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The level set method, based on the classical shape derivative,
is seen to easily handle boundary propagation with topological
changes. However, in practice it does not allow for the nucleation
of new holes (at least in 2-d). For this reason, we couple the
level set method with another approach, known as the bubble or
topological gradient method, which is precisely designed for
introducing new holes in the optimization process. The coupling
of these two method yields an efficient algorithm which can escape
from local minima in a given topological class of shapes.
Both methods relies on a notion of gradient computed through
an adjoint analysis, and have a low CPU cost since they
capture a shape on a fixed Eulerian mesh. The main advantage
of our coupled algorithm is to make the resulting optimal
design largely independent of the initial guess.
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