# Numerical simulation of transport equations in Liquid Composite Molding Processes

Juan-Antonio García, Llanos Gascon, Francisco Chinesta

► To cite this version:

Juan-Antonio García, Llanos Gascon, Francisco Chinesta. Numerical simulation of transport equations in Liquid Composite Molding Processes. International Conference "Numerical Methods for Hyperbolic Equations: Theory and Applications", 2011, Saint Jacques de Compostelle, Spain. hal-01008571

HAL Id: hal-01008571

https://hal.science/hal-01008571

Submitted on 9 May 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Numerical simulation of transport equations in
Liquid Composite Molding Processes†

J.A. García¹, Ll. Gascón¹, F. Chinesta

**Keywords:** Flux limiter, FEM, Liquid Composite Molding, Porous media

**Mathematics Subject Classification 2000:** 65M06, 65N06, 35L40

**Minisymposia:** Multiphase Flow and Porous Media

**Abstract** This work is concerned with the development and application of a numerical procedure for the simulation of the flow phenomena involved in composite manufacturing processes.

Liquid Composite Molding (LCM) is one of most important manufacturing processes to produce reinforced plastics. It is a process where a liquid resin is injected in a closed mold containing a dry fibrous reinforcement. The resin impregnation is usually modelled as a flow in porous media. The mathematical model of injection molding is a free boundary problem described by Darcy law and incompressible flow. An important task in the simulation of the mold filling process is the numerical treatment of the moving boundary defined by the liquid resin flow front. The numerical method used by the authors to solve this problem combines a discretization of an elliptic equation governing the pressure and velocity distributions and a transport hyperbolic equation governing the evolution of the resin volume fraction in a fixed mesh.

The incubation time is another important variable in the LCM process. It is the elapsed time since the reactives are mixed, which for each fluid particle depends on the path travelled throughout the mold, and is directly related with the resin curing conversion. The evolution of both, the volume fraction and the incubation time in the LCM processes, are given by linear advection equations. In general, the LCM process simulation involves an accurate treatment of the advection equation which governs the evolution of different fluid properties: fluid presence function, incubation time, temperature, degree of saturation, etc.

In [2] the authors describe a 2D numerical model for LCM simulations in which the volume fraction has been updated by using a new second-order scheme with flux limiters, that provide excellent results regardless of the position of the injection points. In [3] we propose a general strategy able to carry out an accurate field transport when an Eulerian technique, using a fixed mesh, is adopted. We redefine the flux limiters proposed in [2] in order to account for the numerical difficulties related to the integration of a variable, such as the incubation time, in fixed mesh simulations when it is not defined in the empty part of the mold. In order to obtain an accurate

<footnote>†Financial support: Spanish Ministry of Science and Innovation (Project DPI2010-20333).</footnote>
transport of the correspondent field in these cases, we have assumed that the fluid comes to an element only from its full filled neighbor elements. This condition has been introduced in the formulation of the schemes from the use of a parameter which has to be considered in all stages of the discretization. The numerical method minimizes the numerical diffusion and implicitly transports all fields with the flow front movement. We think this numerical technique can also be useful to update other properties governing by advection equations, by example, to simulate the degree of saturation in LCM filling process with void formation.

The reduction of voids formed during the resin filling process is a major issue in LCM processes [1]. Reducing the void content increases the quality of the composite and improves its mechanical properties. Most of modeling efforts on process simulation of mold filling has been focused on the single phase Darcys law, with resin as the only phase, ignoring the formation and transport of voids. Equations that describe the LCM filling process with void formation are based on a two phase flow model and lead to a coupled system of a nonlinear advection-diffusion equation for saturation and an elliptic equation for pressure and velocity. In general, the saturation equation is a non-linear advection-diffusion equation which includes the capillary pressure effect and it reduces to a purely advection transport equation when capillary effects are neglected. The hyperbolic nature of the saturation equation and its strong coupling through relative permeability represents a challenging numerical issue. In this work, a very accurate numerical approach, based on flux limiters, is proposed to solve this complex flow behavior. The implemented algorithm allows a numerical optimization of the injected flow rate which minimizes the micro/macroscopic void formation during mold filling. Some preliminary numerical results are presented here in order to validate the proposed mathematical model and the numerical scheme. This formulation opens up new opportunities to improve LCM flow simulations and optimize injection molds.

## References

[1] J. BREARD, A. SAOUAB, G. BOUQUET. Numerical simulation of void formation in LCM. *Composites Part A: Applied Science and Manufacturing.* **34**, 517-523, 2003.

[2] J.A. GARCÍA, LL. GASCÓN, F. CHINESTA. A fixed mesh numerical method for modelling the flow in liquid composites molding processes using a volume of fluid technique. *Comput. Methods Appl. Mech. Engrg.* **192**, 877-893, 2003.

[3] F. SÁNCHEZ, J.A. GARCÍA, LL. GASCÓN, F. CHINESTA. Towards an efficient numerical treatment of the transport problems in the Resin Transfer Molding Simulation. *Methods Appl. Mech. Engrg.* **196**, 2300-2312, 2007.

<footnote n="1">¹Instituto de Diseño y Fabricación
Universidad Politécnica de Valencia
Camino de Vera s/n, 46022 Valencia, Spain
llgascon@mat.upv.es, jugarcia@mcm.upv.es</footnote>