9.3 Topological Optimization Problem
339
general statement being based on the linear static analysis of FEM can be presented
in the following way
min f (x, U) in opt ,
for K (x) U = F (x) , g i (x, U) ≤ 0 , 0 ≤ x e ≤ 1 ,
(9.1)
where f -target function; x-vector of design variables coupled with density ρ e , Uvector of displacement, K-global matrix of stiffness, F-force vector and g i stands
for the constraints. Observe that stiffness matrix K and sometimes the force vector
F depend explicitly on the design variables. In the frame of the introduced general
statement of the problem, we consider a series of case problems for various target
functions including flexibility, stress state, frequency, eigenvalue problem, etc.
A crucial role of the methods based on the density plays the choice of the appropriate interpolating functions and penalty functions in order to match the physical
parameters of the problem with continuously designed variables (for instance, density). As we have already mentioned, the variables are coupled with physical density
of each finite element ρ e . The intervals of the density changes are 0 ≤ ρ e ≤ 1 , or
0 < ρ min ≤ ρ e ≤ 1 , where 0 corresponds to an empty element, and 1 corresponds
to the fundamental material, and ρ min denotes the minimal density value, which is
needed to overcome problems implied by singularity of the stiffness matrix. The
mentioned difficulties include singularity of the matrices of the finite elements as
well as lack of ability of the material to appear again in the spaces with zeroth density. The choice of the parameters implies the necessity of rigorous separation of the
space into subspaces with ρ = ρ min and subspaces with ρ = 1. It is usually realized
by employing one of the implicit methods of a penalty, where the most popular is [47,
75]. In the SIMP method, often referred as the step rule or fictions/material model,
the variables of the density are penalized with the use of stepwise principle. This
approach is demonstrated through the following equation
E (ρ e ) = ρ
p
e E 0 ,
(9.2)
where SIMP is employed to the Young modulus of the finite element. Here E (ρ e )variable Young modulus, E 0 -Young modulus of the input material, p-finite penalty
parameter. It should be mentioned that for 0 ≤ ρ e ≤ 1 and positive p (usually it
takes the value of 3), E (ρ e ) is bounded for zero density and by E 0 for ρ e = 1 .
The alternative schemes of interpolation have been worked out mainly for the
removement of disadvantages of the SIMP method. In particular, the developed
Rational Approximation of Material Properties (RAMP) [76] offer, in contrary to
SIMP method, non-zero sensitivity for the zero density. In result, it has been shown
that the RAMP model of material improves some of the occurred difficulties in the
problems dealing with small values of the density and under dependence of the
loading parameters on the design variables.
The second alternative method to SIMP, known as the SINH method, is based
on the hyperbolic sinusoidal functions [77]. In majority of the used methods, the
339
general statement being based on the linear static analysis of FEM can be presented
in the following way
min f (x, U) in opt ,
for K (x) U = F (x) , g i (x, U) ≤ 0 , 0 ≤ x e ≤ 1 ,
(9.1)
where f -target function; x-vector of design variables coupled with density ρ e , Uvector of displacement, K-global matrix of stiffness, F-force vector and g i stands
for the constraints. Observe that stiffness matrix K and sometimes the force vector
F depend explicitly on the design variables. In the frame of the introduced general
statement of the problem, we consider a series of case problems for various target
functions including flexibility, stress state, frequency, eigenvalue problem, etc.
A crucial role of the methods based on the density plays the choice of the appropriate interpolating functions and penalty functions in order to match the physical
parameters of the problem with continuously designed variables (for instance, density). As we have already mentioned, the variables are coupled with physical density
of each finite element ρ e . The intervals of the density changes are 0 ≤ ρ e ≤ 1 , or
0 < ρ min ≤ ρ e ≤ 1 , where 0 corresponds to an empty element, and 1 corresponds
to the fundamental material, and ρ min denotes the minimal density value, which is
needed to overcome problems implied by singularity of the stiffness matrix. The
mentioned difficulties include singularity of the matrices of the finite elements as
well as lack of ability of the material to appear again in the spaces with zeroth density. The choice of the parameters implies the necessity of rigorous separation of the
space into subspaces with ρ = ρ min and subspaces with ρ = 1. It is usually realized
by employing one of the implicit methods of a penalty, where the most popular is [47,
75]. In the SIMP method, often referred as the step rule or fictions/material model,
the variables of the density are penalized with the use of stepwise principle. This
approach is demonstrated through the following equation
E (ρ e ) = ρ
p
e E 0 ,
(9.2)
where SIMP is employed to the Young modulus of the finite element. Here E (ρ e )variable Young modulus, E 0 -Young modulus of the input material, p-finite penalty
parameter. It should be mentioned that for 0 ≤ ρ e ≤ 1 and positive p (usually it
takes the value of 3), E (ρ e ) is bounded for zero density and by E 0 for ρ e = 1 .
The alternative schemes of interpolation have been worked out mainly for the
removement of disadvantages of the SIMP method. In particular, the developed
Rational Approximation of Material Properties (RAMP) [76] offer, in contrary to
SIMP method, non-zero sensitivity for the zero density. In result, it has been shown
that the RAMP model of material improves some of the occurred difficulties in the
problems dealing with small values of the density and under dependence of the
loading parameters on the design variables.
The second alternative method to SIMP, known as the SINH method, is based
on the hyperbolic sinusoidal functions [77]. In majority of the used methods, the