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Structural and Multidisciplinary Optimization

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01-12-2013 | Review Article

Topology optimization approaches

A comparative review

Authors: Ole Sigmund, Kurt Maute

Published in: Structural and Multidisciplinary Optimization | Issue 6/2013

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Abstract

Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsøe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.

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The one material formulation can easily be extended to multiple material phases—see e.g. Sigmund and Torquato (1997); Bendsøe and Sigmund (1999); Sigmund (2001b).

Although the discretized optimization problem (2) is a solid-void optimization problem it is for computational reasons common to treat it as a “solid-almost void” problem, meaning that void is mimicked by a very soft material, hence avoiding to have to remesh or renumber the finite element mesh in between iterations. Hence throughout the paper, unless otherwise noted, \(\rho =0\) must be read as \(\rho =\rho _{min}\), where \(\rho _{min}\) is a small number.

Actually, topology optimization approaches often work best with active volume constraints. Depending on the physical problem considered, superfluous material may create non-physical effects or may obstruct the free movement of structural boundaries in turn restricting convergence to (near)global minima.

Note that there exist approaches that use multiple projections, e.g. multiphase projection (Guest 2009b) and advanced morphology filtering (Sigmund 2007), however, we include them under “three-field approaches” by counting the projection steps as one, no matter how many times they are applied.

The compliance increases until the volume fraction has been reached and decreases after. Hence, if the average energy before and after feasibility becomes equal the algorithm terminates prematurely.

These problems can partially be avoided by performing the optimization on consecutively refined meshes, however, for many physical problems that are more complex than simple compliance minimization (c.f. wave propagation problems as e.g. reviewed in Jensen andSigmund 2011) and electrostatic actuators (Qian and Sigmund 2012) this is not a viable approach.

Note that without filtering the boundaries will not move and hence the design cannot move away from the solid bar starting guess.

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Title: Topology optimization approaches
A comparative review
Authors: Ole Sigmund
Kurt Maute
Publication date: 01-12-2013
Publisher: Springer Berlin Heidelberg
Published in: Structural and Multidisciplinary Optimization / Issue 6/2013
Print ISSN: 1615-147X
Electronic ISSN: 1615-1488
DOI: https://doi.org/10.1007/s00158-013-0978-6

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