FINITE POINTSET METHOD (FPM): A MESHFREE APPROACH FOR INCOMPRESSIBLE FLOW SIMULATIONS APPLIED TO COMPOSITE MATERIALS Alain Trameçon 1, Patrick e Luca 1, Christophe Binetruy 2 an Jörg Kuhnert 3 1 ESI-GROUP Parc affaire SILIC, 99 Rue es Solets, BP 80112 F-94513 RUNGIS, FRANCE: atr@esi-group.com 2 Polymers an Composites Technology & Mechanical Engineering Department Ecole es Mines e Douai, 941 Rue Charles Bourseul, F-59508 DOUAI, France : binetruy@ensm-ouai.fr 3 Fraunhofer Institute for Inustrial Mathematics (ITWM) Gottlieb-Daimler-Straße, D-67663 KAISERSLAUTERN, GERMANY: kuhnert@itwm.fhg.e ABSTRACT: A meshfree particle metho is use to simulate resin flow through a complex network of fibers. Flows are moele by the incompressible Navier-Stokes equations. The particle projection metho is use to solve the Navier-Stokes equations. The spatial erivatives are approximate by the weighte least squares metho (WLS). One application is presente regaring the application of the metho to numerical permeability preiction, relate to LCM processes. KEYWORDS: Mesh-free metho, incompressible Navier-Stokes equations, least squares (LSQ) approximation, LCM, permeability preiction Meshfree techniques INTRODUCTION In this paper we present a mesh-free metho calle FPM (Finite Point Metho) for simulations of resin transfer mouling through a complex network of fibers. This metho was evelope by Dr Jorg Kuhnert at ITWM in Germany. A flui omain is first replace by a iscrete number of points, which are referre to as particles. Each particle carries all flui information, like ensity, velocity, temperature etc. an moves with flui velocity. Therefore, particles themselves can be consiere as geometrical gris of the flui omain. This metho has some avantages over gri base techniques, for example, it can hanle flui omains, which change naturally, whereas gri base techniques require aitional computational effort. Or they are able to simulate the flow in very complicate omains, which woul be impossible or ifficult to mesh. 31
A classical gri free Lagrangian metho is Smoothe Particle Hyroynamics (SPH), which was originally introuce to solve problems in astrophysics (Lucy 1977, Gingol et al. 1977). It has since been extene to simulate the compressible Euler equations in flui ynamics an applie to a wie range of problems, see (Monaghan 92, Monaghan et al. 1983, Morris et al. 1997). The metho has also been extene to simulate invisci incompressible free surface flows (Monaghan 94). The implementation of the bounary conitions is the main problem of the SPH metho. Another approach for solving flui ynamic equations in a gri free framework is the moving least squares or least squares metho (Belytschko et al. 1996, Dilts 1996, Kuhnert 99, Kuhnert 2000, Tiwari et al. 2001 an 2000) which erive in the Finite Pointset Metho (FPM). With this approach bounary conitions can be implemente in a natural way just by placing the particles on bounaries an prescribing bounary conitions on them (Kuhnert 99). The robustness of this metho is shown by the simulation results in the fiel of airbag eployment in car inustry. Here, the membrane (or bounary) of the airbag changes very rapily in time an takes a quite complicate shape (Kuhnert et al. 2000). FPM flui coe FPM is a meshfree CFD finite ifference coe, mainly esigne to overcome several rawbacks of classical CFD methos (Finite Element Metho (FEM), Finite Volume Metho (FVM)). The main rawback of the classical methos (FEM,FVM) is the relatively expansive geometrical mesh-gri require to carry out all numerical computations. The computational cost to establish an maintain these gris becomes more ominant as the consiere geometry becomes complex or moves in time. For several applications, the effort for gri maintenance is beyon acceptance, the computations take too long or fail completely. Thus, FPM opens up new fiels of application in computational structural an flui mechanics, or makes the hanling of several problems much more easy. General Equations FPM is a mesh-free thermal CFD coe for incompressible an compressible flows. FPM inclues newtonian viscosity, natural convection, heat conuction, heat exchange at the bounaries. FPM solves the general Navier Stokes equation as written below in a Lagrange form : ρ + ρ v = 0 + ( ρv) ( ρv ) v + p S = ρ g T ( ρe) + ( ρe) v + ( p v) ( S v) = ( ρ g v) + ( k T) (1) v : flui velocity ρ : ensity p : pressure S : eviatoric stresses g : gravity T : temperature E : specific total energy per unit mass For incompressible flows, these equations can be simplifie as follows : 32
ρ = 0 v T 1 = p + ρ = v = 0 η v + g ρ 1 (k T) ρ c (2) Moving Least Square (MLS) approximation FPM oes not require the support of a mesh an therefore values are known at iscrete interpolation points, which o not have a fixe connection like finite elements between them. The list of neighbor points is etermine for each point at each time step in orer to construct afterwars a proper interpolation function as escribe in Fig.1. For this purpose, a smooth interpolation of the iscrete function values is constructe using polynomial functions, best fitte to the iscrete values using a moving least square metho (Fig.2). Point "i" raius h Fig. 1: Smoothing length f(i) = Point position = Function value Fig. 2 : Moving least square metho : f(i) stans for the value of point I of the function to be interpolate : pressure, ensity, velocities, temperature. x(i) 33
The interpolation function that is constructe is base upon the Moving Least Square (MLS) approximation. The iea is to fin the local polynomial which minimizes the istance between the values at the iscrete points an the approximate values on the function. This is one using a least square fit metho as follows: i= 1, N 2! ( x x ) ( p ( x,x ) min W (3) i f i i = where is the egree of the polynom, >=2 for Navier-Stokes as secon orer erivatives are require. The weight function W(x-xi) ecreases with respect to the istance between the location x of the central point an the neighbor points xi, so that points which are far away from the central point will have less influence than points which are closer. The omain of interpolation is limite by a sphere of raius h, calle the smoothing length, so that points which have a istance greater than h will have no interaction. The interpolation function can afterwars be erive. For Navier Stokes incompressible cases, secon orer erivatives have to be compute. In orer to maintain an even istribution, points are generate or remove automatically uring the simulation. APPLICATION TO NUMERICAL PERMEABILITY PREDICTION In orer to perform accurate LCM filling simulations, physical parameters such as fabric permeability are neee. It is well known that the experimental measurement of that fabric property is very elicate. Usually a flui is injecte at constant pressure (or constant flow rate) through several layers of fabric of cross section A an length L. Then pressure loss p an flow rate Q are measure, an the permeability K is calculate using Darcy s law: QLµ K = (4) A p The simulation of flow through a perioic cell shoul provie a reliable solution an avoi that experimental proceure. Numerically, an injection at constant flow rate coul be performe. Then the pressure loss can be compute by the FPM coe an the permeability is also given by Eqn. 4. Fig.3 shows the fabric cell consiere here for benchmark purpose (Belov et al, 2004). It is boune by a box in contact with yarns that efines the computational omain. Shell elements surrouning the fiber tows prevent the particles from escaping the omain with proper contact. Also particles are not allowe to penetrate the tows. Finally the tows are consiere to be rigi an fixe in space. The porosity of such a omain is 0.36. Starting from an initially unfille cavity, the omain is automatically fille by particles. The inflow velocity is constant an equal to 0.01 m/s. The flui viscosity is constant an equal to 0.01 Pa.s. The compute pressure loss is P= 417,6 Pa giving a saturate permeability of 2.28 10-10 m 2. Computations last aroun 20 minutes with a state of the art PC. 34
Accoring to Belov et al., the saturate permeability of a single layer of such fiber reinforcement is in between 2.6 10-10 m 2 an 3.5 10-10 m 2 using the Lattice Boltzmann Metho (LBM). Measurements provie values between 1.34 10-10 m 2 an 1.49 10-10 m 2. The FPM coe provies results in goo agreement with experimental ata. Fig. 4 : Fabric unit cell an omain use for the calculations Fig.5 : Compute flow patterns an pressure fiel (blue spots are contact zones not wette by the flui) CONCLUSION FPM solves for compressible an incompressible flow problems an can be couple to structural FE coes. An important feature regaring polymer composites manufacturing science is that chemical reactions, heat transfer, temperature epenent viscosity can be hanle. To illustrate the potential of the coe, the case of numerical permeability preiction of a fabric unit cell has been presente an compare with existing experimental ata. The FPM coe provies results in goo agreement with experiments. Current work focuses on flow-inuce fiber eformation. 35
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