Abstract:
We consider shape optimization problems for channel-like domains
occupied by stationary Stokes flow.
The aim is to design the lateral walls of the domain such that a
tracking type functional in terms of the velocity and the pressure
is minimized. The design variables are the Bézier control
points of a composite Bézier curve representation of the
lateral walls. After reformulating the minimization problem within a
primal-dual interior-point framework and considering the associated optimality
conditions, one is lead to a parameter-dependent nonlinear system
which gives rise to the so-called central path. We present a
predictor-corrector based path-following method with an adaptive choice
of the continuation steplength originating from the affine-covariant
convergence theory for Newton-like methods.
This scheme can be extended to a multigrid predictor-corrector method
with nested iterations in the continuation step and a Newton-type
corrector featuring a multigrid solver for the Stokes problems
with respect to the primal and dual variables.
Numerical examples illustrate the efficiency of the multigrid version of
the optimization algorithm.
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