Numerical simulation of reactive flow in liquid composite molding using flux-corrected transport (FCT) based finite element/control volume (FE/CV) method. (English) Zbl 1190.80032

Tan, Hua; Pillai, Krishna M.

Int. J. Heat Mass Transfer 53, No. 9-10, 2256-2271 (2010).

Summary: The mold-filling simulation of liquid composite molding (LCM) is of great importance in optimizing this cost-effective polymer-composites manufacturing process. The flow in LCM is a convection-dominated reactive, non-isothermal flow involving a moving boundary, so the Galerkin finite element method (FEM) has to be adapted with stabilization techniques to solve such problems. The streamline-upwind Petrov-Galerkin (SUPG) method is one of the most popular stabilized methods. However, the use of SUPG still leads to localized numerical wiggles in the vicinity of sharp solution gradients, which is often encountered in the 3D mold-filling simulation of LCM for thick parts. In this study, we propose to use the flux-corrected transport (FCT) based FEM to solve a set of highly convective transport equations. The numerical examples presented in this paper demonstrate the excellent performance of FCT based FEM in suppressing spurious oscillations in the regions of steep solution-gradients as opposed to the numerical instability of SUPG in such regions. For the first time, the FCT based FEM combined with the control-volume method is employed to simulate the non-isothermal mold-filling process in LCM. We have developed a simulation code PORE-FLOW\(\copyright\) based on the scheme proposed in the study. Numerical studies have proven the stability of the FCT based FEM while modeling the mold-filling process of LCM.

MSC:

80A20	Heat and mass transfer, heat flow (MSC2010)
76S05	Flows in porous media; filtration; seepage
76A05	Non-Newtonian fluids
80A30	Chemical kinetics in thermodynamics and heat transfer
80M10	Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
76M10	Finite element methods applied to problems in fluid mechanics
76M12	Finite volume methods applied to problems in fluid mechanics

Keywords:

streamline-upwind Petrov-Galerkin (SUPG); flux-corrected transport (FCT); liquid composite molding; mold filling; PORE-FLOW

Software:

PORE-FLOW; SHASTA

References:

[1]	Chan, A. W.; Hwang, S. T.: Modeling nonisothermal impregnation of fibrous media with reactive polymer resin, Polym. eng. Sci. 32, No. 5, 310-318 (1992)
[2]	Bruschke, M. V.; Advani, S. G.: Numerical approach to model non-isothermal viscous flow through fibrous media with free surfaces, Int. J. Numer. methods fluids 19, No. 7, 575-603 (1994) · Zbl 0825.76439
[3]	Young, W. B.: Three-dimensional nonisothermal mold filling simulations in resin transfer molding, Polym. compos. 15, No. 2, 118-127 (1994)
[4]	Wang, T. J.; Lee, L. J.; Young, W. B.: Control volume finite element method for mold filling simulation, Int. polym. Proc. 10, No. 1, 82-90 (1995)
[5]	Voller, V. R.; Peng, S.; Chen, Y. F.: Numerical solution of transient, free surface problems in porous media, Int. J. Numer. methods eng. 39, No. 17, 2889-2906 (1996) · Zbl 0884.76072
[6]	Mohan, R. V.; Ngo, N. D.; Tamma, K. K.: Three-dimensional resin transfer molding: isothermal process modeling and explicit tracking of moving fronts for thick geometrically complex composites manufacturing applications – part 1, Numer. heat transfer part A 35, No. 8, 815-838 (1999)
[7]	Shojaei, A.; Ghaffarian, S. R.; Karimian, S. M. H.: Numerical simulation of three-dimensional mold filling process in resin transfer molding using quasi-steady state and partial saturation formulations, Compos. sci. Technol. 62, No. 6, 861-879 (2002)
[8]	Young, W. B.: Development of a helicopter landing gear prototype using resin infusion molding, J. reinf. Plast. compos. 28, No. 7, 833-849 (2009)
[9]	Trochu, F.; Gauvin, R.: Limitations of a boundary-fitted finite difference method for the simulation of the resin transfer molding process, J. reinf. Plast. compos. 11, No. 7, 772-786 (1992)
[10]	Yoo, Y. E.; Lee, W. I.: Numerical simulation of the resin transfer mold filling process using the boundary element method, Polym. compos. 17, No. 3, 368-374 (1996)
[11]	Baliga, B. R.; Patankar, S. V.: New finite-element formulation for convection – diffusion problems, Numer. heat transfer 3, No. 4, 393-409 (1980)
[12]	Phelan, F. R. J.R.: Simulation of the injection process in resin transfer molding, Polym. compos. 18, No. 4, 460-476 (1997)
[13]	Brooks, A. N.; Hughes, T. J. R.: Streamline upwind/Petrov – Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier – Stokes equations, Comput. methods appl. Mech. eng. 32, 199-259 (1982) · Zbl 0497.76041
[14]	Hughes, T. J. R.; Franca, L. P.; Hulbert, G. M.: A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations, Comput. methods appl. Mech. eng. 73, No. 2, 173-189 (1989) · Zbl 0697.76100
[15]	Donea, J.: A Taylor – Galerkin method for convective transport problems, Int. J. Numer. methods eng. 20, No. 1, 101-119 (1984) · Zbl 0524.65071
[16]	Löhner, R.; Morgan, K.; Zienkiewicz, O. C.: The solution of non-linear hyperbolic equation systems by the finite element method, Int. J. Numer. methods fluids 4, No. 11, 1043-1063 (1984) · Zbl 0551.76002
[17]	Codina, R.: A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection – diffusion equation, Comput. methods appl. Mech. eng. 110, No. 3 – 4, 325-342 (1993) · Zbl 0844.76048
[18]	Madden, N.; Stynes, M.: Linear enhancements of the streamline diffusion method for convection – diffusion problems, Comput. math. Appl. 32, No. 10, 29-42 (1996) · Zbl 0870.65100
[19]	John, V.; Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection – diffusion equations: part I – a review, Comput. methods appl. Mech. eng. 196, No. 17 – 20, 2197-2215 (2007) · Zbl 1173.76342
[20]	Boris, J. P.; Book, D. L.: Flux-corrected transport. I. shasta, a fluid transport algorithm that works, J. comput. Phys. 11, No. 1, 38-69 (1973) · Zbl 0251.76004
[21]	Zalesak, S. T.: Fully multidimensional flux-corrected transport algorithms for fluids, J. comput. Phys. 31, No. 3, 335-362 (1979) · Zbl 0416.76002
[22]	Löhner, R.; Morgan, K.; Peraire, J.: Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier – Stokes equations, Int. J. Numer. methods fluids 7, No. 10, 1093-1109 (1987) · Zbl 0633.76070
[23]	Kuzmin, D.; Turek, S.: Flux correction tools for finite elements, J. comput. Phys. 175, No. 2, 525-558 (2002) · Zbl 1028.76023
[24]	Kuzmin, D.; Möller, M.; Turek, S.: High-resolution FEM-FCT schemes for multidimensional conservation laws, Comput. methods appl. Mech. eng. 193, No. 45 – 47, 4915-4946 (2004) · Zbl 1112.76393
[25]	Kuzmin, D.: Explicit and implicit FEM-FCT algorithms with flux linearization, J. comput. Phys. 228, No. 7, 2517-2534 (2009) · Zbl 1275.76171
[26]	John, V.; Schmeyer, E.: Finite element methods for time-dependent convection – diffusion-reaction equations with small diffusion, Comput. methods appl. Mech. eng. 198, No. 3 – 4, 475-494 (2008) · Zbl 1228.76088
[27]	Parnas, R. S.; Jr., F. R. Phelan: Effect of heterogeneous porous media on mold filling in resin transfer molding, SAMPE quart. 22, No. 2, 53-60 (1991)
[28]	Pillai, K. M.; Advani, S. G.: A model for unsaturated flow in woven fiber preforms during mold filling in resin transfer molding, J. compos. Mater. 32, No. 19, 1753-1783 (1998)
[29]	Tan, H.; Roy, T.; Pillai, K. M.: Variations in unsaturated flow with flow direction in resin transfer molding: an experimental investigation, Composites part A 38, No. 8, 1872-1892 (2007)
[30]	Pillai, K. M.: Governing equations for unsaturated flow through woven fiber mats. Part 1. Isothermal flows, Composites part A 33, No. 7, 1007-1019 (2002)
[31]	Pillai, K. M.; Munagavalasa, M. S.: Governing equations for unsaturated flow through woven fiber mats. Part 2. Non-isothermal reactive flows, Composites part A 35, No. 4, 403-415 (2004)
[32]	Whitaker, S.: Flow in porous media. I: a theoretical derivation of Darcy’s law, Transp. porous med. 1, No. 1, 3-25 (1986)
[33]	Dessenberger, R. B.; Tucker, C. L.: Thermal dispersion in resin transfer molding, Polym. compos. 16, No. 6, 495-506 (1995)
[34]	Liu, B.; Advani, S. G.: Operator splitting scheme for 3-D temperature solution based on 2-D flow approximation, Comput. mech. 16, No. 2, 74-82 (1995) · Zbl 0825.76514
[35]	Shojaei, A.; Ghaffarian, S. R.; Karimian, S. M. H.: Simulation of the three-dimensional non-isothermal mold filling process in resin transfer molding, Compos. sci. Technol. 63, No. 13, 1931-1948 (2003)
[36]	Young, W. B.: Thermal behaviors of the resin and mold in the process of resin transfer molding, J. reinf. Plast. compos. 14, No. 4, 310 (1995)
[37]	Lin, R. J.; Lee, L. J.; Liou, M. J.: Mold filling and curing analysis liquid composite molding, Polym. compos. 14, No. 1, 71-81 (1993)
[38]	Chiu, H. T.; Yu, B.; Chen, S. C.: Heat transfer during flow and resin reaction through fiber reinforcement, Chem. eng. Sci. 55, No. 17, 3365-3376 (2000)
[39]	Tucker, C. L.; Dessenberger, R. B.: Governing equations for flow through stationary fiber beds, Flow and rheology in polymer composites manufacturing (1994)
[40]	Bear, J.: Dynamics of fluids in porous media, (1971) · Zbl 1191.76002
[41]	Whitaker, S.: The method of volume averaging, (1998) · Zbl 1058.62608
[42]	Kaviany, M.: Principles of heat transfer in porous media. Mechanical engineering series, (1995) · Zbl 0889.76002
[43]	Marcos, P.; Marcelo, L.: Thermal dispersion in porous media as a function of the solid – fluid conductivity ratio, Int. J. Heat mass transfer 51, 5359-5367 (2008) · Zbl 1154.80338
[44]	Lee, L. J.; Young, W. B.; Lin, R. J.: Mold filling and cure modeling of RTM and SRIM processes, Compos. struct. 27, No. 1 – 2, 109-120 (1994)
[45]	Kim, J. H.; Ochoa-Tapia, J. A.; Whitaker, S.: Diffusion in anisotropic porous media, Transp. porous med. 2, 327-356 (1987)
[46]	Han, N. W.; Bhakta, J.; Carbonell, R. G.: Longitudinal and lateral dispersion in packed beds: effect of column length and particle size distribution, Aiche J. 31, No. 2, 277-288 (1985)
[47]	Strong, A. B.: Plastics: materials and processing, (2006)
[48]	Kamal, M. R.; Sourour, S.: Kinetics and thermal characterization of thermoset cure, Polym. eng. Sci. 13, No. 1, 59-64 (1973)
[49]	Castro, J. M.; Macosko, C. W.: Studies of mold filling and curing in the reaction injection molding process, Aiche J. 28, No. 2, 250-260 (1982)
[50]	Reddy, J. N.; Gartling, D. K.: The finite element method in heat transfer and fluid dynamics, (2000) · Zbl 0978.76003
[51]	Hirt, C. W.; Nichols, B. D.: Volume of fluid (VOF) method for the dynamics of free boundaries, J. comput. Phys. 39, No. 1, 201-225 (1981) · Zbl 0462.76020
[52]	Tan, H.; Pillai, K. M.: Effect of fiber-mat anisotropy on 1D mold filling in LCM: a numerical investigation, Polym. compos. 29, No. 8, 869-882 (2008)
[53]	Lundstrom, T. S.: Measurement of void collapse during resin transfer moulding, Composites part A 28A, 201-214 (1997)