Editing Topology optimization (section)

== Implementation methodologies ==
There are various implementation methodologies that have been used to solve topology optimization problems.

=== Solving with discrete/binary variables ===
Solving topology optimization problems in a discrete sense is done by discretizing the design domain into finite elements. The material densities inside these elements are then treated as the problem variables. In this case material density of one indicates the presence of material, while zero indicates an absence of material. Owing to the attainable topological complexity of the design being dependent on the number of elements, a large number is preferred. Large numbers of finite elements increases the attainable topological complexity, but come at a cost. Firstly, solving the FEM system becomes more expensive. Secondly, algorithms that can handle a large number (several thousands of elements is not uncommon) of discrete variables with multiple constraints are unavailable. Moreover, they are impractically sensitive to parameter variations.<ref>{{Cite journal |doi = 10.1007/s00158-013-0978-6|title = Topology optimization approaches |journal = Structural and Multidisciplinary Optimization|volume = 48|issue = 6|pages = 1031–1055|year = 2013 |author2-link=Kurt Maute|last1 = Sigmund|first1 = Ole|last2 = Maute|first2 = Kurt|s2cid = 124426387 }}</ref> In literature problems with up to 30000 variables have been reported.<ref>{{Cite journal |doi = 10.1007/BF01197709|title = Topology optimization using a dual method with discrete variables|journal = Structural Optimization|volume = 17|pages = 14–24|year = 1999|last1 = Beckers|first1 = M. |s2cid = 122845784|url=http://empslocal.ex.ac.uk/people/staff/reverson/uploads/MoodSwings/beckers.pdf}}</ref>

=== Solving the problem with continuous variables ===
The earlier stated complexities with solving topology optimization problems using [[binary data|binary variables]] has caused the community to search for other options. One is the modelling of the densities with continuous variables. The material densities can now also attain values between zero and one. Gradient based algorithms that handle large amounts of continuous variables and multiple constraints are available. But the material properties have to be modelled in a continuous setting. This is done through interpolation. One of the most implemented interpolation methodologies is the ''' Solid Isotropic Material with Penalisation''' method (SIMP).<ref>{{Cite journal |doi = 10.1007/BF01650949|title = Optimal shape design as a material distribution problem|journal = Structural Optimization|volume = 1|issue = 4|pages = 193–202|year = 1989|last1 = Bendsøe|first1 = M. P.|s2cid = 18253872}}</ref><ref name="book">[https://books.google.com/books?id=NGmtmMhVe2sC], a monograph of the subject.</ref> This interpolation is essentially a power law <math> E \;=\; E_0 \,+\, \rho^p (E_1 - E_0) </math>. It interpolates the Young's modulus of the material to the scalar selection field. The value of the penalisation parameter <math>p</math> is generally taken between <math> [1,\, 3]</math>.  This has been shown to confirm the micro-structure of the materials.<ref>{{Cite journal |doi = 10.1007/s004190050248|title = Material interpolation schemes in topology optimization|journal = Archive of Applied Mechanics |volume = 69|issue = 9–10|pages = 635–654|year = 1999|last1 = Bendsøe|first1 = M. P.|last2 = Sigmund|first2 = O.|bibcode = 1999AAM....69..635B |s2cid = 11368603|url=http://www.giref.ulaval.ca/~deteix/bois/documents_references/bendsoe1999.pdf}}</ref> In the SIMP method a lower bound on the Young's modulus is added, <math> E_0 </math>, to make sure the derivatives of the objective function are non-zero when the density becomes zero. The higher the penalisation factor, the more SIMP penalises the algorithm in the use of non-binary densities. Unfortunately, the penalisation parameter also introduces non-convexities.<ref>van Dijk, NP. Langelaar, M. van Keulen, F. ''[http://www1.dem.ist.utl.pt/engopt2010/Book_and_CD/Papers_CD_Final_Version/pdf/03/01270-01.pdf Critical study of design parameterization in topology optimization; The influence of design parameterization on local minima].''. 2nd International Conference on Engineering Optimization, 2010</ref>


=== Commercial software ===
There are several commercial topology optimization software on the market. Most of them use topology optimization as a hint how the optimal design should look like, and manual geometry re-construction is required. There are a few solutions which produce optimal designs ready for Additive Manufacturing.<ref>{{Cite web |date=23 September 2021 |title=Topological optimization software for 3d printing |url=https://www.3dnatives.com/en/topological-optimization-software-for-3d-printing-230920214/}}</ref>