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Topology optimization approaches

A comparative review

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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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Notes

  1. 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).

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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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Authors and Affiliations

  1. Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, 2800 Kgs, Lyngby, Denmark

    Ole Sigmund

  2. Aerospace Engineering Sciences, University of Colorado, Boulder, CO, 80309-0429, USA

    Kurt Maute

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  1. Ole Sigmund
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  2. Kurt Maute
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Correspondence to Ole Sigmund.

Additional information

Grants: The first author appreciates the support from the Villum Foundation through the grant: “NextTop”. The second author acknowledges the support of the National Science Foundation under grant EFRI-1038305. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organization. This work was partially performed during the first authors sabbatical leave at University of Colorado Boulder.

Appendix

Appendix

1.1 A Matlab threshold code

The Matlab script shown below is intended as a post-processing step that converts a grey scale design obtained with the 99-line code (Sigmund 2001a) to a discrete design satisfying the volume fraction constraint.

In the script, the total volume includes the volume taken up by low-density elements. If the discrete approach does not include low-density elements, the third line above can simply be changed to

‘For the more compact 88-line code (Andreassen et al. 2011) the FE-part of above script should be substituted with the following

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Sigmund, O., Maute, K. Topology optimization approaches. Struct Multidisc Optim 48, 1031–1055 (2013). https://doi.org/10.1007/s00158-013-0978-6

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