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The `!natural element method (NEM)`!`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f]`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f] is a `F33f`_`[meshless method`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Meshless_method]`_`f to solve `F33f`_`[partial differential equation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Partial_differential_equation]`_`f, where the `*elements`* do not have a predefined shape as in the `F33f`_`[finite element method`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Finite_element_method]`_`f, but depend on the geometry.`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]`:cite-ref-5[`F5bf`_`[5`#cite-note-5]`_`f]`:cite-ref-6[`F5bf`_`[6`#cite-note-6]`_`f]

A `F33f`_`[Voronoi diagram`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Voronoi_diagram]`_`f partitioning the space is used to create each of these elements.

`F33f`_`[Natural neighbor interpolation functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Natural_neighbor_interpolation]`_`f are then used to model the unknown function within each element.

>>Contents

• `F0af`_`[Applications`#applications]`_`f

• `F0af`_`[References`#references]`_`f

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>>Applications

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

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `:citerefsukumarmoranbelytschko1998`aSukumar, N.; Moran, B.; Belytschko, T. (21 June 1998). "The natural element method in solid mechanics". `*International Journal for Numerical Methods in Engineering`*. `!43`! (5): 839–887. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:1998IJNME..43..839S. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R.

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `:citerefj-yvonnetd-ryckelynckp-lorongf-chinesta2004`aJ. Yvonnet; D. Ryckelynck; P. Lorong; F. Chinesta (2004). "A new extension of the natural element method for non-convex and discontinuous problems: the constrained natural element method (C-NEM)" (PDF). `*International Journal for Numerical Methods in Engineering`*. `!60`! (8): 1451–1474. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:2004IJNME..60.1451Y. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1002/nme.1016. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 122887431.

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `:citerefleecho2019`aLee, Hw; Cho, Jr (April 2019). "Large deformation analysis of elastic bodies by nonlinear Petrov–Galerkin natural element method". `*Advances in Mechanical Engineering`*. `!11`! (4): 168781401984629. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1177/1687814019846293.

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `:citereflushulujiang2017`aLu, Ping; Shu, Yang; Lu, Dahai; Jiang, Kaiyong; Liu, Bin; Huang, Changbiao (2017). "Research on Natural Element Method and the application to simulate metal forming processes". `*Procedia Engineering`*. `!207`!: 1087–1092. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1016/j.proeng.2017.10.1135.

`:cite-note-5`!5.`! `F0af`_`[↑`#cite-ref-5]`_`f "What is the difference between nem (natural element method) and cnem (constrained natural element method)?". `*ResearchGate`*. Retrieved 2019-07-15.

`:cite-note-6`!6.`! `F0af`_`[↑`#cite-ref-6]`_`f `:citerefbotelhomarechalramdane2016`aBotelho, D. P.; Marechal, Y.; Ramdane, B. (November 2016). "Vector interpolation on natural element method: Mesh sensitivity analysis". `*2016 IEEE Conference on Electromagnetic Field Computation (CEFC)`*. `F33f`_`[Institute of Electrical and Electronics Engineers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Institute_of_Electrical_and_Electronics_Engineers]`_`f. p. 1. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1109/CEFC.2016.7816353. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-5090-1032-5. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 27851390.

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