The natural volume method (NVM): Presentation and application to shallow water inviscid flows

International Journal for Numerical Methods in Fluids

Volumn 59, Issue 1, 2009, Pages 19-45

  Ata, R ^a   Soulaimani A ^a   Chinesta, Francisco ^b  

^a ÉCOLE DE TECHNOLOGIE SUPÉRIEURE   (Canada)

^b CNRS   (France)

Author keywords

Collocation; Inviscid flows; Lagrangian description; NEM; Shallow water equations; Stabilization

Indexed keywords

APPLICATIONS; BENCHMARKING; CONTACTS (FLUID MECHANICS); FLUID MECHANICS; FUNCTION EVALUATION; LAGRANGE MULTIPLIERS; STABILIZATION;

ARTIFICIAL VISCOSITIES; BENCHMARK TESTS; COLLOCATION; DIRICHLET BOUNDARY CONDITIONS; DISCONTINUOUS SOLUTIONS; FINITE VOLUMES; FLUX COMPUTATIONS; INTERPOLANT; INVISCID FLOWS; LAGRANGIAN; LAGRANGIAN DESCRIPTION; LAGRANGIAN FORMULATIONS; LARGE DEFORMATIONS; NATURAL ELEMENT METHODS; PARTICLE-BASED METHODS; QUADRILATERAL MESHES; REAL APPLICATIONS; RIEMANN SOLVERS; SHALLOW WATER EQUATIONS; SHALLOW WATER FLOWS; SHALLOW WATERS; SHAPE FUNCTIONS; ST VENANT EQUATIONS; STANDARD ELEMENTS; VOLUME METHODS; VORONOI CELLS;

VISCOUS FLOW;

EID: 61749100910     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.1800     Document Type: Article

References (40)

1. Dorfi EA, Drury LO. Simple adaptive grids for 1-D initial value problems. Journal of Computational Physics 1987; 69:175-195.
2. Miller K, Miller RN. Moving finite elements I. SIAM Journal on Numerical Analysis 1981; 18:1019-1032.
3. Miller K. Moving finite elements II. SIAM Journal on Numerical Analysis1981; 18:1033-1057.
4. Idelsohn SR, Oñate E, Calvo N, Del Pin F. The meshless finite element method. International Journal for Numerical Methods in Engineering 2003; 58(6):893-912.
5. Oñate E, Idelsohn SR, Del Pin F, Aubry R. The particle finite element method. An overview. International Journal of Computational Methods 2004; 1(2):267-307.
6. Huang W, Russell RD. A moving collocation method for solving time dependent partial differential equations. Applied Numerical Mathematics 1996; 20:101-116.
7. Monaghan JJ. Simulating free surface flows with SPH. Journal of Computational Physics 1994; 110(2):399-406.
8. Bonet J, Kulasegaram S, Rodriguez-Paz MX, Profit M. Variational formulation for the smooth particle hydrodynamics (SPH) simulation of fluid and solid problems. Computer Methods in Applied Mechanics and Engineering 2004; 193(12):1245-1257.
9. Lin P, Liu PF. A numerical study of breaking waves in the surf zone. Journal of Fluid Mechanics 1998; 59:239-264.
10. Ata R, Soulaïmani A. A stabilized SPH method for inviscid shallow water flows. International Journal for Numerical Methods in Fluids 2005; 47:139-159.
11. Cueto E, Sukumar N, Calvo B, Martinez MA, Cegoñino J, Doblaré M. Overview and recent advances in natural neighbor Galerkin methods. Archives of Computational Methods in Engineering2003; 10(4):307-384.
12. Cueto E, Cegoñino J, Calvo B, Doblaré M. On the imposition of essential boundary conditions in natural neighbor Galerkin methods. Communications in Numerical Methods in Engineering 2003; 19(5):361-376.
13. Yvonnet J, Ryckelynck D, Lonrong P, Chinesta F. Interpolation naturelle sur les domaines non-convexes par l'utilisation du diagrame de Voronoi contraint. Revue Europeenne des éléments Finis 2003; 12(4):487-509.
14. Sibson R. A brief description of natural neighbor interpolation. In Interpreting Multivariate Data, Barnett V (ed.). Wiley: Chichester, 1981.
15. Sukumar N, Moran B, Belytschko T. The natural elements method in solid mechanics. International Journal for Numerical Methods in Engineering 1998; 43:839-887.
16. Missoum-Benziane D, Ryckelynck D, Chinesta F. A new fully coupled two-scales modelling for mechanical problems involving microstructure: the 95/5 technique. Computer Methods in Applied Mechanics and Engineering 2007; 196:2325-2337.
17. Liang Q, Borthwick AG, Stelling G. Simulation of dam- and dyke-break hydrodynamics on dynamically adaptive quadtree grids. International Journal for Numerical Methods in Fluids 2004; 46:127-162.
18. Del Pin F. A Lagrangian approach based on the natural neighbor interpolation. INRIA Report No. 4090, Sophia Antipolis, France, 2000.
19. Sukumar N, Moran B, Yu Semenov A, Belikov VV. Natural neighbour Galerkin methods. International Journal for Numerical Methods in Engineering 2001; 50:1-27.
20. Farin G. Surfaces over Dirichlet tesselations. Computer Aided Geometric Design 1990; 7(1-4):281-292.
21. Belikov VV, Ivanov VD, Kontorovich VK, Korytnik SA, Semenov AY. The non-Sibsonian interpolation: a new method for interpolation of the values of a function on an arbitrary set of points. Computational Mathematics and Mathematical Physics 1997; 37(1):9-15.
22. Yvonnet J, Ryckelynck D, Lorong P, Chinesta F. A new extension of the natural element method for nonconvex and discontinuous problems: the constrained natural element method (C-NEM). International Journal for Numerical Methods in Engineering 2004; 60(8):1451-1474.
23. Christ NH, Friedberg R, Lee TD. Weights of links and plaquettes in a random lattice. Nuclear Physics B 1982; 210(3):337-346.
24. González D, Cueto E, Martínez A, Doblaré M. Numerical integration in natural neighbour Galerkin methods. International Journal for Numerical Methods in Engineering 2004; 60:2077-2104.
25. Bowyer A. Computing Dirichlet tesselations. Computer Journal 1981; 24(2):162-166.
26. Watson D. Computing the n-dimensional Delaunay tesselation with application to Voronoi polytopes. Computer Journal 1981; 24(2):167-172.
27. Sambridge M, Braun J, McQueen H. Geophysical parametrization and interpolation of irregular data using natural neighbours. Geophysical International Journal 1995; 122:837-857.
28. Cueto E, Doblaré M, Gracia L. Imposing essential boundary conditions in the natural elements method by mean of density-scaled alpha-shapes. International Journal for Numerical Methods in Engineering 2000; 49:519-546.
29. Klein R, Lingas A. A linear-time randomized algorithm for the bounded Voronoi diagram of a simple polygon. International Journal of Computational Geometry and Applications 1996; 6:263-278.
30. Martinez MA, Cueto E, Alfaro I, Doblaré M, Chinesta F. Updated Lagrangian free surface flow simulations with natural neighbour Galerkin methods. International Journal for Numerical Methods in Engineering 2004; 60:2105-2129.
31. Soulaïmani A, Saad Y. An arbitrary Lagrangian-Eulerian finite element method for solving three dimensional free surface flows. Computer Methods in Applied Mechanics and Engineering1998; 162:79-106.
32. González D, Cueto E, Chinesta F, Doblaré M. A natural element updated Lagrangian strategy for free-surface fluid dynamics. Journal of Computational Physics 2007; 223:127-150.
33. Birknes J, Pedersen G. A particle finite element method applied to long wave run-up. International Journal for Numerical Methods in Fluids 2006; 52:237-261.
34. Sukumar N. Voronoi cell finite difference method for the diffusion operator on arbitrary unconstructed grids. International Journal for Numerical Methods in Engineering 2003; 57(1):1-34.
35. Chen JS, Wu CT, Yoon S, You Y. A stabilized conforming nodal integration for Galerkin meshfree methods. International Journal for Numerical Methods in Engineering 2001; 50:435-466.
36. Mishev ID. Finite volume methods on Voronoi meshes. Numerical Methods for Partial Differential Equations 1998; 14(2):193-212.
37. Bardella L, Genna F. Newmark's time integration method from the discretization of extended functionals. Journal of Applied Mechanics 2005; 72(4):527-537.
38. Loukili Y, Soulaïmani A. Numerical tracking of shallow water waves by the unstructured finite volume WAF approximation. International Journal for Computational Methods in Engineering Science and Mechanics 2007; 8(2):75-88.
39. Stoker JJ. Water Waves. Wiley-Interscience: New York, 1957.
40. Toro EF. Shock-capturing Methods for Free Surface Flows. Wiley: New York, 2001.