Fluid flow topology optimization in PolyTop: stability and computational implementation

We present a Matlab implementation of topology optimization for fluid flow problems in the educational computer code PolyTop (Talischi et al. 2012b). The underlying formulation is the well-established porosity approach of Borrvall and Petersson (2003), wherein a dissipative term is introduced to impede the flow in the solid (non-fluid) regions. Polygonal finite elements are used to obtain a stable low-order discretization of the governing Stokes equations for incompressible viscous flow. As a result, the same mesh represents the design field as well as the velocity and pressure fields that characterize its response. Owing to the modular structure of PolyTop, incorporating new physics, in this case modeling fluid flow, involves changes that are limited mainly to the analysis routine. We provide several numerical examples to illustrate the capabilities and use of the code. To illustrate the modularity of the present approach, we extend the implementation to accommodate alternative formulations and cost functions. These include topology optimization formulations where both viscosity and inverse permeability are functions of the design; and flow control where the velocity at a certain location in the domain is maximized in a prescribed direction.