WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Optimal material distribution – topology optimization 5
original design
optimized design variables
xe = 0 xe = 1
FE model w. design variables
optimized design
Figure 1.3 Illustration of the topology optimization method for optimizing the distribution
of two materials in a bandgap structure. Top left: the original design with the dark
color indicating air holes and light/dark gray indicating a dielectric material. Top right:
finite element (FE) model with one continuous design variable xe per element, for which
xe = 0 represent air (light gray) and xe = 1 represent dielectric material (black). Bottom
left: FE model with an optimized set of design variables taking either the value 0 or 1.
Bottom right: the fabricated optimized design.
measure (objective function), e.g. maximizes the wave transmission through the
structure. This procedure is based on the use of analytically computed gradients
(the sensitivity of the objective function wrt. the design variables) and the use of a
mathematical programming tool, as e.g. Krister Svanberg’s MMA (Svanberg, 1987).
The bottom left figure shows the FE model with the optimized element design
variables corresponding to a new distribution of the two materials. Finally, the
bottom right figure shows the fabricated optimized structure.
As a general feature of topology optimization certain numerical techniques must
be applied in order to generate usable optimized structures. Penalization is used to
avoid design variables in the final design with intermediate values (0 < xe < 1) that
do not correspond to any of the specified materials. Regularization is applied to