14.2 Reduction of Size of an Optimization Problem 739the relationship betweenZandXcan be expressed as
Z
12 × 1=[T]
12 × 6X
6 × 1 (14.3)
wherethe matrix [T] is given by
[T]=
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 3
(14.4)
The concept can be extended to many other situations. For example, if the geometry
of the structure is to be varied during optimization (configuration optimization) while
maintaining (1) symmetry about theYaxis and (2) alignment of the three nodes 2, 3,
and 4 (and 6, 7, and 4), we can define the following independent and dependent design
variables:
Independent variables:X 5 , X 6 , Y 6 , Y 7 , Y 4
Dependent variables:
X 1 = −X 5 , X 2 = −X 6 , Y 2 =Y 6 , Y 3 =Y 7 , X 7 =Y 4 −Y 7
Y 4 −Y 6
X 6 ,
X 3 = −X 7 , X 4 = 0 , Y 1 = 0 , Y 5 = 0
Thus the design vectorXis
X=
x 1
x 2
x 3
x 4
x 5
≡
X 5
X 6
Y 6
Y 7
Y 4
(14.5)
The relationship between the dependent and independent variables can be defined more
systematically, by defining a vector of all geometry variables,Z, as
Z= {z 1 z 2... z 14 }T≡ {X 1 Y 1 X 2 Y 2 X 3 Y 3 X 4 Y 4 X 5 Y 5 X 6 Y 6 X 7 Y 7 }Twhich is related toXthrough the relations
zi=fi( X), i= 1 , 2 ,... , 14 (14.6)wherefidenotes a function ofX.