Natural element meshless simulation of flows involving short fiber suspensions

Abstract Numerical modeling of non-Newtonian flows typically involves the coupling between the equations of motion characterized by an elliptic character, and the fluid constitutive equation, which is an advection equation linked to the fluid history. Thus, the numerical modeling of short fiber suspensions flows requires a description of the microstructural evolution (fiber orientation) which affects the flow kinematics and that is itself governed by this kinematics (coupled problem). Some industrial flows involve moving or free boundaries (injection, extrusion, …). Lagrangian descriptions allow an accurate description of the flow front tracking as well as an accurate integration of transport equations along the flow trajectories. However, Lagrangian techniques in the context of finite elements have the important drawback of requiring frequent remeshing in order to avoid large elements distortions. The natural element method (NEM) has the capabilities of Lagrangian models to describe the flow front tracking as well as to treat the convection terms related to the fiber orientation equation without the mesh quality requirement characteristics of the standard finite elements method.

[1]  Charles L. Tucker,et al.  Flow regimes for fiber suspensions in narrow gaps , 1991 .

[2]  John S. Campbell,et al.  Local and global smoothing of discontinuous finite element functions using a least squares method , 1974 .

[3]  B. Moran,et al.  C1natural neighbor interpolant for partial differential equations , 1999 .

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

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

[6]  E. J. Hinch,et al.  Constitutive equations in suspension mechanics. Part 1. General formulation , 1975, Journal of Fluid Mechanics.

[7]  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 .

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

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

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

[11]  Numerically Predicted Fiber Orientations in Dilute Suspensions , 1981 .

[12]  K. Chiba,et al.  Numerical solution of fiber suspension flow through a complex channel , 1998 .

[13]  Numerical simulation of the steady recirculating flows of fiber suspensions , 2000 .

[14]  T. Coupez,et al.  Finite element solution of the 3D mold filling problem for viscous incompressible fluid , 1998 .

[15]  Georges Voronoi.  Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. , 1908 .

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

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

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

[19]  Francisco Chinesta,et al.  On the solution of Fokker–Planck equations in steady recirculating flows involving short fiber suspensions , 2003 .

[20]  David G. Kirkpatrick,et al.  On the shape of a set of points in the plane , 1983, IEEE Trans. Inf. Theory.

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

[22]  G. Batchelor,et al.  Slender-body theory for particles of arbitrary cross-section in Stokes flow , 1970, Journal of Fluid Mechanics.

[23]  Francisco Chinesta,et al.  State-of-the-Art on numerical simulation of fiber-reinforced thermoplastic forming processes , 2002 .

[24]  George L. Hand,et al.  A theory of anisotropic fluids , 1962, Journal of Fluid Mechanics.

[25]  Suresh G. Advani,et al.  Closure approximations for three‐dimensional structure tensors , 1990 .

[26]  S. Güçeri,et al.  Anisotropic channel flow of fiber suspensions , 1992 .

[27]  E. J. Hinch,et al.  Constitutive equations in suspension mechanics. Part 2. Approximate forms for a suspension of rigid particles affected by Brownian rotations , 1976, Journal of Fluid Mechanics.

[28]  Roland Keunings,et al.  Simulation of Non-recirculating Flows of Dilute Fiber Suspensions , 1990 .

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

[30]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .