9.6 Topological Optimization with Maximized Stiffness and Heat Transfer
365
Now the boundary conditions [112] for the characteristic functions T i for a representative cell which take the classical form are shown in Fig. 9.9a, b and have the
following form:
for k
e
11
T 1 (0, y 2 ) = 0, T 1 (1, y 2 ) = 1,
∂ T 1
∂ y 1
(y 1 , 0) =
∂ T 1
∂ y 1
(y 1 , 1) = 0;
for k
e
22
T 2 (y 1 , 0) = 0, T 2 (y 1 , 1) = 1,
∂ T 2
∂ y 2
(0, y 2 ) =
∂ T 2
∂ y 2
(1, y 2 ) = 0.
9.6.4 Topological Optimization
The method of homogenization [44–46] is one of the most effective approaches to
compute the global physical properties of a composite such as the bulk stiffness
modulus, the shear modulus or the heat transfer coefficient.
Having both the elastic and thermal homogenization determined, one can define
an optimization problem aimed at defining the optimal topology for thermoelastic
problems. Such problems are particularly interesting when the elastic and thermal
properties of materials strongly compete with each other and the optimal topologies
for individual problems differ significantly from each other. To solve the problem, it
is necessary to simultaneously take into account both the elastic and thermal material
properties.
Let us assume that the elementary periodic cell is divided into finite elements and
each finite element has its own variable density ρ n (n = 1, . . . , N ). Following the
rule of SIMP [112], where the role of the control variable is played by an artificially
introduced density ρ n (x) the corresponding Young’s modulus E(x) and heat transfer
coefficient k(x) for the two-component composite in the nth finite element can be
written as
E n (x) = E 1 ρ
p
n (x) + E 2 (1 − ρ
p
n (x)),
k n (x) = k 1 ρ
p
n (x) + k 2 (1 − ρ
p
n (x)),
(9.67)
where E 1 , E 2 are Young’s moduli of the materials, and k 1 , k 2 are the heat transfer
coefficients of the materials.
The exponent p ≥ 1 is used as a penalty factor and the increase in p leads to a
clearer decomposition of the material phases (typically p = 4 or p = 5 [113]).
Usually, for the convenience of calculations, optimization problems with respect
to the maximum are reduced to the minimum.
365
Now the boundary conditions [112] for the characteristic functions T i for a representative cell which take the classical form are shown in Fig. 9.9a, b and have the
following form:
for k
e
11
T 1 (0, y 2 ) = 0, T 1 (1, y 2 ) = 1,
∂ T 1
∂ y 1
(y 1 , 0) =
∂ T 1
∂ y 1
(y 1 , 1) = 0;
for k
e
22
T 2 (y 1 , 0) = 0, T 2 (y 1 , 1) = 1,
∂ T 2
∂ y 2
(0, y 2 ) =
∂ T 2
∂ y 2
(1, y 2 ) = 0.
9.6.4 Topological Optimization
The method of homogenization [44–46] is one of the most effective approaches to
compute the global physical properties of a composite such as the bulk stiffness
modulus, the shear modulus or the heat transfer coefficient.
Having both the elastic and thermal homogenization determined, one can define
an optimization problem aimed at defining the optimal topology for thermoelastic
problems. Such problems are particularly interesting when the elastic and thermal
properties of materials strongly compete with each other and the optimal topologies
for individual problems differ significantly from each other. To solve the problem, it
is necessary to simultaneously take into account both the elastic and thermal material
properties.
Let us assume that the elementary periodic cell is divided into finite elements and
each finite element has its own variable density ρ n (n = 1, . . . , N ). Following the
rule of SIMP [112], where the role of the control variable is played by an artificially
introduced density ρ n (x) the corresponding Young’s modulus E(x) and heat transfer
coefficient k(x) for the two-component composite in the nth finite element can be
written as
E n (x) = E 1 ρ
p
n (x) + E 2 (1 − ρ
p
n (x)),
k n (x) = k 1 ρ
p
n (x) + k 2 (1 − ρ
p
n (x)),
(9.67)
where E 1 , E 2 are Young’s moduli of the materials, and k 1 , k 2 are the heat transfer
coefficients of the materials.
The exponent p ≥ 1 is used as a penalty factor and the increase in p leads to a
clearer decomposition of the material phases (typically p = 4 or p = 5 [113]).
Usually, for the convenience of calculations, optimization problems with respect
to the maximum are reduced to the minimum.