Editing Topology optimization (section)

=== 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>