Editing Topology optimization (section)

== Examples ==
[[File:Checkerboards in Topology Optimization.tif|thumb|Checker Board Patterns are shown in this result]]
[[File:Topology Optimization with filtereing.tif|thumb|Topology optimization result when filtering is used]]
[[File:Cantilvr 3d etaopt gsf 050.png|thumb|Topology optimization of a compliance problem]]

=== Structural compliance ===
{{unreferenced section|date=December 2018}}
A stiff structure is one that has the least possible displacement when given certain set of boundary conditions. A global measure of the displacements is the [[strain energy]] (also called [[Stiffness#Compliance|compliance]]) of the structure under the prescribed boundary conditions. The lower the strain energy the higher the stiffness of the structure. So, the objective function of the problem is to minimize the strain energy.

On a broad level, one can visualize that the more the material, the less the deflection as there will be more material to resist the loads. So, the optimization requires an opposing constraint, the volume constraint. This is in reality a cost factor, as we would not want to spend a lot of money on the material. To obtain the total material utilized, an integration of the selection field over the volume can be done.

Finally the elasticity governing differential equations are plugged in so as to get the final problem statement.
:<math>\min_{\rho}\; \int_{\Omega} \frac{1}{2} \mathbf{\sigma}:\mathbf{\varepsilon} \,\mathrm{d}\Omega</math>

subject to:
*<math> \rho \,\in\, [0,\, 1] </math>
*<math> \int_{\Omega} \rho\, \mathrm{d}\Omega \;\leq\; V^*</math>
*<math> \mathbf{\nabla}\cdot\mathbf{\sigma} \,+\, \mathbf{F} \;=\; {\mathbf{0}} </math>
*<math> \mathbf{\sigma} \;=\; \mathsf{C}:\mathbf{\varepsilon}</math>

But, a straightforward implementation in the finite element framework of such a problem is still infeasible owing to issues such as:
# Mesh dependency—Mesh Dependency means that the design obtained on one mesh is not the one that will be obtained on another mesh. The features of the design become more intricate as the mesh gets refined.<ref>{{cite journal |last1=Allaire |first1=Grégoire |last2=Henrot |first2=Antoine |title=On some recent advances in shape optimization |journal=Comptes Rendus de l'Académie des Sciences |date=May 2001 |volume=329 |issue=5 |pages=383–396 |doi=10.1016/S1620-7742(01)01349-6 |url=https://linkinghub.elsevier.com/retrieve/pii/S1620774201013496 |access-date=2021-09-12 |series=Series IIB - Mechanics |publisher=Elsevier |bibcode=2001CRASB.329..383A |language=en |issn=1620-7742}}</ref>
# Numerical instabilities—The selection of region in the form of a chess board.<ref>{{cite journal |last1=Shukla |first1=Avinash |last2=Misra |first2=Anadi |last3=Kumar |first3=Sunil |title=Checkerboard Problem in Finite Element Based Topology Optimization |journal=International Journal of Advances in Engineering & Technology |date=September 2013 |volume=6 |issue=4 |pages=1769–1774|publisher=CiteSeer |citeseerx=10.1.1.670.6771 | url=http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=5BECA0B81B176391016659D59276F7FE?doi=10.1.1.670.6771&rep=rep1&type=pdf |access-date=2022-02-14 |language=en |issn=2231-1963}}</ref>

Some techniques such as [[Kernel_(image_processing)|filtering]] based on image processing<ref>{{cite journal |last1=Bourdin |first1=Blaise |title=Filters in topology optimization |journal=International Journal for Numerical Methods in Engineering |date=2001-03-30 |volume=50 |issue=9 |pages=2143–2158 |doi=10.1002/nme.116 |url=http://doi.wiley.com/10.1002/nme.116 |access-date=2020-08-02 |publisher=Wiley |bibcode=2001IJNME..50.2143B |s2cid=38860291 |language=en |issn=1097-0207}}</ref> are currently being used to alleviate some of these issues. Although it seemed like this was purely a heuristic approach for a long time, theoretical connections to nonlocal elasticity have been made to support the physical sense of these methods.<ref>{{cite journal |last1=Sigmund |first1=Ole |last2=Maute |first2=Kurt |title=Sensitivity filtering from a continuum mechanics perspective |journal=Structural and Multidisciplinary Optimization |date=October 2012 |volume=46 |issue=4 |pages=471–475 |doi=10.1007/s00158-012-0814-4 |url=http://link.springer.com/10.1007/s00158-012-0814-4 |access-date=2021-06-17 |publisher=Springer |s2cid=253680268 |issn=1615-1488}}</ref>

=== Multiphysics problems ===

==== Fluid-structure-interaction ====
[[Fluid–structure interaction|Fluid-structure-interaction]] is a strongly coupled phenomenon and concerns the interaction between a stationary or moving fluid and an elastic structure. Many engineering applications and natural phenomena are subject to fluid-structure-interaction and to take such effects into consideration is therefore critical in the design of many engineering applications. Topology optimisation for fluid structure interaction problems has been studied in e.g. references<ref>{{Cite journal |doi = 10.1002/nme.2777|title = Topology optimization for stationary fluid-structure interaction problems using a new monolithic formulation|journal = International Journal for Numerical Methods in Engineering|volume = 82|issue = 5|pages = 591–616|year = 2010|last1 = Yoon|first1 = Gil Ho|bibcode = 2010IJNME..82..591Y| s2cid=122993997 }}</ref><ref>{{Cite journal |doi = 10.1016/j.finel.2017.07.005|title = Evolutionary topology optimization for structural compliance minimization considering design-dependent FSI loads|journal = Finite Elements in Analysis and Design|volume = 135|pages = 44–55|year = 2017|last1 = Picelli|first1 = R.|last2 = Vicente|first2 = W.M.|last3 = Pavanello|first3 = R.}}</ref><ref>{{Cite journal |doi = 10.1007/s00158-016-1467-5|title = An immersed boundary approach for shape and topology optimization of stationary fluid-structure interaction problems|journal = Structural and Multidisciplinary Optimization|volume = 54|issue = 5|pages = 1191–1208|year = 2016|last1 = Jenkins|first1 = Nicholas|last2 = Maute|first2 = Kurt|s2cid = 124632210}}</ref> and.<ref name=Lundgaard_FSI>{{Cite journal | doi=10.1007/s00158-018-1940-4| title=Revisiting density-based topology optimization for fluid-structure-interaction problems| journal=Structural and Multidisciplinary Optimization| volume=58| issue=3| pages=969–995| year=2018| last1=Lundgaard| first1=Christian| last2=Alexandersen| first2=Joe| last3=Zhou| first3=Mingdong| last4=Andreasen| first4=Casper Schousboe| last5=Sigmund| first5=Ole| s2cid=125798826| url=https://backend.orbit.dtu.dk/ws/files/163153999/grayscale_Lundgaard_C._Alexandersen_J._Zhou_M._Andreasen_C._S._Sigmund_O_2018_.pdf}}</ref> Design solutions solved for different Reynolds numbers are shown below. The design solutions depend on the fluid flow with indicate that the coupling between the fluid and the structure is resolved in the design problems.

{{multiple image
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 | caption1 = Design solution and velocity field for Re=1

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 | caption2  = Design solution and velocity field for Re=5

 | image3 = Fluid-structure-interaction-pressure-field-topology-optimization.png
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 | caption3  = Design solution and pressure field for Re=10

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 | caption4  = Design solution and pressure field for Re=40

 | footer = Design solutions for different Reynolds number for a wall inserted in a channel with a moving fluid.
}}

[[File:Wall-flow-problem-topology-optimization-for-fluid-structure-interaction-problems.png|thumb|Sketch of the well-known wall problem. The objective of the design problem is to minimize the structural compliance.]]

[[File:Fluid-structure-interaction-design-evolution.gif|thumb|Design evolution for a fluid-structure-interaction problem from reference.<ref name=Lundgaard_FSI /> The objective of the design problem is to minimize the structural compliance. The fluid-structure-interaction problem is modelled with Navier-Cauchy and Navier-Stokes equations.]]

==== Thermoelectric energy conversion ====

[[File:Design-sketch.png|thumb|A sketch of the design problem. The aim of the design problem is to spatially distribute two materials, Material A and Material B, to maximise a performance measure such as cooling power or electric power output]]

[[File:Topology-optimization-off-diagonal-design-evolution.gif|thumb|Design evolution for an off-diagonal thermoelectric generator. The design solution of an optimisation problem solved for electric power output. The performance of the device has been optimised by distributing [[Skutterudite]] (yellow) and [[bismuth telluride]] (blue) with a density-based topology optimisation methodology. The aim of the optimisation problem is to maximise the electric power output of the thermoelectric generator.]]

[[File:Evolution-design solution.gif|thumb|Design evolution for a thermoelectric cooler. The aim of the design problem is to maximise the cooling power of the thermoelectric cooler.]]

[[Thermoelectric effect|Thermoelectricity]] is a multi-physic problem which concerns the interaction and coupling between electric and thermal energy in semi conducting materials. Thermoelectric energy conversion can be described by two separately identified effects: The Seebeck effect and the Peltier effect. The Seebeck effect concerns the conversion of thermal energy into electric energy and the Peltier effect concerns the conversion of electric energy into thermal energy.<ref>Rowe, David Michael. [https://books.google.com/books?id=VvCb_deT4kIC&q=Seebeck Thermoelectrics handbook: macro to nano]. CRC press, 2005.</ref> By spatially distributing two thermoelectric materials in a two dimensional design space with a topology optimisation methodology,<ref>{{Cite journal | doi=10.1007/s00158-018-1919-1| title=A density-based topology optimization methodology for thermoelectric energy conversion problems| journal=Structural and Multidisciplinary Optimization| volume=57| issue=4| pages=1427–1442| year=2018| last1=Lundgaard| first1=Christian| last2=Sigmund| first2=Ole| s2cid=126031362| url=https://backend.orbit.dtu.dk/ws/files/163153924/grayscale_Lundgaard_C._Sigmund_O_2018_.pdf}}</ref> it is possible to exceed performance of the constitutive thermoelectric materials for [[Thermoelectric cooling|thermoelectric coolers]] and [[thermoelectric generator]]s.<ref>{{Cite journal |doi = 10.1007/s11664-018-6606-x|title = Topology Optimization of Segmented Thermoelectric Generators|journal = Journal of Electronic Materials|volume = 47|issue = 12|pages = 6959–6971|year = 2018|last1 = Lundgaard|first1 = Christian|last2 = Sigmund|first2 = Ole|last3 = Bjørk|first3 = Rasmus|bibcode = 2018JEMat..47.6959L |s2cid = 105113187|url=https://www.researchgate.net/publication/323143969}}</ref>

===3F3D Form Follows Force 3D Printing===
The current proliferation of 3D printer technology has allowed designers and engineers to use topology optimization techniques when designing new products. Topology optimization combined with 3D printing can result in less weight, improved structural performance and shortened design-to-manufacturing cycle. As the designs, while efficient, might not be realisable with more traditional manufacturing techniques.{{citation needed|date=November 2018}}

=== Internal contact ===
[[File:3D Topology Optimization with Internal Contact.webm|thumb|3D Topology Optimization with Internal Contact for a hook mechanism.]]
[[File:Topology optimization of contact problem problem with the third medium approach..gif|thumb|Design development and deformation of self-engaging hooks resulting from topology optimization of a contact problem using the TMC method <ref name=":0" />.]]
Internal contact can be included in topology optimization by applying the [[third medium contact method]].<ref>{{Cite journal |last1=Wriggers |first1=P. |last2=Schröder |first2=J. |last3=Schwarz |first3=A. |date=2013-03-30 |title=A finite element method for contact using a third medium |url=http://dx.doi.org/10.1007/s00466-013-0848-5 |journal=Computational Mechanics |volume=52 |issue=4 |pages=837–847 |doi=10.1007/s00466-013-0848-5 |bibcode=2013CompM..52..837W |s2cid=254032357 |issn=0178-7675}}</ref><ref>{{Cite journal |last1=Frederiksen |first1=Andreas H. |last2=Rokoš |first2=Ondřej |last3=Poulios |first3=Konstantinos |last4=Sigmund |first4=Ole |last5=Geers |first5=Marc G. D. |date=2024-12-01 |title=Adding friction to Third Medium Contact: A crystal plasticity inspired approach |journal=Computer Methods in Applied Mechanics and Engineering |volume=432 |pages=117412 |doi=10.1016/j.cma.2024.117412 |issn=0045-7825|doi-access=free |bibcode=2024CMAME.43217412F }}</ref> The third medium contact (TMC) method is an implicit contact formulation that is continuous and differentiable. This makes TMC suitable for use with gradient-based approaches to topology optimization. Monolithic<ref>{{Cite journal |last1=Bluhm |first1=Gore Lukas |last2=Sigmund |first2=Ole |last3=Poulios |first3=Konstantinos |date=2021-03-04 |title=Internal contact modeling for finite strain topology optimization |url=http://dx.doi.org/10.1007/s00466-021-01974-x |journal=Computational Mechanics |volume=67 |issue=4 |pages=1099–1114 |arxiv=2010.14277 |bibcode=2021CompM..67.1099B |doi=10.1007/s00466-021-01974-x |issn=0178-7675 |s2cid=225076340}}</ref> as well as staggerede approaches,<ref name=":0">{{Cite journal |last1=Frederiksen |first1=Andreas Henrik |last2=Sigmund |first2=Ole |last3=Poulios |first3=Konstantinos |date=2023-10-07 |title=Topology optimization of self-contacting structures |url=https://doi.org/10.1007/s00466-023-02396-7 |journal=Computational Mechanics |language=en |volume=73 |issue=4 |pages=967–981 |arxiv=2305.06750 |bibcode=2023CompM..73..967F |doi=10.1007/s00466-023-02396-7 |issn=1432-0924}}</ref><ref>{{Cite journal |last1=Frederiksen |first1=Andreas H. |last2=Dalklint |first2=Anna |last3=Sigmund |first3=Ole |last4=Poulios |first4=Konstantinos |date=2025-03-01 |title=Improved third medium formulation for 3D topology optimization with contact |journal=Computer Methods in Applied Mechanics and Engineering |volume=436 |pages=117595 |doi=10.1016/j.cma.2024.117595 |issn=0045-7825|doi-access=free |bibcode=2025CMAME.43617595F }}</ref> which are more common in topology optimization, have been used to create various design with internal contact. Recently, thermal contact has been included in the TMC topology optmization framework. <ref>{{Cite journal |last1=Dalklint |first1=Anna |last2=Alexandersen |first2=Joe |last3=Frederiksen |first3=Andreas Henrik |last4=Poulios |first4=Konstantinos |last5=Sigmund |first5=Ole |date=2025 |title=Topology Optimization of Contact-Aided Thermo-Mechanical Regulators |url=https://onlinelibrary.wiley.com/doi/full/10.1002/nme.7661 |journal=International Journal for Numerical Methods in Engineering |language=en |volume=126 |issue=2 |pages=e7661 |doi=10.1002/nme.7661 |issn=1097-0207|arxiv=2406.00865 }}</ref>