousands to million. This obviously in-

depending on the specific nature of the optimiza-
tion problem. This is a potential problem in it-
erative optimization process and warns the ap-
plicability of the LSM based RSM approach for
RDO of complex dynamic system under stochas-
tic load. Furthermore, to study the suitability of
the proposed adaptive RSM, the statistical metrics
as described below have been also computed
(Bouazizi, Ghanmi, & Bouhaddi, 2009):
The Root Mean Square Error (RMSE) is de-
fined as:

( )

2

y − y

∑ ˆ

p

i i

RMSE

= (31)

p i =

1

In the above, y i is the predicted response ob-
tained by the considered LSM or MLSM based
metamodel and y i is the actual response obtained
by the direct MCS for ith sample. The sample
size, p is taken as 1000 for the present numerical
investigation.