360
9 Topologic Optimization of Vibrations of Size-Dependent Beams
and a foam core. The used multi-objective optimization technique allowed them to
minimize both the weight and the cost of the hybrid sandwich plate simultaneously.
The Pareto frontier trade off the curve was generated by optimizing a sequence of
combining weight and cost objective functions.
Seretis et al. [110] carried out the multi-parameter design of experiments based
on the multi-objective curing cycle optimization for glass fabric/epoxy composites
by using the Poisson regression and generic algorithm. The authors achieved the
estimation of the curing parameters for optimum tensile and flexural performance
with high accuracy, and they demonstrated that the Poisson regression theoretical
model can predict the tensile and flexural response of the cured composites accurately.
Passos and Luersen [111] employed the multi-objective optimization of curvilinear fibre composites in two cases: for a square plate and a fuselage-like section.
The considered problems consisted of three to twelve variables in order to detect the
resulting Pareto front properly. To overcome the long computational time needed by
the finite element method, the Kriging-based approaches were used.
In this section, we define and solve the problem of topological optimization of
the composite microstructure in order to construct a material with the largest effective values of the Young modulus and thermal conductivity. The studied two-phase
composite consists of two materials: the first one has high Young modulus and low
thermal conductivity whereas the second one is characterized by low Young modulus
and high thermal conductivity.
Here we study the influence of two competing material properties, i.e. mechanical
and thermal, using linear weight functions. In particular, the problem of construction
of the Pareto spaces yielding an insight into the role of competing material properties
in the optimal topologies is addressed (see [100, 103]).
9.6.2 Determination of the Effective Tensor of Elastic
Properties
The following assumptions are introduced: the composite is linearly elastic and
macroscopically transversely isotropic with respect to both mechanical and heat
parameters. Initial stresses are absent and the introduced inclusions are homogeneous, linearly elastic, isotropic and regularly packed. The matrix is homogeneous,
linearly elastic and isotropic with respect to mechanical and thermal parameters. For
composites consisting of linear elastic materials, the governing differential equations
of a homogeneous and microstructural element are composed of linear elasticity
equations.
In the elastic regime, the microscopic behaviour of the elementary periodic cell
made from an anisotropic material can be described by the effective stress tensor σ i j
and deformation tensor ε i j of a homogenized medium. The tensors are coupled by
means of the effective elasticity tensor C
e
i jkl
9 Topologic Optimization of Vibrations of Size-Dependent Beams
and a foam core. The used multi-objective optimization technique allowed them to
minimize both the weight and the cost of the hybrid sandwich plate simultaneously.
The Pareto frontier trade off the curve was generated by optimizing a sequence of
combining weight and cost objective functions.
Seretis et al. [110] carried out the multi-parameter design of experiments based
on the multi-objective curing cycle optimization for glass fabric/epoxy composites
by using the Poisson regression and generic algorithm. The authors achieved the
estimation of the curing parameters for optimum tensile and flexural performance
with high accuracy, and they demonstrated that the Poisson regression theoretical
model can predict the tensile and flexural response of the cured composites accurately.
Passos and Luersen [111] employed the multi-objective optimization of curvilinear fibre composites in two cases: for a square plate and a fuselage-like section.
The considered problems consisted of three to twelve variables in order to detect the
resulting Pareto front properly. To overcome the long computational time needed by
the finite element method, the Kriging-based approaches were used.
In this section, we define and solve the problem of topological optimization of
the composite microstructure in order to construct a material with the largest effective values of the Young modulus and thermal conductivity. The studied two-phase
composite consists of two materials: the first one has high Young modulus and low
thermal conductivity whereas the second one is characterized by low Young modulus
and high thermal conductivity.
Here we study the influence of two competing material properties, i.e. mechanical
and thermal, using linear weight functions. In particular, the problem of construction
of the Pareto spaces yielding an insight into the role of competing material properties
in the optimal topologies is addressed (see [100, 103]).
9.6.2 Determination of the Effective Tensor of Elastic
Properties
The following assumptions are introduced: the composite is linearly elastic and
macroscopically transversely isotropic with respect to both mechanical and heat
parameters. Initial stresses are absent and the introduced inclusions are homogeneous, linearly elastic, isotropic and regularly packed. The matrix is homogeneous,
linearly elastic and isotropic with respect to mechanical and thermal parameters. For
composites consisting of linear elastic materials, the governing differential equations
of a homogeneous and microstructural element are composed of linear elasticity
equations.
In the elastic regime, the microscopic behaviour of the elementary periodic cell
made from an anisotropic material can be described by the effective stress tensor σ i j
and deformation tensor ε i j of a homogenized medium. The tensors are coupled by
means of the effective elasticity tensor C
e
i jkl