Maria Bayard Dühring - Solid Mechanics

16 Chapter 3 Topology optimization applied to time-harmonic propagating waves
3.4 Sensitivity analysis
The design variables are updated by a gradient based optimization algorithm, see
section 3.5, and the derivatives with respect to the design variables of the objective
and the constraint functions must be evaluated. They indicate how much the
function will change if a certain design variable is changed an infinitesimal quantity.
These derivatives can be calculated as the design variable is introduced as a
continuous field. The design variable field ξ is discretized using finite element basis
functions {φJ+1,n(r)} in a similar way as for the dependent variables
Nd
ξ(r) = ξnφJ+1,n(r). (3.8)
n=1
The degrees of freedom are assembled in the vector ξ = {ξ1, ξ2, ...ξNd }T , and typically
zero or first order Lagrange elements are used.
The complex field vector uk is via (2.5) an implicit function of the design vari-
denote the
ables, which is written as uk(ξ) = uR k (ξ) + iuIk (ξ), where uR k and uIk real and the imaginary parts of uk. Thus the derivative of the objective function
Φ = Φ(uR k (ξ), uIk (ξ), ξ) is given by the following expression found by the chain rule
dΦ
dξ
= ∂Φ
∂ξ +
J
∂Φ
k=1
∂u R k
∂u R k
∂ξ
+ ∂Φ
∂u I k
∂uI
k
. (3.9)
∂ξ
As uk is an implicit function of ξ the derivatives ∂uR k /∂ξ and ∂uIk /∂ξ are not known
directly. The sensitivity analysis is therefore done by employing an adjoint method
where the unknown derivatives are eliminated at the expense of determining adjoint
and complex variable fields [65]. From equation (2.5) it is known that J j=1 Skjuj −
fk = 0, and therefore equation J j=1 ¯ Skjūj −¯fk = 0 is also valid. The overbar denotes
the complex conjugate. It is assumed that the load does not depend on the design
variables such that the derivative of J
j=1 Skjuj and J
j=1 ¯ Skjūj with respect to ξ
are also zero. When they are multiplied with the adjoint variable fields 1
2 λk and
1
2 ¯ λk, respectively, and added to (3.9) it does not change the value of the derivative
dΦ
dξ =∂Φ
∂ξ +
J
∂Φ
∂u
k=1
R ∂u
k
R k ∂Φ
+
∂ξ ∂uI ∂u
k
I
k
∂ξ
J
1
+
2 λT
J
d
k Skjuj +
dξ
j=1
1
2 ¯ λ T
J
d
¯Skjūj
k
. (3.10)
dξ
j=1
k=1