## Abstract

Computational procedures for topology optimization of continuum
problems using a material distribution method are typically based
on the application of the finite element method (FEM) (see, e.g. [1]). In the present work we study a computational
framework based on the finite volume method (FVM, see, e.g.
[2]) in order to develop methods for topology design for
applications where conservation laws are critical such that
element--wise conservation in the discretized models has a high
priority. This encompasses
problems involving for example mass and heat transport.
The work described in this presentation is focused on a
prototype model for topology optimization of steady heat diffusion.
This allows for a study of the basic ingredients in working with FVM
methods when dealing with topology optimization problems.
The FVM and FEM based formulations differ both in how one
computes the design derivative of the system matrix $\mathbf K$ and
in how one computes the discretized version of certain objective
functions. Thus for a cost function for minimum dissipated energy
(like minimum compliance for an elastic structure) one obtains an
expression $ c = \mathbf u^\T \tilde{\mathbf K} \mathbf u $, where
$\tilde{\mathbf K}$ is different from $\mathbf K $; in a FEM scheme
these matrices are equal following the principle of virtual work.
Using a staggered mesh and averaging procedures consistent
with the FVM the checkerboard problem is eliminated. Two averages
are compared to FE solutions, being the arithmetic and harmonic
average with the latter being the well known Reuss lower bound.
[1] Bendsøe, MP and Sigmund, O 2004: Topology Optimization - Theory, Methods, and Applications. Berlin Heidelberg: Springer Verlag
[2] Versteeg, HK and Malalasekera, W 1995: An introduction to Computational Fluid Dynamics: the Finite Volume Method. London: Longman Scientific Technical

Original language | English |
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Publication date | 2005 |

Publication status | Published - 2005 |

Event | M.I.T. conference - Boston, USA Duration: 1 Jan 2005 → … Conference number: 3 |

### Conference

Conference | M.I.T. conference |
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Number | 3 |

City | Boston, USA |

Period | 01/01/2005 → … |