PCG solver and its computational complexity for implicit control-volume finite-element method of RTM mold filling simulation

The numerical simulation of a mass and heat transfer model for the curing stage of the resin transfer molding (RTM) process is known as a useful method to analyze the process before the mold is actually built. The optimization of non-isothermal mold filling simulation time without losing the efficiency remains an important challenge in the RTM process. These were some reasons that motivate our work, in which we are interested in the amelioration of the performance of the RTM simulation code in terms of execution time and memory space occupation. A modified control volume/finite element method (CV/FEM) is developed to solve the resin flow problem. Full advantage is taken of some of the intrinsic characteristics of this method, in particular, its capability of eliminating the need to re-mesh continuously the resin-filled domain at each time step. Our developed model leads to the numerical prediction of temperature, pressure distribution, and flow front position with great accuracy, together with a precise representation of the thermal (spatiotemporal) behavior of the resin inside the mold. Furthermore, we have improved new methods to optimize the cycle time, since despite the intense interest in the modeling and simulation of RTM process, minimization of mold filling time without losing the part quality remains an important issue in the resin transfer molding process. Thereby, the effects of several parameters on the filling process are deeply investigated. Moreover, we tried to identify the thermal conductivity of a composite material by inverse analysis of the heat conduction phenomenon in resin transfer molding process. The Gauss–Newton–Levenberg–Marquardt method was privileged. The validity of our approaches is evaluated with analytical and experimental results where an excellent agreement was found.

This is the second of two companion papers dealing with the evaluation and enhancements of some control volume finite-element methods (CVFEMs) for fluid flow and heat transfer. One unequal-order CVFEM and two versions of an equal-order CVFEM, for the prediction of two-dimensional incompressible fluid flow are examined in this paper. In the unequal-order CVFEM, six-node macroelements and three-node subelements are used to discretize the calculation domain, and the velocity components are interpolated by flow-oriented upwind-type functions. In the equal-order CVFEM, the calculation domain is discretized by three-node triangular elements, the dependent variables are colocated, and the velocity components are interpolated by flow-oriented upwind-type functions which explicitly account for the influence of the pressure gradient on the velocity distribution in each element. The SIMPLEC procedure is used to solve the discretized equations in all three CVFEMs. The proposed equal-order CVFEM is more accurate than ...

A two-step finite volume method for the simulation of multiphase fluid flow in heterogeneous and anisotropic reservoirs

An implicit control-volume finite element method and its time step strategies for injection molding simulation

Simulation of Three-dimensional Flow in Compression Resin Transfer Molding by the Control Volume/Finite Element Method

The physical variables involved in a flow field are of very different scales to those of the aeroacoustic field within the flow. Thus, direct aeroacoustic computations (DACs) require extremely small time steps to capture the fluctuations in the aeroacoustic field and stabilize the numerical scheme. As a result, the computation time for DACs can be very long. This paper describes an implicit Adaptively Switched Time Stepping (ASTS) scheme that enables the use of larger time steps when solving DAC problems. ASTS is based on the adaptive selection of the time stepping scheme according to the value of the residual. Early in the iteration process, an artificial time term is used to afford larger time steps while ensuring that the numerical scheme remains stable. Once the residual is less than some preset reference value, the artificial time term is removed, thus eliminating the Newton linearization error. Simulation results show that ASTS can reduce the computational cost of DACs by a factor of more than eight while maintaining the required level of accuracy.

Asymptotic expansion homogenization of permeability tensor for plain woven fabrics

Resin transfer moulding is characterized by the permeability tensor, which is a measure of the resistance to resin flow through the preform. Complete prediction of the second order permeability tensor for three dimensional circular braided preforms is critical to an understanding of the resin transfer moulding process. The permeability can be predicted by considering resin flow through the multi-axial fibre structure. In this study, the permeability tensor for a 3-D circular braided preform was calculated by solving a boundary problem of a periodic unit cell. The flow field through the unit cell was obtained by using a 3-D control volume finite element method (CVFEM) and Darcy's law was utilized to obtain the permeability tensor. The flow analyses were carried out for two cases, one in which the fibre tow was regarded as a permeable porous medium, and one in which it was regarded as an impermeable solid. It was found that the flow within the intra-tow region of the braided preform was negligible if the inter-tow porosity was relatively high, but flow through the tow, especially flow in the thickness direction must be considered when the porosity is low. The permeability of the braided preform was measured by a radial flow experiment and compared with the predicted permeability.

Resin transfer molding is one of the most common fabrication methods for composite materials. It is attractive due to its high volume, high performance, and low cost manufacturing of polymer composites. In this process, a dry preform of reinforcing fibers is placed inside a closed mold; then the resin is injected into the mold cavity. The composite part can be removed from the mold after the resin cures. This paper surveys current issues in the resin transfer molding process and focuses on mold filling and simulation. The Control Volume Finite Element Method is applied to simulate the process. The 2-D and 3-D computational methods are presented. The simulation program was developed with C language by the authors. The 2-D and 3-D models were integrated in the program. In real applications, the 2-D elements and 3-D elements can be applied simultaneously according to the specific requirements. MSC/PATRAN 3 software (MacNeal-Schwendler Corporation) was used to generate the finite element mesh and display the results. Some case studies are conducted to demonstrate the application of the computer simulation to RTM process design.

The presence of air bubbles impacts the quality of the produced composite part, by reducing its mechanical properties, and also it might degrade its surface finish. The modelling of air bubbles entrapment requires the consideration of three phenomena: air bubble’s creation, compression and transport. Very few studies have been conducted on this latter phenomenon. The model developed in this work is proposed for a unidirectional reinforcement. It is integrated into a simulation code of resin transfer moulding process, via the control volume finite element method. That model takes into account the dual scale pores in fibrous media, and simulates the three said phenomena highlighting the migration phenomenon and the coexistence of micro and macro air bubbles. As a result, the spatial distribution of created, compressed and transported air bubbles as well as its macro and micro remaining quantities, in the end of the injection are estimated.

The application of the implicit scheme of the multiscale discrete velocity method for three-dimensional flows of diatomic gas in all flow regimes

In molecular dynamics simulations, the limited time step size has been a barrier to simulating long-time behaviors. Implicit time integration methods allow markedly larger time steps than the standard explicit time method, although they have major drawbacks such as overheads solving linear systems and instability of Newton iterations. To overcome these issues, we propose a semi-implicit time integration scheme, the semi-implicit Hessian correction (SimHec) scheme, for overdamped Langevin dynamics. The method focuses on the Hessian matrices of bonded and nonbonded interactions, where components with large negative Hessian eigenvalues are cut off in the linear approximation of momentum equations to avoid instability. The narrow band Hessian matrix enables an efficient parallelized linear solution with an overlapping approximation. We tested SimHec for the interdomain fluctuations in adenylate kinase and the powerstroke transition of myosin II using a coarse-grained protein model. SimHec reproduced the same dynamics as the explicit method, although the transition dynamics tended to be accelerated and fluctuations in bonded potentials were slightly reduced. These deviations were corrected using a hybrid method, SimHec-H, which adds explicit time steps after the semi-implicit time step. The proposed scheme allowed us to use time steps 50-200 times larger than those in explicit time integration, which resulted in a speedup factor of 7-30 taking the overhead into account.

In injection moulding processes such as Resin Transfer Moulding (RTM) for example, numerical simulations are usually performed with a fixed mesh, on which the displacement of the flow front is predicted by the numerical algorithm. During the injection, special physical phenomena occur on the front, such as capillary effects inside the fibre tows or heat transfer when the fluid is injected at a different temperature than the mould. In order to approximate these phenomena accurately, it is always better to adapt the mesh to the shape of the flow front. This can be achieved by implementing re-meshing algorithms, which will provide not only more accurate solutions, but also faster calculations. In order to represent precisely the shape of the saturated domain in the cavity, the mesh needs to be non-isotropic in the vicinity of the flow front. The size of the elements along the front is connected to the overall accuracy needed for the simulation; the size in the perpendicular direction governs the accuracy on the position of the moving boundary in time. Since these two constraints on element size are not related, the need for non-isotropic mesh refinement is crucial. In the approach proposed here, the mesh is changed at each time step from a background isotropic mesh used as starting point in the refinement algorithm. The solution needs to be projected on the new mesh after each re-meshing. This amounts to adopting a new filling algorithm, which will be validated by comparison to a standard simulation (without re-meshing) and with experimental data.

This paper describes three different time integration methods for unsteady incompressible Navier–Stokes equations. Explicit Euler and fractional‐step Adams–Bashford methods are compared with an implicit three‐level method based on a steady‐state SIMPLE method. The implicit solver employs a dual time stepping and an iteration within the time step. The spatial discretization is based on a co‐located finite‐volume technique. The influence of the convergence limits and the time‐step size on the accuracy of the predictions are studied. The efficiency of the different solvers is compared in a vortex‐shedding flow over a cylinder in the Reynolds number range of 100–1600. A high‐Reynolds‐number flow over a biconvex airfoil profile is also computed. The computations are performed in two dimensions. At the low‐Reynolds‐number range the explicit methods appear to be faster by a factor from 5 to 10. In the high‐Reynolds‐number case, the explicit Adams–Bashford method and the implicit method appear to be approximately equally fast while yielding similar results. Copyright © 2002 John Wiley & Sons, Ltd.

We present implicit time integration schemes suitable for modeling free surface Stokes flow dynamics with marker in cell (MIC) based spatial discretization. Our target is for example thermal convection surrounded by deformable surface boundaries to simulate the long term planetary formation process. The numerical system becomes stiff when the dynamical balancing time scale for the increasing/decreasing load by surface deformation is very short compared with the time scale associated with thermal convection. Any explicit time integration scheme will require very small time steps; otherwise, serious numerical oscillation (spurious solutions) will occur. The implicit time integration scheme possesses a wider stability region than the explicit method; therefore, it is suitable for stiff problems. To investigate an efficient solution method for the stiff Stokes flow system, we apply first (backward Euler (BE)) and second order (trapezoidal method (TR) and trapezoidal rule—backward difference formula (TR-BDF2)) accurate implicit methods for the MIC solution scheme. The introduction of implicit time integration schemes results in nonlinear systems of equations. We utilize a Jacobian free Newton Krylov (JFNK) based Newton framework to solve the resulting nonlinear equations. In this work we also investigate two efficient implicit solution strategies to reduce the computational cost when solving stiff nonlinear systems. The two methods differ in how the advective term in the material transport evolution equation is treated. We refer to the method that employs Lagrangian update as “fully implicit” (Imp), whilst the method that employs Eulerian update is referred to as “semi-implicit” (SImp). Using a finite difference (FD) method, we have performed a series of numerical experiments which clarify the accuracy of solutions and trade-off between the computational cost associated with the nonlinear solver and time step size. In comparison with the general explicit Euler method, the second order accurate Imp methods reduce total computational cost successfully through the utilization of a large time step without sacrificing accuracy and stability. Moreover, the proposed SImp method is effective in reducing the computational cost associated with evaluating the nonlinear residual while obtaining a solution similar to the Imp method.