Appl Math Optim 52:263–277 (2005)
2005 Springer Science+Business Media, Inc.
The Limits of Porous Materials in the
Topology Optimization of Stokes Flows
Center for Aerospace Structures, Department of Aerospace Engineering Sciences,
University of Colorado, Boulder, CO 80309-0429, USA
Abstract. We consider a problem concerning the distribution of a solid material
in a given bounded control volume with the goal to minimize the potential power
of the Stokes ﬂow with given velocities at the boundary through the material-free
part of the domain. We also study the relaxed problem of the optimal distribution of
the porous material with a spatially varying Darcy permeability tensor, where the
governing equations are known as the Darcy–Stokes, or Brinkman, equations. We
show that the introduction of the requirement of zero power dissipation due to the
ﬂow through the porous material into the relaxed problem results in it becoming a
well-posed mathematical problem, which admits optimal solutions that have extreme
permeability properties (i.e., assume only zero or inﬁnite permeability); thus, they
are also optimal in the original (non-relaxed) problem.
Two numerical techniques are presented for the solution of the constrained
problem. One is based on a sequence of optimal Brinkman ﬂows with increasing
viscosities, from the mathematical point of view nothing but the exterior penalty
approach applied to the problem. Another technique is more special, and is based
on the “sizing” approximation of the problem using a mix of two different porous
materials with high and low permeabilities, respectively.
This paper thus complements the study of Borrvall and Petersson (Internat.
J. Numer. Methods Fluids, vol. 41, no. 1, pp. 77–107, 2003), where only sizing
optimization problems are treated.
Key Words. Topology optimization, Fluid mechanics, Stokes ﬂow.
AMS Classiﬁcation. 49J20, 49J45, 76D55, 62K05.
This research was supported by the Swedish Research Council (Grant 621-2002-5780).