Natural element method: Difference between revisions

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[[Image:Euclidean Voronoi diagram.svg|thumb|20 points and their Voronoi cells]]

{{AFC submission|t||ts=20190621055005|u=Hassan Hadji|ns=118|demo=}}<!-- Important, do not remove this line before article has been created. -->

The '''natural element method (NEM)'''<ref>{{cite journal|title=The natural element method in solid mechanics|first1=N.|last1=Sukumar|first2=B.|last2=Moran|first3=T.|last3=Belytschko|date=21 June 1998|journal=International Journal for Numerical Methods in Engineering|volume=43|issue=5|pages=839–887|doi=10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R|bibcode=1998IJNME..43..839S}}</ref><ref>{{cite journal| title=A new extension of the natural element method for non-convex and discontinuous problems: the constrained natural element method (C-NEM)| author1=J. Yvonnet| author2=D. Ryckelynck| author3=P. Lorong| author4=F. Chinesta| journal=International Journal for Numerical Methods in Engineering| year=2004| volume=60| issue=8| pages=1451–1474| doi=10.1002/nme.1016| bibcode=2004IJNME..60.1451Y| s2cid=122887431| url=https://hal.archives-ouvertes.fr/hal-01508695/file/YRLC.pdf}}</ref><ref>{{cite journal |last1=Lee |first1=Hw |last2=Cho |first2=Jr |title=Large deformation analysis of elastic bodies by nonlinear Petrov–Galerkin natural element method |journal=Advances in Mechanical Engineering |date=April 2019 |volume=11 |issue=4 |pages=168781401984629 |doi=10.1177/1687814019846293 |doi-access=free }}</ref> is a [[meshless method]] to solve [[partial differential equation]], where the ''elements'' do not have a predefined shape as in the [[finite element method]], but depend on the geometry.<ref>{{cite journal |last1=Lu |first1=Ping |last2=Shu |first2=Yang |last3=Lu |first3=Dahai |last4=Jiang |first4=Kaiyong |last5=Liu |first5=Bin |last6=Huang |first6=Changbiao |title=Research on Natural Element Method and the application to simulate metal forming processes |journal=Procedia Engineering |date=2017 |volume=207 |pages=1087–1092 |doi=10.1016/j.proeng.2017.10.1135 |doi-access=free }}</ref><ref>{{Cite web |url=https://www.researchgate.net/post/What_is_the_difference_between_nem_natural_element_method_and_cnem_constrained_natural_element_method |title=What is the difference between nem (natural element method) and cnem (constrained natural element method)? |website=ResearchGate |language=en |access-date=2019-07-15}}{{rs|date=May 2024}}</ref><ref>{{Cite book |chapter=Vector interpolation on natural element method: Mesh sensitivity analysis |last1=Botelho |first1=D. P. |last2=Marechal |first2=Y. |date=November 2016 |publisher=[[Institute of Electrical and Electronics Engineers]] |page=1 |doi=10.1109/CEFC.2016.7816353 |last3=Ramdane |first3=B.|title=2016 IEEE Conference on Electromagnetic Field Computation (CEFC) |isbn=978-1-5090-1032-5 |s2cid=27851390 }}</ref>

A constrained [[Voronoi diagram]] partitioning the space will beis used to create each of these elements.

The Natural Element Method (NEM) <ref>{{cite journal|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/%28SICI%291097-0207%2819981115%2943%3A5%3C839%3A%3AAID-NME423%3E3.0.CO%3B2-R|title=The natural element method in solid mechanics|first1=N.|last1=Sukumar|first2=B.|last2=Moran|first3=T.|last3=Belytschko|date=21 June 1998|publisher=|journal=International Journal for Numerical Methods in Engineering|volume=43|issue=5|pages=839–887|accessdate=21 June 2019|via=Wiley Online Library|doi=10.1002/(SICI)1097-0207(19981115)43:53.0.CO;2-R}}</ref> is a meshfree method to solve [[partial differential equation]], where the ''elements'' don't have predefined nodes like the [[Finite Element Method]], but depend on the geometry.

Laplace[[Natural_neighbor_interpolation|Natural neighbor interpolation functions]] are then used to model the unknown function within each element.

A constrained [[Voronoi diagram]] partitioning the space will be used to create each of these elements.

== Applications ==

Laplace interpolation functions are then used to model the unknown function within each element.

When the simulation is dynamic, this method prevents the elements to be ill-formed, having the possibility to easily redefine them at each time step depending on the geometry.

== References ==

{{reflist}}

{{Numerical PDE}}

{{AFC submission|||ts=20190621060734|u=Hassan Hadji|ns=118}}

[[Category:Numerical differential equations]]

[[Category:Numerical analysis]]

[[Category:Computational fluid dynamics]]

[[Category:Computational mathematics]]

[[Category:Simulation]]