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I am studying the different topology optimization methods. There are numerous resources out there but when it comes to comparing different algorithms, in terms of strengths and weaknesses most of these resources remain very ambiguous and don't give a clear feedback on why a method is more advantageous compared to one or more other methods.

Right now I am working on level set method for topology optimization and found some very good articles about it but again I don't see a clear explanation on why level set could be a potentially good alternative.

What are the advantages that we can't find in all other methods and is exclusive to level set methods?

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First off, let me make a subtle distinction in the implementation of level set methods in topology optimization. In the literature, you will see this method implemented using shape derivatives as in here or using material derivatives (through the use of a Heaviside function) as in here. In my experience, using shape derivatives works better, but material derivatives allow you to switch optimization algorithms more easily. Using shape derivatives rely more on the Hamilton-Jacobi equation as optimizer.

In topology optimization, the other method commonly used is the density/volume-fraction method (link). Advantages:

  • Easy to implement.
  • It can change topologies more arbitrarily

Disadvantages:

  • Harder to obtain clear material boundaries.
  • Slower convergence due to nonlinearities induced by the penalization techniques.

Advantages of level set method:

  • Material boundary clearly defined.
  • No need for penalization techniques to obtain a discrete solution.
  • Faster convergence when implemented correctly (using shape derivatives)

Disadvantages:

  • It cannot nucleate new holes from scratch (it can just merge holes). There are some methods to nucleate new holes, but are physics dependent (topological derivatives).
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