Maria Bayard Dühring - Solid Mechanics

The degrees of freedom (dofs) corresponding to the two fields are assembled in the vectors ^p ¼
f ^p 1; ^p 2; ...; ^p Ng T and n ¼fx1; x2; ...; xNd gT . For the model in 2D a triangular element mesh is employed and
tetrahedral elements are used in 3D. Quadratic Lagrange elements are used for the complex pressure
amplitude ^p to obtain high accuracy in the solution and for the design variable x linear Lagrange elements
are utilized.
The commercial program Comsol Multiphysics with Matlab [25] is employed for the finite element analysis.
This results in the discretized equation
S^p ¼ f, (12)
where S is the system matrix and f is the load vector which are both complex valued.
To update the design variables in the optimization algorithm the derivatives with respect to the design
variables of the objective and the constraint function must be evaluated. This is possible as the design variable
is introduced as a continuous field. The complex sound pressure vector ^p is via Eq. (12) an implicit function of
the design variables, which is written as ^pðnÞ ¼^p RðnÞþi^p IðnÞ, where ^p R and ^p I denote the real and the
imaginary part of ^p. Thus the derivative of the objective function F ¼ Fð^p RðnÞ; ^p IðnÞ; nÞ is given by the
following expression found by the chain rule
dF
dn
qF qF q^p R qF q^p I
¼ þ þ . (13)
qn q^p R qn q^p I qn
As ^p is an implicit function of n the derivatives q^p R=qn and q^p I=qn 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 an adjoint and complex variable field k from the adjoint equation
where
qF
q^p R
i qF
q^p I
S T k ¼
qF
q^p R
i qF
q^p I
T
, (14)
¼ 1
R
Oop dr
Z
ð2 ^p R i2 ^p IÞf1;n dr. (15)
Oop
The sensitivity analysis follows the standard adjoint sensitivity approach [26]. For further details of the adjoint
sensitivity method applied to wave propagation problems, the reader is referred to Ref. [27]. Eq. (13) for the
derivative of the objective function then reduces to
dF
dn
¼ qF
qn
qS
þ Re kT ^p . (16)
qn
Finally, the derivative of the constraint function with respect to one of the design variables is
q 1
R
qxn Od dr
Z
!
xðrÞ dr b ¼
Od
1
R
Od dr
Z
f2;nðrÞ dr. (17)
Od
The vectors qF=qn, R
Oop ð2 ^p R i2 ^p IÞf1;n dr and R
Od f2;nðrÞ dr as well as the matrix qS=qn are assembled in
Comsol Multiphysics as described in Eq. [28].
2.5. Practical implementation
ARTICLE IN PRESS
M.B. Dühring et al. / Journal of Sound and Vibration 317 (2008) 557–575 561
The optimization problem (8)–(10) is solved using the Method of Moving Asymptotes, MMA [29] which is
an algorithm that uses information from the previous iteration steps and gradient information. To fulfill the
volume constraint from the first iteration of the optimization procedure the initial design is usually chosen as a
uniform distribution of material with the volume fraction b. It should be emphasized that the final design
depends both on the initial design and the allowable amount of material to be placed and is therefore
dependent on b. Here the best solutions out of several tries will be presented. To make the model more realistic
and to minimize local resonance effects a small amount of mass-proportional damping is added.