Computational Methods for Debonding in Composites

246 E. Lund and L.S. Johansen
12.2.1 Parametrization for Single Layered Laminate Structures
As in traditional topology optimization the parametrization of the DMO formulation
is invoked at the finite element level. The element constitutive matrix, C e ,fora
single layered laminate structure may in general be expressed as a sum over the
element number of candidate material configurations, n e :
n e
Ce = ∑ wiCi = w1C1 + w2C2 + ··· + wn
i=1
eCne, 0 ≤ wi ≤ 1 (12.1)
where each candidate material is characterized by a constitutive matrix Ci. The
weight functions wi must all have values between 0 and 1 in order to be physically
allowable. Furthermore, in case of solving buckling problems or having a
mass constraint as in the optimization problems studied here, it is necessary that
the sum of the weight functions is 1.0, i.e., ∑ ne
i=1 wi = 1.0. If this demand is not
fulfilled, physically meaningless results will be obtained for the buckling load factor
for intermediate values of the weight functions. Similarly, the mass density ρ is
computed using the weight functions, and if the sum of the weight functions does
not add up to unity, the computed mass M cannot be compared to the prescribed
mass constraint M.
For details about different parametrization schemes the reader is referred to [30,
31], and only the most effective implementation is briefly outlined here.
For each element a number of design variables xe i , i = 1,...,ne is introduced, and
the weight functions wi used are defined as
wi =
ˆwi
∑ ne
k=1 ˆwk
, i = 1,...,n e
where ˆwi =(x e i )p
ne � e
∏ 1 − (x j )
j=1; j�=i
p� (12.2)
As an example, in case of four candidate materials the weight functions ˆwi are given
as
ˆw1 =(xe 1 )p (1 − (xe 2 )p )(1− (xe 3 )p )(1− (xe 4 )p )
ˆw2 =(xe 2 )p (1 − (xe 1 )p )(1− (xe 3 )p )(1− (xe 4 )p )
ˆw3 =(xe 3 )p (1 − (xe 1 )p )(1− (xe 2 )p )(1− (xe 4 )p )
ˆw4 =(xe 4 )p (1 − (xe 1 )p )(1− (xe 2 )p )(1− (xe 3 )p (12.3)
)
The SIMP method known from topology optimization has been adopted by introducing
the power, p, to penalize intermediate values of xe i , such that the design
variables xe i are pushed towards 0 or 1. The power p is typically set to 1 or 2 in the
beginning of the optimization process and then increased by 1 for every 10 design
iterations until p is 3 or 4. Moreover, the term (1 − xe j ) j�=i is introduced such that an
increase in xe i results in a decrease of all other weight functions. Finally, the weights
have been normalized in order to satisfy the constraint that the sum of the weight
functions is 1.0 (this is in general not the case for the weight functions ˆwi).