Natural element method

20 points and their Voronoi cells

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

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. doi:10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R. Bibcode: 1998IJNME..43..839S.
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)". International Journal for Numerical Methods in Engineering 60 (8): 1451–1474. doi:10.1002/nme.1016. Bibcode: 2004IJNME..60.1451Y. https://hal.archives-ouvertes.fr/hal-01508695/file/YRLC.pdf.
3. ↑ "Large deformation analysis of elastic bodies by nonlinear Petrov–Galerkin natural element method". Advances in Mechanical Engineering. April 2019. https://www.researchgate.net/publication/332678382.
4. ↑ Lu, Ping; Shu, Yang; Lu, Dahai; Jiang, Kaiyong; Liu, Bin; Huang, Changbiao (2017-01-01). "Research on Natural Element Method and the application to simulate metal forming processes". Procedia Engineering. International Conference on the Technology of Plasticity, ICTP 2017, 17-22 September 2017, Cambridge, United Kingdom (ScienceDirect) 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)?" (in en). https://www.researchgate.net/post/What_is_the_difference_between_nem_natural_element_method_and_cnem_constrained_natural_element_method.
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.

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Numerical partial differential equations by method

Finite difference|

| Parabolic|

* Forward-time central-space (FTCS)
* Crank–Nicolson

|

Hyperbolic|

* Lax–Friedrichs
* Lax–Wendroff
* MacCormack
* Upwind
* Method of characteristics

Others|

* Alternating direction-implicit (ADI)
* Finite-difference time-domain (FDTD)

Finite volume|

* Godunov
* High-resolution
* Monotonic upstream-centered (MUSCL)
* Advection upstream-splitting (AUSM)
* Riemann solver
* essentially non-oscillatory (ENO)
* weighted essentially non-oscillatory (WENO)

Finite element|

* hp-FEM
* Extended (XFEM)
* Discontinuous Galerkin (DG)
* Spectral element (SEM)
* Mortar
* Gradient discretisation (GDM)
* Loubignac iteration
* Smoothed (S-FEM)

Meshless/Meshfree|

* Smoothed-particle hydrodynamics (SPH)
* Moving particle semi-implicit method (MPS)
* Material point method (MPM)
* Particle-in-cell (PIC)

Domain decomposition|

* Schur complement
* Fictitious domain
* Schwarz alternating
* additive
* abstract additive
* Neumann–Dirichlet
* Neumann–Neumann
* Poincaré–Steklov operator
* Balancing (BDD)
* Balancing by constraints (BDDC)
* Tearing and interconnect (FETI)
* FETI-DP

Others|

* Spectral
* Pseudospectral (DVR)
* Method of lines
* Multigrid
* Collocation
* Level-set
* Boundary element
* Immersed boundary
* Analytic element
* Particle-in-cell
* Isogeometric analysis
* Infinite difference method
* Infinite element method
* Galerkin method
* Petrov–Galerkin method
* Validated numerics
* Computer-assisted proof

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