Nicolas Moulin

Monolithic Approach of Stokes-Darcy Coupling for LCM Process Modelling

The present contribution is devoted to developing robust finite element solutions for coupling flows in both purely fluid region, ruled by Stokes equations, and fibrous preform region governed by a Darcys law. Particularly the cases of low permeability of preform, down to 10−15 m, are of interest to model LCM processes. Flows are solved using mixed finite elements stabilized with a sub-grid scale stabilization technique (ASGS). A special attention is paid to the interface conditions, namely normal stress and velocity continuity and tangential velocity constraint similar to a Beaver-JosephSaffman’s condition. The originality of the model consists in using one single mesh to represents the Stokes and the Darcy sub-domains (monolithic approach). A level set context is used to represent Stokes-Darcy interface and to capture the moving flow front. It is shown that provided a very special attention is paid to the coupling conditions, very precise results can be obtained. This monolithic approach is now perfectly robust due to the introduction of the ASGS sub-grid scale stabilization and leads to perform 3D complex shapes for manufacturing process by resin infusion. Introduction Manufacturing processes by resin infusion are competitive routes to elaborate composite structures with organic matrix, especially for large pieces in aeronautics. They consist in infusing a liquid resin through the thickness of the reinforcements rather than in their plane. Under the effect of a mechanical pressure applied on the top of the mold, the resin flows and infuses into the preforms. However, although these processes are efficient, they still remain hard to control. Indeed the physical and mechanical properties of the final part (the final thickness and the fiber volume) are hardly predictable. To control this process, we developed a model based on the coupling between the resin flow within the porous domain (Darcy), and the purely fluid domain (Stokes). The Stokes-Darcy coupled problem has been studied by many researchers in many field of engineering. Both a decoupled approach as proposed by [1] and a monolithic approach as proposed by [2] are investigated in severe regimes. The decoupled strategy consists of using two different element spaces to solve the Stokes and Darcy equations, whereas the unified strategy consists in using the same finite element space. In literature, flows are solved using mixed finite elements stabilized with P1+/P1 elements in Stokes and Darcys domain [1]. Another approach consists in using P1+/P1 elements in Stokes and HVM (Hughes Variational Multiscale) Method for stabilization in Darcy [2]. Due to consistency errors and spurious oscillations that appear in this previous approach, we use a robust approach which yields improvements compared to the latest one. The robustness of the approach which is assessed in this paper, is ensured by using ASGS method (Algebraic Subgrid Scale) to stabilize velocity and pressure approximated by linear and continuous elements in Stokes and Darcy domains. Signed functions are used to represent the Stokes-Darcy interface and to capture the moving flow front. The paper is organized as follows. The first section presents the mathematical modelling for the Stokes-Darcy coupled problem. The next section introduces both the velocity-pressure mixed formulation for the Stokes-Darcy problem and the variational multiscale method used for stabilization. The last section shows numerical validation and results in severe regimes (low permeability, down to 10−15m2, complex 2D and 3D geometries with curved interfaces) to illustrate the capability of modelling manufacturing processes by resin infusion. Key Engineering Materials Online: 2013-06-13 ISSN: 1662-9795, Vols. 554-557, pp 447-455 doi:10.4028/www.scientific.net/KEM.554-557.447

Stokes–Darcy coupling in severe regimes using multiscale stabilisation for mixed finite elements: monolithic approach versus decoupled approach

The article exposes robust finite element solutions for coupling flows in both purely fluid region, ruled by Stokes equations, and a porous region of low permeability (down to 10−15 m2) governed by Darcy’s equations. Relying on stabilised FE formulations, two different numerical strategies are investigated for coupling Stokes–Darcy flows: a decoupled strategy, based on the use of two matching meshes and two finite element spaces for discretising the Stokes–Darcy coupled system; a unified or monolithic strategy, consisting in defining one single mesh for discretising the computational domain, associated with one single finite element space. In the first case, P1+/P1 mixed finite element is used for both Stokes and Darcy (primal form), while P1/P1 approximation is used in the second case with the dual form of the Stokes–Darcy coupled problem stabilised by a variational multi-scale method. The method of manufactured solution is used to evaluate the convergence rates of the solutions and the code robustness. Then, cases of flows normal and tangential to the Stokes–Darcy interface are investigated, and a comparison with available analytical solutions is carried out. Capabilities of both approaches are then demonstrated in solving problems with complex geometry and 3D cases.

Modelling and simulating the forming of new dry automated lay-up reinforcements for primary structures

While weight has been so far the main driver for the development of prepreg based-composites solutions for aeronautics, a new weight-cost trade-off tends to drive choices for next-generation aircrafts. As a response, Hexcel has designed a new dry reinforcement type for aircraft primary structures, which combines the benefits of automation, out-of-autoclave process cost-effectiveness, and mechanical performances competitive to prepreg solutions: HiTape® is a unidirectional (UD) dry carbon reinforcement with thermoplastic veil on each side designed for aircraft primary structures [1-3]. One privileged process route for HiTape® in high volume automated processes consists in forming initially flat dry reinforcement stacks, before resin infusion [4] or injection. Simulation of the forming step aims at predicting the geometry and mechanical properties of the formed stack (so-called preform) for process optimisation. Extensive work has been carried out on prepreg and dry woven fabrics forming behaviour and simul...

Numerical approach for modelling across scales infusion-based processing of aircraft primary structures

This study aims to establish a numerical strategy allowing to take into account the capillary and wetting issues, considered on the macroscopic scale as a discontinuity of pressure at the fluid-gas interface, and surface tension force balance at the local scale. This modelling is based on the Brinkman/Darcy and Stokes equations solved by a finite element stabilized method. Specific numerical methods are implemented to deal with the discontinuity of pressure field across the flow front. One of the challenges lies in modelling across scales capillary force effects in infusion-based processes to scale-up rules for flows at the process scale, because the computation cost of numerical simulations at local scales is too not tractable industrially.