9.4 Applying Topological Cell Optimization
351
where ε
th
e stands for the vector of thermal deformation for an element of the form
ε
th
e (ρ e ) = α (ρ e ) T e ϕ
T
.
(9.26)
Here the following notation is employed: α (ρ e )—temperature coefficient of linear expansion which depends on the element density; T e —temperature change of
element e (here understood as the averaged value of mode temperatures); ϕ—vector
[110] or [111000] corresponding to 2D and 3D problems, respectively.
Substitution of (9.24) and (9.26) into (9.25) yields
f
th
e (ρ e ) =
e
E (ρ e ) α (ρ e ) B
T
e
¯
C T e ϕ
T d.
(9.27)
Observe that E (ρ e ) and α (ρ e ) in (9.27) depend on the density variables, and consequently, require interpolation of those properties. In order to simplify the problem,
we unify those parameters within the following one coefficient of the heat load
β (ρ e ) = E (ρ e ) α (ρ e ) .
(9.28)
In what follows, the mentioned parameter uniquely characterizes the material
property m and f
th
e (ρ e ) can be recast to the following form:
f
th
e (ρ e ) =
e
β (ρ e ) B
T
e
¯
CT e ϕ
T d.
(9.29)
Finally, the global vector of the heating load includes contributions of all elements,
and hence
F
th
(ρ) =
N
e=1
f
th
e (ρ e ).
(9.30)
As earlier, the vector of heat load of an element for the non-optimized subspace
can be obtained from the previous relations by taking ρ e = 1.
9.4.5 Interpolating Schemes
As it has been mentioned in Sect. 9.3.1, in the case of the load being design depended
and particularly the thermal load, there appear difficulties with proper distribution
of material into phases under the SIMP interpolation scheme. Therefore, RAMP has
been recommended. The stiffness E (ρ e ) and heat expansion coefficient β (ρ e ) are
interpolated, respectively, by the following relations:
351
where ε
th
e stands for the vector of thermal deformation for an element of the form
ε
th
e (ρ e ) = α (ρ e ) T e ϕ
T
.
(9.26)
Here the following notation is employed: α (ρ e )—temperature coefficient of linear expansion which depends on the element density; T e —temperature change of
element e (here understood as the averaged value of mode temperatures); ϕ—vector
[110] or [111000] corresponding to 2D and 3D problems, respectively.
Substitution of (9.24) and (9.26) into (9.25) yields
f
th
e (ρ e ) =
e
E (ρ e ) α (ρ e ) B
T
e
¯
C T e ϕ
T d.
(9.27)
Observe that E (ρ e ) and α (ρ e ) in (9.27) depend on the density variables, and consequently, require interpolation of those properties. In order to simplify the problem,
we unify those parameters within the following one coefficient of the heat load
β (ρ e ) = E (ρ e ) α (ρ e ) .
(9.28)
In what follows, the mentioned parameter uniquely characterizes the material
property m and f
th
e (ρ e ) can be recast to the following form:
f
th
e (ρ e ) =
e
β (ρ e ) B
T
e
¯
CT e ϕ
T d.
(9.29)
Finally, the global vector of the heating load includes contributions of all elements,
and hence
F
th
(ρ) =
N
e=1
f
th
e (ρ e ).
(9.30)
As earlier, the vector of heat load of an element for the non-optimized subspace
can be obtained from the previous relations by taking ρ e = 1.
9.4.5 Interpolating Schemes
As it has been mentioned in Sect. 9.3.1, in the case of the load being design depended
and particularly the thermal load, there appear difficulties with proper distribution
of material into phases under the SIMP interpolation scheme. Therefore, RAMP has
been recommended. The stiffness E (ρ e ) and heat expansion coefficient β (ρ e ) are
interpolated, respectively, by the following relations: