Composite Materials Research Progress

Optimization of Laminated Composite Structures… 81
maximization (7.14) is replaced by a sequence of quadratic sub-problems. Each sub-problem
is itself partially solved by a first order maximization algorithm in the dual space.
In the case of a multi-objective formulation the optimization problem writes :
min max gl0
( x)
X l = 1,...,
nc
g j (x ) ≤ g j j = 1,...,
m
(7.15)
where nc is the number of load cases. Using the bound formulation (Olhoff, 1989) the
problem (7.15) can be written as:
g l0
( x)
≤ β
1 2
min β
2
l = 1,...,
nc
(7.16)
g j (x ) ≤ g j
j = 1,...,
m
where β is the multiobjective factor, that is an additional design variable in the optimization
problem. Instead of solving (7.16) problem (7.17) is considered where a new variable δ is
introduced for the possible relaxation of the set of constraints.
( ) ( ) 2
1 2 1 2 C ( k −1)
min β + δ + p + ∑ xi
− xi
2 2 2 i
g j0
( x ) ≤ β j g j0
j = 1,...,
nobj
(7.17)
g j ( x ) ≤ g j ( 1+
δ )
j = 1,...,
m
g j0
are target values on the objective functions. The dual approach described for monoobjective
optimisation problems is then applied to (7.17).
8. Applications of the Optimization Solution Procedure
In the following examples (except the simple laminate designs and the topology optimization
problem), the structural and semi-analytical sensitivity analyses are carried out with
SAMCEF (http://www.samcef.com). The Boss Quattro optimisation tool box
(http://www.samcef.com) is used for defining and solving the optimisation problem
(Radovcic and Remouchamps 2002).
8.1. Laminate Subjected to in- and out-of-plane Loadings
A symmetric 4 plies laminate made of carbon/epoxy is considered. The load case and the
initial configuration are provided in Table 8.1. The fibers orientations of each ply are the