Natural element method

The natural element method (NEM)^[1]^[2]^[3] 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.^[4]^[5]^[6]

A Voronoi diagram partitioning the space is used to create each of these elements.

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

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

^ Sukumar, 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. Bibcode:1998IJNME..43..839S. doi:10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R.
^ J. 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. Bibcode:2004IJNME..60.1451Y. doi:10.1002/nme.1016. S2CID 122887431.
^ Lee, 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. doi:10.1177/1687814019846293.
^ Lu, 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. doi:10.1016/j.proeng.2017.10.1135.
^ "What is the difference between nem (natural element method) and cnem (constrained natural element method)?". ResearchGate. Retrieved 2019-07-15.^{[unreliable source?]}
^ Botelho, 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). Institute of Electrical and Electronics Engineers. p. 1. doi:10.1109/CEFC.2016.7816353. ISBN 978-1-5090-1032-5. S2CID 27851390.

[1] Sukumar, 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. Bibcode:1998IJNME..43..839S. doi:10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R.

[2] J. 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. Bibcode:2004IJNME..60.1451Y. doi:10.1002/nme.1016. S2CID 122887431.

[3] Lee, 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. doi:10.1177/1687814019846293.

[4] Lu, 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. doi:10.1016/j.proeng.2017.10.1135.

[5] "What is the difference between nem (natural element method) and cnem (constrained natural element method)?". ResearchGate. Retrieved 2019-07-15.^{[unreliable source?]}

[6] Botelho, 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). Institute of Electrical and Electronics Engineers. p. 1. doi:10.1109/CEFC.2016.7816353. ISBN 978-1-5090-1032-5. S2CID 27851390.

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