Editing Topology optimization (section)

=== 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>