A natural element updated Lagrangian strategy for free-surface fluid dynamics

We present a novel algorithm to simulate free-surface fluid dynamics phenomena at low Reynolds numbers in an updated Lagrangian framework. It is based on the use of one of the most recent meshless methods, the so-called natural element method. Free-surface tracking is performed by employing a particular instance of ''shape constructors'' called @a-shapes. This means that at each time step the geometry of the domain is extracted by employing a particular member of the finite set of shapes described by the nodal cloud. The Lagrangian framework allows us to integrate the inertial terms of the Navier-Stokes equations by employing the method of characteristics which are, precisely, the nodal pathlines. A theoretical description of the method is included together with some examples showing its performance.

[1]  Adrian Bowyer,et al.  Computing Dirichlet Tessellations , 1981, Comput. J..

[2]  R. Sibson.  A vector identity for the Dirichlet tessellation , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  Ted Belytschko,et al.  Numerical integration of the Galerkin weak form in meshfree methods , 1999 .

[4]  Kokichi Sugihara,et al.  Two Generalizations of an Interpolant Based on Voronoi Diagrams , 1999, Int. J. Shape Model..

[5]  S. Nagel,et al.  Drop splashing on a dry smooth surface. , 2005, Physical review letters.

[6]  Elías Cueto,et al.  On the imposition of essential boundary conditions in natural neighbour Galerkin methods , 2003 .

[7]  Manuel Doblaré,et al.  Imposing essential boundary conditions in the natural element method by means of density-scaled?-shapes , 2000 .

[8]  F. Chinesta,et al.  A new extension of the natural element method for non‐convex and discontinuous problems: the constrained natural element method (C‐NEM) , 2004 .

[9]  Elías Cueto,et al.  Modelling three‐dimensional piece‐wise homogeneous domains using the α‐shape‐based natural element method , 2002 .

[10]  C. K. Thornhill,et al.  Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane , 1952, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[11]  A. D. Gordon,et al.  Interpreting multivariate data , 1982 .

[12]  Hyun Gyu Kim,et al.  A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods , 1999 .

[13]  Elías Cueto,et al.  Volumetric locking in natural neighbour Galerkin methods , 2004 .

[14]  Roland W. Lewis,et al.  A mixed Lagrangian-Eulerian approach to modelling fluid flow during mould filling , 1997 .

[15]  Jiun-Shyan Chen,et al.  A stabilized conforming nodal integration for Galerkin mesh-free methods , 2001 .

[16]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[17]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[18]  Francis H. Harlow,et al.  The Splash of a Liquid Drop , 1967 .

[19]  Balasubramaniam Ramaswamy,et al.  Numerical simulation of unsteady viscous free surface flow , 1990 .

[20]  O. Pironneau.  Finite Element Methods for Fluids , 1990 .

[21]  Qingwei Ma,et al.  Meshless local Petrov-Galerkin method for two-dimensional nonlinear water wave problems , 2005 .

[22]  Stéphane Zaleski,et al.  Jet Formation in Bubbles Bursting at a Free Surface , 2002 .

[23]  A. H. Thiessen.  PRECIPITATION AVERAGES FOR LARGE AREAS , 1911 .

[24]  V. D. Ivanov,et al.  The non-Sibsonian interpolation : A new method of interpolation of the values of a function on an arbitrary set of points , 1997 .

[25]  I. Babuska,et al.  ON THE ANGLE CONDITION IN THE FINITE ELEMENT METHOD , 1976 .

[26]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[27]  D. F. Watson.  Computing the n-Dimensional Delaunay Tesselation with Application to Voronoi Polytopes , 1981, Comput. J..

[28]  Peter Wriggers,et al.  Arbitrary Lagrangian Eulerian finite element analysis of free surface flow , 2000 .

[29]  Tamal K. Dey,et al.  Alpha-shapes and flow shapes are homotopy equivalent , 2003, STOC '03.

[30]  B. Moran,et al.  Natural neighbour Galerkin methods , 2001 .

[31]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[32]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[33]  G. Tryggvason,et al.  A front-tracking method for viscous, incompressible, multi-fluid flows , 1992 .

[34]  Wing Kam Liu,et al.  Reproducing kernel particle methods for structural dynamics , 1995 .

[35]  T. Belytschko,et al.  THE NATURAL ELEMENT METHOD IN SOLID MECHANICS , 1998 .

[36]  J. Hyvärinen,et al.  An Arbitrary Lagrangian-Eulerian finite element method , 1998 .

[37]  Elías Cueto,et al.  Numerical integration in Natural Neighbour Galerkin methods , 2004 .

[38]  Icíar Alfaro,et al.  Updated Lagrangian free surface flow simulations with natural neighbour Galerkin methods , 2004 .

[39]  K. Bathe.  Finite Element Procedures , 1995 .

[40]  Herbert Edelsbrunner,et al.  Three-dimensional alpha shapes , 1994, ACM Trans. Graph..

[41]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[42]  Eugenio Oñate,et al.  The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves , 2004 .

[43]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[44]  Eugenio Oñate,et al.  The meshless finite element method , 2003 .

[45]  Miguel Ángel Martínez,et al.  Overview and recent advances in natural neighbour galerkin methods , 2003 .

[46]  S. Zaleski,et al.  Jet formation in bubbles bursting at a free surface , 2002 .

[47]  E. Laitone,et al.  The second approximation to cnoidal and solitary waves , 1960, Journal of Fluid Mechanics.

[48]  K. Bathe,et al.  The inf-sup test , 1993 .