Maria Bayard Dühring - Solid Mechanics

18 Chapter 3 Topology optimization applied to time-harmonic propagating waves
The two expressions (3.16) and (3.17) are equivalent, which is seen by taking the
complex conjugate on both sides of expression (3.17). This means that when λj is
computed by equation (3.16) it will also fulfill equation (3.17). Equation (3.16) is
transposed and it is used that Skj is symmetric
J
T ∂Φ
j=1
Skjλj =
∂u R k
− i ∂Φ
∂u I k
. (3.18)
The expression in (3.18) can be solved as one system in a similar way as the discretized
problem in (2.5). In the case where wk in the objective function (3.3) is
equal to the field variable uk the right hand side of equation (3.18) is found by the
expression
∂Φ
∂u R k,n
− i ∂Φ
∂u I k,n
=
=
Ωo
Ωo
∂Φ
∂u R k
∂uR k
∂uR k,n
In the case where wk indicates the derivative ∂uk
∂xk
(3.18) is found by
∂Φ
∂u R k,n
=
Ωo
− i ∂Φ
∂u I k,n
=
2w R k − i2w I k
∂Φ
Ωo ∂wR k
∂φk,n
∂xk
− i ∂Φ
∂u I k
∂u I k
∂u I k,n
dr
(2u R k − i2u I k)φk,ndr. (3.19)
− i ∂Φ
dr =
∂w I k
Ωo
∂φk,n
∂xk
2 ∂uR k
∂xk
the right hand side of equation
dr
− i2 ∂uI
k ∂φk,n
dr. (3.20)
∂xk ∂xk
The derivatives ∂uk can be calculated automatically by Comsol Multiphysics. The
∂xk
adjoint variable fields are now determined such that the two last terms in (3.13)
vanish and the derivative of the objective function reduces to
dΦ ∂Φ
=
dξ ∂ξ +
J
1
2 λT
J ∂Skj
k
∂ξ uj + 1
2 ¯ λ T
J ∂
k
¯ Skj
∂ξ ūj
, (3.21)
k=1
and can be rewritten in the form
dΦ
dξ
= ∂Φ
∂ξ +
j=1
J
ℜ
k=1
λ T
k
J
j=1
j=1
∂Skj
∂ξ uj
. (3.22)
Finally, the derivative of the constraint function with respect to one of the design
variables is
∂ 1
∂ξn Ωd dr
ξ(r)dr − β =
Ωd
1
Ωd dr
φJ+1,n(r)dr. (3.23)
Ωd
The mentioned vectors ∂Φ/∂ξ,
Ωo (2uRk − i2uI
k )φk,ndr, 2 Ωo
∂uR k
∂xk − i2 ∂uI
k
∂φk,n
∂xk ∂xk dr
and
Ωd φJ+1,n(r)dr as well as the matrix ∂Skj/∂ξ are assembled in Comsol Multiphysics
as described in [66].